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 = 0.03, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}
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Mathematica [B] time = 3.76, 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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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\coth \left (b d \log \left (c x^{n}\right ) + a d\right )}{x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\coth \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.13, size = 0, normalized size = 0.00 \[ \int \frac {\coth \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{2 \, x^{2}} - \int \frac {1}{c^{b d} x^{3} e^{\left (b d \log \left (x^{n}\right ) + a d\right )} + x^{3}}\,{d x} + \int \frac {1}{c^{b d} x^{3} e^{\left (b d \log \left (x^{n}\right ) + a d\right )} - x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\mathrm {coth}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\coth {\left (a d + b d \log {\left (c x^{n} \right )} \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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