Optimal. Leaf size=23 \[ \frac{2}{3} (\sin (x)+1)^{3/2}-2 \sqrt{\sin (x)+1} \]
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Rubi [A] time = 0.0367756, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2833, 43} \[ \frac{2}{3} (\sin (x)+1)^{3/2}-2 \sqrt{\sin (x)+1} \]
Antiderivative was successfully verified.
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Rule 2833
Rule 43
Rubi steps
\begin{align*} \int \frac{\cos (x) \sin (x)}{\sqrt{1+\sin (x)}} \, dx &=\operatorname{Subst}\left (\int \frac{x}{\sqrt{1+x}} \, dx,x,\sin (x)\right )\\ &=\operatorname{Subst}\left (\int \left (-\frac{1}{\sqrt{1+x}}+\sqrt{1+x}\right ) \, dx,x,\sin (x)\right )\\ &=-2 \sqrt{1+\sin (x)}+\frac{2}{3} (1+\sin (x))^{3/2}\\ \end{align*}
Mathematica [A] time = 0.0224933, size = 31, normalized size = 1.35 \[ \frac{2 (\sin (x)-2) \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^2}{3 \sqrt{\sin (x)+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 18, normalized size = 0.8 \begin{align*}{\frac{2}{3} \left ( 1+\sin \left ( x \right ) \right ) ^{{\frac{3}{2}}}}-2\,\sqrt{1+\sin \left ( x \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.958319, size = 23, normalized size = 1. \begin{align*} \frac{2}{3} \,{\left (\sin \left (x\right ) + 1\right )}^{\frac{3}{2}} - 2 \, \sqrt{\sin \left (x\right ) + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.04638, size = 47, normalized size = 2.04 \begin{align*} \frac{2}{3} \, \sqrt{\sin \left (x\right ) + 1}{\left (\sin \left (x\right ) - 2\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.322213, size = 26, normalized size = 1.13 \begin{align*} \frac{2 \sqrt{\sin{\left (x \right )} + 1} \sin{\left (x \right )}}{3} - \frac{4 \sqrt{\sin{\left (x \right )} + 1}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05914, size = 23, normalized size = 1. \begin{align*} \frac{2}{3} \,{\left (\sin \left (x\right ) + 1\right )}^{\frac{3}{2}} - 2 \, \sqrt{\sin \left (x\right ) + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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