3.244 \(\int \sqrt{-\sec (e+f x)} \sqrt{a-a \sec (e+f x)} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0633156, 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} \[ \frac{2 \sqrt{a} \sinh ^{-1}\left (\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a-a \sec (e+f x)}}\right )}{f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[a]*ArcSinh[(Sqrt[a]*Tan[e + f*x])/Sqrt[a - a*Sec[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{-\sec (e+f x)} \sqrt{a-a \sec (e+f x)} \, dx &=\frac{2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{a}}} \, dx,x,\frac{a \tan (e+f x)}{\sqrt{a-a \sec (e+f x)}}\right )}{f}\\ &=\frac{2 \sqrt{a} \sinh ^{-1}\left (\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a-a \sec (e+f x)}}\right )}{f}\\ \end{align*}

Mathematica [C]  time = 1.8488, size = 299, normalized size = 7.87 \[ \frac{\csc \left (\frac{1}{2} (e+f x)\right ) \sqrt{a-a \sec (e+f x)} \left (\log \left (-\sqrt{2} \sin \left (\frac{1}{2} (e+f x)\right )-\sqrt{2} \cos \left (\frac{1}{2} (e+f x)\right )+2\right )-\log \left (-\sqrt{2} \sin \left (\frac{1}{2} (e+f x)\right )+\sqrt{2} \cos \left (\frac{1}{2} (e+f x)\right )+2\right )-2 i \tan ^{-1}\left (\frac{\cos \left (\frac{1}{4} (e+f x)\right )-\left (\sqrt{2}-1\right ) \sin \left (\frac{1}{4} (e+f x)\right )}{\left (1+\sqrt{2}\right ) \cos \left (\frac{1}{4} (e+f x)\right )-\sin \left (\frac{1}{4} (e+f x)\right )}\right )+2 i \tan ^{-1}\left (\frac{\cos \left (\frac{1}{4} (e+f x)\right )-\left (1+\sqrt{2}\right ) \sin \left (\frac{1}{4} (e+f x)\right )}{\left (\sqrt{2}-1\right ) \cos \left (\frac{1}{4} (e+f x)\right )-\sin \left (\frac{1}{4} (e+f x)\right )}\right )-4 \tanh ^{-1}\left (\sqrt{2} \sec \left (\frac{f x}{4}\right ) \cos \left (\frac{1}{4} (2 e+f x)\right )+\tan \left (\frac{f x}{4}\right )\right )\right )}{2 \sqrt{2} f \sqrt{-\sec (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Csc[(e + f*x)/2]*((-2*I)*ArcTan[(Cos[(e + f*x)/4] - (-1 + Sqrt[2])*Sin[(e + f*x)/4])/((1 + Sqrt[2])*Cos[(e +
f*x)/4] - Sin[(e + f*x)/4])] + (2*I)*ArcTan[(Cos[(e + f*x)/4] - (1 + Sqrt[2])*Sin[(e + f*x)/4])/((-1 + Sqrt[2]
)*Cos[(e + f*x)/4] - Sin[(e + f*x)/4])] - 4*ArcTanh[Sqrt[2]*Cos[(2*e + f*x)/4]*Sec[(f*x)/4] + Tan[(f*x)/4]] +
Log[2 - Sqrt[2]*Cos[(e + f*x)/2] - Sqrt[2]*Sin[(e + f*x)/2]] - Log[2 + Sqrt[2]*Cos[(e + f*x)/2] - Sqrt[2]*Sin[
(e + f*x)/2]])*Sqrt[a - a*Sec[e + f*x]])/(2*Sqrt[2]*f*Sqrt[-Sec[e + f*x]])

