3.4.48 \(\int \frac {\cos (x) \sin (x)}{\sqrt {1+\sin (x)}} \, dx\) [348]

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

[Out]

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

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Rubi [A]
time = 0.02, 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 = {2912, 45} \begin {gather*} \frac {2}{3} (\sin (x)+1)^{3/2}-2 \sqrt {\sin (x)+1} \end {gather*}

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 45

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])

Rule 2912

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/b)*x)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[
{a, b, c, d, e, f, m, n}, x]

Rubi steps

\begin {align*} \int \frac {\cos (x) \sin (x)}{\sqrt {1+\sin (x)}} \, dx &=\text {Subst}\left (\int \frac {x}{\sqrt {1+x}} \, dx,x,\sin (x)\right )\\ &=\text {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 = 0.01, size = 31, normalized size = 1.35 \begin {gather*} \frac {2 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^2 (-2+\sin (x))}{3 \sqrt {1+\sin (x)}} \end {gather*}

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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Mathics [A]
time = 1.85, size = 12, normalized size = 0.52 \begin {gather*} \frac {2 \left (-2+\text {Sin}\left [x\right ]\right ) \sqrt {1+\text {Sin}\left [x\right ]}}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[(Cos[x]*Sin[x])/Sqrt[1 + Sin[x]],x]')

[Out]

2 (-2 + Sin[x]) Sqrt[1 + Sin[x]] / 3

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Maple [A]
time = 0.04, size = 18, normalized size = 0.78

method	result	size
derivativedivides	\(\frac {2 \left (\sin \left (x \right )+1\right )^{\frac {3}{2}}}{3}-2 \sqrt {\sin \left (x \right )+1}\)	\(18\)
default	\(\frac {2 \left (\sin \left (x \right )+1\right )^{\frac {3}{2}}}{3}-2 \sqrt {\sin \left (x \right )+1}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)*sin(x)/(sin(x)+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.26, size = 17, normalized size = 0.74 \begin {gather*} \frac {2}{3} \, {\left (\sin \left (x\right ) + 1\right )}^{\frac {3}{2}} - 2 \, \sqrt {\sin \left (x\right ) + 1} \end {gather*}

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 = 0.32, size = 12, normalized size = 0.52 \begin {gather*} \frac {2}{3} \, \sqrt {\sin \left (x\right ) + 1} {\left (\sin \left (x\right ) - 2\right )} \end {gather*}

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 = 0.13, size = 26, normalized size = 1.13 \begin {gather*} \frac {2 \sqrt {\sin {\left (x \right )} + 1} \sin {\left (x \right )}}{3} - \frac {4 \sqrt {\sin {\left (x \right )} + 1}}{3} \end {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (17) = 34\).
time = 0.01, size = 61, normalized size = 2.65 \begin {gather*} \frac {-25165824 \tan ^{4}\left (\frac {x}{4}\right )-33554432 \tan ^{3}\left (\frac {x}{4}\right )-8388608}{\sqrt {2}\cdot 1572864 \left (\sqrt {2} \left (\tan ^{2}\left (\frac {x}{4}\right )+1\right )^{3} \mathrm {sign}\left (\cos \left (-\frac {\pi }{4}+\frac {x}{2}\right )\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*sin(x)/(1+sin(x))^(1/2),x)

[Out]

-8/3*(3*tan(1/4*x)^4 + 4*tan(1/4*x)^3 + 1)/((tan(1/4*x)^2 + 1)^3*sgn(cos(-1/4*pi + 1/2*x)))

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Mupad [B]
time = 0.10, size = 12, normalized size = 0.52 \begin {gather*} \frac {2\,\sqrt {\sin \left (x\right )+1}\,\left (\sin \left (x\right )-2\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(x)*sin(x))/(sin(x) + 1)^(1/2),x)

[Out]

(2*(sin(x) + 1)^(1/2)*(sin(x) - 2))/3

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