Optimal. Leaf size=158 \[ -\frac{\sin ^3\left (a+b \log \left (c x^n\right )\right )}{x \left (9 b^2 n^2+1\right )}-\frac{6 b^2 n^2 \sin \left (a+b \log \left (c x^n\right )\right )}{x \left (9 b^4 n^4+10 b^2 n^2+1\right )}-\frac{6 b^3 n^3 \cos \left (a+b \log \left (c x^n\right )\right )}{x \left (9 b^4 n^4+10 b^2 n^2+1\right )}-\frac{3 b n \sin ^2\left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{x \left (9 b^2 n^2+1\right )} \]
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Rubi [A] time = 0.0468833, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {4487, 4485} \[ -\frac{\sin ^3\left (a+b \log \left (c x^n\right )\right )}{x \left (9 b^2 n^2+1\right )}-\frac{6 b^2 n^2 \sin \left (a+b \log \left (c x^n\right )\right )}{x \left (9 b^4 n^4+10 b^2 n^2+1\right )}-\frac{6 b^3 n^3 \cos \left (a+b \log \left (c x^n\right )\right )}{x \left (9 b^4 n^4+10 b^2 n^2+1\right )}-\frac{3 b n \sin ^2\left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{x \left (9 b^2 n^2+1\right )} \]
Antiderivative was successfully verified.
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Rule 4487
Rule 4485
Rubi steps
\begin{align*} \int \frac{\sin ^3\left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx &=-\frac{3 b n \cos \left (a+b \log \left (c x^n\right )\right ) \sin ^2\left (a+b \log \left (c x^n\right )\right )}{\left (1+9 b^2 n^2\right ) x}-\frac{\sin ^3\left (a+b \log \left (c x^n\right )\right )}{\left (1+9 b^2 n^2\right ) x}+\frac{\left (6 b^2 n^2\right ) \int \frac{\sin \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx}{1+9 b^2 n^2}\\ &=-\frac{6 b^3 n^3 \cos \left (a+b \log \left (c x^n\right )\right )}{\left (1+10 b^2 n^2+9 b^4 n^4\right ) x}-\frac{6 b^2 n^2 \sin \left (a+b \log \left (c x^n\right )\right )}{\left (1+10 b^2 n^2+9 b^4 n^4\right ) x}-\frac{3 b n \cos \left (a+b \log \left (c x^n\right )\right ) \sin ^2\left (a+b \log \left (c x^n\right )\right )}{\left (1+9 b^2 n^2\right ) x}-\frac{\sin ^3\left (a+b \log \left (c x^n\right )\right )}{\left (1+9 b^2 n^2\right ) x}\\ \end{align*}
Mathematica [A] time = 0.334706, size = 125, normalized size = 0.79 \[ \frac{-3 b n \left (9 b^2 n^2+1\right ) \cos \left (a+b \log \left (c x^n\right )\right )+3 \left (b^3 n^3+b n\right ) \cos \left (3 \left (a+b \log \left (c x^n\right )\right )\right )+2 \sin \left (a+b \log \left (c x^n\right )\right ) \left (\left (b^2 n^2+1\right ) \cos \left (2 \left (a+b \log \left (c x^n\right )\right )\right )-13 b^2 n^2-1\right )}{4 x \left (9 b^4 n^4+10 b^2 n^2+1\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.063, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \sin \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{3}}{{x}^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.28572, size = 1343, normalized size = 8.5 \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 = 0.509677, size = 321, normalized size = 2.03 \begin{align*} \frac{3 \,{\left (b^{3} n^{3} + b n\right )} \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} - 3 \,{\left (3 \, b^{3} n^{3} + b n\right )} \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) -{\left (7 \, b^{2} n^{2} -{\left (b^{2} n^{2} + 1\right )} \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 1\right )} \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{{\left (9 \, b^{4} n^{4} + 10 \, b^{2} n^{2} + 1\right )} x} \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (b \log \left (c x^{n}\right ) + a\right )^{3}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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