Optimal. Leaf size=157 \[ -\frac {2 i \, _2F_1\left (1,\frac {i}{2 b d n};1+\frac {i}{2 b d n};-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d n x}+\frac {i \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d n x \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}+\frac {1+\frac {i}{b d n}}{x} \]
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Rubi [F] time = 0.06, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}
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Mathematica [A] time = 4.29, size = 184, normalized size = 1.17 \[ \frac {(2 b d n+i) \left (-i \, _2F_1\left (1,\frac {i}{2 b d n};1+\frac {i}{2 b d n};-e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+\tan \left (d \left (a+b \log \left (c x^n\right )\right )\right )+b d n\right )-e^{2 i d \left (a+b \log \left (c x^n\right )\right )} \, _2F_1\left (1,1+\frac {i}{2 b d n};2+\frac {i}{2 b d n};-e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )}{b d n x (2 b d n+i)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\tan \left (b d \log \left (c x^{n}\right ) + a d\right )^{2}}{x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.23, size = 0, normalized size = 0.00 \[ \int \frac {\tan ^{2}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {tan}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}^2}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tan ^{2}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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