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Maple [B]  time = 0.252, size = 127, normalized size = 3.3 \begin{align*} -{\frac{\cos \left ( fx+e \right ) }{f\sin \left ( fx+e \right ) } \left ({\it Artanh} \left ({\frac{-\cos \left ( fx+e \right ) -1+\sin \left ( fx+e \right ) }{2}\sqrt{ \left ( 1+\cos \left ( fx+e \right ) \right ) ^{-1}}} \right ) -{\it Artanh} \left ({\frac{\cos \left ( fx+e \right ) +1+\sin \left ( fx+e \right ) }{2}\sqrt{ \left ( 1+\cos \left ( fx+e \right ) \right ) ^{-1}}} \right ) \right ) \sqrt{- \left ( \cos \left ( fx+e \right ) \right ) ^{-1}}\sqrt{{\frac{a \left ( -1+\cos \left ( fx+e \right ) \right ) }{\cos \left ( fx+e \right ) }}}{\frac{1}{\sqrt{ \left ( 1+\cos \left ( fx+e \right ) \right ) ^{-1}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/f*(arctanh(1/2*(1/(1+cos(f*x+e)))^(1/2)*(-cos(f*x+e)-1+sin(f*x+e)))-arctanh(1/2*(1/(1+cos(f*x+e)))^(1/2)*(c
os(f*x+e)+1+sin(f*x+e))))*cos(f*x+e)*(-1/cos(f*x+e))^(1/2)*(a*(-1+cos(f*x+e))/cos(f*x+e))^(1/2)/sin(f*x+e)/(1/
(1+cos(f*x+e)))^(1/2)

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Maxima [B]  time = 2.43655, size = 477, normalized size = 12.55 \begin{align*} -\frac{\sqrt{a}{\left (\log \left (2 \, \cos \left (\frac{1}{2} \, \arctan \left (\sin \left (f x + e\right ), \cos \left (f x + e\right )\right )\right )^{2} + 2 \, \sin \left (\frac{1}{2} \, \arctan \left (\sin \left (f x + e\right ), \cos \left (f x + e\right )\right )\right )^{2} + 2 \, \sqrt{2} \cos \left (\frac{1}{2} \, \arctan \left (\sin \left (f x + e\right ), \cos \left (f x + e\right )\right )\right ) + 2 \, \sqrt{2} \sin \left (\frac{1}{2} \, \arctan \left (\sin \left (f x + e\right ), \cos \left (f x + e\right )\right )\right ) + 2\right ) + \log \left (2 \, \cos \left (\frac{1}{2} \, \arctan \left (\sin \left (f x + e\right ), \cos \left (f x + e\right )\right )\right )^{2} + 2 \, \sin \left (\frac{1}{2} \, \arctan \left (\sin \left (f x + e\right ), \cos \left (f x + e\right )\right )\right )^{2} + 2 \, \sqrt{2} \cos \left (\frac{1}{2} \, \arctan \left (\sin \left (f x + e\right ), \cos \left (f x + e\right )\right )\right ) - 2 \, \sqrt{2} \sin \left (\frac{1}{2} \, \arctan \left (\sin \left (f x + e\right ), \cos \left (f x + e\right )\right )\right ) + 2\right ) - \log \left (2 \, \cos \left (\frac{1}{2} \, \arctan \left (\sin \left (f x + e\right ), \cos \left (f x + e\right )\right )\right )^{2} + 2 \, \sin \left (\frac{1}{2} \, \arctan \left (\sin \left (f x + e\right ), \cos \left (f x + e\right )\right )\right )^{2} - 2 \, \sqrt{2} \cos \left (\frac{1}{2} \, \arctan \left (\sin \left (f x + e\right ), \cos \left (f x + e\right )\right )\right ) + 2 \, \sqrt{2} \sin \left (\frac{1}{2} \, \arctan \left (\sin \left (f x + e\right ), \cos \left (f x + e\right )\right )\right ) + 2\right ) - \log \left (2 \, \cos \left (\frac{1}{2} \, \arctan \left (\sin \left (f x + e\right ), \cos \left (f x + e\right )\right )\right )^{2} + 2 \, \sin \left (\frac{1}{2} \, \arctan \left (\sin \left (f x + e\right ), \cos \left (f x + e\right )\right )\right )^{2} - 2 \, \sqrt{2} \cos \left (\frac{1}{2} \, \arctan \left (\sin \left (f x + e\right ), \cos \left (f x + e\right )\right )\right ) - 2 \, \sqrt{2} \sin \left (\frac{1}{2} \, \arctan \left (\sin \left (f x + e\right ), \cos \left (f x + e\right )\right )\right ) + 2\right )\right )}}{2 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(a)*(log(2*cos(1/2*arctan2(sin(f*x + e), cos(f*x + e)))^2 + 2*sin(1/2*arctan2(sin(f*x + e), cos(f*x +
