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]

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

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Rubi [A]  time = 0.0390379, antiderivative size = 17, normalized size of antiderivative = 1., 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 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]

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]

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*}

Mathematica [C]  time = 2.43993, 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] + Ellipti
cPi[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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Maple [B]  time = 0.086, size = 52, normalized size = 3.1 \begin{align*} -2\,{\frac{ \left ( 1-\cos \left ( x \right ) +\sin \left ( x \right ) \right ) \sqrt{\sin \left ( x \right ) }}{\sqrt{1+\sin \left ( x \right ) } \left ( -1+\cos \left ( x \right ) \right ) }\sqrt{-{\frac{-1+\cos \left ( x \right ) }{\sin \left ( x \right ) }}}\arctan \left ( \sqrt{-{\frac{-1+\cos \left ( x \right ) }{\sin \left ( x \right ) }}} \right ) } \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{\sin \left (x\right ) + 1} \sqrt{\sin \left (x\right )}}\,{d x} \end{align*}

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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Fricas [A]  time = 1.68873, size = 107, normalized size = 6.29 \begin{align*} 2 \, \sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{\sin \left (x\right ) + 1} \sqrt{\sin \left (x\right )}}{\cos \left (x\right ) + \sin \left (x\right ) + 1}\right ) \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{\sin{\left (x \right )} + 1} \sqrt{\sin{\left (x \right )}}}\, dx \end{align*}

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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{\sin \left (x\right ) + 1} \sqrt{\sin \left (x\right )}}\,{d x} \end{align*}

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)