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

Optimal. Leaf size=31 \[ \sqrt {2} \tanh ^{-1}\left (\frac {\cos (x)}{\sqrt {2} \sqrt {1-\sin (x)} \sqrt {\sin (x)}}\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 31, normalized size of antiderivative = 1.00, 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, 206} \[ \sqrt {2} \tanh ^{-1}\left (\frac {\cos (x)}{\sqrt {2} \sqrt {1-\sin (x)} \sqrt {\sin (x)}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 206

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

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

________________________________________________________________________________________

Mathematica [C]  time = 2.52, size = 125, normalized size = 4.03 \[ \frac {2 \sin (x) \sec ^2\left (\frac {x}{4}\right ) \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right ) \left (F\left (\left .\sin ^{-1}\left (\frac {1}{\sqrt {\tan \left (\frac {x}{4}\right )}}\right )\right |-1\right )-\Pi \left (-1-\sqrt {2};\left .\sin ^{-1}\left (\frac {1}{\sqrt {\tan \left (\frac {x}{4}\right )}}\right )\right |-1\right )-\Pi \left (-1+\sqrt {2};\left .\sin ^{-1}\left (\frac {1}{\sqrt {\tan \left (\frac {x}{4}\right )}}\right )\right |-1\right )\right )}{\sqrt {-((\sin (x)-1) \sin (x))} \tan ^{\frac {3}{2}}\left (\frac {x}{4}\right ) \sqrt {1-\cot ^2\left (\frac {x}{4}\right )}} \]

Antiderivative was successfully verified.

[In]

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

________________________________________________________________________________________

fricas [A]  time = 0.49, size = 31, normalized size = 1.00 \[ \sqrt {2} \log \left (\frac {\sqrt {2} \sqrt {-\sin \relax (x) + 1} \sqrt {\sin \relax (x)} + \cos \relax (x)}{\sin \relax (x) - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(2)*log((sqrt(2)*sqrt(-sin(x) + 1)*sqrt(sin(x)) + cos(x))/(sin(x) - 1))

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

maple [B]  time = 0.16, size = 52, normalized size = 1.68 \[ -\frac {2 \sqrt {-\frac {-1+\cos \relax (x )}{\sin \relax (x )}}\, \left (-1+\cos \relax (x )+\sin \relax (x )\right ) \left (\sqrt {\sin }\relax (x )\right ) \arctanh \left (\sqrt {-\frac {-1+\cos \relax (x )}{\sin \relax (x )}}\right )}{\sqrt {1-\sin \relax (x )}\, \left (-1+\cos \relax (x )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1-sin(x))^(1/2)/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))/(1-sin(x))
^(1/2)/(-1+cos(x))

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{\sqrt {\sin \relax (x)}\,\sqrt {1-\sin \relax (x)}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________