Optimal. Leaf size=168 \[ -\frac {2 (e x)^{m+1} \, _2F_1\left (1,\frac {m+1}{2 b d n};\frac {m+1}{2 b d n}+1;e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{b d e n}+\frac {(e x)^{m+1} \left (e^{2 a d} \left (c x^n\right )^{2 b d}+1\right )}{b d e n \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}+\frac {(e x)^{m+1} (b d n+m+1)}{b d e (m+1) n} \]
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Rubi [F] time = 0.08, 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 = {} \[ \int (e x)^m \coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int (e x)^m \coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\int (e x)^m \coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\\ \end {align*}
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Mathematica [A] time = 14.88, size = 312, normalized size = 1.86 \[ (e x)^m \left (\frac {x}{m+1}-\frac {x^{-2 m} \exp \left (-\frac {(2 m+1) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{b n}\right ) \left ((m+1) x^{2 b d n+2 m+1} \exp \left (\frac {(2 b d n+2 m+1) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{b n}\right ) \, _2F_1\left (1,\frac {m+2 b d n+1}{2 b d n};\frac {m+4 b d n+1}{2 b d n};e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )+(2 b d n+m+1) e^{\frac {(2 m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}} \, _2F_1\left (1,\frac {m+1}{2 b d n};\frac {m+1}{2 b d n}+1;e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )+(2 b d n+m+1) e^{\frac {(2 m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}} \coth \left (d \left (a+b \log \left (c x^n\right )\right )\right )\right )}{b d n (2 b d n+m+1)}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (e x\right )^{m} \coth \left (b d \log \left (c x^{n}\right ) + a d\right )^{2}, x\right ) \]
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 \[ \int \left (e x\right )^{m} \coth \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.56, size = 0, normalized size = 0.00 \[ \int \left (e x \right )^{m} \left (\coth ^{2}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\right )\, dx \]
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 \[ -e^{m} {\left (m + 1\right )} \int \frac {x^{m}}{b c^{b d} d n e^{\left (b d \log \left (x^{n}\right ) + a d\right )} + b d n}\,{d x} + e^{m} {\left (m + 1\right )} \int \frac {x^{m}}{b c^{b d} d n e^{\left (b d \log \left (x^{n}\right ) + a d\right )} - b d n}\,{d x} + \frac {b c^{2 \, b d} d e^{m} n x e^{\left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d + m \log \relax (x)\right )} - {\left (b d e^{m} n + 2 \, e^{m} {\left (m + 1\right )}\right )} x x^{m}}{{\left (m n + n\right )} b c^{2 \, b d} d e^{\left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right )} - {\left (m n + n\right )} b d} \]
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 \[ \int {\mathrm {coth}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}^2\,{\left (e\,x\right )}^m \,d x \]
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 \[ \int \left (e x\right )^{m} \coth ^{2}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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