Optimal. Leaf size=99 \[ \frac {(e x)^{m+1} \left (-e^{2 a} x^{2 b}-1\right )^p \left (e^{2 a} x^{2 b}+1\right )^{-p} F_1\left (\frac {m+1}{2 b};p,-p;\frac {m+1}{2 b}+1;e^{2 a} x^{2 b},-e^{2 a} x^{2 b}\right )}{e (m+1)} \]
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Rubi [F] time = 0.12, 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 = {} \[ \int (e x)^m \coth ^p(a+b \log (x)) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int (e x)^m \coth ^p(a+b \log (x)) \, dx &=\int (e x)^m \coth ^p(a+b \log (x)) \, dx\\ \end {align*}
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Mathematica [A] time = 3.23, size = 126, normalized size = 1.27 \[ \frac {x (e x)^m \left (1-e^{2 a} x^{2 b}\right )^p \left (e^{2 a} x^{2 b}+1\right )^{-p} \left (\frac {e^{2 a} x^{2 b}+1}{e^{2 a} x^{2 b}-1}\right )^p F_1\left (\frac {m+1}{2 b};p,-p;\frac {m+1}{2 b}+1;e^{2 a} x^{2 b},-e^{2 a} x^{2 b}\right )}{m+1} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (e x\right )^{m} \coth \left (b \log \relax (x) + a\right )^{p}, x\right ) \]
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 \[ \int \left (e x\right )^{m} \coth \left (b \log \relax (x) + a\right )^{p}\,{d x} \]
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 \[ \int \left (e x \right )^{m} \left (\coth ^{p}\left (a +b \ln \relax (x )\right )\right )\, dx \]
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 \[ \int \left (e x\right )^{m} \coth \left (b \log \relax (x) + a\right )^{p}\,{d x} \]
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 \[ \int {\mathrm {coth}\left (a+b\,\ln \relax (x)\right )}^p\,{\left (e\,x\right )}^m \,d x \]
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 \[ \int \left (e x\right )^{m} \coth ^{p}{\left (a + b \log {\relax (x )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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