Optimal. Leaf size=111 \[ -\frac{2 \cosh \left (a+b \log \left (c x^n\right )\right )}{3 b n \sinh ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}+\frac{2 i \sqrt{i \sinh \left (a+b \log \left (c x^n\right )\right )} \text{EllipticF}\left (\frac{1}{2} \left (i a+i b \log \left (c x^n\right )-\frac{\pi }{2}\right ),2\right )}{3 b n \sqrt{\sinh \left (a+b \log \left (c x^n\right )\right )}} \]
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Rubi [A] time = 0.0610743, antiderivative size = 111, 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, 2642, 2641} \[ -\frac{2 \cosh \left (a+b \log \left (c x^n\right )\right )}{3 b n \sinh ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}+\frac{2 i \sqrt{i \sinh \left (a+b \log \left (c x^n\right )\right )} F\left (\left .\frac{1}{2} \left (i a+i b \log \left (c x^n\right )-\frac{\pi }{2}\right )\right |2\right )}{3 b n \sqrt{\sinh \left (a+b \log \left (c x^n\right )\right )}} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int \frac{1}{x \sinh ^{\frac{5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\sinh ^{\frac{5}{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 )}{3 b n \sinh ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{\sinh (a+b x)}} \, dx,x,\log \left (c x^n\right )\right )}{3 n}\\ &=-\frac{2 \cosh \left (a+b \log \left (c x^n\right )\right )}{3 b n \sinh ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}-\frac{\sqrt{i \sinh \left (a+b \log \left (c x^n\right )\right )} \operatorname{Subst}\left (\int \frac{1}{\sqrt{i \sinh (a+b x)}} \, dx,x,\log \left (c x^n\right )\right )}{3 n \sqrt{\sinh \left (a+b \log \left (c x^n\right )\right )}}\\ &=-\frac{2 \cosh \left (a+b \log \left (c x^n\right )\right )}{3 b n \sinh ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}+\frac{2 i F\left (\left .\frac{1}{2} \left (i a-\frac{\pi }{2}+i b \log \left (c x^n\right )\right )\right |2\right ) \sqrt{i \sinh \left (a+b \log \left (c x^n\right )\right )}}{3 b n \sqrt{\sinh \left (a+b \log \left (c x^n\right )\right )}}\\ \end{align*}
Mathematica [C] time = 0.127231, size = 122, normalized size = 1.1 \[ -\frac{2 \left (\sinh \left (a+b \log \left (c x^n\right )\right ) \sqrt{-\sinh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )-\cosh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};\cosh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+\sinh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )\right )+\cosh \left (a+b \log \left (c x^n\right )\right )\right )}{3 b n \sinh ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.055, size = 144, normalized size = 1.3 \begin{align*} -{\frac{1}{3\,bn\cosh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) } \left ( i\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 ) \sinh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) +2\, \left ( \cosh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2} \right ) \left ( \sinh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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{5}{2}}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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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{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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