\(\int \frac {(a+b \text {csch}^{-1}(c+d x))^2}{e+f x} \, dx\) [11]

Optimal result

Integrand size = 20, antiderivative size = 475 \[ \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{e+f x} \, dx=-\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2 \log \left (1-e^{2 \text {csch}^{-1}(c+d x)}\right )}{f}+\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2 \log \left (1+\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{f}+\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2 \log \left (1+\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{f}-\frac {b \left (a+b \text {csch}^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (2,e^{2 \text {csch}^{-1}(c+d x)}\right )}{f}+\frac {2 b \left (a+b \text {csch}^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (2,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{f}+\frac {2 b \left (a+b \text {csch}^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (2,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{f}+\frac {b^2 \operatorname {PolyLog}\left (3,e^{2 \text {csch}^{-1}(c+d x)}\right )}{2 f}-\frac {2 b^2 \operatorname {PolyLog}\left (3,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{f}-\frac {2 b^2 \operatorname {PolyLog}\left (3,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{f} \] Output:

-(a+b*arccsch(d*x+c))^2*ln(1-(1/(d*x+c)+(1+1/(d*x+c)^2)^(1/2))^2)/f+(a+b*a 
rccsch(d*x+c))^2*ln(1+(1/(d*x+c)+(1+1/(d*x+c)^2)^(1/2))*(-c*f+d*e)/(f-(d^2 
*e^2-2*c*d*e*f+(c^2+1)*f^2)^(1/2)))/f+(a+b*arccsch(d*x+c))^2*ln(1+(1/(d*x+ 
c)+(1+1/(d*x+c)^2)^(1/2))*(-c*f+d*e)/(f+(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)^(1 
/2)))/f-b*(a+b*arccsch(d*x+c))*polylog(2,(1/(d*x+c)+(1+1/(d*x+c)^2)^(1/2)) 
^2)/f+2*b*(a+b*arccsch(d*x+c))*polylog(2,-(1/(d*x+c)+(1+1/(d*x+c)^2)^(1/2) 
)*(-c*f+d*e)/(f-(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)^(1/2)))/f+2*b*(a+b*arccsch 
(d*x+c))*polylog(2,-(1/(d*x+c)+(1+1/(d*x+c)^2)^(1/2))*(-c*f+d*e)/(f+(d^2*e 
^2-2*c*d*e*f+(c^2+1)*f^2)^(1/2)))/f+1/2*b^2*polylog(3,(1/(d*x+c)+(1+1/(d*x 
+c)^2)^(1/2))^2)/f-2*b^2*polylog(3,-(1/(d*x+c)+(1+1/(d*x+c)^2)^(1/2))*(-c* 
f+d*e)/(f-(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)^(1/2)))/f-2*b^2*polylog(3,-(1/(d 
*x+c)+(1+1/(d*x+c)^2)^(1/2))*(-c*f+d*e)/(f+(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2) 
^(1/2)))/f
 

Mathematica [F]

\[ \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{e+f x} \, dx=\int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{e+f x} \, dx \] Input:

Integrate[(a + b*ArcCsch[c + d*x])^2/(e + f*x),x]
 

Output:

Integrate[(a + b*ArcCsch[c + d*x])^2/(e + f*x), x]
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 2.60 (sec) , antiderivative size = 568, normalized size of antiderivative = 1.20, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6876, 6130, 6103, 3042, 26, 4199, 25, 2620, 3011, 2720, 6095, 2620, 3011, 2720, 7143}

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.

\(\displaystyle \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{e+f x} \, dx\)

\(\Big \downarrow \) 6876

\(\displaystyle -\int \frac {(c+d x)^2 \sqrt {1+\frac {1}{(c+d x)^2}} \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{d e-c f+f (c+d x)}d\text {csch}^{-1}(c+d x)\)

\(\Big \downarrow \) 6130

\(\displaystyle -\int \frac {(c+d x) \sqrt {1+\frac {1}{(c+d x)^2}} \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{f+\frac {d e-c f}{c+d x}}d\text {csch}^{-1}(c+d x)\)

