Optimal. Leaf size=23 \[ \frac {2}{3} (\sin (x)+1)^{3/2}-2 \sqrt {\sin (x)+1} \]
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Rubi [A] time = 0.04, antiderivative size = 23, normalized size of antiderivative = 1.00, 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 43
Rule 2833
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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 0.43, size = 12, normalized size = 0.52 \[ \frac {2}{3} \, \sqrt {\sin \relax (x) + 1} {\left (\sin \relax (x) - 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.89, size = 17, normalized size = 0.74 \[ \frac {2}{3} \, {\left (\sin \relax (x) + 1\right )}^{\frac {3}{2}} - 2 \, \sqrt {\sin \relax (x) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 18, normalized size = 0.78 \[ \frac {2 \left (\sin \relax (x )+1\right )^{\frac {3}{2}}}{3}-2 \sqrt {\sin \relax (x )+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 17, normalized size = 0.74 \[ \frac {2}{3} \, {\left (\sin \relax (x) + 1\right )}^{\frac {3}{2}} - 2 \, \sqrt {\sin \relax (x) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 12, normalized size = 0.52 \[ \frac {2\,\sqrt {\sin \relax (x)+1}\,\left (\sin \relax (x)-2\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.36, size = 26, normalized size = 1.13 \[ \frac {2 \sqrt {\sin {\relax (x )} + 1} \sin {\relax (x )}}{3} - \frac {4 \sqrt {\sin {\relax (x )} + 1}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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