e)))^2 + 2*sqrt(2)*cos(1/2*arctan2(sin(f*x + e), cos(f*x + e))) + 2*sqrt(2)*sin(1/2*arctan2(sin(f*x + e), cos
(f*x + e))) + 2) + log(2*cos(1/2*arctan2(sin(f*x + e), cos(f*x + e)))^2 + 2*sin(1/2*arctan2(sin(f*x + e), cos(
f*x + e)))^2 + 2*sqrt(2)*cos(1/2*arctan2(sin(f*x + e), cos(f*x + e))) - 2*sqrt(2)*sin(1/2*arctan2(sin(f*x + e)
, cos(f*x + e))) + 2) - log(2*cos(1/2*arctan2(sin(f*x + e), cos(f*x + e)))^2 + 2*sin(1/2*arctan2(sin(f*x + e),
cos(f*x + e)))^2 - 2*sqrt(2)*cos(1/2*arctan2(sin(f*x + e), cos(f*x + e))) + 2*sqrt(2)*sin(1/2*arctan2(sin(f*x
+ e), cos(f*x + e))) + 2) - log(2*cos(1/2*arctan2(sin(f*x + e), cos(f*x + e)))^2 + 2*sin(1/2*arctan2(sin(f*x
+ e), cos(f*x + e)))^2 - 2*sqrt(2)*cos(1/2*arctan2(sin(f*x + e), cos(f*x + e))) - 2*sqrt(2)*sin(1/2*arctan2(si
n(f*x + e), cos(f*x + e))) + 2))/f

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Fricas [B]  time = 1.80491, size = 545, normalized size = 14.34 \begin{align*} \left [\frac{\sqrt{a} \log \left (\frac{4 \,{\left (\cos \left (f x + e\right )^{3} + 3 \, \cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right )\right )} \sqrt{a} \sqrt{\frac{a \cos \left (f x + e\right ) - a}{\cos \left (f x + e\right )}} \sqrt{-\frac{1}{\cos \left (f x + e\right )}} +{\left (a \cos \left (f x + e\right )^{2} + 8 \, a \cos \left (f x + e\right ) + 8 \, a\right )} \sin \left (f x + e\right )}{\cos \left (f x + e\right )^{2} \sin \left (f x + e\right )}\right )}{2 \, f}, -\frac{\sqrt{-a} \arctan \left (\frac{2 \,{\left (\cos \left (f x + e\right )^{2} + \cos \left (f x + e\right )\right )} \sqrt{-a} \sqrt{\frac{a \cos \left (f x + e\right ) - a}{\cos \left (f x + e\right )}} \sqrt{-\frac{1}{\cos \left (f x + e\right )}}}{{\left (a \cos \left (f x + e\right ) + 2 \, a\right )} \sin \left (f x + e\right )}\right )}{f}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 2.41662, size = 124, normalized size = 3.26 \begin{align*} \frac{\sqrt{2}{\left (\frac{\sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + a}}{2 \, \sqrt{-a}}\right )}{\sqrt{-a}} - \frac{\sqrt{2} \arctan \left (\frac{\sqrt{a}}{\sqrt{-a}}\right )}{\sqrt{-a}}\right )}{\left | a \right |} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )\right )}{f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(2)*(sqrt(2)*arctan(1/2*sqrt(2)*sqrt(a*tan(1/2*f*x + 1/2*e)^2 + a)/sqrt(-a))/sqrt(-a) - sqrt(2)*arctan(sqr
t(a)/sqrt(-a))/sqrt(-a))*abs(a)*sgn(tan(1/2*f*x + 1/2*e)^3 + tan(1/2*f*x + 1/2*e))/f