\(\Big \downarrow \) 6103

\(\displaystyle \frac {(d e-c f) \int \frac {\sqrt {1+\frac {1}{(c+d x)^2}} \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{f+\frac {d e-c f}{c+d x}}d\text {csch}^{-1}(c+d x)}{f}-\frac {\int (c+d x) \sqrt {1+\frac {1}{(c+d x)^2}} \left (a+b \text {csch}^{-1}(c+d x)\right )^2d\text {csch}^{-1}(c+d x)}{f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {(d e-c f) \int \frac {\sqrt {1+\frac {1}{(c+d x)^2}} \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{f+\frac {d e-c f}{c+d x}}d\text {csch}^{-1}(c+d x)}{f}-\frac {\int -i \left (a+b \text {csch}^{-1}(c+d x)\right )^2 \tan \left (i \text {csch}^{-1}(c+d x)+\frac {\pi }{2}\right )d\text {csch}^{-1}(c+d x)}{f}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {(d e-c f) \int \frac {\sqrt {1+\frac {1}{(c+d x)^2}} \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{f+\frac {d e-c f}{c+d x}}d\text {csch}^{-1}(c+d x)}{f}+\frac {i \int \left (a+b \text {csch}^{-1}(c+d x)\right )^2 \tan \left (i \text {csch}^{-1}(c+d x)+\frac {\pi }{2}\right )d\text {csch}^{-1}(c+d x)}{f}\)

\(\Big \downarrow \) 4199

\(\displaystyle \frac {(d e-c f) \int \frac {\sqrt {1+\frac {1}{(c+d x)^2}} \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{f+\frac {d e-c f}{c+d x}}d\text {csch}^{-1}(c+d x)}{f}+\frac {i \left (2 i \int -\frac {e^{2 \text {csch}^{-1}(c+d x)} \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{1-e^{2 \text {csch}^{-1}(c+d x)}}d\text {csch}^{-1}(c+d x)-\frac {i \left (a+b \text {csch}^{-1}(c+d x)\right )^3}{3 b}\right )}{f}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {(d e-c f) \int \frac {\sqrt {1+\frac {1}{(c+d x)^2}} \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{f+\frac {d e-c f}{c+d x}}d\text {csch}^{-1}(c+d x)}{f}+\frac {i \left (-2 i \int \frac {e^{2 \text {csch}^{-1}(c+d x)} \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{1-e^{2 \text {csch}^{-1}(c+d x)}}d\text {csch}^{-1}(c+d x)-\frac {i \left (a+b \text {csch}^{-1}(c+d x)\right )^3}{3 b}\right )}{f}\)

\(\Big \downarrow \) 2620

\(\displaystyle \frac {(d e-c f) \int \frac {\sqrt {1+\frac {1}{(c+d x)^2}} \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{f+\frac {d e-c f}{c+d x}}d\text {csch}^{-1}(c+d x)}{f}+\frac {i \left (-2 i \left (b \int \left (a+b \text {csch}^{-1}(c+d x)\right ) \log \left (1-e^{2 \text {csch}^{-1}(c+d x)}\right )d\text {csch}^{-1}(c+d x)-\frac {1}{2} \log \left (1-e^{2 \text {csch}^{-1}(c+d x)}\right ) \left (a+b \text {csch}^{-1}(c+d x)\right )^2\right )-\frac {i \left (a+b \text {csch}^{-1}(c+d x)\right )^3}{3 b}\right )}{f}\)

\(\Big \downarrow \) 3011

\(\displaystyle \frac {(d e-c f) \int \frac {\sqrt {1+\frac {1}{(c+d x)^2}} \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{f+\frac {d e-c f}{c+d x}}d\text {csch}^{-1}(c+d x)}{f}+\frac {i \left (-2 i \left (b \left (\frac {1}{2} b \int \operatorname {PolyLog}\left (2,e^{2 \text {csch}^{-1}(c+d x)}\right )d\text {csch}^{-1}(c+d x)-\frac {1}{2} \operatorname {PolyLog}\left (2,e^{2 \text {csch}^{-1}(c+d x)}\right ) \left (a+b \text {csch}^{-1}(c+d x)\right )\right )-\frac {1}{2} \log \left (1-e^{2 \text {csch}^{-1}(c+d x)}\right ) \left (a+b \text {csch}^{-1}(c+d x)\right )^2\right )-\frac {i \left (a+b \text {csch}^{-1}(c+d x)\right )^3}{3 b}\right )}{f}\)

