Optimal. Leaf size=135 \[ \frac {(e x)^{1+m} \left (-1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )^p \left (1+e^{2 a d} \left (c x^n\right )^{2 b d}\right )^{-p} F_1\left (\frac {1+m}{2 b d n};p,-p;1+\frac {1+m}{2 b d n};e^{2 a d} \left (c x^n\right )^{2 b d},-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{e (1+m)} \]
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Rubi [A]
time = 0.11, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {5659, 5657,
525, 524} \begin {gather*} \frac {(e x)^{m+1} \left (-e^{2 a d} \left (c x^n\right )^{2 b d}-1\right )^p \left (e^{2 a d} \left (c x^n\right )^{2 b d}+1\right )^{-p} F_1\left (\frac {m+1}{2 b d n};p,-p;\frac {m+1}{2 b d n}+1;e^{2 a d} \left (c x^n\right )^{2 b d},-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{e (m+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 524
Rule 525
Rule 5657
Rule 5659
Rubi steps
\begin {align*} \int (e x)^m \coth ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\int (e x)^m \coth ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\\ \end {align*}
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Mathematica [A]
time = 3.47, size = 174, normalized size = 1.29 \begin {gather*} \frac {x (e x)^m \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )^p \left (1+e^{2 a d} \left (c x^n\right )^{2 b d}\right )^{-p} \left (\frac {1+e^{2 a d} \left (c x^n\right )^{2 b d}}{-1+e^{2 a d} \left (c x^n\right )^{2 b d}}\right )^p F_1\left (\frac {1+m}{2 b d n};p,-p;1+\frac {1+m}{2 b d n};e^{2 a d} \left (c x^n\right )^{2 b d},-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{1+m} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 1.06, size = 0, normalized size = 0.00 \[\int \left (e x \right )^{m} \left (\coth ^{p}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\right )\, dx\]
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 \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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 (d\,\left (a+b\,\ln \left (c\,x^n\right )\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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