Optimal. Leaf size=210 \[ \frac{3 b^2 n^2 x^2 \sin ^2\left (a+b \log \left (c x^n\right )\right )}{2 \left (4 b^4 n^4+5 b^2 n^2+1\right )}+\frac{x^2 \sin ^4\left (a+b \log \left (c x^n\right )\right )}{2 \left (4 b^2 n^2+1\right )}-\frac{b n x^2 \sin ^3\left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{4 b^2 n^2+1}-\frac{3 b^3 n^3 x^2 \sin \left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{2 \left (4 b^4 n^4+5 b^2 n^2+1\right )}+\frac{3 b^4 n^4 x^2}{4 \left (4 b^4 n^4+5 b^2 n^2+1\right )} \]
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Rubi [A] time = 0.0609191, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {4487, 30} \[ \frac{3 b^2 n^2 x^2 \sin ^2\left (a+b \log \left (c x^n\right )\right )}{2 \left (4 b^4 n^4+5 b^2 n^2+1\right )}+\frac{x^2 \sin ^4\left (a+b \log \left (c x^n\right )\right )}{2 \left (4 b^2 n^2+1\right )}-\frac{b n x^2 \sin ^3\left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{4 b^2 n^2+1}-\frac{3 b^3 n^3 x^2 \sin \left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{2 \left (4 b^4 n^4+5 b^2 n^2+1\right )}+\frac{3 b^4 n^4 x^2}{4 \left (4 b^4 n^4+5 b^2 n^2+1\right )} \]
Antiderivative was successfully verified.
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Rule 4487
Rule 30
Rubi steps
\begin{align*} \int x \sin ^4\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 ) \sin ^3\left (a+b \log \left (c x^n\right )\right )}{1+4 b^2 n^2}+\frac{x^2 \sin ^4\left (a+b \log \left (c x^n\right )\right )}{2 \left (1+4 b^2 n^2\right )}+\frac{\left (3 b^2 n^2\right ) \int x \sin ^2\left (a+b \log \left (c x^n\right )\right ) \, dx}{1+4 b^2 n^2}\\ &=-\frac{3 b^3 n^3 x^2 \cos \left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{2 \left (1+5 b^2 n^2+4 b^4 n^4\right )}+\frac{3 b^2 n^2 x^2 \sin ^2\left (a+b \log \left (c x^n\right )\right )}{2 \left (1+5 b^2 n^2+4 b^4 n^4\right )}-\frac{b n x^2 \cos \left (a+b \log \left (c x^n\right )\right ) \sin ^3\left (a+b \log \left (c x^n\right )\right )}{1+4 b^2 n^2}+\frac{x^2 \sin ^4\left (a+b \log \left (c x^n\right )\right )}{2 \left (1+4 b^2 n^2\right )}+\frac{\left (3 b^4 n^4\right ) \int x \, dx}{2 \left (1+5 b^2 n^2+4 b^4 n^4\right )}\\ &=\frac{3 b^4 n^4 x^2}{4 \left (1+5 b^2 n^2+4 b^4 n^4\right )}-\frac{3 b^3 n^3 x^2 \cos \left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{2 \left (1+5 b^2 n^2+4 b^4 n^4\right )}+\frac{3 b^2 n^2 x^2 \sin ^2\left (a+b \log \left (c x^n\right )\right )}{2 \left (1+5 b^2 n^2+4 b^4 n^4\right )}-\frac{b n x^2 \cos \left (a+b \log \left (c x^n\right )\right ) \sin ^3\left (a+b \log \left (c x^n\right )\right )}{1+4 b^2 n^2}+\frac{x^2 \sin ^4\left (a+b \log \left (c x^n\right )\right )}{2 \left (1+4 b^2 n^2\right )}\\ \end{align*}
Mathematica [A] time = 0.435803, size = 169, normalized size = 0.8 \[ \frac{x^2 \left (-16 b^3 n^3 \sin \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+2 b^3 n^3 \sin \left (4 \left (a+b \log \left (c x^n\right )\right )\right )-4 \left (4 b^2 n^2+1\right ) \cos \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+\left (b^2 n^2+1\right ) \cos \left (4 \left (a+b \log \left (c x^n\right )\right )\right )-4 b n \sin \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+2 b n \sin \left (4 \left (a+b \log \left (c x^n\right )\right )\right )+12 b^4 n^4+15 b^2 n^2+3\right )}{16 \left (4 b^4 n^4+5 b^2 n^2+1\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.079, size = 0, normalized size = 0. \begin{align*} \int x \left ( \sin \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{4}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.41093, size = 1465, normalized size = 6.98 \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 = 0.518291, size = 443, normalized size = 2.11 \begin{align*} \frac{2 \,{\left (b^{2} n^{2} + 1\right )} x^{2} \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{4} - 2 \,{\left (5 \, b^{2} n^{2} + 2\right )} x^{2} \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} +{\left (3 \, b^{4} n^{4} + 8 \, b^{2} n^{2} + 2\right )} x^{2} + 2 \,{\left (2 \,{\left (b^{3} n^{3} + b n\right )} x^{2} \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} -{\left (5 \, b^{3} n^{3} + 2 \, b n\right )} x^{2} \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )} \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{4 \,{\left (4 \, b^{4} n^{4} + 5 \, b^{2} n^{2} + 1\right )}} \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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