\(\int \frac {e^{n \text {arctanh}(a x)}}{(c-\frac {c}{a^2 x^2})^{5/2}} \, dx\) [785]

Optimal result

Integrand size = 24, antiderivative size = 1027 \[ \int \frac {e^{n \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx =\text {Too large to display} \] Output:

-(2+n)*(4+n)*(-a*x+1)^(-3/2-1/2*n)*(a*x+1)^(-3/2+1/2*n)*(-a^2*x^2+1)^(5/2) 
/a^6/(3+n)/(c-c/a^2/x^2)^(5/2)/x^5+(4+n)*(-a*x+1)^(-3/2-1/2*n)*(a*x+1)^(-3 
/2+1/2*n)*(-a^2*x^2+1)^(5/2)/a^4/(c-c/a^2/x^2)^(5/2)/x^3-(-a*x+1)^(-3/2-1/ 
2*n)*(a*x+1)^(-3/2+1/2*n)*(-a^2*x^2+1)^(5/2)/a^2/(c-c/a^2/x^2)^(5/2)/x+n*( 
-a*x+1)^(-1/2-1/2*n)*(a*x+1)^(-3/2+1/2*n)*(-a^2*x^2+1)^(5/2)/a^6/(1+n)/(c- 
c/a^2/x^2)^(5/2)/x^5+n*(4+n)*(-a*x+1)^(-1/2-1/2*n)*(a*x+1)^(-3/2+1/2*n)*(- 
a^2*x^2+1)^(5/2)/a^6/(-n^2+9)/(c-c/a^2/x^2)^(5/2)/x^5-2*n*(-a*x+1)^(1/2-1/ 
2*n)*(a*x+1)^(-3/2+1/2*n)*(-a^2*x^2+1)^(5/2)/a^6/(-n^2+1)/(c-c/a^2/x^2)^(5 
/2)/x^5+2*n*(-a*x+1)^(3/2-1/2*n)*(a*x+1)^(-3/2+1/2*n)*(-a^2*x^2+1)^(5/2)/a 
^6/(1+n)/(n^2-4*n+3)/(c-c/a^2/x^2)^(5/2)/x^5-3*n*(-a*x+1)^(-1/2-1/2*n)*(a* 
x+1)^(-1/2+1/2*n)*(-a^2*x^2+1)^(5/2)/a^6/(1+n)/(c-c/a^2/x^2)^(5/2)/x^5+n*( 
4+n)*(-n^2+7)*(-a*x+1)^(-1/2-1/2*n)*(a*x+1)^(-1/2+1/2*n)*(-a^2*x^2+1)^(5/2 
)/a^6/(3-n)/(1+n)/(3+n)/(c-c/a^2/x^2)^(5/2)/x^5+3*n*(-a*x+1)^(1/2-1/2*n)*( 
a*x+1)^(-1/2+1/2*n)*(-a^2*x^2+1)^(5/2)/a^6/(-n^2+1)/(c-c/a^2/x^2)^(5/2)/x^ 
5-n*(4+n)*(-n^2+7)*(-a*x+1)^(1/2-1/2*n)*(a*x+1)^(-1/2+1/2*n)*(-a^2*x^2+1)^ 
(5/2)/a^6/(n^4-10*n^2+9)/(c-c/a^2/x^2)^(5/2)/x^5+3*n*(-a*x+1)^(-1/2-1/2*n) 
*(a*x+1)^(1/2+1/2*n)*(-a^2*x^2+1)^(5/2)/a^6/(1+n)/(c-c/a^2/x^2)^(5/2)/x^5- 
2^(3/2+1/2*n)*n*(-a*x+1)^(-1/2-1/2*n)*(-a^2*x^2+1)^(5/2)*hypergeom([-1/2-1 
/2*n, -1/2-1/2*n],[1/2-1/2*n],-1/2*a*x+1/2)/a^6/(1+n)/(c-c/a^2/x^2)^(5/2)/ 
x^5
                                                                                    
