Optimal. Leaf size=135 \[ -\frac {1-\frac {1}{b d n}}{x}+\frac {1-e^{2 a d} \left (c x^n\right )^{2 b d}}{b d n x \left (1+e^{2 a d} \left (c x^n\right )^{2 b d}\right )}-\frac {2 \, _2F_1\left (1,-\frac {1}{2 b d n};1-\frac {1}{2 b d n};-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{b d n x} \]
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Rubi [A]
time = 0.12, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {5658, 5656,
516, 470, 371} \begin {gather*} -\frac {2 \, _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 x}+\frac {1-e^{2 a d} \left (c x^n\right )^{2 b d}}{b d n x \left (e^{2 a d} \left (c x^n\right )^{2 b d}+1\right )}-\frac {1-\frac {1}{b d n}}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 470
Rule 516
Rule 5656
Rule 5658
Rubi steps
\begin {align*} \int \frac {\tanh ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx &=\int \frac {\tanh ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx\\ \end {align*}
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Mathematica [A]
time = 2.43, size = 162, normalized size = 1.20 \begin {gather*} -\frac {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+\, _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 )+\tanh \left (d \left (a+b \log \left (c x^n\right )\right )\right )\right )}{b d n (-1+2 b d n) x} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.37, size = 0, normalized size = 0.00 \[\int \frac {\tanh ^{2}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{x^{2}}\, dx\]
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 \frac {\tanh ^{2}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}}{x^{2}}\, 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 \frac {{\mathrm {tanh}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}^2}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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