Optimal. Leaf size=30 \[ \frac{x^3}{3}+\frac{x}{2}+2 \sin (x)-2 x \cos (x)-\frac{1}{2} \sin (x) \cos (x) \]
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Rubi [A] time = 0.0340623, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.833, Rules used = {6742, 3296, 2637, 2635, 8} \[ \frac{x^3}{3}+\frac{x}{2}+2 \sin (x)-2 x \cos (x)-\frac{1}{2} \sin (x) \cos (x) \]
Antiderivative was successfully verified.
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Rule 6742
Rule 3296
Rule 2637
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int (x+\sin (x))^2 \, dx &=\int \left (x^2+2 x \sin (x)+\sin ^2(x)\right ) \, dx\\ &=\frac{x^3}{3}+2 \int x \sin (x) \, dx+\int \sin ^2(x) \, dx\\ &=\frac{x^3}{3}-2 x \cos (x)-\frac{1}{2} \cos (x) \sin (x)+\frac{\int 1 \, dx}{2}+2 \int \cos (x) \, dx\\ &=\frac{x}{2}+\frac{x^3}{3}-2 x \cos (x)+2 \sin (x)-\frac{1}{2} \cos (x) \sin (x)\\ \end{align*}
Mathematica [A] time = 0.0578726, size = 30, normalized size = 1. \[ \frac{1}{6} x \left (2 x^2+3\right )+2 \sin (x)-\frac{1}{4} \sin (2 x)-2 x \cos (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 25, normalized size = 0.8 \begin{align*}{\frac{x}{2}}+{\frac{{x}^{3}}{3}}-2\,x\cos \left ( x \right ) +2\,\sin \left ( x \right ) -{\frac{\cos \left ( x \right ) \sin \left ( x \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.935999, size = 32, normalized size = 1.07 \begin{align*} \frac{1}{3} \, x^{3} - 2 \, x \cos \left (x\right ) + \frac{1}{2} \, x - \frac{1}{4} \, \sin \left (2 \, x\right ) + 2 \, \sin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.17985, size = 76, normalized size = 2.53 \begin{align*} \frac{1}{3} \, x^{3} - 2 \, x \cos \left (x\right ) - \frac{1}{2} \,{\left (\cos \left (x\right ) - 4\right )} \sin \left (x\right ) + \frac{1}{2} \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.192206, size = 41, normalized size = 1.37 \begin{align*} \frac{x^{3}}{3} + \frac{x \sin ^{2}{\left (x \right )}}{2} + \frac{x \cos ^{2}{\left (x \right )}}{2} - 2 x \cos{\left (x \right )} - \frac{\sin{\left (x \right )} \cos{\left (x \right )}}{2} + 2 \sin{\left (x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05383, size = 32, normalized size = 1.07 \begin{align*} \frac{1}{3} \, x^{3} - 2 \, x \cos \left (x\right ) + \frac{1}{2} \, x - \frac{1}{4} \, \sin \left (2 \, x\right ) + 2 \, \sin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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