Optimal. Leaf size=177 \[ -\frac {\left (m^2+2 m+9\right ) (e x)^{m+1} \, _2F_1\left (1,\frac {m+1}{4};\frac {m+5}{4};e^{2 a} x^4\right )}{4 e (m+1)}-\frac {\left (e^{2 a} x^4+1\right )^2 (e x)^{m+1}}{4 e \left (1-e^{2 a} x^4\right )^2}-\frac {e^{-2 a} \left (e^{2 a} (3-m)-e^{4 a} (m+5) x^4\right ) (e x)^{m+1}}{8 e \left (1-e^{2 a} x^4\right )}+\frac {(m+3) (m+5) (e x)^{m+1}}{8 e (m+1)} \]
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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 = {} \[ \int (e x)^m \coth ^3(a+2 \log (x)) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int (e x)^m \coth ^3(a+2 \log (x)) \, dx &=\int (e x)^m \coth ^3(a+2 \log (x)) \, dx\\ \end {align*}
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Mathematica [A] time = 0.23, size = 108, normalized size = 0.61 \[ -\frac {x (e x)^m \left (6 \, _2F_1\left (1,\frac {m+1}{4};\frac {m+5}{4};x^4 (\cosh (2 a)+\sinh (2 a))\right )-12 \, _2F_1\left (2,\frac {m+1}{4};\frac {m+5}{4};x^4 (\cosh (2 a)+\sinh (2 a))\right )+8 \, _2F_1\left (3,\frac {m+1}{4};\frac {m+5}{4};x^4 (\cosh (2 a)+\sinh (2 a))\right )-1\right )}{m+1} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.40, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (e x\right )^{m} \coth \left (a + 2 \, \log \relax (x)\right )^{3}, x\right ) \]
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 \[ \int \left (e x\right )^{m} \coth \left (a + 2 \, \log \relax (x)\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.15, size = 0, normalized size = 0.00 \[ \int \left (e x \right )^{m} \left (\coth ^{3}\left (a +2 \ln \relax (x )\right )\right )\, dx \]
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 \[ \int \left (e x\right )^{m} \coth \left (a + 2 \, \log \relax (x)\right )^{3}\,{d x} \]
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 \[ \int {\mathrm {coth}\left (a+2\,\ln \relax (x)\right )}^3\,{\left (e\,x\right )}^m \,d x \]
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 \[ \int \left (e x\right )^{m} \coth ^{3}{\left (a + 2 \log {\relax (x )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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