Optimal. Leaf size=149 \[ \frac{6 b^3 n^3 x \sin \left (a+b \log \left (c x^n\right )\right )}{9 b^4 n^4+10 b^2 n^2+1}+\frac{x \cos ^3\left (a+b \log \left (c x^n\right )\right )}{9 b^2 n^2+1}+\frac{6 b^2 n^2 x \cos \left (a+b \log \left (c x^n\right )\right )}{9 b^4 n^4+10 b^2 n^2+1}+\frac{3 b n x \sin \left (a+b \log \left (c x^n\right )\right ) \cos ^2\left (a+b \log \left (c x^n\right )\right )}{9 b^2 n^2+1} \]
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Rubi [A] time = 0.036033, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {4478, 4476} \[ \frac{6 b^3 n^3 x \sin \left (a+b \log \left (c x^n\right )\right )}{9 b^4 n^4+10 b^2 n^2+1}+\frac{x \cos ^3\left (a+b \log \left (c x^n\right )\right )}{9 b^2 n^2+1}+\frac{6 b^2 n^2 x \cos \left (a+b \log \left (c x^n\right )\right )}{9 b^4 n^4+10 b^2 n^2+1}+\frac{3 b n x \sin \left (a+b \log \left (c x^n\right )\right ) \cos ^2\left (a+b \log \left (c x^n\right )\right )}{9 b^2 n^2+1} \]
Antiderivative was successfully verified.
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Rule 4478
Rule 4476
Rubi steps
\begin{align*} \int \cos ^3\left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{x \cos ^3\left (a+b \log \left (c x^n\right )\right )}{1+9 b^2 n^2}+\frac{3 b n x \cos ^2\left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{1+9 b^2 n^2}+\frac{\left (6 b^2 n^2\right ) \int \cos \left (a+b \log \left (c x^n\right )\right ) \, dx}{1+9 b^2 n^2}\\ &=\frac{6 b^2 n^2 x \cos \left (a+b \log \left (c x^n\right )\right )}{1+10 b^2 n^2+9 b^4 n^4}+\frac{x \cos ^3\left (a+b \log \left (c x^n\right )\right )}{1+9 b^2 n^2}+\frac{6 b^3 n^3 x \sin \left (a+b \log \left (c x^n\right )\right )}{1+10 b^2 n^2+9 b^4 n^4}+\frac{3 b n x \cos ^2\left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{1+9 b^2 n^2}\\ \end{align*}
Mathematica [A] time = 0.41392, size = 117, normalized size = 0.79 \[ \frac{x \left (3 \left (9 b^2 n^2+1\right ) \cos \left (a+b \log \left (c x^n\right )\right )+\left (b^2 n^2+1\right ) \cos \left (3 \left (a+b \log \left (c x^n\right )\right )\right )+6 b n \sin \left (a+b \log \left (c x^n\right )\right ) \left (\left (b^2 n^2+1\right ) \cos \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+5 b^2 n^2+1\right )\right )}{36 b^4 n^4+40 b^2 n^2+4} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.095, size = 0, normalized size = 0. \begin{align*} \int \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 = 1.22745, size = 1335, normalized size = 8.96 \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 = 0.500971, size = 308, normalized size = 2.07 \begin{align*} \frac{6 \, b^{2} n^{2} x \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) +{\left (b^{2} n^{2} + 1\right )} x \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + 3 \,{\left (2 \, b^{3} n^{3} x +{\left (b^{3} n^{3} + b n\right )} x \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} + 10 \, b^{2} n^{2} + 1} \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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