Optimal. Leaf size=55 \[ \frac{\, _2F_1\left (1,-\frac{1}{b d n};1-\frac{1}{b d n};e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{x^2}-\frac{1}{2 x^2} \]
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Rubi [F] time = 0.0296067, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\coth \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\coth \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx &=\int \frac{\coth \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx\\ \end{align*}
Mathematica [B] time = 3.67831, size = 191, normalized size = 3.47 \[ \frac{-\frac{e^{2 d \left (a+b \log \left (c x^n\right )\right )} \, _2F_1\left (1,1-\frac{1}{b d n};2-\frac{1}{b d n};e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )}{b d n-1}+\, _2F_1\left (1,-\frac{1}{b d n};1-\frac{1}{b d n};e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )+\coth \left (d \left (a+b \log \left (c x^n\right )\right )\right )-\coth \left (d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )+\sinh (b d n \log (x)) \text{csch}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \text{csch}\left (d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )}{2 x^2} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 1.026, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\rm coth} \left (d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right )}{{x}^{3}}}\, dx \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*} -\frac{1}{2 \, x^{2}} - \int \frac{1}{c^{b d} x^{3} e^{\left (b d \log \left (x^{n}\right ) + a d\right )} + x^{3}}\,{d x} + \int \frac{1}{c^{b d} x^{3} e^{\left (b d \log \left (x^{n}\right ) + a d\right )} - x^{3}}\,{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{\coth \left (b d \log \left (c x^{n}\right ) + a d\right )}{x^{3}}, 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{\coth \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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