Optimal. Leaf size=126 \[ \left (1+\frac {1}{b d n}\right ) x+\frac {x \left (1+e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{b d n \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}-\frac {2 x \, _2F_1\left (1,\frac {1}{2 b d n};1+\frac {1}{2 b d n};e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{b d n} \]
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Rubi [A]
time = 0.11, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5655, 5657,
516, 470, 371} \begin {gather*} -\frac {2 x \, _2F_1\left (1,\frac {1}{2 b d n};1+\frac {1}{2 b d n};e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{b d n}+\frac {x \left (e^{2 a d} \left (c x^n\right )^{2 b d}+1\right )}{b d n \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}+x \left (\frac {1}{b d n}+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 470
Rule 516
Rule 5655
Rule 5657
Rubi steps
\begin {align*} \int \coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\int \coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\\ \end {align*}
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Mathematica [A]
time = 5.70, size = 160, normalized size = 1.27 \begin {gather*} \frac {x \left (-e^{2 d \left (a+b \log \left (c x^n\right )\right )} \, _2F_1\left (1,1+\frac {1}{2 b d n};2+\frac {1}{2 b d n};e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )+(1+2 b d n) \left (b d n-\coth \left (d \left (a+b \log \left (c x^n\right )\right )\right )-\, _2F_1\left (1,\frac {1}{2 b d n};1+\frac {1}{2 b d n};e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )\right )\right )}{b d n (1+2 b d n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.54, size = 0, normalized size = 0.00 \[\int \coth ^{2}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\, dx\]
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 \coth ^{2}{\left (d \left (a + b \log {\left (c x^{n} \right )}\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 )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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