3.20 \(\int \sqrt{-\csc (e+f x)} \sqrt{a-a \csc (e+f x)} \, dx\)

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

[Out]

(-2*Sqrt[a]*ArcSinh[(Sqrt[a]*Cot[e + f*x])/Sqrt[a - a*Csc[e + f*x]]])/f

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Rubi [A]  time = 0.0708915, 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 = {3801, 215} \[ -\frac{2 \sqrt{a} \sinh ^{-1}\left (\frac{\sqrt{a} \cot (e+f x)}{\sqrt{a-a \csc (e+f x)}}\right )}{f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[a]*ArcSinh[(Sqrt[a]*Cot[e + f*x])/Sqrt[a - a*Csc[e + f*x]]])/f

Rule 3801

Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[(-2*a*Sq
rt[(a*d)/b])/(b*f), Subst[Int[1/Sqrt[1 + x^2/a], x], x, (b*Cot[e + f*x])/Sqrt[a + b*Csc[e + f*x]]], x] /; Free
Q[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && GtQ[(a*d)/b, 0]

Rule 215

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

Rubi steps

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

Mathematica [B]  time = 0.775479, size = 101, normalized size = 2.66 \[ \frac{2 \tan \left (\frac{1}{2} (e+f x)\right ) \sqrt{-\csc (e+f x)} \sqrt{a-a \csc (e+f x)} \left (\tanh ^{-1}\left (\sqrt{\sec ^2\left (\frac{1}{2} (e+f x)\right )}\right )+\sinh ^{-1}\left (\tan \left (\frac{1}{2} (e+f x)\right )\right )\right )}{f \left (\tan \left (\frac{1}{2} (e+f x)\right )-1\right ) \sqrt{\sec ^2\left (\frac{1}{2} (e+f x)\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(ArcSinh[Tan[(e + f*x)/2]] + ArcTanh[Sqrt[Sec[(e + f*x)/2]^2]])*Sqrt[-Csc[e + f*x]]*Sqrt[a - a*Csc[e + f*x]
]*Tan[(e + f*x)/2])/(f*Sqrt[Sec[(e + f*x)/2]^2]*(-1 + Tan[(e + f*x)/2]))

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Maple [B]  time = 0.333, size = 117, normalized size = 3.1 \begin{align*} 2\,{\frac{-1+\cos \left ( fx+e \right ) }{f \left ( -1+\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) \right ) }\sqrt{- \left ( \sin \left ( fx+e \right ) \right ) ^{-1}}\sqrt{{\frac{a \left ( \sin \left ( fx+e \right ) -1 \right ) }{\sin \left ( fx+e \right ) }}} \left ( \arctan \left ( 1/2\,\sin \left ( fx+e \right ) \sqrt{-2\, \left ( 1+\cos \left ( fx+e \right ) \right ) ^{-1}} \right ) -\arctan \left ({\frac{1}{\sqrt{-2\, \left ( 1+\cos \left ( fx+e \right ) \right ) ^{-1}}}} \right ) \right ){\frac{1}{\sqrt{-2\, \left ( 1+\cos \left ( fx+e \right ) \right ) ^{-1}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 0.52994, size = 764, normalized size = 20.11 \begin{align*} \left [\frac{\sqrt{a} \log \left (\frac{a \cos \left (f x + e\right )^{3} - 7 \, a \cos \left (f x + e\right )^{2} - 4 \,{\left (\cos \left (f x + e\right )^{3} + 3 \, \cos \left (f x + e\right )^{2} +{\left (\cos \left (f x + e\right )^{2} - 2 \, \cos \left (f x + e\right ) - 3\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 3\right )} \sqrt{a} \sqrt{\frac{a \sin \left (f x + e\right ) - a}{\sin \left (f x + e\right )}} \sqrt{-\frac{1}{\sin \left (f x + e\right )}} - 9 \, a \cos \left (f x + e\right ) -{\left (a \cos \left (f x + e\right )^{2} + 8 \, a \cos \left (f x + e\right ) - a\right )} \sin \left (f x + e\right ) - a}{\cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )^{2} -{\left (\cos \left (f x + e\right )^{2} - 1\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 1}\right )}{2 \, f}, \frac{\sqrt{-a} \arctan \left (-\frac{{\left (\cos \left (f x + e\right )^{2} - 2 \, \sin \left (f x + e\right ) - 1\right )} \sqrt{-a} \sqrt{\frac{a \sin \left (f x + e\right ) - a}{\sin \left (f x + e\right )}} \sqrt{-\frac{1}{\sin \left (f x + e\right )}}}{2 \, a \cos \left (f x + e\right )}\right )}{f}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*sqrt(a)*log((a*cos(f*x + e)^3 - 7*a*cos(f*x + e)^2 - 4*(cos(f*x + e)^3 + 3*cos(f*x + e)^2 + (cos(f*x + e)
^2 - 2*cos(f*x + e) - 3)*sin(f*x + e) - cos(f*x + e) - 3)*sqrt(a)*sqrt((a*sin(f*x + e) - a)/sin(f*x + e))*sqrt
(-1/sin(f*x + e)) - 9*a*cos(f*x + e) - (a*cos(f*x + e)^2 + 8*a*cos(f*x + e) - a)*sin(f*x + e) - a)/(cos(f*x +
e)^3 + cos(f*x + e)^2 - (cos(f*x + e)^2 - 1)*sin(f*x + e) - cos(f*x + e) - 1))/f, sqrt(-a)*arctan(-1/2*(cos(f*
x + e)^2 - 2*sin(f*x + e) - 1)*sqrt(-a)*sqrt((a*sin(f*x + e) - a)/sin(f*x + e))*sqrt(-1/sin(f*x + e))/(a*cos(f
*x + e)))/f]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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