List of integrals of irrational functions

Integrals involving r = √a2 + x2

∫ r d x = 1 2 ( x r + a 2 ln ⁡ ( x + r ) ) ∫ r 3 d x = 1 4 x r 3 + 3 8 a 2 x r + 3 8 a 4 ln ⁡ ( x + r ) ∫ r 5 d x = 1 6 x r 5 + 5 24 a 2 x r 3 + 5 16 a 4 x r + 5 16 a 6 ln ⁡ ( x + r ) ∫ x r d x = r 3 3 ∫ x r 3 d x = r 5 5 ∫ x r 2 n + 1 d x = r 2 n + 3 2 n + 3 ∫ x 2 r d x = x r 3 4 − a 2 x r 8 − a 4 8 ln ⁡ ( x + r ) ∫ x 2 r 3 d x = x r 5 6 − a 2 x r 3 24 − a 4 x r 16 − a 6 16 ln ⁡ ( x + r ) ∫ x 3 r d x = r 5 5 − a 2 r 3 3 ∫ x 3 r 3 d x = r 7 7 − a 2 r 5 5 ∫ x 3 r 2 n + 1 d x = r 2 n + 5 2 n + 5 − a 2 r 2 n + 3 2 n + 3 ∫ x 4 r d x = x 3 r 3 6 − a 2 x r 3 8 + a 4 x r 16 + a 6 16 ln ⁡ ( x + r ) ∫ x 4 r 3 d x = x 3 r 5 8 − a 2 x r 5 16 + a 4 x r 3 64 + 3 a 6 x r 128 + 3 a 8 128 ln ⁡ ( x + r ) ∫ x 5 r d x = r 7 7 − 2 a 2 r 5 5 + a 4 r 3 3 ∫ x 5 r 3 d x = r 9 9 − 2 a 2 r 7 7 + a 4 r 5 5 ∫ x 5 r 2 n + 1 d x = r 2 n + 7 2 n + 7 − 2 a 2 r 2 n + 5 2 n + 5 + a 4 r 2 n + 3 2 n + 3 ∫ r d x x = r − a ln ⁡ | a + r x | = r − a arsinh ⁡ a x ∫ r 3 d x x = r 3 3 + a 2 r − a 3 ln ⁡ | a + r x | ∫ r 5 d x x = r 5 5 + a 2 r 3 3 + a 4 r − a 5 ln ⁡ | a + r x | ∫ r 7 d x x = r 7 7 + a 2 r 5 5 + a 4 r 3 3 + a 6 r − a 7 ln ⁡ | a + r x | ∫ d x r = arsinh ⁡ x a = ln ⁡ ( x + r a ) ∫ d x r 3 = x a 2 r ∫ x d x r = r ∫ x d x r 3 = − 1 r ∫ x 2 d x r = x 2 r − a 2 2 arsinh ⁡ x a = x 2 r − a 2 2 ln ⁡ ( x + r a ) ∫ d x x r = − 1 a arsinh ⁡ a x = − 1 a ln ⁡ | a + r x |

Integrals involving s = √x2 − a2

Assume x² > a² (for x² < a², see next section):

∫ s d x = 1 2 ( x s − a 2 ln ⁡ ( x + s ) ) ∫ x s d x = 1 3 s 3 ∫ s d x x = s − a arccos ⁡ | a x | ∫ d x s = ln ⁡ | x + s a |

Here ln ⁡ | x + s a | = s g n ( x ) arcosh ⁡ | x a | = 1 2 ln ⁡ ( x + s x − s ) , where the positive value of arcosh ⁡ | x a | is to be taken.

∫ x d x s = s ∫ x d x s 3 = − 1 s ∫ x d x s 5 = − 1 3 s 3 ∫ x d x s 7 = − 1 5 s 5 ∫ x d x s 2 n + 1 = − 1 ( 2 n − 1 ) s 2 n − 1 ∫ x 2 m d x s 2 n + 1 = − 1 2 n − 1 x 2 m − 1 s 2 n − 1 + 2 m − 1 2 n − 1 ∫ x 2 m − 2 d x s 2 n − 1 ∫ x 2 d x s = x s 2 + a 2 2 ln ⁡ | x + s a | ∫ x 2 d x s 3 = − x s + ln ⁡ | x + s a | ∫ x 4 d x s = x 3 s 4 + 3 8 a 2 x s + 3 8 a 4 ln ⁡ | x + s a | ∫ x 4 d x s 3 = x s 2 − a 2 x s + 3 2 a 2 ln ⁡ | x + s a | ∫ x 4 d x s 5 = − x s − 1 3 x 3 s 3 + ln ⁡ | x + s a | ∫ x 2 m d x s 2 n + 1 = ( − 1 ) n − m 1 a 2 ( n − m ) ∑ i = 0 n − m − 1 1 2 ( m + i ) + 1 ( n − m − 1 i ) x 2 ( m + i ) + 1 s 2 ( m + i ) + 1 ( n > m ≥ 0 ) ∫ d x s 3 = − 1 a 2 x s ∫ d x s 5 = 1 a 4 [ x s − 1 3 x 3 s 3 ] ∫ d x s 7 = − 1 a 6 [ x s − 2 3 x 3 s 3 + 1 5 x 5 s 5 ] ∫ d x s 9 = 1 a 8 [ x s − 3 3 x 3 s 3 + 3 5 x 5 s 5 − 1 7 x 7 s 7 ] ∫ x 2 d x s 5 = − 1 a 2 x 3 3 s 3 ∫ x 2 d x s 7 = 1 a 4 [ 1 3 x 3 s 3 − 1 5 x 5 s 5 ] ∫ x 2 d x s 9 = − 1 a 6 [ 1 3 x 3 s 3 − 2 5 x 5 s 5 + 1 7 x 7 s 7 ]

