3.86 \(\int \frac{\sqrt{a-a \sin (e+f x)}}{\sqrt{-\sin (e+f x)}} \, dx\)

Optimal. Leaf size=38 \[ \frac{2 \sqrt{a} \sin ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a-a \sin (e+f x)}}\right )}{f} \]

[Out]

(2*Sqrt[a]*ArcSin[(Sqrt[a]*Cos[e + f*x])/Sqrt[a - a*Sin[e + f*x]]])/f

________________________________________________________________________________________

Rubi [A]  time = 0.0701887, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {2774, 216} \[ \frac{2 \sqrt{a} \sin ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a-a \sin (e+f x)}}\right )}{f} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a - a*Sin[e + f*x]]/Sqrt[-Sin[e + f*x]],x]

[Out]

(2*Sqrt[a]*ArcSin[(Sqrt[a]*Cos[e + f*x])/Sqrt[a - a*Sin[e + f*x]]])/f

Rule 2774

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

Rule 216

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

Rubi steps

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

Mathematica [C]  time = 0.468421, size = 119, normalized size = 3.13 \[ -\frac{\sqrt{-1+e^{2 i (e+f x)}} \sqrt{a-a \sin (e+f x)} \left (\tan ^{-1}\left (\sqrt{-1+e^{2 i (e+f x)}}\right )+i \tanh ^{-1}\left (\frac{e^{i (e+f x)}}{\sqrt{-1+e^{2 i (e+f x)}}}\right )\right )}{f \left (e^{i (e+f x)}-i\right ) \sqrt{-\sin (e+f x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a - a*Sin[e + f*x]]/Sqrt[-Sin[e + f*x]],x]

[Out]

-((Sqrt[-1 + E^((2*I)*(e + f*x))]*(ArcTan[Sqrt[-1 + E^((2*I)*(e + f*x))]] + I*ArcTanh[E^(I*(e + f*x))/Sqrt[-1
+ E^((2*I)*(e + f*x))]])*Sqrt[a - a*Sin[e + f*x]])/((-I + E^(I*(e + f*x)))*f*Sqrt[-Sin[e + f*x]]))

________________________________________________________________________________________

Maple [B]  time = 0.112, size = 271, normalized size = 7.1 \begin{align*}{\frac{\sqrt{2}\sin \left ( fx+e \right ) }{2\,f \left ( -1+\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) \right ) }\sqrt{-{\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}\sqrt{-a \left ( -1+\sin \left ( fx+e \right ) \right ) } \left ( \ln \left ( -{ \left ( \sqrt{2}\sqrt{-{\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}\sin \left ( fx+e \right ) +\sin \left ( fx+e \right ) -\cos \left ( fx+e \right ) +1 \right ) \left ( \sqrt{2}\sqrt{-{\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}\sin \left ( fx+e \right ) -\sin \left ( fx+e \right ) +\cos \left ( fx+e \right ) -1 \right ) ^{-1}} \right ) -\ln \left ( -{ \left ( \sqrt{2}\sqrt{-{\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}\sin \left ( fx+e \right ) -\sin \left ( fx+e \right ) +\cos \left ( fx+e \right ) -1 \right ) \left ( \sqrt{2}\sqrt{-{\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}\sin \left ( fx+e \right ) +\sin \left ( fx+e \right ) -\cos \left ( fx+e \right ) +1 \right ) ^{-1}} \right ) \right ){\frac{1}{\sqrt{-\sin \left ( fx+e \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/f*2^(1/2)*(-(-1+cos(f*x+e))/sin(f*x+e))^(1/2)*(-a*(-1+sin(f*x+e)))^(1/2)*sin(f*x+e)*(ln(-(2^(1/2)*(-(-1+co
s(f*x+e))/sin(f*x+e))^(1/2)*sin(f*x+e)+sin(f*x+e)-cos(f*x+e)+1)/(2^(1/2)*(-(-1+cos(f*x+e))/sin(f*x+e))^(1/2)*s
in(f*x+e)-sin(f*x+e)+cos(f*x+e)-1))-ln(-(2^(1/2)*(-(-1+cos(f*x+e))/sin(f*x+e))^(1/2)*sin(f*x+e)-sin(f*x+e)+cos
(f*x+e)-1)/(2^(1/2)*(-(-1+cos(f*x+e))/sin(f*x+e))^(1/2)*sin(f*x+e)+sin(f*x+e)-cos(f*x+e)+1)))/(-sin(f*x+e))^(1
/2)/(-1+cos(f*x+e)+sin(f*x+e))

