Optimal. Leaf size=306 \[ -\frac {(e x)^{m+1} \left (2 b^2 d^2 n^2+m^2+2 m+1\right ) \, _2F_1\left (1,\frac {m+1}{2 b d n};\frac {m+1}{2 b d n}+1;e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{b^2 d^2 e (m+1) n^2}+\frac {e^{-2 a d} (e x)^{m+1} \left (\frac {e^{4 a d} (2 b d n+m+1) \left (c x^n\right )^{2 b d}}{n}+\frac {e^{2 a d} (-2 b d n+m+1)}{n}\right )}{2 b^2 d^2 e n \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}-\frac {(e x)^{m+1} \left (e^{2 a d} \left (c x^n\right )^{2 b d}+1\right )^2}{2 b d e n \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )^2}+\frac {(e x)^{m+1} (b d n+m+1) (2 b d n+m+1)}{2 b^2 d^2 e (m+1) n^2} \]
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Rubi [F] time = 0.07, 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 = {} \[ \int (e x)^m \coth ^3\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int (e x)^m \coth ^3\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\int (e x)^m \coth ^3\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\\ \end {align*}
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Mathematica [A] time = 16.83, size = 600, normalized size = 1.96 \[ -\frac {x^{-m} (e x)^m \left (2 b^2 d^2 n^2+m^2+2 m+1\right ) \text {csch}\left (d \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right ) \left (\frac {\sinh \left (d \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right ) \exp \left (-\frac {(2 m+1) \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )}{b n}\right ) \left ((2 b d n+m+1) \exp \left (\frac {2 a m+a+b (2 m+1) \left (\log \left (c x^n\right )-n \log (x)\right )+b (m+1) n \log (x)}{b n}\right ) \, _2F_1\left (1,\frac {m+1}{2 b d n};\frac {m+1}{2 b d n}+1;e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )+(m+1) \exp \left (\frac {a (2 b d n+2 m+1)}{b n}+\frac {(2 b d n+2 m+1) \left (\log \left (c x^n\right )-n \log (x)\right )}{n}+\log (x) (2 b d n+m+1)\right ) \, _2F_1\left (1,\frac {m+2 b d n+1}{2 b d n};\frac {m+4 b d n+1}{2 b d n};e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )+(2 b d n+m+1) \coth \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \exp \left (\frac {2 a m+a+b (2 m+1) \left (\log \left (c x^n\right )-n \log (x)\right )+b (m+1) n \log (x)}{b n}\right )\right )}{(m+1) (2 b d n+m+1)}+\frac {x^{m+1} \sinh (b d n \log (x)) \text {csch}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{m+1}\right )}{2 b^2 d^2 n^2}+\frac {(m+1) x (e x)^m \sinh (b d n \log (x)) \text {csch}\left (d \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right ) \text {csch}\left (d \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )+b d n \log (x)\right )}{2 b^2 d^2 n^2}+\frac {x (e x)^m \coth \left (d \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right )}{m+1}-\frac {x (e x)^m \text {csch}^2\left (d \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )+b d n \log (x)\right )}{2 b d n} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (e x\right )^{m} \coth \left (b d \log \left (c x^{n}\right ) + a d\right )^{3}, x\right ) \]
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 \[ \int \left (e x\right )^{m} \coth \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.69, size = 0, normalized size = 0.00 \[ \int \left (e x \right )^{m} \left (\coth ^{3}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\right )\, dx \]
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 \[ -{\left (2 \, b^{2} d^{2} e^{m} n^{2} + {\left (m^{2} + 2 \, m + 1\right )} e^{m}\right )} \int \frac {x^{m}}{2 \, {\left (b^{2} c^{b d} d^{2} n^{2} e^{\left (b d \log \left (x^{n}\right ) + a d\right )} + b^{2} d^{2} n^{2}\right )}}\,{d x} + {\left (2 \, b^{2} d^{2} e^{m} n^{2} + {\left (m^{2} + 2 \, m + 1\right )} e^{m}\right )} \int \frac {x^{m}}{2 \, {\left (b^{2} c^{b d} d^{2} n^{2} e^{\left (b d \log \left (x^{n}\right ) + a d\right )} - b^{2} d^{2} n^{2}\right )}}\,{d x} + \frac {b^{2} c^{4 \, b d} d^{2} e^{m} n^{2} x e^{\left (4 \, b d \log \left (x^{n}\right ) + 4 \, a d + m \log \relax (x)\right )} + {\left (b^{2} d^{2} e^{m} n^{2} + {\left (m^{2} + 2 \, m + 1\right )} e^{m}\right )} x x^{m} - {\left (2 \, b^{2} c^{2 \, b d} d^{2} e^{m} n^{2} e^{\left (2 \, a d\right )} + 2 \, {\left (m n + n\right )} b c^{2 \, b d} d e^{m} e^{\left (2 \, a d\right )} + {\left (m^{2} + 2 \, m + 1\right )} c^{2 \, b d} e^{m} e^{\left (2 \, a d\right )}\right )} x e^{\left (2 \, b d \log \left (x^{n}\right ) + m \log \relax (x)\right )}}{{\left (m n^{2} + n^{2}\right )} b^{2} c^{4 \, b d} d^{2} e^{\left (4 \, b d \log \left (x^{n}\right ) + 4 \, a d\right )} - 2 \, {\left (m n^{2} + n^{2}\right )} b^{2} c^{2 \, b d} d^{2} e^{\left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right )} + {\left (m n^{2} + n^{2}\right )} b^{2} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\mathrm {coth}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}^3\,{\left (e\,x\right )}^m \,d x \]
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 \[ \int \left (e x\right )^{m} \coth ^{3}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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