Optimal. Leaf size=56 \[ \frac{x^4}{4}+\frac{3 x^2}{4}+3 x^2 \sin (x)-\frac{\sin ^3(x)}{3}-5 \sin (x)+\frac{3 \cos ^2(x)}{4}+6 x \cos (x)+\frac{3}{2} x \sin (x) \cos (x) \]
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Rubi [A] time = 0.0694327, 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, 2637, 3310, 30, 2633} \[ \frac{x^4}{4}+\frac{3 x^2}{4}+3 x^2 \sin (x)-\frac{\sin ^3(x)}{3}-5 \sin (x)+\frac{3 \cos ^2(x)}{4}+6 x \cos (x)+\frac{3}{2} x \sin (x) \cos (x) \]
Antiderivative was successfully verified.
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Rule 6742
Rule 3296
Rule 2637
Rule 3310
Rule 30
Rule 2633
Rubi steps
\begin{align*} \int (x+\cos (x))^3 \, dx &=\int \left (x^3+3 x^2 \cos (x)+3 x \cos ^2(x)+\cos ^3(x)\right ) \, dx\\ &=\frac{x^4}{4}+3 \int x^2 \cos (x) \, dx+3 \int x \cos ^2(x) \, dx+\int \cos ^3(x) \, dx\\ &=\frac{x^4}{4}+\frac{3 \cos ^2(x)}{4}+3 x^2 \sin (x)+\frac{3}{2} x \cos (x) \sin (x)+\frac{3 \int x \, dx}{2}-6 \int x \sin (x) \, dx-\operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (x)\right )\\ &=\frac{3 x^2}{4}+\frac{x^4}{4}+6 x \cos (x)+\frac{3 \cos ^2(x)}{4}+\sin (x)+3 x^2 \sin (x)+\frac{3}{2} x \cos (x) \sin (x)-\frac{\sin ^3(x)}{3}-6 \int \cos (x) \, dx\\ &=\frac{3 x^2}{4}+\frac{x^4}{4}+6 x \cos (x)+\frac{3 \cos ^2(x)}{4}-5 \sin (x)+3 x^2 \sin (x)+\frac{3}{2} x \cos (x) \sin (x)-\frac{\sin ^3(x)}{3}\\ \end{align*}
Mathematica [A] time = 0.103939, size = 51, normalized size = 0.91 \[ \frac{1}{12} \left (3 x^4+9 x^2+9 \left (4 x^2-7\right ) \sin (x)+9 x \sin (2 x)+\sin (3 x)\right )+6 x \cos (x)+\frac{3}{8} \cos (2 x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.032, size = 57, normalized size = 1. \begin{align*}{\frac{ \left ( 2+ \left ( \cos \left ( x \right ) \right ) ^{2} \right ) \sin \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}\sin \left ( x \right ) -6\,\sin \left ( x \right ) +6\,x\cos \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 = 1.1555, size = 62, normalized size = 1.11 \begin{align*} \frac{1}{4} \, x^{4} - \frac{1}{3} \, \sin \left (x\right )^{3} + \frac{3}{4} \, x^{2} + 6 \, x \cos \left (x\right ) + \frac{3}{4} \, x \sin \left (2 \, x\right ) + 3 \,{\left (x^{2} - 2\right )} \sin \left (x\right ) + \frac{3}{8} \, \cos \left (2 \, x\right ) + \sin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.32278, size = 135, normalized size = 2.41 \begin{align*} \frac{1}{4} \, x^{4} + \frac{3}{4} \, x^{2} + 6 \, x \cos \left (x\right ) + \frac{3}{4} \, \cos \left (x\right )^{2} + \frac{1}{6} \,{\left (18 \, x^{2} + 9 \, x \cos \left (x\right ) + 2 \, \cos \left (x\right )^{2} - 32\right )} \sin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.5201, size = 85, normalized size = 1.52 \begin{align*} \frac{x^{4}}{4} + \frac{3 x^{2} \sin ^{2}{\left (x \right )}}{4} + 3 x^{2} \sin{\left (x \right )} + \frac{3 x^{2} \cos ^{2}{\left (x \right )}}{4} + \frac{3 x \sin{\left (x \right )} \cos{\left (x \right )}}{2} + 6 x \cos{\left (x \right )} + \frac{2 \sin ^{3}{\left (x \right )}}{3} + \sin{\left (x \right )} \cos ^{2}{\left (x \right )} - 6 \sin{\left (x \right )} + \frac{3 \cos ^{2}{\left (x \right )}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12398, size = 62, normalized size = 1.11 \begin{align*} \frac{1}{4} \, x^{4} + \frac{3}{4} \, x^{2} + 6 \, x \cos \left (x\right ) + \frac{3}{4} \, x \sin \left (2 \, x\right ) + \frac{3}{4} \,{\left (4 \, x^{2} - 7\right )} \sin \left (x\right ) + \frac{3}{8} \, \cos \left (2 \, x\right ) + \frac{1}{12} \, \sin \left (3 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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