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 = 0.07, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, 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 30
Rule 2633
Rule 2638
Rule 3296
Rule 3310
Rule 6742
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*}
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Mathematica [A] time = 0.09, 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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fricas [A] time = 0.49, size = 46, normalized size = 0.82 \[ \frac {1}{4} \, x^{4} + \frac {1}{3} \, \cos \relax (x)^{3} + \frac {3}{4} \, x^{2} - {\left (3 \, x^{2} - 5\right )} \cos \relax (x) - \frac {3}{4} \, \cos \relax (x)^{2} - \frac {3}{2} \, {\left (x \cos \relax (x) - 4 \, x\right )} \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 46, normalized size = 0.82 \[ \frac {1}{4} \, x^{4} + \frac {3}{4} \, x^{2} - \frac {3}{4} \, {\left (4 \, x^{2} - 7\right )} \cos \relax (x) - \frac {3}{4} \, x \sin \left (2 \, x\right ) + 6 \, x \sin \relax (x) + \frac {1}{12} \, \cos \left (3 \, x\right ) - \frac {3}{8} \, \cos \left (2 \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 57, normalized size = 1.02 \[ -\frac {\left (2+\sin ^{2}\relax (x )\right ) \cos \relax (x )}{3}+3 x \left (-\frac {\cos \relax (x ) \sin \relax (x )}{2}+\frac {x}{2}\right )-\frac {3 x^{2}}{4}+\frac {3 \left (\sin ^{2}\relax (x )\right )}{4}-3 x^{2} \cos \relax (x )+6 \cos \relax (x )+6 x \sin \relax (x )+\frac {x^{4}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 48, normalized size = 0.86 \[ \frac {1}{4} \, x^{4} + \frac {1}{3} \, \cos \relax (x)^{3} + \frac {3}{4} \, x^{2} - 3 \, {\left (x^{2} - 2\right )} \cos \relax (x) - \frac {3}{4} \, x \sin \left (2 \, x\right ) + 6 \, x \sin \relax (x) - \frac {3}{8} \, \cos \left (2 \, x\right ) - \cos \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.31, size = 46, normalized size = 0.82 \[ 5\,\cos \relax (x)-3\,x^2\,\cos \relax (x)-\frac {3\,{\cos \relax (x)}^2}{4}+\frac {{\cos \relax (x)}^3}{3}+6\,x\,\sin \relax (x)+\frac {3\,x^2}{4}+\frac {x^4}{4}-\frac {3\,x\,\cos \relax (x)\,\sin \relax (x)}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.30, size = 85, normalized size = 1.52 \[ \frac {x^{4}}{4} + \frac {3 x^{2} \sin ^{2}{\relax (x )}}{4} + \frac {3 x^{2} \cos ^{2}{\relax (x )}}{4} - 3 x^{2} \cos {\relax (x )} - \frac {3 x \sin {\relax (x )} \cos {\relax (x )}}{2} + 6 x \sin {\relax (x )} - \sin ^{2}{\relax (x )} \cos {\relax (x )} + \frac {3 \sin ^{2}{\relax (x )}}{4} - \frac {2 \cos ^{3}{\relax (x )}}{3} + 6 \cos {\relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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