3.57 \(\int \sqrt {a+b \cos ^2(x)} \, dx\)

Optimal. Leaf size=42 \[ \frac {\sqrt {a+b \cos ^2(x)} E\left (x+\frac {\pi }{2}|-\frac {b}{a}\right )}{\sqrt {\frac {b \cos ^2(x)}{a}+1}} \]

[Out]

-(sin(x)^2)^(1/2)/sin(x)*EllipticE(cos(x),(-b/a)^(1/2))*(a+b*cos(x)^2)^(1/2)/(1+b*cos(x)^2/a)^(1/2)

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Rubi [A]  time = 0.04, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3178, 3177} \[ \frac {\sqrt {a+b \cos ^2(x)} E\left (x+\frac {\pi }{2}|-\frac {b}{a}\right )}{\sqrt {\frac {b \cos ^2(x)}{a}+1}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*Cos[x]^2],x]

[Out]

(Sqrt[a + b*Cos[x]^2]*EllipticE[Pi/2 + x, -(b/a)])/Sqrt[1 + (b*Cos[x]^2)/a]

Rule 3177

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[e + f*x, -(b/a)])/f, x]
/; FreeQ[{a, b, e, f}, x] && GtQ[a, 0]

Rule 3178

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Dist[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[1 + (b*Sin
[e + f*x]^2)/a], Int[Sqrt[1 + (b*Sin[e + f*x]^2)/a], x], x] /; FreeQ[{a, b, e, f}, x] &&  !GtQ[a, 0]

Rubi steps

\begin {align*} \int \sqrt {a+b \cos ^2(x)} \, dx &=\frac {\sqrt {a+b \cos ^2(x)} \int \sqrt {1+\frac {b \cos ^2(x)}{a}} \, dx}{\sqrt {1+\frac {b \cos ^2(x)}{a}}}\\ &=\frac {\sqrt {a+b \cos ^2(x)} E\left (\frac {\pi }{2}+x|-\frac {b}{a}\right )}{\sqrt {1+\frac {b \cos ^2(x)}{a}}}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 46, normalized size = 1.10 \[ \frac {\sqrt {2 a+b \cos (2 x)+b} E\left (x\left |\frac {b}{a+b}\right .\right )}{\sqrt {\frac {2 a+b \cos (2 x)+b}{a+b}}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b*Cos[x]^2],x]

[Out]

(Sqrt[2*a + b + b*Cos[2*x]]*EllipticE[x, b/(a + b)])/Sqrt[(2*a + b + b*Cos[2*x])/(a + b)]

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fricas [F]  time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {b \cos \relax (x)^{2} + a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(b*cos(x)^2 + a), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \cos \relax (x)^{2} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(b*cos(x)^2 + a), x)

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maple [A]  time = 0.97, size = 49, normalized size = 1.17 \[ -\frac {a \sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, \sqrt {\frac {a +b \left (\cos ^{2}\relax (x )\right )}{a}}\, \EllipticE \left (\cos \relax (x ), \sqrt {-\frac {b}{a}}\right )}{\sin \relax (x ) \sqrt {a +b \left (\cos ^{2}\relax (x )\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*(sin(x)^2)^(1/2)*((a+b*cos(x)^2)/a)^(1/2)*EllipticE(cos(x),(-1/a*b)^(1/2))/sin(x)/(a+b*cos(x)^2)^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \cos \relax (x)^{2} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(b*cos(x)^2 + a), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \sqrt {b\,{\cos \relax (x)}^2+a} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*cos(x)^2)^(1/2),x)

[Out]

int((a + b*cos(x)^2)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a + b \cos ^{2}{\relax (x )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a + b*cos(x)**2), x)

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