Optimal. Leaf size=10 \[ \frac{1}{4} (x+\sin (x))^4 \]
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Rubi [A] time = 0.0378183, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {6686} \[ \frac{1}{4} (x+\sin (x))^4 \]
Antiderivative was successfully verified.
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Rule 6686
Rubi steps
\begin{align*} \int (1+\cos (x)) (x+\sin (x))^3 \, dx &=\frac{1}{4} (x+\sin (x))^4\\ \end{align*}
Mathematica [A] time = 0.0184468, size = 10, normalized size = 1. \[ \frac{1}{4} (x+\sin (x))^4 \]
Antiderivative was successfully verified.
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Maple [B] time = 0.036, size = 65, normalized size = 6.5 \begin{align*} \sin \left ( x \right ){x}^{3}-{\frac{3\, \left ( \cos \left ( x \right ) \right ) ^{2}{x}^{2}}{2}}+3\,x \left ( 1/2\,\cos \left ( x \right ) \sin \left ( x \right ) +x/2 \right ) -{\frac{3\,{x}^{2}}{2}}+x \left ( \sin \left ( x \right ) \right ) ^{3}+{\frac{ \left ( \sin \left ( x \right ) \right ) ^{4}}{4}}+{\frac{{x}^{4}}{4}}+3\,x \left ( -1/2\,\cos \left ( x \right ) \sin \left ( x \right ) +x/2 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.954098, size = 11, normalized size = 1.1 \begin{align*} \frac{1}{4} \,{\left (x + \sin \left (x\right )\right )}^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.03644, size = 126, normalized size = 12.6 \begin{align*} \frac{1}{4} \, x^{4} + \frac{1}{4} \, \cos \left (x\right )^{4} - \frac{1}{2} \,{\left (3 \, x^{2} + 1\right )} \cos \left (x\right )^{2} + \frac{3}{2} \, x^{2} +{\left (x^{3} - x \cos \left (x\right )^{2} + x\right )} \sin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.61257, size = 36, normalized size = 3.6 \begin{align*} \frac{x^{4}}{4} + x^{3} \sin{\left (x \right )} + \frac{3 x^{2} \sin ^{2}{\left (x \right )}}{2} + x \sin ^{3}{\left (x \right )} + \frac{\sin ^{4}{\left (x \right )}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.08073, size = 82, normalized size = 8.2 \begin{align*} \frac{1}{4} \, x^{4} + \frac{3}{4} \, x^{2} - \frac{1}{4} \,{\left (3 \, x^{2} - 1\right )} \cos \left (2 \, x\right ) - \frac{1}{4} \, x \sin \left (3 \, x\right ) + \frac{1}{4} \,{\left (4 \, x^{3} - 21 \, x\right )} \sin \left (x\right ) + 6 \, x \sin \left (x\right ) + \frac{1}{32} \, \cos \left (4 \, 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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