Optimal. Leaf size=135 \[ -\frac {2 \, _2F_1\left (1,-\frac {1}{b d n};1-\frac {1}{b d n};e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{b d n x^2}+\frac {e^{2 a d} \left (c x^n\right )^{2 b d}+1}{b d n x^2 \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}+\frac {2-b d n}{2 b d n x^2} \]
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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 = {} \[ \int \frac {\coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx \]
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^3} \, dx &=\int \frac {\coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx\\ \end {align*}
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Mathematica [A] time = 3.59, size = 156, normalized size = 1.16 \[ \frac {2 e^{2 d \left (a+b \log \left (c x^n\right )\right )} \, _2F_1\left (1,1-\frac {1}{b d n};2-\frac {1}{b d n};e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )-(b d n-1) \left (2 \, _2F_1\left (1,-\frac {1}{b d n};1-\frac {1}{b d n};e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )+2 \coth \left (d \left (a+b \log \left (c x^n\right )\right )\right )+b d n\right )}{2 b d n x^2 (b d n-1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\coth \left (b d \log \left (c x^{n}\right ) + a d\right )^{2}}{x^{3}}, x\right ) \]
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 \[ \int \frac {\coth \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{2}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.15, size = 0, normalized size = 0.00 \[ \int \frac {\coth ^{2}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{x^{3}}\, dx \]
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 \[ -\frac {b c^{2 \, b d} d n e^{\left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right )} - b d n + 4}{2 \, {\left (b c^{2 \, b d} d n x^{2} e^{\left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right )} - b d n x^{2}\right )}} + 2 \, \int \frac {1}{b c^{b d} d n x^{3} e^{\left (b d \log \left (x^{n}\right ) + a d\right )} + b d n x^{3}}\,{d x} - 2 \, \int \frac {1}{b c^{b d} d n x^{3} e^{\left (b d \log \left (x^{n}\right ) + a d\right )} - b d n x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {coth}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}^2}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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