Optimal. Leaf size=163 \[ \frac {(3 i-b d n) x^3}{3 b d n}+\frac {i x^3 \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d n \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}-\frac {2 i x^3 \, _2F_1\left (1,-\frac {3 i}{2 b d n};1-\frac {3 i}{2 b d n};-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d n} \]
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Rubi [A]
time = 0.12, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {4593, 4591,
516, 470, 371} \begin {gather*} -\frac {2 i x^3 \, _2F_1\left (1,-\frac {3 i}{2 b d n};1-\frac {3 i}{2 b d n};-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d n}+\frac {i x^3 \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d n \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}+\frac {x^3 (-b d n+3 i)}{3 b d n} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 470
Rule 516
Rule 4591
Rule 4593
Rubi steps
\begin {align*} \int x^2 \tan ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\int x^2 \tan ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\\ \end {align*}
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Mathematica [A]
time = 6.87, size = 189, normalized size = 1.16 \begin {gather*} -\frac {x^3 \left (-9 e^{2 i d \left (a+b \log \left (c x^n\right )\right )} \, _2F_1\left (1,1-\frac {3 i}{2 b d n};2-\frac {3 i}{2 b d n};-e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+(-3 i+2 b d n) \left (b d n+3 i \, _2F_1\left (1,-\frac {3 i}{2 b d n};1-\frac {3 i}{2 b d n};-e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )-3 \tan \left (d \left (a+b \log \left (c x^n\right )\right )\right )\right )\right )}{3 b d n (-3 i+2 b d n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int x^{2} \left (\tan ^{2}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int x^{2} \tan ^{2}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \int x^2\,{\mathrm {tan}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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