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

Optimal. Leaf size=37 \[ \frac{2 \sqrt{a} \sinh ^{-1}\left (\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right )}{f} \]

[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 = 0.0576645, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {3801, 215} \[ \frac{2 \sqrt{a} \sinh ^{-1}\left (\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right )}{f} \]

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 [A]  time = 0.13311, size = 54, normalized size = 1.46 \[ -\frac{2 \tan \left (\frac{1}{2} (e+f x)\right ) \sqrt{a (\sec (e+f x)+1)} \sin ^{-1}\left (\sqrt{\sec (e+f x)}\right )}{f \sqrt{1-\sec (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*ArcSin[Sqrt[Sec[e + f*x]]]*Sqrt[a*(1 + Sec[e + f*x])]*Tan[(e + f*x)/2])/(f*Sqrt[1 - Sec[e + f*x]])

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Maple [B]  time = 0.222, size = 147, normalized size = 4. \begin{align*}{\frac{\sqrt{2}\cos \left ( fx+e \right ) \left ( -1+\cos \left ( fx+e \right ) \right ) }{f \left ( \sin \left ( fx+e \right ) \right ) ^{2}}\sqrt{{\frac{a \left ( 1+\cos \left ( fx+e \right ) \right ) }{\cos \left ( fx+e \right ) }}}\sqrt{ \left ( \cos \left ( fx+e \right ) \right ) ^{-1}} \left ( \arctan \left ({\frac{\sqrt{2} \left ( \cos \left ( fx+e \right ) +1-\sin \left ( fx+e \right ) \right ) }{4}\sqrt{-2\, \left ( 1+\cos \left ( fx+e \right ) \right ) ^{-1}}} \right ) -\arctan \left ({\frac{\sqrt{2} \left ( \cos \left ( fx+e \right ) +1+\sin \left ( fx+e \right ) \right ) }{4}\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(sec(f*x+e)^(1/2)*(a+a*sec(f*x+e))^(1/2),x)

[Out]

1/f*2^(1/2)*(1/cos(f*x+e)*a*(1+cos(f*x+e)))^(1/2)*(1/cos(f*x+e))^(1/2)*cos(f*x+e)*(-1+cos(f*x+e))*(arctan(1/4*
2^(1/2)*(-2/(1+cos(f*x+e)))^(1/2)*(cos(f*x+e)+1-sin(f*x+e)))-arctan(1/4*2^(1/2)*(-2/(1+cos(f*x+e)))^(1/2)*(cos
(f*x+e)+1+sin(f*x+e))))/sin(f*x+e)^2/(-2/(1+cos(f*x+e)))^(1/2)

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Maxima [B]  time = 2.24869, size = 325, normalized size = 8.78 \begin{align*} \frac{\sqrt{a}{\left (\log \left (2 \, \cos \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 2 \, \sin \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 2 \, \sqrt{2} \cos \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 2 \, \sqrt{2} \sin \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 2\right ) - \log \left (2 \, \cos \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 2 \, \sin \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 2 \, \sqrt{2} \cos \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 2 \, \sqrt{2} \sin \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 2\right ) + \log \left (2 \, \cos \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 2 \, \sin \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 2 \, \sqrt{2} \cos \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 2 \, \sqrt{2} \sin \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 2\right ) - \log \left (2 \, \cos \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 2 \, \sin \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 2 \, \sqrt{2} \cos \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 2 \, \sqrt{2} \sin \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\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*f*x + 1/2*e)^2 + 2*sin(1/2*f*x + 1/2*e)^2 + 2*sqrt(2)*cos(1/2*f*x + 1/2*e) + 2*sqrt
(2)*sin(1/2*f*x + 1/2*e) + 2) - log(2*cos(1/2*f*x + 1/2*e)^2 + 2*sin(1/2*f*x + 1/2*e)^2 + 2*sqrt(2)*cos(1/2*f*
x + 1/2*e) - 2*sqrt(2)*sin(1/2*f*x + 1/2*e) + 2) + log(2*cos(1/2*f*x + 1/2*e)^2 + 2*sin(1/2*f*x + 1/2*e)^2 - 2
*sqrt(2)*cos(1/2*f*x + 1/2*e) + 2*sqrt(2)*sin(1/2*f*x + 1/2*e) + 2) - log(2*cos(1/2*f*x + 1/2*e)^2 + 2*sin(1/2
*f*x + 1/2*e)^2 - 2*sqrt(2)*cos(1/2*f*x + 1/2*e) - 2*sqrt(2)*sin(1/2*f*x + 1/2*e) + 2))/f

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Fricas [B]  time = 2.00403, size = 495, normalized size = 13.38 \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} - \frac{4 \,{\left (\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 )}} \sin \left (f x + e\right )}{\sqrt{\cos \left (f x + e\right )}} + 8 \, a}{\cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )^{2}}\right )}{2 \, f}, \frac{\sqrt{-a} \arctan \left (\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt{\cos \left (f x + e\right )} \sin \left (f x + e\right )}{a \cos \left (f x + e\right )^{2} - a \cos \left (f x + e\right ) - 2 \, a}\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((a*cos(f*x + e)^3 - 7*a*cos(f*x + e)^2 - 4*(cos(f*x + e)^2 - 2*cos(f*x + e))*sqrt(a)*sqrt((a*
cos(f*x + e) + a)/cos(f*x + e))*sin(f*x + e)/sqrt(cos(f*x + e)) + 8*a)/(cos(f*x + e)^3 + cos(f*x + e)^2))/f, s
qrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sqrt(cos(f*x + e))*sin(f*x + e)/(a*cos(f*x +
e)^2 - a*cos(f*x + e) - 2*a))/f]

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

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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]

Timed out