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

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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.

[In]

Int[(Cos[x]*Sin[x])/Sqrt[1 + Sin[x]],x]

[Out]

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

Rule 2833

Int[cos[(e_.) + (f_.)*(x_)]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)
])^(n_.), x_Symbol] :> Dist[1/(b*f), Subst[Int[(a + x)^m*(c + (d*x)/b)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[
{a, b, c, d, e, f, m, n}, x]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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.

[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]])

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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.

[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)

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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.

[In]

integrate(cos(x)*sin(x)/(1+sin(x))^(1/2),x, algorithm="maxima")

[Out]

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

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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.

[In]

integrate(cos(x)*sin(x)/(1+sin(x))^(1/2),x, algorithm="fricas")

[Out]

2/3*sqrt(sin(x) + 1)*(sin(x) - 2)

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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.

[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

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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.

[In]

integrate(cos(x)*sin(x)/(1+sin(x))^(1/2),x, algorithm="giac")

[Out]

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