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

Optimal. Leaf size=42 \[ -\frac{\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{a} \cos (x)}{\sqrt{2} \sqrt{\sin (x)} \sqrt{a \sin (x)+a}}\right )}{\sqrt{a}} \]

[Out]

-((Sqrt[2]*ArcTan[(Sqrt[a]*Cos[x])/(Sqrt[2]*Sqrt[Sin[x]]*Sqrt[a + a*Sin[x]])])/Sqrt[a])

________________________________________________________________________________________

Rubi [A]  time = 0.0574363, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2782, 205} \[ -\frac{\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{a} \cos (x)}{\sqrt{2} \sqrt{\sin (x)} \sqrt{a \sin (x)+a}}\right )}{\sqrt{a}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((Sqrt[2]*ArcTan[(Sqrt[a]*Cos[x])/(Sqrt[2]*Sqrt[Sin[x]]*Sqrt[a + a*Sin[x]])])/Sqrt[a])

Rule 2782

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

Rule 205

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

Rubi steps

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

Mathematica [C]  time = 0.0899767, size = 125, normalized size = 2.98 \[ \frac{2 \sqrt{\sin (x)} \sec ^2\left (\frac{x}{4}\right ) \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right ) \left (F\left (\left .\sin ^{-1}\left (\frac{1}{\sqrt{\tan \left (\frac{x}{4}\right )}}\right )\right |-1\right )+\Pi \left (1-\sqrt{2};\left .-\sin ^{-1}\left (\frac{1}{\sqrt{\tan \left (\frac{x}{4}\right )}}\right )\right |-1\right )+\Pi \left (1+\sqrt{2};\left .-\sin ^{-1}\left (\frac{1}{\sqrt{\tan \left (\frac{x}{4}\right )}}\right )\right |-1\right )\right )}{\tan ^{\frac{3}{2}}\left (\frac{x}{4}\right ) \sqrt{1-\cot ^2\left (\frac{x}{4}\right )} \sqrt{a (\sin (x)+1)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[Sin[x]]*Sqrt[a + a*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[a*(1 + Sin[x])]*Tan[x/4]^(3/2))

________________________________________________________________________________________

Maple [A]  time = 0.089, size = 54, normalized size = 1.3 \begin{align*} -2\,{\frac{ \left ( 1-\cos \left ( x \right ) +\sin \left ( x \right ) \right ) \sqrt{\sin \left ( x \right ) }}{\sqrt{a \left ( 1+\sin \left ( x \right ) \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)/(a+a*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))/(a*(1+sin(x)
))^(1/2)/(-1+cos(x))

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a \sin \left (x\right ) + a} \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)/(a+a*sin(x))^(1/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 2.00485, size = 539, normalized size = 12.83 \begin{align*} \left [\frac{1}{4} \, \sqrt{2} \sqrt{-\frac{1}{a}} \log \left (\frac{17 \, \cos \left (x\right )^{3} - 4 \, \sqrt{2}{\left (3 \, \cos \left (x\right )^{2} +{\left (3 \, \cos \left (x\right ) + 4\right )} \sin \left (x\right ) - \cos \left (x\right ) - 4\right )} \sqrt{a \sin \left (x\right ) + a} \sqrt{-\frac{1}{a}} \sqrt{\sin \left (x\right )} + 3 \, \cos \left (x\right )^{2} +{\left (17 \, \cos \left (x\right )^{2} + 14 \, \cos \left (x\right ) - 4\right )} \sin \left (x\right ) - 18 \, \cos \left (x\right ) - 4}{\cos \left (x\right )^{3} + 3 \, \cos \left (x\right )^{2} +{\left (\cos \left (x\right )^{2} - 2 \, \cos \left (x\right ) - 4\right )} \sin \left (x\right ) - 2 \, \cos \left (x\right ) - 4}\right ), \frac{\sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{a \sin \left (x\right ) + a}{\left (3 \, \sin \left (x\right ) - 1\right )}}{4 \, \sqrt{a} \cos \left (x\right ) \sqrt{\sin \left (x\right )}}\right )}{2 \, \sqrt{a}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*sqrt(2)*sqrt(-1/a)*log((17*cos(x)^3 - 4*sqrt(2)*(3*cos(x)^2 + (3*cos(x) + 4)*sin(x) - cos(x) - 4)*sqrt(a*
sin(x) + a)*sqrt(-1/a)*sqrt(sin(x)) + 3*cos(x)^2 + (17*cos(x)^2 + 14*cos(x) - 4)*sin(x) - 18*cos(x) - 4)/(cos(
x)^3 + 3*cos(x)^2 + (cos(x)^2 - 2*cos(x) - 4)*sin(x) - 2*cos(x) - 4)), 1/2*sqrt(2)*arctan(1/4*sqrt(2)*sqrt(a*s
in(x) + a)*(3*sin(x) - 1)/(sqrt(a)*cos(x)*sqrt(sin(x))))/sqrt(a)]

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a \left (\sin{\left (x \right )} + 1\right )} \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)/(a+a*sin(x))**(1/2),x)

[Out]

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a \sin \left (x\right ) + a} \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)/(a+a*sin(x))^(1/2),x, algorithm="giac")

[Out]

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