Optimal. Leaf size=59 \[ \frac {(e x)^{m+1}}{e (m+1)}-\frac {2 (e x)^{m+1} \, _2F_1\left (1,\frac {m+1}{4};\frac {m+5}{4};e^{2 a} x^4\right )}{e (m+1)} \]
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Rubi [F] time = 0.04, 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 = {} \[ \int (e x)^m \coth (a+2 \log (x)) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int (e x)^m \coth (a+2 \log (x)) \, dx &=\int (e x)^m \coth (a+2 \log (x)) \, dx\\ \end {align*}
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Mathematica [A] time = 0.10, size = 46, normalized size = 0.78 \[ -\frac {x (e x)^m \left (2 \, _2F_1\left (1,\frac {m+1}{4};\frac {m+5}{4};x^4 (\cosh (2 a)+\sinh (2 a))\right )-1\right )}{m+1} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (e x\right )^{m} \coth \left (a + 2 \, \log \relax (x)\right ), x\right ) \]
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 \[ \int \left (e x\right )^{m} \coth \left (a + 2 \, \log \relax (x)\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.11, size = 0, normalized size = 0.00 \[ \int \left (e x \right )^{m} \coth \left (a +2 \ln \relax (x )\right )\, dx \]
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 \[ \int \left (e x\right )^{m} \coth \left (a + 2 \, \log \relax (x)\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \mathrm {coth}\left (a+2\,\ln \relax (x)\right )\,{\left (e\,x\right )}^m \,d x \]
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 \[ \int \left (e x\right )^{m} \coth {\left (a + 2 \log {\relax (x )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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