Optimal. Leaf size=99 \[ \frac {(e x)^{1+m} \left (-1-e^{2 a} x^{2 b}\right )^p \left (1+e^{2 a} x^{2 b}\right )^{-p} F_1\left (\frac {1+m}{2 b};p,-p;1+\frac {1+m}{2 b};e^{2 a} x^{2 b},-e^{2 a} x^{2 b}\right )}{e (1+m)} \]
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Rubi [A]
time = 0.06, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5657, 525, 524}
\begin {gather*} \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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 524
Rule 525
Rule 5657
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 = 2.15, size = 126, normalized size = 1.27 \begin {gather*} \frac {x (e x)^m \left (1-e^{2 a} x^{2 b}\right )^p \left (1+e^{2 a} x^{2 b}\right )^{-p} \left (\frac {1+e^{2 a} x^{2 b}}{-1+e^{2 a} x^{2 b}}\right )^p F_1\left (\frac {1+m}{2 b};p,-p;1+\frac {1+m}{2 b};e^{2 a} x^{2 b},-e^{2 a} x^{2 b}\right )}{1+m} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.45, size = 0, normalized size = 0.00 \[\int \left (e x \right )^{m} \left (\coth ^{p}\left (a +b \ln \left (x \right )\right )\right )\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \left (e x\right )^{m} \coth ^{p}{\left (a + b \log {\left (x \right )} \right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\mathrm {coth}\left (a+b\,\ln \left (x\right )\right )}^p\,{\left (e\,x\right )}^m \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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