Optimal. Leaf size=131 \[ -\frac{2 x^2 \, _2F_1\left (1,\frac{1}{b d n};1+\frac{1}{b d n};-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{b d n}+\frac{x^2 \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{b d n \left (e^{2 a d} \left (c x^n\right )^{2 b d}+1\right )}+\frac{1}{2} x^2 \left (\frac{2}{b d n}+1\right ) \]
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Rubi [F] time = 0.0427686, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int x \tanh ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int x \tanh ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\int x \tanh ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\\ \end{align*}
Mathematica [A] time = 7.73318, size = 155, normalized size = 1.18 \[ \frac{x^2 \left (2 e^{2 d \left (a+b \log \left (c x^n\right )\right )} \, _2F_1\left (1,1+\frac{1}{b d n};2+\frac{1}{b d n};-e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )+(b d n+1) \left (-2 \, _2F_1\left (1,\frac{1}{b d n};1+\frac{1}{b d n};-e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )-2 \tanh \left (d \left (a+b \log \left (c x^n\right )\right )\right )+b d n\right )\right )}{2 b d n (b d n+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.098, size = 0, normalized size = 0. \begin{align*} \int x \left ( \tanh \left ( d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{b c^{2 \, b d} d n x^{2} e^{\left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right )} +{\left (b d n + 4\right )} x^{2}}{2 \,{\left (b c^{2 \, b d} d n e^{\left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right )} + b d n\right )}} - 4 \, \int \frac{x}{b c^{2 \, b d} d n e^{\left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right )} + b d n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x \tanh \left (b d \log \left (c x^{n}\right ) + a d\right )^{2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \tanh ^{2}{\left (a d + b d \log{\left (c x^{n} \right )} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \tanh \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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