Optimal. Leaf size=53 \[ x-2 x \, _2F_1\left (1,\frac{1}{2 b d n};1+\frac{1}{2 b d n};-e^{2 a d} \left (c x^n\right )^{2 b d}\right ) \]
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Rubi [F] time = 0.0112735, 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 = {} \[ \int \tanh \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \tanh \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\int \tanh \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\\ \end{align*}
Mathematica [B] time = 8.55814, size = 126, normalized size = 2.38 \[ \frac{x e^{2 d \left (a+b \log \left (c x^n\right )\right )} \, _2F_1\left (1,1+\frac{1}{2 b d n};2+\frac{1}{2 b d n};-e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )}{2 b d n+1}-x \, _2F_1\left (1,\frac{1}{2 b d n};1+\frac{1}{2 b d n};-e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.717, size = 0, normalized size = 0. \begin{align*} \int \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*} x - 2 \, \int \frac{1}{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 (\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 \tanh{\left (d \left (a + b \log{\left (c x^{n} \right )}\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 \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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