\(\Big \downarrow \) 2720

\(\displaystyle \frac {(d e-c f) \int \frac {\sqrt {1+\frac {1}{(c+d x)^2}} \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{f+\frac {d e-c f}{c+d x}}d\text {csch}^{-1}(c+d x)}{f}+\frac {i \left (-2 i \left (b \left (\frac {1}{4} b \int e^{-2 \text {csch}^{-1}(c+d x)} \operatorname {PolyLog}\left (2,e^{2 \text {csch}^{-1}(c+d x)}\right )de^{2 \text {csch}^{-1}(c+d x)}-\frac {1}{2} \operatorname {PolyLog}\left (2,e^{2 \text {csch}^{-1}(c+d x)}\right ) \left (a+b \text {csch}^{-1}(c+d x)\right )\right )-\frac {1}{2} \log \left (1-e^{2 \text {csch}^{-1}(c+d x)}\right ) \left (a+b \text {csch}^{-1}(c+d x)\right )^2\right )-\frac {i \left (a+b \text {csch}^{-1}(c+d x)\right )^3}{3 b}\right )}{f}\)

\(\Big \downarrow \) 6095

\(\displaystyle \frac {(d e-c f) \left (\int \frac {e^{\text {csch}^{-1}(c+d x)} \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{f+e^{\text {csch}^{-1}(c+d x)} (d e-c f)-\sqrt {d^2 e^2-2 c d f e+c^2 f^2+f^2}}d\text {csch}^{-1}(c+d x)+\int \frac {e^{\text {csch}^{-1}(c+d x)} \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{f+e^{\text {csch}^{-1}(c+d x)} (d e-c f)+\sqrt {d^2 e^2-2 c d f e+c^2 f^2+f^2}}d\text {csch}^{-1}(c+d x)-\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^3}{3 b (d e-c f)}\right )}{f}+\frac {i \left (-2 i \left (b \left (\frac {1}{4} b \int e^{-2 \text {csch}^{-1}(c+d x)} \operatorname {PolyLog}\left (2,e^{2 \text {csch}^{-1}(c+d x)}\right )de^{2 \text {csch}^{-1}(c+d x)}-\frac {1}{2} \operatorname {PolyLog}\left (2,e^{2 \text {csch}^{-1}(c+d x)}\right ) \left (a+b \text {csch}^{-1}(c+d x)\right )\right )-\frac {1}{2} \log \left (1-e^{2 \text {csch}^{-1}(c+d x)}\right ) \left (a+b \text {csch}^{-1}(c+d x)\right )^2\right )-\frac {i \left (a+b \text {csch}^{-1}(c+d x)\right )^3}{3 b}\right )}{f}\)

\(\Big \downarrow \) 2620

\(\displaystyle \frac {(d e-c f) \left (-\frac {2 b \int \left (a+b \text {csch}^{-1}(c+d x)\right ) \log \left (\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt {d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}}+1\right )d\text {csch}^{-1}(c+d x)}{d e-c f}-\frac {2 b \int \left (a+b \text {csch}^{-1}(c+d x)\right ) \log \left (\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt {d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}}+1\right )d\text {csch}^{-1}(c+d x)}{d e-c f}+\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2 \log \left (\frac {(d e-c f) e^{\text {csch}^{-1}(c+d x)}}{f-\sqrt {\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}}+1\right )}{d e-c f}+\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2 \log \left (\frac {(d e-c f) e^{\text {csch}^{-1}(c+d x)}}{\sqrt {\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}+f}+1\right )}{d e-c f}-\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^3}{3 b (d e-c f)}\right )}{f}+\frac {i \left (-2 i \left (b \left (\frac {1}{4} b \int e^{-2 \text {csch}^{-1}(c+d x)} \operatorname {PolyLog}\left (2,e^{2 \text {csch}^{-1}(c+d x)}\right )de^{2 \text {csch}^{-1}(c+d x)}-\frac {1}{2} \operatorname {PolyLog}\left (2,e^{2 \text {csch}^{-1}(c+d x)}\right ) \left (a+b \text {csch}^{-1}(c+d x)\right )\right )-\frac {1}{2} \log \left (1-e^{2 \text {csch}^{-1}(c+d x)}\right ) \left (a+b \text {csch}^{-1}(c+d x)\right )^2\right )-\frac {i \left (a+b \text {csch}^{-1}(c+d x)\right )^3}{3 b}\right )}{f}\)