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-a \sin \left (f x + e\right ) + a}}{\sqrt{-\sin \left (f x + e\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a-a*sin(f*x+e))^(1/2)/(-sin(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(-a*sin(f*x + e) + a)/sqrt(-sin(f*x + e)), x)

________________________________________________________________________________________

Fricas [B]  time = 2.63137, size = 941, normalized size = 24.76 \begin{align*} \left [\frac{\sqrt{-a} \log \left (\frac{128 \, a \cos \left (f x + e\right )^{5} - 128 \, a \cos \left (f x + e\right )^{4} - 416 \, a \cos \left (f x + e\right )^{3} + 128 \, a \cos \left (f x + e\right )^{2} + 8 \,{\left (16 \, \cos \left (f x + e\right )^{4} - 24 \, \cos \left (f x + e\right )^{3} - 66 \, \cos \left (f x + e\right )^{2} -{\left (16 \, \cos \left (f x + e\right )^{3} + 40 \, \cos \left (f x + e\right )^{2} - 26 \, \cos \left (f x + e\right ) - 51\right )} \sin \left (f x + e\right ) + 25 \, \cos \left (f x + e\right ) + 51\right )} \sqrt{-a \sin \left (f x + e\right ) + a} \sqrt{-a} \sqrt{-\sin \left (f x + e\right )} + 289 \, a \cos \left (f x + e\right ) -{\left (128 \, a \cos \left (f x + e\right )^{4} + 256 \, a \cos \left (f x + e\right )^{3} - 160 \, a \cos \left (f x + e\right )^{2} - 288 \, a \cos \left (f x + e\right ) + a\right )} \sin \left (f x + e\right ) + a}{\cos \left (f x + e\right ) - \sin \left (f x + e\right ) + 1}\right )}{4 \, f}, -\frac{\sqrt{a} \arctan \left (\frac{{\left (8 \, \cos \left (f x + e\right )^{2} - 8 \, \sin \left (f x + e\right ) - 9\right )} \sqrt{-a \sin \left (f x + e\right ) + a} \sqrt{a} \sqrt{-\sin \left (f x + e\right )}}{4 \,{\left (2 \, a \cos \left (f x + e\right )^{3} - a \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 2 \, a \cos \left (f x + e\right )\right )}}\right )}{2 \, f}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*sqrt(-a)*log((128*a*cos(f*x + e)^5 - 128*a*cos(f*x + e)^4 - 416*a*cos(f*x + e)^3 + 128*a*cos(f*x + e)^2 +
8*(16*cos(f*x + e)^4 - 24*cos(f*x + e)^3 - 66*cos(f*x + e)^2 - (16*cos(f*x + e)^3 + 40*cos(f*x + e)^2 - 26*co
s(f*x + e) - 51)*sin(f*x + e) + 25*cos(f*x + e) + 51)*sqrt(-a*sin(f*x + e) + a)*sqrt(-a)*sqrt(-sin(f*x + e)) +
289*a*cos(f*x + e) - (128*a*cos(f*x + e)^4 + 256*a*cos(f*x + e)^3 - 160*a*cos(f*x + e)^2 - 288*a*cos(f*x + e)
+ a)*sin(f*x + e) + a)/(cos(f*x + e) - sin(f*x + e) + 1))/f, -1/2*sqrt(a)*arctan(1/4*(8*cos(f*x + e)^2 - 8*si
n(f*x + e) - 9)*sqrt(-a*sin(f*x + e) + a)*sqrt(a)*sqrt(-sin(f*x + e))/(2*a*cos(f*x + e)^3 - a*cos(f*x + e)*sin
(f*x + e) - 2*a*cos(f*x + e)))/f]

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a-a*sin(f*x+e))**(1/2)/(-sin(f*x+e))**(1/2),x)

[Out]

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-a \sin \left (f x + e\right ) + a}}{\sqrt{-\sin \left (f x + e\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a-a*sin(f*x+e))^(1/2)/(-sin(f*x+e))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(-a*sin(f*x + e) + a)/sqrt(-sin(f*x + e)), x)