Optimal. Leaf size=87 \[ \frac {(e x)^{1+m}}{e (1+m)}-\frac {2 (e x)^{1+m} \, _2F_1\left (1,\frac {1+m}{2 b d n};1+\frac {1+m}{2 b d n};e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{e (1+m)} \]
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Rubi [A]
time = 0.05, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {5659, 5657,
470, 371} \begin {gather*} \frac {(e x)^{m+1}}{e (m+1)}-\frac {2 (e x)^{m+1} \, _2F_1\left (1,\frac {m+1}{2 b d n};\frac {m+1}{2 b d n}+1;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 371
Rule 470
Rule 5657
Rule 5659
Rubi steps
\begin {align*} \int (e x)^m \coth \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\int (e x)^m \coth \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\\ \end {align*}
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Mathematica [A]
time = 9.10, size = 158, normalized size = 1.82 \begin {gather*} \frac {x (e x)^m \left (-\, _2F_1\left (1,\frac {1+m}{2 b d n};1+\frac {1+m}{2 b d n};e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )-\frac {e^{2 a d} (1+m) \left (c x^n\right )^{2 b d} \, _2F_1\left (1,\frac {1+m+2 b d n}{2 b d n};\frac {1+m+4 b d n}{2 b d n};e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{1+m+2 b d n}\right )}{1+m} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.57, size = 0, normalized size = 0.00 \[\int \left (e x \right )^{m} \coth \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\, dx\]
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 {\left (a d + b d \log {\left (c x^{n} \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 (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )\,{\left (e\,x\right )}^m \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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