Optimal. Leaf size=23 \[ \frac{2}{3} (\sin (x)+1)^{3/2}-2 \sqrt{\sin (x)+1} \]
[Out]
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Rubi [A] time = 0.0585358, 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 \[ \frac{2}{3} (\sin (x)+1)^{3/2}-2 \sqrt{\sin (x)+1} \]
Antiderivative was successfully verified.
[In] Int[(Cos[x]*Sin[x])/Sqrt[1 + Sin[x]],x]
[Out]
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Rubi in Sympy [A] time = 2.5498, size = 20, normalized size = 0.87 \[ \frac{2 \left (\sin{\left (x \right )} + 1\right )^{\frac{3}{2}}}{3} - 2 \sqrt{\sin{\left (x \right )} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(cos(x)*sin(x)/(1+sin(x))**(1/2),x)
[Out]
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Mathematica [A] time = 0.0370521, 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.
[In] Integrate[(Cos[x]*Sin[x])/Sqrt[1 + Sin[x]],x]
[Out]
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Maple [A] time = 0.011, size = 18, normalized size = 0.8 \[{\frac{2}{3} \left ( 1+\sin \left ( x \right ) \right ) ^{{\frac{3}{2}}}}-2\,\sqrt{1+\sin \left ( x \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(cos(x)*sin(x)/(1+sin(x))^(1/2),x)
[Out]
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Maxima [A] time = 1.36845, size = 23, normalized size = 1. \[ \frac{2}{3} \,{\left (\sin \left (x\right ) + 1\right )}^{\frac{3}{2}} - 2 \, \sqrt{\sin \left (x\right ) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(cos(x)*sin(x)/sqrt(sin(x) + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.210375, size = 16, normalized size = 0.7 \[ \frac{2}{3} \, \sqrt{\sin \left (x\right ) + 1}{\left (\sin \left (x\right ) - 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(cos(x)*sin(x)/sqrt(sin(x) + 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.470907, size = 26, normalized size = 1.13 \[ \frac{2 \sqrt{\sin{\left (x \right )} + 1} \sin{\left (x \right )}}{3} - \frac{4 \sqrt{\sin{\left (x \right )} + 1}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(cos(x)*sin(x)/(1+sin(x))**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.210148, size = 23, normalized size = 1. \[ \frac{2}{3} \,{\left (\sin \left (x\right ) + 1\right )}^{\frac{3}{2}} - 2 \, \sqrt{\sin \left (x\right ) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(cos(x)*sin(x)/sqrt(sin(x) + 1),x, algorithm="giac")
[Out]