Optimal. Leaf size=107 \[ -\frac{2 \cosh \left (a+b \log \left (c x^n\right )\right )}{b n \sqrt{\sinh \left (a+b \log \left (c x^n\right )\right )}}-\frac{2 i \sqrt{\sinh \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac{1}{2} \left (i a+i b \log \left (c x^n\right )-\frac{\pi }{2}\right )\right |2\right )}{b n \sqrt{i \sinh \left (a+b \log \left (c x^n\right )\right )}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0566298, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2636, 2640, 2639} \[ -\frac{2 \cosh \left (a+b \log \left (c x^n\right )\right )}{b n \sqrt{\sinh \left (a+b \log \left (c x^n\right )\right )}}-\frac{2 i \sqrt{\sinh \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac{1}{2} \left (i a+i b \log \left (c x^n\right )-\frac{\pi }{2}\right )\right |2\right )}{b n \sqrt{i \sinh \left (a+b \log \left (c x^n\right )\right )}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2636
Rule 2640
Rule 2639
Rubi steps
\begin{align*} \int \frac{1}{x \sinh ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\sinh ^{\frac{3}{2}}(a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac{2 \cosh \left (a+b \log \left (c x^n\right )\right )}{b n \sqrt{\sinh \left (a+b \log \left (c x^n\right )\right )}}+\frac{\operatorname{Subst}\left (\int \sqrt{\sinh (a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac{2 \cosh \left (a+b \log \left (c x^n\right )\right )}{b n \sqrt{\sinh \left (a+b \log \left (c x^n\right )\right )}}+\frac{\sqrt{\sinh \left (a+b \log \left (c x^n\right )\right )} \operatorname{Subst}\left (\int \sqrt{i \sinh (a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{n \sqrt{i \sinh \left (a+b \log \left (c x^n\right )\right )}}\\ &=-\frac{2 \cosh \left (a+b \log \left (c x^n\right )\right )}{b n \sqrt{\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 ) \sqrt{\sinh \left (a+b \log \left (c x^n\right )\right )}}{b n \sqrt{i \sinh \left (a+b \log \left (c x^n\right )\right )}}\\ \end{align*}
Mathematica [A] time = 0.0622233, size = 80, normalized size = 0.75 \[ -\frac{2 \left (\cosh \left (a+b \log \left (c x^n\right )\right )-\sqrt{i \sinh \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac{1}{4} \left (-2 i a-2 i b \log \left (c x^n\right )+\pi \right )\right |2\right )\right )}{b n \sqrt{\sinh \left (a+b \log \left (c x^n\right )\right )}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.013, size = 212, normalized size = 2. \begin{align*}{\frac{1}{bn\cosh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) } \left ( 2\,\sqrt{1-i\sinh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }\sqrt{2}\sqrt{1+i\sinh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }\sqrt{i\sinh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }{\it EllipticE} \left ( \sqrt{1-i\sinh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) },1/2\,\sqrt{2} \right ) -\sqrt{1-i\sinh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }\sqrt{2}\sqrt{1+i\sinh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }\sqrt{i\sinh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }{\it EllipticF} \left ( \sqrt{1-i\sinh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) },{\frac{\sqrt{2}}{2}} \right ) -2\, \left ( \cosh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2} \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 \sinh \left (b \log \left (c x^{n}\right ) + a\right )^{\frac{3}{2}}}\,{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 \sinh \left (b \log \left (c x^{n}\right ) + a\right )^{\frac{3}{2}}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 \sinh \left (b \log \left (c x^{n}\right ) + a\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]