                                                                                    
 

Mathematica [A] (warning: unable to verify)

Time = 7.46 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.22 \[ \int \frac {e^{n \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx=\frac {\left (-1+a^2 x^2\right )^2 \left (-\frac {8 e^{n \text {arctanh}(a x)} (-1+a n x)}{-1+n^2}-\frac {e^{n \text {arctanh}(a x)} \left (-9+n^2+10 a n x-2 a n^3 x-2 a n \left (-1+n^2\right ) x \cosh (2 \text {arctanh}(a x))+3 \left (-1+n^2\right ) \sqrt {1-a^2 x^2} \cosh (3 \text {arctanh}(a x))\right )}{9-10 n^2+n^4}-\frac {4 \left (-1+a^2 x^2\right ) \left (-e^{n \text {arctanh}(a x)} (1+n)+\frac {2 e^{(1+n) \text {arctanh}(a x)} n \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2},\frac {3+n}{2},-e^{2 \text {arctanh}(a x)}\right )}{\sqrt {1-a^2 x^2}}\right )}{1+n}\right )}{4 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5} \] Input:

Integrate[E^(n*ArcTanh[a*x])/(c - c/(a^2*x^2))^(5/2),x]
 

Output:

((-1 + a^2*x^2)^2*((-8*E^(n*ArcTanh[a*x])*(-1 + a*n*x))/(-1 + n^2) - (E^(n 
*ArcTanh[a*x])*(-9 + n^2 + 10*a*n*x - 2*a*n^3*x - 2*a*n*(-1 + n^2)*x*Cosh[ 
2*ArcTanh[a*x]] + 3*(-1 + n^2)*Sqrt[1 - a^2*x^2]*Cosh[3*ArcTanh[a*x]]))/(9 
 - 10*n^2 + n^4) - (4*(-1 + a^2*x^2)*(-(E^(n*ArcTanh[a*x])*(1 + n)) + (2*E 
^((1 + n)*ArcTanh[a*x])*n*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, -E^(2 
*ArcTanh[a*x])])/Sqrt[1 - a^2*x^2]))/(1 + n)))/(4*a^6*(c - c/(a^2*x^2))^(5 
/2)*x^5)
 

Rubi [A] (verified)

Time = 1.70 (sec) , antiderivative size = 650, normalized size of antiderivative = 0.63, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {6710, 6700, 111, 25, 177, 105, 101, 25, 88, 55, 48, 137, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {e^{n \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 6710

\(\displaystyle \frac {\left (1-a^2 x^2\right )^{5/2} \int \frac {e^{n \text {arctanh}(a x)} x^5}{\left (1-a^2 x^2\right )^{5/2}}dx}{x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}\)

\(\Big \downarrow \) 6700

\(\displaystyle \frac {\left (1-a^2 x^2\right )^{5/2} \int x^5 (1-a x)^{\frac {1}{2} (-n-5)} (a x+1)^{\frac {n-5}{2}}dx}{x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}\)

\(\Big \downarrow \) 111

\(\displaystyle \frac {\left (1-a^2 x^2\right )^{5/2} \left (-\frac {\int -x^3 (1-a x)^{\frac {1}{2} (-n-5)} (a x+1)^{\frac {n-5}{2}} (a n x+4)dx}{a^2}-\frac {x^4 (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {1}{2} (-n-3)}}{a^2}\right )}{x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\left (1-a^2 x^2\right )^{5/2} \left (\frac {\int x^3 (1-a x)^{\frac {1}{2} (-n-5)} (a x+1)^{\frac {n-5}{2}} (a n x+4)dx}{a^2}-\frac {x^4 (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{a^2}\right )}{x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}\)

