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

Optimal. Leaf size=17 \[ -\sqrt {2} \sin ^{-1}\left (\frac {\cos (x)}{\sin (x)+1}\right ) \]

[Out]

-arcsin(cos(x)/(1+sin(x)))*2^(1/2)

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Rubi [A]  time = 0.04, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2781, 216} \[ -\sqrt {2} \sin ^{-1}\left (\frac {\cos (x)}{\sin (x)+1}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 2781

Int[1/(Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> -Dist[Sqr
t[2]/(Sqrt[a]*f), Subst[Int[1/Sqrt[1 - x^2], x], x, (b*Cos[e + f*x])/(a + b*Sin[e + f*x])], x] /; FreeQ[{a, b,
d, e, f}, x] && EqQ[a^2 - b^2, 0] && EqQ[d, a/b] && GtQ[a, 0]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {\sin (x)} \sqrt {1+\sin (x)}} \, dx &=-\left (\sqrt {2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,\frac {\cos (x)}{1+\sin (x)}\right )\right )\\ &=-\sqrt {2} \sin ^{-1}\left (\frac {\cos (x)}{1+\sin (x)}\right )\\ \end {align*}

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Mathematica [C]  time = 2.57, size = 123, normalized size = 7.24 \[ \frac {2 \sqrt {\sin (x)} \sec ^2\left (\frac {x}{4}\right ) \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right ) \left (F\left (\left .\sin ^{-1}\left (\frac {1}{\sqrt {\tan \left (\frac {x}{4}\right )}}\right )\right |-1\right )-\Pi \left (1-\sqrt {2};\left .\sin ^{-1}\left (\frac {1}{\sqrt {\tan \left (\frac {x}{4}\right )}}\right )\right |-1\right )-\Pi \left (1+\sqrt {2};\left .\sin ^{-1}\left (\frac {1}{\sqrt {\tan \left (\frac {x}{4}\right )}}\right )\right |-1\right )\right )}{\sqrt {\sin (x)+1} \tan ^{\frac {3}{2}}\left (\frac {x}{4}\right ) \sqrt {1-\cot ^2\left (\frac {x}{4}\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(EllipticF[ArcSin[1/Sqrt[Tan[x/4]]], -1] - EllipticPi[1 - Sqrt[2], ArcSin[1/Sqrt[Tan[x/4]]], -1] - Elliptic
Pi[1 + Sqrt[2], ArcSin[1/Sqrt[Tan[x/4]]], -1])*Sec[x/4]^2*(Cos[x/2] + Sin[x/2])*Sqrt[Sin[x]])/(Sqrt[1 - Cot[x/
4]^2]*Sqrt[1 + Sin[x]]*Tan[x/4]^(3/2))

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fricas [A]  time = 0.50, size = 28, normalized size = 1.65 \[ 2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {\sin \relax (x) + 1} \sqrt {\sin \relax (x)}}{\cos \relax (x) + \sin \relax (x) + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(2)*arctan(sqrt(2)*sqrt(sin(x) + 1)*sqrt(sin(x))/(cos(x) + sin(x) + 1))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {\sin \relax (x) + 1} \sqrt {\sin \relax (x)}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(sin(x) + 1)*sqrt(sin(x))), x)

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maple [B]  time = 0.13, size = 52, normalized size = 3.06 \[ -\frac {2 \sqrt {-\frac {-1+\cos \relax (x )}{\sin \relax (x )}}\, \left (1-\cos \relax (x )+\sin \relax (x )\right ) \left (\sqrt {\sin }\relax (x )\right ) \arctan \left (\sqrt {-\frac {-1+\cos \relax (x )}{\sin \relax (x )}}\right )}{\sqrt {1+\sin \relax (x )}\, \left (-1+\cos \relax (x )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(-(-1+cos(x))/sin(x))^(1/2)*(1-cos(x)+sin(x))*sin(x)^(1/2)*arctan((-(-1+cos(x))/sin(x))^(1/2))/(1+sin(x))^(
1/2)/(-1+cos(x))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {\sin \relax (x) + 1} \sqrt {\sin \relax (x)}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(sin(x) + 1)*sqrt(sin(x))), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.06 \[ \int \frac {1}{\sqrt {\sin \relax (x)}\,\sqrt {\sin \relax (x)+1}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {\sin {\relax (x )} + 1} \sqrt {\sin {\relax (x )}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(sqrt(sin(x) + 1)*sqrt(sin(x))), x)

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