Optimal. Leaf size=157 \[ -\frac{2 i \text{Hypergeometric2F1}\left (1,\frac{i}{2 b d n},1+\frac{i}{2 b d n},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d n x}+\frac{i \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d n x \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}+\frac{1+\frac{i}{b d n}}{x} \]
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Rubi [F] time = 0.0551413, 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 = {} \[ \int \frac{\tan ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\tan ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx &=\int \frac{\tan ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx\\ \end{align*}
Mathematica [A] time = 4.31487, size = 184, normalized size = 1.17 \[ \frac{(2 b d n+i) \left (-i \text{Hypergeometric2F1}\left (1,\frac{i}{2 b d n},1+\frac{i}{2 b d n},-e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+\tan \left (d \left (a+b \log \left (c x^n\right )\right )\right )+b d n\right )-e^{2 i d \left (a+b \log \left (c x^n\right )\right )} \text{Hypergeometric2F1}\left (1,1+\frac{i}{2 b d n},2+\frac{i}{2 b d n},-e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )}{b d n x (2 b d n+i)} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.37, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \tan \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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{\tan \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{\tan ^{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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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