\(\Big \downarrow \) 177

\(\displaystyle \frac {\left (1-a^2 x^2\right )^{5/2} \left (\frac {(n+4) \int x^3 (1-a x)^{\frac {1}{2} (-n-5)} (a x+1)^{\frac {n-5}{2}}dx-n \int x^3 (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-5}{2}}dx}{a^2}-\frac {x^4 (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{a^2}\right )}{x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}\)

\(\Big \downarrow \) 105

\(\displaystyle \frac {\left (1-a^2 x^2\right )^{5/2} \left (\frac {(n+4) \left (\frac {x^3 (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{a (n+3)}-\frac {3 \int x^2 (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-5}{2}}dx}{a (n+3)}\right )-n \int x^3 (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-5}{2}}dx}{a^2}-\frac {x^4 (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{a^2}\right )}{x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}\)

\(\Big \downarrow \) 101

\(\displaystyle \frac {\left (1-a^2 x^2\right )^{5/2} \left (\frac {(n+4) \left (\frac {x^3 (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{a (n+3)}-\frac {3 \left (\frac {\int -(1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-5}{2}} (1-a (1-n) x)dx}{a^2}+\frac {x (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {1}{2} (-n-1)}}{a^2}\right )}{a (n+3)}\right )-n \int x^3 (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-5}{2}}dx}{a^2}-\frac {x^4 (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{a^2}\right )}{x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\left (1-a^2 x^2\right )^{5/2} \left (\frac {(n+4) \left (\frac {x^3 (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{a (n+3)}-\frac {3 \left (\frac {x (1-a x)^{\frac {1}{2} (-n-1)} (a x+1)^{\frac {n-3}{2}}}{a^2}-\frac {\int (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-5}{2}} (1-a (1-n) x)dx}{a^2}\right )}{a (n+3)}\right )-n \int x^3 (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-5}{2}}dx}{a^2}-\frac {x^4 (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{a^2}\right )}{x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}\)

\(\Big \downarrow \) 88

\(\displaystyle \frac {\left (1-a^2 x^2\right )^{5/2} \left (\frac {(n+4) \left (\frac {x^3 (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{a (n+3)}-\frac {3 \left (\frac {x (1-a x)^{\frac {1}{2} (-n-1)} (a x+1)^{\frac {n-3}{2}}}{a^2}-\frac {\frac {\left (-n^2+2 n+1\right ) \int (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}dx}{3-n}-\frac {(2-n) (1-a x)^{\frac {1}{2} (-n-1)} (a x+1)^{\frac {n-3}{2}}}{a (3-n)}}{a^2}\right )}{a (n+3)}\right )-n \int x^3 (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-5}{2}}dx}{a^2}-\frac {x^4 (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{a^2}\right )}{x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}\)

\(\Big \downarrow \) 55

\(\displaystyle \frac {\left (1-a^2 x^2\right )^{5/2} \left (\frac {(n+4) \left (\frac {x^3 (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{a (n+3)}-\frac {3 \left (\frac {x (1-a x)^{\frac {1}{2} (-n-1)} (a x+1)^{\frac {n-3}{2}}}{a^2}-\frac {\frac {\left (-n^2+2 n+1\right ) \left (\frac {\int (1-a x)^{\frac {1}{2} (-n-1)} (a x+1)^{\frac {n-3}{2}}dx}{n+1}+\frac {(a x+1)^{\frac {n-1}{2}} (1-a x)^{\frac {1}{2} (-n-1)}}{a (n+1)}\right )}{3-n}-\frac {(2-n) (1-a x)^{\frac {1}{2} (-n-1)} (a x+1)^{\frac {n-3}{2}}}{a (3-n)}}{a^2}\right )}{a (n+3)}\right )-n \int x^3 (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-5}{2}}dx}{a^2}-\frac {x^4 (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{a^2}\right )}{x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}\)

