Optimal. Leaf size=134 \[ -\frac {2 \, _2F_1\left (1,-\frac {1}{2 b d n};1-\frac {1}{2 b d n};e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{b d n x}+\frac {e^{2 a d} \left (c x^n\right )^{2 b d}+1}{b d n x \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}-\frac {1-\frac {1}{b d n}}{x} \]
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Rubi [F] time = 0.05, 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 = {} \[ \int \frac {\coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx &=\int \frac {\coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx\\ \end {align*}
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Mathematica [A] time = 3.63, size = 158, normalized size = 1.18 \[ \frac {e^{2 d \left (a+b \log \left (c x^n\right )\right )} \, _2F_1\left (1,1-\frac {1}{2 b d n};2-\frac {1}{2 b d n};e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )-(2 b d n-1) \left (\, _2F_1\left (1,-\frac {1}{2 b d n};1-\frac {1}{2 b d n};e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )+\coth \left (d \left (a+b \log \left (c x^n\right )\right )\right )+b d n\right )}{b d n x (2 b d n-1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\coth \left (b d \log \left (c x^{n}\right ) + a d\right )^{2}}{x^{2}}, x\right ) \]
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 \[ \int \frac {\coth \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{2}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.12, size = 0, normalized size = 0.00 \[ \int \frac {\coth ^{2}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{x^{2}}\, dx \]
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 \[ -\frac {b c^{2 \, b d} d n e^{\left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right )} - b d n + 2}{b c^{2 \, b d} d n x e^{\left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right )} - b d n x} + \int \frac {1}{b c^{b d} d n x^{2} e^{\left (b d \log \left (x^{n}\right ) + a d\right )} + b d n x^{2}}\,{d x} - \int \frac {1}{b c^{b d} d n x^{2} e^{\left (b d \log \left (x^{n}\right ) + a d\right )} - b d n x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {coth}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}^2}{x^2} \,d x \]
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 \[ \int \frac {\coth ^{2}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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