Optimal. Leaf size=135 \[ -\frac{2 \, _2F_1\left (1,-\frac{1}{2 b d n};1-\frac{1}{2 b d n};-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{b d n x}+\frac{1-e^{2 a d} \left (c x^n\right )^{2 b d}}{b d n x \left (e^{2 a d} \left (c x^n\right )^{2 b d}+1\right )}-\frac{1-\frac{1}{b d n}}{x} \]
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Rubi [F] time = 0.0523026, 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 = {} \[ \int \frac{\tanh ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx \]
Verification is Not applicable to the result.
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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*}
Mathematica [A] time = 3.41635, size = 162, normalized size = 1.2 \[ -\frac{e^{2 d \left (a+b \log \left (c x^n\right )\right )} \, _2F_1\left (1,1-\frac{1}{2 b d n};2-\frac{1}{2 b d n};-e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )+(2 b d n-1) \left (\, _2F_1\left (1,-\frac{1}{2 b d n};1-\frac{1}{2 b d n};-e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )+\tanh \left (d \left (a+b \log \left (c x^n\right )\right )\right )+b d n\right )}{b d n x (2 b d n-1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.102, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \tanh \left ( d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) \right ) ^{2}}{{x}^{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 e^{\left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right )} + b d n - 2}{b c^{2 \, b d} d n x e^{\left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right )} + b d n x} + 2 \, \int \frac{1}{b c^{2 \, b d} d n x^{2} e^{\left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right )} + b d n x^{2}}\,{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 (\frac{\tanh \left (b d \log \left (c x^{n}\right ) + a d\right )^{2}}{x^{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 \frac{\tanh ^{2}{\left (a d + b d \log{\left (c x^{n} \right )} \right )}}{x^{2}}\, 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 \frac{\tanh \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{2}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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