\(\Big \downarrow \) 3011

\(\displaystyle \frac {(d e-c f) \left (-\frac {2 b \left (b \int \operatorname {PolyLog}\left (2,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt {d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}}\right )d\text {csch}^{-1}(c+d x)-\left (a+b \text {csch}^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (2,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt {d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}}\right )\right )}{d e-c f}-\frac {2 b \left (b \int \operatorname {PolyLog}\left (2,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt {d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}}\right )d\text {csch}^{-1}(c+d x)-\left (a+b \text {csch}^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (2,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt {d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}}\right )\right )}{d e-c f}+\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2 \log \left (\frac {(d e-c f) e^{\text {csch}^{-1}(c+d x)}}{f-\sqrt {\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}}+1\right )}{d e-c f}+\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2 \log \left (\frac {(d e-c f) e^{\text {csch}^{-1}(c+d x)}}{\sqrt {\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}+f}+1\right )}{d e-c f}-\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^3}{3 b (d e-c f)}\right )}{f}+\frac {i \left (-2 i \left (b \left (\frac {1}{4} b \int e^{-2 \text {csch}^{-1}(c+d x)} \operatorname {PolyLog}\left (2,e^{2 \text {csch}^{-1}(c+d x)}\right )de^{2 \text {csch}^{-1}(c+d x)}-\frac {1}{2} \operatorname {PolyLog}\left (2,e^{2 \text {csch}^{-1}(c+d x)}\right ) \left (a+b \text {csch}^{-1}(c+d x)\right )\right )-\frac {1}{2} \log \left (1-e^{2 \text {csch}^{-1}(c+d x)}\right ) \left (a+b \text {csch}^{-1}(c+d x)\right )^2\right )-\frac {i \left (a+b \text {csch}^{-1}(c+d x)\right )^3}{3 b}\right )}{f}\)

\(\Big \downarrow \) 2720

\(\displaystyle \frac {(d e-c f) \left (-\frac {2 b \left (b \int e^{-\text {csch}^{-1}(c+d x)} \operatorname {PolyLog}\left (2,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt {d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}}\right )de^{\text {csch}^{-1}(c+d x)}-\left (a+b \text {csch}^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (2,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt {d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}}\right )\right )}{d e-c f}-\frac {2 b \left (b \int e^{-\text {csch}^{-1}(c+d x)} \operatorname {PolyLog}\left (2,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt {d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}}\right )de^{\text {csch}^{-1}(c+d x)}-\left (a+b \text {csch}^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (2,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt {d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}}\right )\right )}{d e-c f}+\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2 \log \left (\frac {(d e-c f) e^{\text {csch}^{-1}(c+d x)}}{f-\sqrt {\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}}+1\right )}{d e-c f}+\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2 \log \left (\frac {(d e-c f) e^{\text {csch}^{-1}(c+d x)}}{\sqrt {\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}+f}+1\right )}{d e-c f}-\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^3}{3 b (d e-c f)}\right )}{f}+\frac {i \left (-2 i \left (b \left (\frac {1}{4} b \int e^{-2 \text {csch}^{-1}(c+d x)} \operatorname {PolyLog}\left (2,e^{2 \text {csch}^{-1}(c+d x)}\right )de^{2 \text {csch}^{-1}(c+d x)}-\frac {1}{2} \operatorname {PolyLog}\left (2,e^{2 \text {csch}^{-1}(c+d x)}\right ) \left (a+b \text {csch}^{-1}(c+d x)\right )\right )-\frac {1}{2} \log \left (1-e^{2 \text {csch}^{-1}(c+d x)}\right ) \left (a+b \text {csch}^{-1}(c+d x)\right )^2\right )-\frac {i \left (a+b \text {csch}^{-1}(c+d x)\right )^3}{3 b}\right )}{f}\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {(d e-c f) \left (-\frac {2 b \left (b \operatorname {PolyLog}\left (3,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt {d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}}\right )-\left (a+b \text {csch}^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (2,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt {d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}}\right )\right )}{d e-c f}-\frac {2 b \left (b \operatorname {PolyLog}\left (3,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt {d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}}\right )-\left (a+b \text {csch}^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (2,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt {d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}}\right )\right )}{d e-c f}+\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2 \log \left (\frac {(d e-c f) e^{\text {csch}^{-1}(c+d x)}}{f-\sqrt {\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}}+1\right )}{d e-c f}+\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2 \log \left (\frac {(d e-c f) e^{\text {csch}^{-1}(c+d x)}}{\sqrt {\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}+f}+1\right )}{d e-c f}-\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^3}{3 b (d e-c f)}\right )}{f}+\frac {i \left (-2 i \left (b \left (\frac {1}{4} b \operatorname {PolyLog}\left (3,e^{2 \text {csch}^{-1}(c+d x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,e^{2 \text {csch}^{-1}(c+d x)}\right ) \left (a+b \text {csch}^{-1}(c+d x)\right )\right )-\frac {1}{2} \log \left (1-e^{2 \text {csch}^{-1}(c+d x)}\right ) \left (a+b \text {csch}^{-1}(c+d x)\right )^2\right )-\frac {i \left (a+b \text {csch}^{-1}(c+d x)\right )^3}{3 b}\right )}{f}\)