\(\Big \downarrow \) 48

\(\displaystyle \frac {\left (1-a^2 x^2\right )^{5/2} \left (\frac {(n+4) \left (\frac {x^3 (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{a (n+3)}-\frac {3 \left (\frac {x (1-a x)^{\frac {1}{2} (-n-1)} (a x+1)^{\frac {n-3}{2}}}{a^2}-\frac {\frac {\left (-n^2+2 n+1\right ) \left (\frac {(1-a x)^{\frac {1}{2} (-n-1)} (a x+1)^{\frac {n-1}{2}}}{a (n+1)}-\frac {(1-a x)^{\frac {1-n}{2}} (a x+1)^{\frac {n-1}{2}}}{a (1-n) (n+1)}\right )}{3-n}-\frac {(2-n) (1-a x)^{\frac {1}{2} (-n-1)} (a x+1)^{\frac {n-3}{2}}}{a (3-n)}}{a^2}\right )}{a (n+3)}\right )-n \int x^3 (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-5}{2}}dx}{a^2}-\frac {x^4 (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{a^2}\right )}{x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}\)

\(\Big \downarrow \) 137

\(\displaystyle \frac {\left (1-a^2 x^2\right )^{5/2} \left (\frac {(n+4) \left (\frac {x^3 (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{a (n+3)}-\frac {3 \left (\frac {x (1-a x)^{\frac {1}{2} (-n-1)} (a x+1)^{\frac {n-3}{2}}}{a^2}-\frac {\frac {\left (-n^2+2 n+1\right ) \left (\frac {(1-a x)^{\frac {1}{2} (-n-1)} (a x+1)^{\frac {n-1}{2}}}{a (n+1)}-\frac {(1-a x)^{\frac {1-n}{2}} (a x+1)^{\frac {n-1}{2}}}{a (1-n) (n+1)}\right )}{3-n}-\frac {(2-n) (1-a x)^{\frac {1}{2} (-n-1)} (a x+1)^{\frac {n-3}{2}}}{a (3-n)}}{a^2}\right )}{a (n+3)}\right )-n \int \left (\frac {3 (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-5}{2}+1}}{a^3}-\frac {3 (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-5}{2}+2}}{a^3}+\frac {(1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-5}{2}+3}}{a^3}-\frac {(1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-5}{2}}}{a^3}\right )dx}{a^2}-\frac {x^4 (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{a^2}\right )}{x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\left (1-a^2 x^2\right )^{5/2} \left (\frac {(n+4) \left (\frac {x^3 (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{a (n+3)}-\frac {3 \left (\frac {x (1-a x)^{\frac {1}{2} (-n-1)} (a x+1)^{\frac {n-3}{2}}}{a^2}-\frac {\frac {\left (-n^2+2 n+1\right ) \left (\frac {(1-a x)^{\frac {1}{2} (-n-1)} (a x+1)^{\frac {n-1}{2}}}{a (n+1)}-\frac {(1-a x)^{\frac {1-n}{2}} (a x+1)^{\frac {n-1}{2}}}{a (1-n) (n+1)}\right )}{3-n}-\frac {(2-n) (1-a x)^{\frac {1}{2} (-n-1)} (a x+1)^{\frac {n-3}{2}}}{a (3-n)}}{a^2}\right )}{a (n+3)}\right )-n \left (\frac {2^{\frac {n+3}{2}} (1-a x)^{\frac {1}{2} (-n-1)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-n-1),\frac {1}{2} (-n-1),\frac {1-n}{2},\frac {1}{2} (1-a x)\right )}{a^4 (n+1)}+\frac {2 (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {1-n}{2}}}{a^4 \left (1-n^2\right )}-\frac {3 (a x+1)^{\frac {n-1}{2}} (1-a x)^{\frac {1-n}{2}}}{a^4 \left (1-n^2\right )}-\frac {2 (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {3-n}{2}}}{a^4 (n+1) \left (n^2-4 n+3\right )}-\frac {(a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {1}{2} (-n-1)}}{a^4 (n+1)}+\frac {3 (a x+1)^{\frac {n-1}{2}} (1-a x)^{\frac {1}{2} (-n-1)}}{a^4 (n+1)}-\frac {3 (a x+1)^{\frac {n+1}{2}} (1-a x)^{\frac {1}{2} (-n-1)}}{a^4 (n+1)}\right )}{a^2}-\frac {x^4 (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{a^2}\right )}{x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}\)

