3.90 \(\int \frac{1}{\sqrt{\sin (x)} \sqrt{a-a \sin (x)}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.064463, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2782, 208} \[ \frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (x)}{\sqrt{2} \sqrt{\sin (x)} \sqrt{a-a \sin (x)}}\right )}{\sqrt{a}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[2]*ArcTanh[(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 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[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} \tanh ^{-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 = 0.10248, size = 128, normalized size = 3.05 \[ \frac{2 \sqrt{\sin (x)} \sec ^2\left (\frac{x}{4}\right ) \left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right ) \left (F\left (\left .\sin ^{-1}\left (\frac{1}{\sqrt{\tan \left (\frac{x}{4}\right )}}\right )\right |-1\right )+\Pi \left (-1-\sqrt{2};\left .-\sin ^{-1}\left (\frac{1}{\sqrt{\tan \left (\frac{x}{4}\right )}}\right )\right |-1\right )+\Pi \left (-1+\sqrt{2};\left .-\sin ^{-1}\left (\frac{1}{\sqrt{\tan \left (\frac{x}{4}\right )}}\right )\right |-1\right )\right )}{\tan ^{\frac{3}{2}}\left (\frac{x}{4}\right ) \sqrt{1-\cot ^2\left (\frac{x}{4}\right )} \sqrt{a-a \sin (x)}} \]

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] + Ellipt
icPi[-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 - Co
t[x/4]^2]*Sqrt[a - a*Sin[x]]*Tan[x/4]^(3/2))

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Maple [A]  time = 0.093, size = 53, 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 ) }}}{\it Artanh} \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)*arctanh((-(-1+cos(x))/sin(x))^(1/2))/(-a*(-1+si
n(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{-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)

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Fricas [A]  time = 2.09568, size = 543, normalized size = 12.93 \begin{align*} \left [\frac{\sqrt{2} \log \left (\frac{17 \, \cos \left (x\right )^{3} + 3 \, \cos \left (x\right )^{2} + \frac{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{\sin \left (x\right )}}{\sqrt{a}} -{\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 )}{4 \, \sqrt{a}}, -\frac{1}{2} \, \sqrt{2} \sqrt{-\frac{1}{a}} \arctan \left (\frac{\sqrt{2} \sqrt{-a \sin \left (x\right ) + a} \sqrt{-\frac{1}{a}}{\left (3 \, \sin \left (x\right ) + 1\right )}}{4 \, \cos \left (x\right ) \sqrt{\sin \left (x\right )}}\right )\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)*log((17*cos(x)^3 + 3*cos(x)^2 + 4*sqrt(2)*(3*cos(x)^2 - (3*cos(x) + 4)*sin(x) - cos(x) - 4)*sqrt(
-a*sin(x) + a)*sqrt(sin(x))/sqrt(a) - (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))/sqrt(a), -1/2*sqrt(2)*sqrt(-1/a)*arctan(1/4*sqrt(2)*s
qrt(-a*sin(x) + a)*sqrt(-1/a)*(3*sin(x) + 1)/(cos(x)*sqrt(sin(x))))]

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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)

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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)