Input:

Int[(a + b*ArcCsch[c + d*x])^2/(e + f*x),x]
 

Output:

(I*(((-1/3*I)*(a + b*ArcCsch[c + d*x])^3)/b - (2*I)*(-1/2*((a + b*ArcCsch[ 
c + d*x])^2*Log[1 - E^(2*ArcCsch[c + d*x])]) + b*(-1/2*((a + b*ArcCsch[c + 
 d*x])*PolyLog[2, E^(2*ArcCsch[c + d*x])]) + (b*PolyLog[3, E^(2*ArcCsch[c 
+ d*x])])/4))))/f + ((d*e - c*f)*(-1/3*(a + b*ArcCsch[c + d*x])^3/(b*(d*e 
- c*f)) + ((a + b*ArcCsch[c + d*x])^2*Log[1 + (E^ArcCsch[c + d*x]*(d*e - c 
*f))/(f - Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2])])/(d*e - c*f) + ((a + 
 b*ArcCsch[c + d*x])^2*Log[1 + (E^ArcCsch[c + d*x]*(d*e - c*f))/(f + Sqrt[ 
d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2])])/(d*e - c*f) - (2*b*(-((a + b*ArcCs 
ch[c + d*x])*PolyLog[2, -((E^ArcCsch[c + d*x]*(d*e - c*f))/(f - Sqrt[d^2*e 
^2 - 2*c*d*e*f + (1 + c^2)*f^2]))]) + b*PolyLog[3, -((E^ArcCsch[c + d*x]*( 
d*e - c*f))/(f - Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2]))]))/(d*e - c*f 
) - (2*b*(-((a + b*ArcCsch[c + d*x])*PolyLog[2, -((E^ArcCsch[c + d*x]*(d*e 
 - c*f))/(f + Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2]))]) + b*PolyLog[3, 
 -((E^ArcCsch[c + d*x]*(d*e - c*f))/(f + Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c 
^2)*f^2]))]))/(d*e - c*f)))/f
 

Defintions of rubi rules used

rule 25

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

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 2620

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

rule 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

rule 3011

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) 
*(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + 
b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F]))   Int[(f + g*x)^( 
m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e 
, f, g, n}, x] && GtQ[m, 0]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4199

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_ 
.)*(x_)], x_Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp 
[2*I   Int[((c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x 
))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && In 
tegerQ[4*k] && IGtQ[m, 0]
 

rule 6095

Int[(Cosh[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin 
h[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[-(e + f*x)^(m + 1)/(b*f*(m + 1)), 
 x] + (Int[(e + f*x)^m*(E^(c + d*x)/(a - Rt[a^2 + b^2, 2] + b*E^(c + d*x))) 
, x] + Int[(e + f*x)^m*(E^(c + d*x)/(a + Rt[a^2 + b^2, 2] + b*E^(c + d*x))) 
, x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 0]
 

rule 6103

Int[(Coth[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_ 
.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[1/a   Int[(e + f*x)^m*Coth[ 
c + d*x]^n, x], x] - Simp[b/a   Int[(e + f*x)^m*Cosh[c + d*x]*(Coth[c + d*x 
]^(n - 1)/(a + b*Sinh[c + d*x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && 
 IGtQ[m, 0] && IGtQ[n, 0]
 

