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]

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

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Rubi [A]  time = 0.06, antiderivative size = 41, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.02, 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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fricas [A]  time = 0.97, size = 105, normalized size = 2.56 \[ \left [\frac {1}{2} \, \sqrt {2} \sqrt {-\frac {1}{a}} \log \left (-\frac {2 \, \sqrt {2} \sqrt {a \cos \relax (x) + a} \sqrt {-\frac {1}{a}} \sqrt {\cos \relax (x)} \sin \relax (x) - 3 \, \cos \relax (x)^{2} - 2 \, \cos \relax (x) + 1}{\cos \relax (x)^{2} + 2 \, \cos \relax (x) + 1}\right ), \frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {a \cos \relax (x) + a} \sqrt {\cos \relax (x)} \sin \relax (x)}{2 \, {\left (\cos \relax (x)^{2} + \cos \relax (x)\right )} \sqrt {a}}\right )}{\sqrt {a}}\right ] \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a \cos \relax (x) + a} \sqrt {\cos \relax (x)}}\,{d x} \]

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)

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maple [A]  time = 0.06, size = 42, normalized size = 1.02 \[ -\frac {\sqrt {\frac {\cos \relax (x )}{\cos \relax (x )+1}}\, \sqrt {a \left (\cos \relax (x )+1\right )}\, \arcsin \left (\frac {-1+\cos \relax (x )}{\sin \relax (x )}\right ) \sqrt {2}}{\sqrt {\cos \relax (x )}\, a} \]

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]

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

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

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: RuntimeError >> ECL says: Error executing code in Maxima: sign: argument cannot be imaginary
; found %i

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {\cos \relax (x)}\,\sqrt {a+a\,\cos \relax (x)}} \,d x \]

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]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a \left (\cos {\relax (x )} + 1\right )} \sqrt {\cos {\relax (x )}}}\, dx \]

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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