Optimal. Leaf size=56 \[ \frac {3 x^3 \cos \left (a+b \log \left (c x^n\right )\right )}{9+b^2 n^2}+\frac {b n x^3 \sin \left (a+b \log \left (c x^n\right )\right )}{9+b^2 n^2} \]
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Rubi [A]
time = 0.01, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {4574}
\begin {gather*} \frac {b n x^3 \sin \left (a+b \log \left (c x^n\right )\right )}{b^2 n^2+9}+\frac {3 x^3 \cos \left (a+b \log \left (c x^n\right )\right )}{b^2 n^2+9} \end {gather*}
Antiderivative was successfully verified.
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Rule 4574
Rubi steps
\begin {align*} \int x^2 \cos \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {3 x^3 \cos \left (a+b \log \left (c x^n\right )\right )}{9+b^2 n^2}+\frac {b n x^3 \sin \left (a+b \log \left (c x^n\right )\right )}{9+b^2 n^2}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 43, normalized size = 0.77 \begin {gather*} \frac {x^3 \left (3 \cos \left (a+b \log \left (c x^n\right )\right )+b n \sin \left (a+b \log \left (c x^n\right )\right )\right )}{9+b^2 n^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int x^{2} \cos \left (a +b \ln \left (c \,x^{n}\right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 218 vs.
\(2 (56) = 112\).
time = 0.30, size = 218, normalized size = 3.89 \begin {gather*} \frac {{\left ({\left (b \cos \left (b \log \left (c\right )\right ) \sin \left (2 \, b \log \left (c\right )\right ) - b \cos \left (2 \, b \log \left (c\right )\right ) \sin \left (b \log \left (c\right )\right ) + b \sin \left (b \log \left (c\right )\right )\right )} n + 3 \, \cos \left (2 \, b \log \left (c\right )\right ) \cos \left (b \log \left (c\right )\right ) + 3 \, \sin \left (2 \, b \log \left (c\right )\right ) \sin \left (b \log \left (c\right )\right ) + 3 \, \cos \left (b \log \left (c\right )\right )\right )} x^{3} \cos \left (b \log \left (x^{n}\right ) + a\right ) + {\left ({\left (b \cos \left (2 \, b \log \left (c\right )\right ) \cos \left (b \log \left (c\right )\right ) + b \sin \left (2 \, b \log \left (c\right )\right ) \sin \left (b \log \left (c\right )\right ) + b \cos \left (b \log \left (c\right )\right )\right )} n - 3 \, \cos \left (b \log \left (c\right )\right ) \sin \left (2 \, b \log \left (c\right )\right ) + 3 \, \cos \left (2 \, b \log \left (c\right )\right ) \sin \left (b \log \left (c\right )\right ) - 3 \, \sin \left (b \log \left (c\right )\right )\right )} x^{3} \sin \left (b \log \left (x^{n}\right ) + a\right )}{2 \, {\left ({\left (b^{2} \cos \left (b \log \left (c\right )\right )^{2} + b^{2} \sin \left (b \log \left (c\right )\right )^{2}\right )} n^{2} + 9 \, \cos \left (b \log \left (c\right )\right )^{2} + 9 \, \sin \left (b \log \left (c\right )\right )^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.07, size = 48, normalized size = 0.86 \begin {gather*} \frac {b n x^{3} \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + 3 \, x^{3} \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{b^{2} n^{2} + 9} \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*} \begin {cases} \int x^{2} \cos {\left (a - \frac {3 i \log {\left (c x^{n} \right )}}{n} \right )}\, dx & \text {for}\: b = - \frac {3 i}{n} \\\int x^{2} \cos {\left (a + \frac {3 i \log {\left (c x^{n} \right )}}{n} \right )}\, dx & \text {for}\: b = \frac {3 i}{n} \\\frac {b n x^{3} \sin {\left (a + b \log {\left (c x^{n} \right )} \right )}}{b^{2} n^{2} + 9} + \frac {3 x^{3} \cos {\left (a + b \log {\left (c x^{n} \right )} \right )}}{b^{2} n^{2} + 9} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 923 vs.
\(2 (56) = 112\).
time = 0.45, size = 923, normalized size = 16.48 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.45, size = 43, normalized size = 0.77 \begin {gather*} \frac {x^3\,\left (3\,\cos \left (a+b\,\ln \left (c\,x^n\right )\right )+b\,n\,\sin \left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{b^2\,n^2+9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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