rule 6130

Int[(((e_.) + (f_.)*(x_))^(m_.)*(F_)[(c_.) + (d_.)*(x_)]^(n_.)*(G_)[(c_.) + 
 (d_.)*(x_)]^(p_.))/(Csch[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> I 
nt[(e + f*x)^m*Sinh[c + d*x]*F[c + d*x]^n*(G[c + d*x]^p/(b + a*Sinh[c + d*x 
])), x] /; FreeQ[{a, b, c, d, e, f}, x] && HyperbolicQ[F] && HyperbolicQ[G] 
 && IntegersQ[m, n, p]
 

rule 6876

Int[((a_.) + ArcCsch[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^( 
m_.), x_Symbol] :> Simp[-(d^(m + 1))^(-1)   Subst[Int[(a + b*x)^p*Csch[x]*C 
oth[x]*(d*e - c*f + f*Csch[x])^m, x], x, ArcCsch[c + d*x]], x] /; FreeQ[{a, 
 b, c, d, e, f}, x] && IGtQ[p, 0] && IntegerQ[m]
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

Maple [F]

\[\int \frac {\left (a +b \,\operatorname {arccsch}\left (d x +c \right )\right )^{2}}{f x +e}d x\]

Input:

int((a+b*arccsch(d*x+c))^2/(f*x+e),x)
 

Output:

int((a+b*arccsch(d*x+c))^2/(f*x+e),x)
 

Fricas [F]

\[ \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{e+f x} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (d x + c\right ) + a\right )}^{2}}{f x + e} \,d x } \] Input:

integrate((a+b*arccsch(d*x+c))^2/(f*x+e),x, algorithm="fricas")
 

Output:

integral((b^2*arccsch(d*x + c)^2 + 2*a*b*arccsch(d*x + c) + a^2)/(f*x + e) 
, x)
 

Sympy [F]

\[ \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{e+f x} \, dx=\int \frac {\left (a + b \operatorname {acsch}{\left (c + d x \right )}\right )^{2}}{e + f x}\, dx \] Input:

integrate((a+b*acsch(d*x+c))**2/(f*x+e),x)
 

Output:

Integral((a + b*acsch(c + d*x))**2/(e + f*x), x)
                                                                                    
                                                                                    
 

Maxima [F]

\[ \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{e+f x} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (d x + c\right ) + a\right )}^{2}}{f x + e} \,d x } \] Input:

integrate((a+b*arccsch(d*x+c))^2/(f*x+e),x, algorithm="maxima")
 

Output:

a^2*log(f*x + e)/f + integrate(b^2*log(sqrt(1/(d*x + c)^2 + 1) + 1/(d*x + 
c))^2/(f*x + e) + 2*a*b*log(sqrt(1/(d*x + c)^2 + 1) + 1/(d*x + c))/(f*x + 
e), x)
 

Giac [F]

\[ \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{e+f x} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (d x + c\right ) + a\right )}^{2}}{f x + e} \,d x } \] Input:

integrate((a+b*arccsch(d*x+c))^2/(f*x+e),x, algorithm="giac")
 

Output:

integrate((b*arccsch(d*x + c) + a)^2/(f*x + e), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{e+f x} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (\frac {1}{c+d\,x}\right )\right )}^2}{e+f\,x} \,d x \] Input:

int((a + b*asinh(1/(c + d*x)))^2/(e + f*x),x)
 

Output:

int((a + b*asinh(1/(c + d*x)))^2/(e + f*x), x)
 

Reduce [F]

\[ \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{e+f x} \, dx=\frac {2 \left (\int \frac {\mathit {acsch} \left (d x +c \right )}{f x +e}d x \right ) a b f +\left (\int \frac {\mathit {acsch} \left (d x +c \right )^{2}}{f x +e}d x \right ) b^{2} f +\mathrm {log}\left (f x +e \right ) a^{2}}{f} \] Input:

int((a+b*acsch(d*x+c))^2/(f*x+e),x)
 

Output:

(2*int(acsch(c + d*x)/(e + f*x),x)*a*b*f + int(acsch(c + d*x)**2/(e + f*x) 
,x)*b**2*f + log(e + f*x)*a**2)/f