Optimal. Leaf size=63 \[ \frac{x^3}{3}-\frac{2}{3} x^3 \, _2F_1\left (1,\frac{3}{2 b d n};1+\frac{3}{2 b d n};-e^{2 a d} \left (c x^n\right )^{2 b d}\right ) \]
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Rubi [F] time = 0.0296561, 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 = {} \[ \int x^2 \tanh \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int x^2 \tanh \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\int x^2 \tanh \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\\ \end{align*}
Mathematica [B] time = 7.56023, size = 136, normalized size = 2.16 \[ \frac{x^3 \left (3 e^{2 d \left (a+b \log \left (c x^n\right )\right )} \, _2F_1\left (1,1+\frac{3}{2 b d n};2+\frac{3}{2 b d n};-e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )-(2 b d n+3) \, _2F_1\left (1,\frac{3}{2 b d n};1+\frac{3}{2 b d n};-e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )\right )}{6 b d n+9} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.784, size = 0, normalized size = 0. \begin{align*} \int{x}^{2}\tanh \left ( d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) \, 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{1}{3} \, x^{3} - 2 \, \int \frac{x^{2}}{c^{2 \, b d} e^{\left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right )} + 1}\,{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^{2} \tanh \left (b d \log \left (c x^{n}\right ) + a d\right ), 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^{2} \tanh{\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^{2} \tanh \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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