Input:

Int[E^(n*ArcTanh[a*x])/(c - c/(a^2*x^2))^(5/2),x]
 

Output:

((1 - a^2*x^2)^(5/2)*(-((x^4*(1 - a*x)^((-3 - n)/2)*(1 + a*x)^((-3 + n)/2) 
)/a^2) + ((4 + n)*((x^3*(1 - a*x)^((-3 - n)/2)*(1 + a*x)^((-3 + n)/2))/(a* 
(3 + n)) - (3*((x*(1 - a*x)^((-1 - n)/2)*(1 + a*x)^((-3 + n)/2))/a^2 - (-( 
((2 - n)*(1 - a*x)^((-1 - n)/2)*(1 + a*x)^((-3 + n)/2))/(a*(3 - n))) + ((1 
 + 2*n - n^2)*(((1 - a*x)^((-1 - n)/2)*(1 + a*x)^((-1 + n)/2))/(a*(1 + n)) 
 - ((1 - a*x)^((1 - n)/2)*(1 + a*x)^((-1 + n)/2))/(a*(1 - n)*(1 + n))))/(3 
 - n))/a^2))/(a*(3 + n))) - n*(-(((1 - a*x)^((-1 - n)/2)*(1 + a*x)^((-3 + 
n)/2))/(a^4*(1 + n))) + (2*(1 - a*x)^((1 - n)/2)*(1 + a*x)^((-3 + n)/2))/( 
a^4*(1 - n^2)) - (2*(1 - a*x)^((3 - n)/2)*(1 + a*x)^((-3 + n)/2))/(a^4*(1 
+ n)*(3 - 4*n + n^2)) + (3*(1 - a*x)^((-1 - n)/2)*(1 + a*x)^((-1 + n)/2))/ 
(a^4*(1 + n)) - (3*(1 - a*x)^((1 - n)/2)*(1 + a*x)^((-1 + n)/2))/(a^4*(1 - 
 n^2)) - (3*(1 - a*x)^((-1 - n)/2)*(1 + a*x)^((1 + n)/2))/(a^4*(1 + n)) + 
(2^((3 + n)/2)*(1 - a*x)^((-1 - n)/2)*Hypergeometric2F1[(-1 - n)/2, (-1 - 
n)/2, (1 - n)/2, (1 - a*x)/2])/(a^4*(1 + n))))/a^2))/((c - c/(a^2*x^2))^(5 
/2)*x^5)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 48

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp 
[(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ 
a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
 

rule 55

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(S 
implify[m + n + 2]/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^Simplify[m + 1]*( 
c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 
 2], 0] && NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ 
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] ||  !SumSimp 
lerQ[n, 1])
 

rule 88

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^Simplify[p + 1], 
 x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] &&  !RationalQ[p] && SumSimpl 
erQ[p, 1]
 

rule 101

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + 
 p + 3))), x] + Simp[1/(d*f*(n + p + 3))   Int[(c + d*x)^n*(e + f*x)^p*Simp 
[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f 
*(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, 
 c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]
 

rule 105

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 
1)*(b*e - a*f))), x] - Simp[n*((d*e - c*f)/((m + 1)*(b*e - a*f)))   Int[(a 
+ b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, 
e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] 
 ||  !SumSimplerQ[p, 1]) && NeQ[m, -1]
 

rule 111

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(a + b*x)^(m - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 
)/(d*f*(m + n + p + 1))), x] + Simp[1/(d*f*(m + n + p + 1))   Int[(a + b*x) 
^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m 
 - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m 
 + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] & 
& GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]
 

rule 137

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (IGtQ[m, 0] || (ILtQ[m, 0] && 
ILtQ[n, 0]))
 

rule 177

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h/b   Int[(a + b*x)^(m + 1)*(c + d 
*x)^n*(e + f*x)^p, x], x] + Simp[(b*g - a*h)/b   Int[(a + b*x)^m*(c + d*x)^ 
n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, p}, x] && (Su 
mSimplerQ[m, 1] || ( !SumSimplerQ[n, 1] &&  !SumSimplerQ[p, 1]))
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6700

