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]

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

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Rubi [A]  time = 0.0459595, antiderivative size = 31, 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, 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 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 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])

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 = 2.402, 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] + Ellipt
icPi[-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))

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Maple [B]  time = 0.098, size = 52, normalized size = 1.7 \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 ) }}}{\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/(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))

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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/(1-sin(x))^(1/2)/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 = 1.67378, size = 104, normalized size = 3.35 \begin{align*} \sqrt{2} \log \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/(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))

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

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)

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

[Out]

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