3.262 \(\int \frac{1}{\sqrt{\cos (x)} \sqrt{a+a \cos (x)}} \, dx\)

Optimal. Leaf size=41 \[ \frac{\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{a} \sin (x)}{\sqrt{2} \sqrt{\cos (x)} \sqrt{a \cos (x)+a}}\right )}{\sqrt{a}} \]

[Out]

(Sqrt[2]*ArcTan[(Sqrt[a]*Sin[x])/(Sqrt[2]*Sqrt[Cos[x]]*Sqrt[a + a*Cos[x]])])/Sqrt[a]

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Rubi [A]  time = 0.060865, antiderivative size = 41, 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, 205} \[ \frac{\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{a} \sin (x)}{\sqrt{2} \sqrt{\cos (x)} \sqrt{a \cos (x)+a}}\right )}{\sqrt{a}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[Cos[x]]*Sqrt[a + a*Cos[x]]),x]

[Out]

(Sqrt[2]*ArcTan[(Sqrt[a]*Sin[x])/(Sqrt[2]*Sqrt[Cos[x]]*Sqrt[a + a*Cos[x]])])/Sqrt[a]

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 205

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

Rubi steps

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

Mathematica [A]  time = 0.0193084, size = 32, normalized size = 0.78 \[ \frac{2 \cos \left (\frac{x}{2}\right ) \tan ^{-1}\left (\frac{\sin \left (\frac{x}{2}\right )}{\sqrt{\cos (x)}}\right )}{\sqrt{a (\cos (x)+1)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[Cos[x]]*Sqrt[a + a*Cos[x]]),x]

[Out]

(2*ArcTan[Sin[x/2]/Sqrt[Cos[x]]]*Cos[x/2])/Sqrt[a*(1 + Cos[x])]

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Maple [A]  time = 0.13, size = 42, normalized size = 1. \begin{align*} -{\frac{\sqrt{2}}{a}\sqrt{{\frac{\cos \left ( x \right ) }{\cos \left ( x \right ) +1}}}\sqrt{a \left ( \cos \left ( x \right ) +1 \right ) }\arcsin \left ({\frac{-1+\cos \left ( x \right ) }{\sin \left ( x \right ) }} \right ){\frac{1}{\sqrt{\cos \left ( x \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/cos(x)^(1/2)/(a+a*cos(x))^(1/2),x)

[Out]

-2^(1/2)/a/cos(x)^(1/2)*(cos(x)/(cos(x)+1))^(1/2)*(a*(cos(x)+1))^(1/2)*arcsin((-1+cos(x))/sin(x))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/cos(x)^(1/2)/(a+a*cos(x))^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.07373, size = 346, normalized size = 8.44 \begin{align*} \left [\frac{1}{2} \, \sqrt{2} \sqrt{-\frac{1}{a}} \log \left (-\frac{2 \, \sqrt{2} \sqrt{a \cos \left (x\right ) + a} \sqrt{-\frac{1}{a}} \sqrt{\cos \left (x\right )} \sin \left (x\right ) - 3 \, \cos \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1}{\cos \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1}\right ), \frac{\sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{a \cos \left (x\right ) + a} \sqrt{\cos \left (x\right )} \sin \left (x\right )}{2 \,{\left (\cos \left (x\right )^{2} + \cos \left (x\right )\right )} \sqrt{a}}\right )}{\sqrt{a}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/cos(x)^(1/2)/(a+a*cos(x))^(1/2),x, algorithm="fricas")

[Out]

[1/2*sqrt(2)*sqrt(-1/a)*log(-(2*sqrt(2)*sqrt(a*cos(x) + a)*sqrt(-1/a)*sqrt(cos(x))*sin(x) - 3*cos(x)^2 - 2*cos
(x) + 1)/(cos(x)^2 + 2*cos(x) + 1)), sqrt(2)*arctan(1/2*sqrt(2)*sqrt(a*cos(x) + a)*sqrt(cos(x))*sin(x)/((cos(x
)^2 + cos(x))*sqrt(a)))/sqrt(a)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/cos(x)**(1/2)/(a+a*cos(x))**(1/2),x)

[Out]

Integral(1/(sqrt(a*(cos(x) + 1))*sqrt(cos(x))), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a \cos \left (x\right ) + a} \sqrt{\cos \left (x\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/cos(x)^(1/2)/(a+a*cos(x))^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(a*cos(x) + a)*sqrt(cos(x))), x)