Integrals involving u = √a2 − x2

∫ u d x = 1 2 ( x u + a 2 arcsin ⁡ x a ) ( | x | ≤ | a | ) ∫ x u d x = − 1 3 u 3 ( | x | ≤ | a | ) ∫ x 2 u d x = − x 4 u 3 + a 2 8 ( x u + a 2 arcsin ⁡ x a ) ( | x | ≤ | a | ) ∫ u d x x = u − a ln ⁡ | a + u x | ( | x | ≤ | a | ) ∫ d x u = arcsin ⁡ x a ( | x | ≤ | a | ) ∫ x 2 d x u = 1 2 ( − x u + a 2 arcsin ⁡ x a ) ( | x | ≤ | a | ) ∫ u d x = 1 2 ( x u − sgn ⁡ x arcosh ⁡ | x a | ) (for  | x | ≥ | a | ) ∫ x u d x = − u ( | x | ≤ | a | )

Integrals involving R = √ax2 + bx + c

Assume (ax² + bx + c) cannot be reduced to the following expression (px + q)² for some p and q.

∫ d x R = 1 a ln ⁡ | 2 a R + 2 a x + b | (for  a > 0 ) ∫ d x R = 1 a arsinh ⁡ 2 a x + b 4 a c − b 2 (for  a > 0 ,  4 a c − b 2 > 0 ) ∫ d x R = 1 a ln ⁡ | 2 a x + b | (for  a > 0 ,  4 a c − b 2 = 0 ) ∫ d x R = − 1 − a arcsin ⁡ 2 a x + b b 2 − 4 a c (for  a < 0 ,  4 a c − b 2 < 0 ,  | 2 a x + b | < b 2 − 4 a c ) ∫ d x R 3 = 4 a x + 2 b ( 4 a c − b 2 ) R ∫ d x R 5 = 4 a x + 2 b 3 ( 4 a c − b 2 ) R ( 1 R 2 + 8 a 4 a c − b 2 ) ∫ d x R 2 n + 1 = 2 ( 2 n − 1 ) ( 4 a c − b 2 ) ( 2 a x + b R 2 n − 1 + 4 a ( n − 1 ) ∫ d x R 2 n − 1 ) ∫ x R d x = R a − b 2 a ∫ d x R ∫ x R 3 d x = − 2 b x + 4 c ( 4 a c − b 2 ) R ∫ x R 2 n + 1 d x = − 1 ( 2 n − 1 ) a R 2 n − 1 − b 2 a ∫ d x R 2 n + 1 ∫ d x x R = − 1 c ln ⁡ | 2 c R + b x + 2 c x | ,   c > 0 ∫ d x x R = − 1 c arsinh ⁡ ( b x + 2 c | x | 4 a c − b 2 ) ,   c < 0 ∫ d x x R = 1 − c arcsin ⁡ ( b x + 2 c | x | b 2 − 4 a c ) ,   c < 0 , b 2 − 4 a c > 0 ∫ d x x R = − 2 b x ( a x 2 + b x ) ,   c = 0 ∫ x 2 R d x = 2 a x − 3 b 4 a 2 R + 3 b 2 − 4 a c 8 a 2 ∫ d x R ∫ d x x 2 R = − R c x − b 2 c ∫ d x x R ∫ R d x = 2 a x + b 4 a R + 4 a c − b 2 8 a ∫ d x R ∫ x R d x = R 3 3 a − b ( 2 a x + b ) 8 a 2 R − b ( 4 a c − b 2 ) 16 a 2 ∫ d x R ∫ x 2 R d x = 6 a x − 5 b 24 a 2 R 3 + 5 b 2 − 4 a c 16 a 2 ∫ R d x ∫ R x d x = R + b 2 ∫ d x R + c ∫ d x x R ∫ R x 2 d x = − R x + a ∫ d x R + b 2 ∫ d x x R ∫ x 2 d x R 3 = ( 2 b 2 − 4 a c ) x + 2 b c a ( 4 a c − b 2 ) R + 1 a ∫ d x R

Integrals involving S = √ax + b

∫ S d x = 2 S 3 3 a ∫ d x S = 2 S a ∫ d x x S = { − 2 b a r c o t h ( S b ) (for  b > 0 , a x > 0 ) − 2 b a r t a n h ( S b ) (for  b > 0 , a x < 0 ) 2 − b arctan ⁡ ( S − b ) (for  b < 0 ) ∫ S x d x = { 2 ( S − b a r c o t h ( S b ) ) (for  b > 0 , a x > 0 ) 2 ( S − b a r t a n h ( S b ) ) (for  b > 0 , a x < 0 ) 2 ( S − − b arctan ⁡ ( S − b ) ) (for  b < 0 ) ∫ x n S d x = 2 a ( 2 n + 1 ) ( x n S − b n ∫ x n − 1 S d x ) ∫ x n S d x = 2 a ( 2 n + 3 ) ( x n S 3 − n b ∫ x n − 1 S d x ) ∫ 1 x n S d x = − 1 b ( n − 1 ) ( S x n − 1 + ( n − 3 2 ) a ∫ d x x n − 1 S )