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]

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

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Rubi [A]  time = 0.0350783, antiderivative size = 42, normalized size of antiderivative = 1., 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 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]

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]

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*}

Mathematica [A]  time = 0.0722043, size = 46, normalized size = 1.1 \[ \frac{\sqrt{2 a+b \cos (2 x)+b} E\left (x\left |\frac{b}{a+b}\right .\right )}{\sqrt{\frac{2 a+b \cos (2 x)+b}{a+b}}} \]

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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Maple [A]  time = 0.336, size = 49, normalized size = 1.2 \begin{align*} -{\frac{a}{\sin \left ( x \right ) }\sqrt{ \left ( \sin \left ( x \right ) \right ) ^{2}}\sqrt{{\frac{a+b \left ( \cos \left ( x \right ) \right ) ^{2}}{a}}}{\it EllipticE} \left ( \cos \left ( x \right ) ,\sqrt{-{\frac{b}{a}}} \right ){\frac{1}{\sqrt{a+b \left ( \cos \left ( x \right ) \right ) ^{2}}}}} \end{align*}

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \cos \left (x\right )^{2} + a}\,{d x} \end{align*}

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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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{b \cos \left (x\right )^{2} + a}, x\right ) \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a + b \cos ^{2}{\left (x \right )}}\, dx \end{align*}

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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \cos \left (x\right )^{2} + a}\,{d x} \end{align*}

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)