Optimal. Leaf size=72 \[ -\frac{2 i E\left (\left .\frac{1}{2} \left (i a+i b \log \left (c x^n\right )-\frac{\pi }{2}\right )\right |2\right )}{b n \sqrt{i \sinh \left (a+b \log \left (c x^n\right )\right )} \sqrt{\text{csch}\left (a+b \log \left (c x^n\right )\right )}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0444987, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {3771, 2639} \[ -\frac{2 i E\left (\left .\frac{1}{2} \left (i a+i b \log \left (c x^n\right )-\frac{\pi }{2}\right )\right |2\right )}{b n \sqrt{i \sinh \left (a+b \log \left (c x^n\right )\right )} \sqrt{\text{csch}\left (a+b \log \left (c x^n\right )\right )}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3771
Rule 2639
Rubi steps
\begin{align*} \int \frac{1}{x \sqrt{\text{csch}\left (a+b \log \left (c x^n\right )\right )}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{\text{csch}(a+b x)}} \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \sqrt{i \sinh (a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{n \sqrt{\text{csch}\left (a+b \log \left (c x^n\right )\right )} \sqrt{i \sinh \left (a+b \log \left (c x^n\right )\right )}}\\ &=-\frac{2 i E\left (\left .\frac{1}{2} \left (i a-\frac{\pi }{2}+i b \log \left (c x^n\right )\right )\right |2\right )}{b n \sqrt{\text{csch}\left (a+b \log \left (c x^n\right )\right )} \sqrt{i \sinh \left (a+b \log \left (c x^n\right )\right )}}\\ \end{align*}
Mathematica [A] time = 0.0755923, size = 68, normalized size = 0.94 \[ \frac{2 \sqrt{i \sinh \left (a+b \log \left (c x^n\right )\right )} \sqrt{\text{csch}\left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac{1}{2} \left (\frac{\pi }{2}-i \left (a+b \log \left (c x^n\right )\right )\right )\right |2\right )}{b n} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.027, size = 146, normalized size = 2. \begin{align*}{\frac{\sqrt{2}}{n\cosh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) b}\sqrt{-i \left ( i+\sinh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) }\sqrt{-i \left ( -\sinh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) +i \right ) }\sqrt{i\sinh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) } \left ( 2\,{\it EllipticE} \left ( \sqrt{1-i\sinh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) },1/2\,\sqrt{2} \right ) -{\it EllipticF} \left ( \sqrt{1-i\sinh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) },{\frac{\sqrt{2}}{2}} \right ) \right ){\frac{1}{\sqrt{\sinh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \sqrt{\operatorname{csch}\left (b \log \left (c x^{n}\right ) + a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{x \sqrt{\operatorname{csch}\left (b \log \left (c x^{n}\right ) + a\right )}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \sqrt{\operatorname{csch}{\left (a + b \log{\left (c x^{n} \right )} \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \sqrt{\operatorname{csch}\left (b \log \left (c x^{n}\right ) + a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]