3.348 \(\int \frac{\cos (x) \sin (x)}{\sqrt{1+\sin (x)}} \, dx\)

Optimal. Leaf size=23 \[ \frac{2}{3} (\sin (x)+1)^{3/2}-2 \sqrt{\sin (x)+1} \]

[Out]

-2*Sqrt[1 + Sin[x]] + (2*(1 + Sin[x])^(3/2))/3

_______________________________________________________________________________________

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]

-2*Sqrt[1 + Sin[x]] + (2*(1 + Sin[x])^(3/2))/3

_______________________________________________________________________________________

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]

2*(sin(x) + 1)**(3/2)/3 - 2*sqrt(sin(x) + 1)

_______________________________________________________________________________________

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]

(2*(Cos[x/2] + Sin[x/2])^2*(-2 + Sin[x]))/(3*Sqrt[1 + Sin[x]])

_______________________________________________________________________________________

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]

2/3*(1+sin(x))^(3/2)-2*(1+sin(x))^(1/2)

_______________________________________________________________________________________

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]

2/3*(sin(x) + 1)^(3/2) - 2*sqrt(sin(x) + 1)

_______________________________________________________________________________________

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]

2/3*sqrt(sin(x) + 1)*(sin(x) - 2)

_______________________________________________________________________________________

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]

2*sqrt(sin(x) + 1)*sin(x)/3 - 4*sqrt(sin(x) + 1)/3

_______________________________________________________________________________________

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]

2/3*(sin(x) + 1)^(3/2) - 2*sqrt(sin(x) + 1)