Optimal. Leaf size=158 \[ \frac{6 b^3 n^3 x^2 \sin \left (a+b \log \left (c x^n\right )\right )}{9 b^4 n^4+40 b^2 n^2+16}+\frac{2 x^2 \cos ^3\left (a+b \log \left (c x^n\right )\right )}{9 b^2 n^2+4}+\frac{12 b^2 n^2 x^2 \cos \left (a+b \log \left (c x^n\right )\right )}{9 b^4 n^4+40 b^2 n^2+16}+\frac{3 b n x^2 \sin \left (a+b \log \left (c x^n\right )\right ) \cos ^2\left (a+b \log \left (c x^n\right )\right )}{9 b^2 n^2+4} \]
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Rubi [A] time = 0.0451337, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {4488, 4486} \[ \frac{6 b^3 n^3 x^2 \sin \left (a+b \log \left (c x^n\right )\right )}{9 b^4 n^4+40 b^2 n^2+16}+\frac{2 x^2 \cos ^3\left (a+b \log \left (c x^n\right )\right )}{9 b^2 n^2+4}+\frac{12 b^2 n^2 x^2 \cos \left (a+b \log \left (c x^n\right )\right )}{9 b^4 n^4+40 b^2 n^2+16}+\frac{3 b n x^2 \sin \left (a+b \log \left (c x^n\right )\right ) \cos ^2\left (a+b \log \left (c x^n\right )\right )}{9 b^2 n^2+4} \]
Antiderivative was successfully verified.
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Rule 4488
Rule 4486
Rubi steps
\begin{align*} \int x \cos ^3\left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{2 x^2 \cos ^3\left (a+b \log \left (c x^n\right )\right )}{4+9 b^2 n^2}+\frac{3 b n x^2 \cos ^2\left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{4+9 b^2 n^2}+\frac{\left (6 b^2 n^2\right ) \int x \cos \left (a+b \log \left (c x^n\right )\right ) \, dx}{4+9 b^2 n^2}\\ &=\frac{12 b^2 n^2 x^2 \cos \left (a+b \log \left (c x^n\right )\right )}{16+40 b^2 n^2+9 b^4 n^4}+\frac{2 x^2 \cos ^3\left (a+b \log \left (c x^n\right )\right )}{4+9 b^2 n^2}+\frac{6 b^3 n^3 x^2 \sin \left (a+b \log \left (c x^n\right )\right )}{16+40 b^2 n^2+9 b^4 n^4}+\frac{3 b n x^2 \cos ^2\left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{4+9 b^2 n^2}\\ \end{align*}
Mathematica [A] time = 0.487056, size = 123, normalized size = 0.78 \[ \frac{x^2 \left (6 \left (9 b^2 n^2+4\right ) \cos \left (a+b \log \left (c x^n\right )\right )+2 \left (b^2 n^2+4\right ) \cos \left (3 \left (a+b \log \left (c x^n\right )\right )\right )+6 b n \sin \left (a+b \log \left (c x^n\right )\right ) \left (\left (b^2 n^2+4\right ) \cos \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+5 b^2 n^2+4\right )\right )}{4 \left (9 b^4 n^4+40 b^2 n^2+16\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.065, size = 0, normalized size = 0. \begin{align*} \int x \left ( \cos \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.23019, size = 1370, normalized size = 8.67 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.507962, size = 327, normalized size = 2.07 \begin{align*} \frac{12 \, b^{2} n^{2} x^{2} \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + 2 \,{\left (b^{2} n^{2} + 4\right )} x^{2} \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + 3 \,{\left (2 \, b^{3} n^{3} x^{2} +{\left (b^{3} n^{3} + 4 \, b n\right )} x^{2} \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2}\right )} \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{9 \, b^{4} n^{4} + 40 \, b^{2} n^{2} + 16} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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