Optimal. Leaf size=76 \[ \frac {x^3}{6}-\frac {1}{24} e^{-2 a \sqrt {-\frac {1}{n^2}} n} x^3 \left (c x^n\right )^{3/n}-\frac {1}{4} e^{2 a \sqrt {-\frac {1}{n^2}} n} x^3 \left (c x^n\right )^{-3/n} \log (x) \]
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Rubi [A]
time = 0.05, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {4581, 4577}
\begin {gather*} -\frac {1}{24} x^3 e^{-2 a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{3/n}-\frac {1}{4} x^3 e^{2 a \sqrt {-\frac {1}{n^2}} n} \log (x) \left (c x^n\right )^{-3/n}+\frac {x^3}{6} \end {gather*}
Antiderivative was successfully verified.
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Rule 4577
Rule 4581
Rubi steps
\begin {align*} \int x^2 \sin ^2\left (a+\frac {3}{2} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx &=\frac {\left (x^3 \left (c x^n\right )^{-3/n}\right ) \text {Subst}\left (\int x^{-1+\frac {3}{n}} \sin ^2\left (a+\frac {3}{2} \sqrt {-\frac {1}{n^2}} \log (x)\right ) \, dx,x,c x^n\right )}{n}\\ &=-\frac {\left (x^3 \left (c x^n\right )^{-3/n}\right ) \text {Subst}\left (\int \left (\frac {e^{2 a \sqrt {-\frac {1}{n^2}} n}}{x}-2 x^{-1+\frac {3}{n}}+e^{-2 a \sqrt {-\frac {1}{n^2}} n} x^{-1+\frac {6}{n}}\right ) \, dx,x,c x^n\right )}{4 n}\\ &=\frac {x^3}{6}-\frac {1}{24} e^{-2 a \sqrt {-\frac {1}{n^2}} n} x^3 \left (c x^n\right )^{3/n}-\frac {1}{4} e^{2 a \sqrt {-\frac {1}{n^2}} n} x^3 \left (c x^n\right )^{-3/n} \log (x)\\ \end {align*}
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Mathematica [F]
time = 0.29, size = 0, normalized size = 0.00 \begin {gather*} \int x^2 \sin ^2\left (a+\frac {3}{2} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.07, size = 0, normalized size = 0.00 \[\int x^{2} \left (\sin ^{2}\left (a +\frac {3 \ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}}{2}\right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 47, normalized size = 0.62 \begin {gather*} -\frac {c^{\frac {6}{n}} x^{6} \cos \left (2 \, a\right ) - 4 \, c^{\frac {3}{n}} x^{3} + 6 \, \cos \left (2 \, a\right ) \log \left (x\right )}{24 \, c^{\frac {3}{n}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains complex when optimal does not.
time = 1.48, size = 59, normalized size = 0.78 \begin {gather*} -\frac {1}{24} \, {\left (x^{6} - 4 \, x^{3} e^{\left (\frac {2 i \, a n - 3 \, \log \left (c\right )}{n}\right )} + 6 \, e^{\left (\frac {2 \, {\left (2 i \, a n - 3 \, \log \left (c\right )\right )}}{n}\right )} \log \left (x\right )\right )} e^{\left (-\frac {2 i \, a n - 3 \, \log \left (c\right )}{n}\right )} \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} \sin ^{2}{\left (a + \frac {3 \sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )}}{2} \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.81, size = 1, normalized size = 0.01 \begin {gather*} +\infty \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.97, size = 92, normalized size = 1.21 \begin {gather*} \frac {x^3}{6}-\frac {x^3\,{\mathrm {e}}^{-a\,2{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{\sqrt {-\frac {1}{n^2}}\,3{}\mathrm {i}}}\,1{}\mathrm {i}}{12\,n\,\sqrt {-\frac {1}{n^2}}+12{}\mathrm {i}}+\frac {x^3\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{\sqrt {-\frac {1}{n^2}}\,3{}\mathrm {i}}\,1{}\mathrm {i}}{12\,n\,\sqrt {-\frac {1}{n^2}}-12{}\mathrm {i}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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