Integrand size = 20, antiderivative size = 475 \[ \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{e+f x} \, dx=-\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2 \log \left (1-e^{2 \text {csch}^{-1}(c+d x)}\right )}{f}+\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2 \log \left (1+\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{f}+\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2 \log \left (1+\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{f}-\frac {b \left (a+b \text {csch}^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (2,e^{2 \text {csch}^{-1}(c+d x)}\right )}{f}+\frac {2 b \left (a+b \text {csch}^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (2,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{f}+\frac {2 b \left (a+b \text {csch}^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (2,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{f}+\frac {b^2 \operatorname {PolyLog}\left (3,e^{2 \text {csch}^{-1}(c+d x)}\right )}{2 f}-\frac {2 b^2 \operatorname {PolyLog}\left (3,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{f}-\frac {2 b^2 \operatorname {PolyLog}\left (3,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{f} \]
-(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
\[ \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{e+f x} \, dx=\int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{e+f x} \, dx \]
Result contains complex when optimal does not.
Time = 2.61 (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 |
\(\displaystyle \frac {(d e-c f) \left (-\frac {2 b \left (b \operatorname {PolyLog}\left (3,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt {d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}}\right )-\left (a+b \text {csch}^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (2,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt {d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}}\right )\right )}{d e-c f}-\frac {2 b \left (b \operatorname {PolyLog}\left (3,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt {d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}}\right )-\left (a+b \text {csch}^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (2,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt {d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}}\right )\right )}{d e-c f}+\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2 \log \left (\frac {(d e-c f) e^{\text {csch}^{-1}(c+d x)}}{f-\sqrt {\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}}+1\right )}{d e-c f}+\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2 \log \left (\frac {(d e-c f) e^{\text {csch}^{-1}(c+d x)}}{\sqrt {\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}+f}+1\right )}{d e-c f}-\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^3}{3 b (d e-c f)}\right )}{f}+\frac {i \left (-2 i \left (b \left (\frac {1}{4} b \operatorname {PolyLog}\left (3,e^{2 \text {csch}^{-1}(c+d x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,e^{2 \text {csch}^{-1}(c+d x)}\right ) \left (a+b \text {csch}^{-1}(c+d x)\right )\right )-\frac {1}{2} \log \left (1-e^{2 \text {csch}^{-1}(c+d x)}\right ) \left (a+b \text {csch}^{-1}(c+d x)\right )^2\right )-\frac {i \left (a+b \text {csch}^{-1}(c+d x)\right )^3}{3 b}\right )}{f}\) |
(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
3.1.11.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
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]
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]]]
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]
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]
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]
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]
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]
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]
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]
\[\int \frac {\left (a +b \,\operatorname {arccsch}\left (d x +c \right )\right )^{2}}{f x +e}d x\]
\[ \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{e+f x} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (d x + c\right ) + a\right )}^{2}}{f x + e} \,d x } \]
\[ \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{e+f x} \, dx=\int \frac {\left (a + b \operatorname {acsch}{\left (c + d x \right )}\right )^{2}}{e + f x}\, dx \]
\[ \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{e+f x} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (d x + c\right ) + a\right )}^{2}}{f x + e} \,d x } \]
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)
\[ \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{e+f x} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (d x + c\right ) + a\right )}^{2}}{f x + e} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{e+f x} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (\frac {1}{c+d\,x}\right )\right )}^2}{e+f\,x} \,d x \]