Optimal. Leaf size=56 \[ \frac{x^4}{4}+\frac{3 x^2}{4}-3 x^2 \cos (x)+\frac{3 \sin ^2(x)}{4}+6 x \sin (x)+\frac{\cos ^3(x)}{3}+5 \cos (x)-\frac{3}{2} x \sin (x) \cos (x) \]
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Rubi [A] time = 0.0669768, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used = {6742, 3296, 2638, 3310, 30, 2633} \[ \frac{x^4}{4}+\frac{3 x^2}{4}-3 x^2 \cos (x)+\frac{3 \sin ^2(x)}{4}+6 x \sin (x)+\frac{\cos ^3(x)}{3}+5 \cos (x)-\frac{3}{2} x \sin (x) \cos (x) \]
Antiderivative was successfully verified.
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Rule 6742
Rule 3296
Rule 2638
Rule 3310
Rule 30
Rule 2633
Rubi steps
\begin{align*} \int (x+\sin (x))^3 \, dx &=\int \left (x^3+3 x^2 \sin (x)+3 x \sin ^2(x)+\sin ^3(x)\right ) \, dx\\ &=\frac{x^4}{4}+3 \int x^2 \sin (x) \, dx+3 \int x \sin ^2(x) \, dx+\int \sin ^3(x) \, dx\\ &=\frac{x^4}{4}-3 x^2 \cos (x)-\frac{3}{2} x \cos (x) \sin (x)+\frac{3 \sin ^2(x)}{4}+\frac{3 \int x \, dx}{2}+6 \int x \cos (x) \, dx-\operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (x)\right )\\ &=\frac{3 x^2}{4}+\frac{x^4}{4}-\cos (x)-3 x^2 \cos (x)+\frac{\cos ^3(x)}{3}+6 x \sin (x)-\frac{3}{2} x \cos (x) \sin (x)+\frac{3 \sin ^2(x)}{4}-6 \int \sin (x) \, dx\\ &=\frac{3 x^2}{4}+\frac{x^4}{4}+5 \cos (x)-3 x^2 \cos (x)+\frac{\cos ^3(x)}{3}+6 x \sin (x)-\frac{3}{2} x \cos (x) \sin (x)+\frac{3 \sin ^2(x)}{4}\\ \end{align*}
Mathematica [A] time = 0.0884516, size = 48, normalized size = 0.86 \[ \frac{1}{24} \left (6 x \left (x^3+3 x+24 \sin (x)-3 \sin (2 x)\right )-18 \left (4 x^2-7\right ) \cos (x)-9 \cos (2 x)+2 \cos (3 x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.037, size = 57, normalized size = 1. \begin{align*} -{\frac{ \left ( 2+ \left ( \sin \left ( x \right ) \right ) ^{2} \right ) \cos \left ( x \right ) }{3}}+3\,x \left ( -1/2\,\cos \left ( x \right ) \sin \left ( x \right ) +x/2 \right ) -{\frac{3\,{x}^{2}}{4}}+{\frac{3\, \left ( \sin \left ( x \right ) \right ) ^{2}}{4}}-3\,{x}^{2}\cos \left ( x \right ) +6\,\cos \left ( x \right ) +6\,x\sin \left ( x \right ) +{\frac{{x}^{4}}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.1164, size = 65, normalized size = 1.16 \begin{align*} \frac{1}{4} \, x^{4} + \frac{1}{3} \, \cos \left (x\right )^{3} + \frac{3}{4} \, x^{2} - 3 \,{\left (x^{2} - 2\right )} \cos \left (x\right ) - \frac{3}{4} \, x \sin \left (2 \, x\right ) + 6 \, x \sin \left (x\right ) - \frac{3}{8} \, \cos \left (2 \, x\right ) - \cos \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.02615, size = 135, normalized size = 2.41 \begin{align*} \frac{1}{4} \, x^{4} + \frac{1}{3} \, \cos \left (x\right )^{3} + \frac{3}{4} \, x^{2} -{\left (3 \, x^{2} - 5\right )} \cos \left (x\right ) - \frac{3}{4} \, \cos \left (x\right )^{2} - \frac{3}{2} \,{\left (x \cos \left (x\right ) - 4 \, x\right )} \sin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.384255, size = 85, normalized size = 1.52 \begin{align*} \frac{x^{4}}{4} + \frac{3 x^{2} \sin ^{2}{\left (x \right )}}{4} + \frac{3 x^{2} \cos ^{2}{\left (x \right )}}{4} - 3 x^{2} \cos{\left (x \right )} - \frac{3 x \sin{\left (x \right )} \cos{\left (x \right )}}{2} + 6 x \sin{\left (x \right )} - \sin ^{2}{\left (x \right )} \cos{\left (x \right )} - \frac{2 \cos ^{3}{\left (x \right )}}{3} - \frac{3 \cos ^{2}{\left (x \right )}}{4} + 6 \cos{\left (x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18936, size = 62, normalized size = 1.11 \begin{align*} \frac{1}{4} \, x^{4} + \frac{3}{4} \, x^{2} - \frac{3}{4} \,{\left (4 \, x^{2} - 7\right )} \cos \left (x\right ) - \frac{3}{4} \, x \sin \left (2 \, x\right ) + 6 \, x \sin \left (x\right ) + \frac{1}{12} \, \cos \left (3 \, x\right ) - \frac{3}{8} \, \cos \left (2 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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