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x 
_Symbol] :> Simp[c^p   Int[x^m*(1 - a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x], 
 x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || 
 GtQ[c, 0])
 

rule 6710

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_), x_Symbo 
l] :> Simp[x^(2*p)*((c + d/x^2)^p/(1 - a^2*x^2)^p)   Int[(u/x^(2*p))*(1 - a 
^2*x^2)^p*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c 
+ a^2*d, 0] &&  !IntegerQ[p] &&  !IntegerQ[n/2]
 

Maple [F]

\[\int \frac {{\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )}}{\left (c -\frac {c}{a^{2} x^{2}}\right )^{\frac {5}{2}}}d x\]

Input:

int(exp(n*arctanh(a*x))/(c-c/a^2/x^2)^(5/2),x)
 

Output:

int(exp(n*arctanh(a*x))/(c-c/a^2/x^2)^(5/2),x)
 

Fricas [F]

\[ \int \frac {e^{n \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx=\int { \frac {\left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(exp(n*arctanh(a*x))/(c-c/a^2/x^2)^(5/2),x, algorithm="fricas")
 

Output:

integral(a^6*x^6*(-(a*x + 1)/(a*x - 1))^(1/2*n)*sqrt((a^2*c*x^2 - c)/(a^2* 
x^2))/(a^6*c^3*x^6 - 3*a^4*c^3*x^4 + 3*a^2*c^3*x^2 - c^3), x)
 

Sympy [F(-2)]

Exception generated. \[ \int \frac {e^{n \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:

integrate(exp(n*atanh(a*x))/(c-c/a**2/x**2)**(5/2),x)
 

Output:

Exception raised: HeuristicGCDFailed >> no luck
                                                                                    
                                                                                    
 

Maxima [F]

\[ \int \frac {e^{n \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx=\int { \frac {\left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(exp(n*arctanh(a*x))/(c-c/a^2/x^2)^(5/2),x, algorithm="maxima")
 

Output:

integrate((-(a*x + 1)/(a*x - 1))^(1/2*n)/(c - c/(a^2*x^2))^(5/2), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{n \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(exp(n*arctanh(a*x))/(c-c/a^2/x^2)^(5/2),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{n \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx=\int \frac {{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}}{{\left (c-\frac {c}{a^2\,x^2}\right )}^{5/2}} \,d x \] Input:

int(exp(n*atanh(a*x))/(c - c/(a^2*x^2))^(5/2),x)
 

Output:

int(exp(n*atanh(a*x))/(c - c/(a^2*x^2))^(5/2), x)
 

Reduce [F]

\[ \int \frac {e^{n \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx=\frac {\left (\int \frac {e^{\mathit {atanh} \left (a x \right ) n} x^{5}}{\sqrt {a^{2} x^{2}-1}\, a^{4} x^{4}-2 \sqrt {a^{2} x^{2}-1}\, a^{2} x^{2}+\sqrt {a^{2} x^{2}-1}}d x \right ) a^{5}}{\sqrt {c}\, c^{2}} \] Input:

int(exp(n*atanh(a*x))/(c-c/a^2/x^2)^(5/2),x)
 

Output:

(int((e**(atanh(a*x)*n)*x**5)/(sqrt(a**2*x**2 - 1)*a**4*x**4 - 2*sqrt(a**2 
*x**2 - 1)*a**2*x**2 + sqrt(a**2*x**2 - 1)),x)*a**5)/(sqrt(c)*c**2)