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]

arctanh(1/2*cos(x)*a^(1/2)*2^(1/2)/sin(x)^(1/2)/(a-a*sin(x))^(1/2))*2^(1/2)/a^(1/2)

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Rubi [A]  time = 0.06, antiderivative size = 42, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [C]  time = 0.10, 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] - 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 - a*Sin[x]]*Tan[x/4]^(3/2))

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fricas [A]  time = 0.62, size = 168, normalized size = 4.00 \[ \left [\frac {\sqrt {2} \log \left (\frac {17 \, \cos \relax (x)^{3} + 3 \, \cos \relax (x)^{2} + \frac {4 \, \sqrt {2} {\left (3 \, \cos \relax (x)^{2} - {\left (3 \, \cos \relax (x) + 4\right )} \sin \relax (x) - \cos \relax (x) - 4\right )} \sqrt {-a \sin \relax (x) + a} \sqrt {\sin \relax (x)}}{\sqrt {a}} - {\left (17 \, \cos \relax (x)^{2} + 14 \, \cos \relax (x) - 4\right )} \sin \relax (x) - 18 \, \cos \relax (x) - 4}{\cos \relax (x)^{3} + 3 \, \cos \relax (x)^{2} - {\left (\cos \relax (x)^{2} - 2 \, \cos \relax (x) - 4\right )} \sin \relax (x) - 2 \, \cos \relax (x) - 4}\right )}{4 \, \sqrt {a}}, -\frac {1}{2} \, \sqrt {2} \sqrt {-\frac {1}{a}} \arctan \left (\frac {\sqrt {2} \sqrt {-a \sin \relax (x) + a} \sqrt {-\frac {1}{a}} {\left (3 \, \sin \relax (x) + 1\right )}}{4 \, \cos \relax (x) \sqrt {\sin \relax (x)}}\right )\right ] \]

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

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)

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maple [A]  time = 0.15, size = 53, normalized size = 1.26 \[ -\frac {2 \sqrt {-\frac {-1+\cos \relax (x )}{\sin \relax (x )}}\, \left (-1+\cos \relax (x )+\sin \relax (x )\right ) \left (\sqrt {\sin }\relax (x )\right ) \arctanh \left (\sqrt {-\frac {-1+\cos \relax (x )}{\sin \relax (x )}}\right )}{\sqrt {-a \left (-1+\sin \relax (x )\right )}\, \left (-1+\cos \relax (x )\right )} \]

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

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {\sin \relax (x)}\,\sqrt {a-a\,\sin \relax (x)}} \,d x \]

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]

int(1/(sin(x)^(1/2)*(a - a*sin(x))^(1/2)), x)

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

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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