Optimal. Leaf size=19 \[ \frac{\sin ^5(x)}{5}+\frac{2 \sin ^3(x)}{3}+\sin (x) \]
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Rubi [A] time = 0.0191771, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {3190, 194} \[ \frac{\sin ^5(x)}{5}+\frac{2 \sin ^3(x)}{3}+\sin (x) \]
Antiderivative was successfully verified.
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Rule 3190
Rule 194
Rubi steps
\begin{align*} \int \cos (x) \left (1+\sin ^2(x)\right )^2 \, dx &=\operatorname{Subst}\left (\int \left (1+x^2\right )^2 \, dx,x,\sin (x)\right )\\ &=\operatorname{Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,\sin (x)\right )\\ &=\sin (x)+\frac{2 \sin ^3(x)}{3}+\frac{\sin ^5(x)}{5}\\ \end{align*}
Mathematica [A] time = 0.0035859, size = 19, normalized size = 1. \[ \frac{\sin ^5(x)}{5}+\frac{2 \sin ^3(x)}{3}+\sin (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 16, normalized size = 0.8 \begin{align*} \sin \left ( x \right ) +{\frac{2\, \left ( \sin \left ( x \right ) \right ) ^{3}}{3}}+{\frac{ \left ( \sin \left ( x \right ) \right ) ^{5}}{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.955533, size = 20, normalized size = 1.05 \begin{align*} \frac{1}{5} \, \sin \left (x\right )^{5} + \frac{2}{3} \, \sin \left (x\right )^{3} + \sin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63509, size = 61, normalized size = 3.21 \begin{align*} \frac{1}{15} \,{\left (3 \, \cos \left (x\right )^{4} - 16 \, \cos \left (x\right )^{2} + 28\right )} \sin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.09388, size = 17, normalized size = 0.89 \begin{align*} \frac{\sin ^{5}{\left (x \right )}}{5} + \frac{2 \sin ^{3}{\left (x \right )}}{3} + \sin{\left (x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06927, size = 20, normalized size = 1.05 \begin{align*} \frac{1}{5} \, \sin \left (x\right )^{5} + \frac{2}{3} \, \sin \left (x\right )^{3} + \sin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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