Optimal. Leaf size=57 \[ \frac{2 x^2 \sin \left (a+b \log \left (c x^n\right )\right )}{b^2 n^2+4}-\frac{b n x^2 \cos \left (a+b \log \left (c x^n\right )\right )}{b^2 n^2+4} \]
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Rubi [A] time = 0.0126992, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {4485} \[ \frac{2 x^2 \sin \left (a+b \log \left (c x^n\right )\right )}{b^2 n^2+4}-\frac{b n x^2 \cos \left (a+b \log \left (c x^n\right )\right )}{b^2 n^2+4} \]
Antiderivative was successfully verified.
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Rule 4485
Rubi steps
\begin{align*} \int x \sin \left (a+b \log \left (c x^n\right )\right ) \, dx &=-\frac{b n x^2 \cos \left (a+b \log \left (c x^n\right )\right )}{4+b^2 n^2}+\frac{2 x^2 \sin \left (a+b \log \left (c x^n\right )\right )}{4+b^2 n^2}\\ \end{align*}
Mathematica [A] time = 0.0587051, size = 44, normalized size = 0.77 \[ -\frac{x^2 \left (b n \cos \left (a+b \log \left (c x^n\right )\right )-2 \sin \left (a+b \log \left (c x^n\right )\right )\right )}{b^2 n^2+4} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.033, size = 0, normalized size = 0. \begin{align*} \int x\sin \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.23481, size = 296, normalized size = 5.19 \begin{align*} -\frac{{\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 - 2 \, \cos \left (b \log \left (c\right )\right ) \sin \left (2 \, b \log \left (c\right )\right ) + 2 \, \cos \left (2 \, b \log \left (c\right )\right ) \sin \left (b \log \left (c\right )\right ) - 2 \, \sin \left (b \log \left (c\right )\right )\right )} x^{2} \cos \left (b \log \left (x^{n}\right ) + a\right ) -{\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 + 2 \, \cos \left (2 \, b \log \left (c\right )\right ) \cos \left (b \log \left (c\right )\right ) + 2 \, \sin \left (2 \, b \log \left (c\right )\right ) \sin \left (b \log \left (c\right )\right ) + 2 \, \cos \left (b \log \left (c\right )\right )\right )} x^{2} \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} + 4 \, \cos \left (b \log \left (c\right )\right )^{2} + 4 \, \sin \left (b \log \left (c\right )\right )^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.482559, size = 130, normalized size = 2.28 \begin{align*} -\frac{b n x^{2} \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) - 2 \, x^{2} \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{b^{2} n^{2} + 4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.26222, size = 1246, normalized size = 21.86 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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