3.55 \(\int \sqrt {1+\cos ^2(x)} \, dx\)

Optimal. Leaf size=9 \[ E\left (\left .x+\frac {\pi }{2}\right |-1\right ) \]

[Out]

-(sin(x)^2)^(1/2)/sin(x)*EllipticE(cos(x),I)

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Rubi [A]  time = 0.01, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {3177} \[ E\left (\left .x+\frac {\pi }{2}\right |-1\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + Cos[x]^2],x]

[Out]

EllipticE[Pi/2 + x, -1]

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

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Mathematica [A]  time = 0.02, size = 11, normalized size = 1.22 \[ \sqrt {2} E\left (x\left |\frac {1}{2}\right .\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 + Cos[x]^2],x]

[Out]

Sqrt[2]*EllipticE[x, 1/2]

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fricas [F]  time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {\cos \relax (x)^{2} + 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(cos(x)^2 + 1), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\cos \relax (x)^{2} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(cos(x)^2 + 1), x)

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maple [B]  time = 1.56, size = 41, normalized size = 4.56 \[ -\frac {\sqrt {\left (1+\cos ^{2}\relax (x )\right ) \left (\sin ^{2}\relax (x )\right )}\, \sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, \EllipticE \left (\cos \relax (x ), i\right )}{\sqrt {1-\left (\cos ^{4}\relax (x )\right )}\, \sin \relax (x )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((1+cos(x)^2)*sin(x)^2)^(1/2)*(sin(x)^2)^(1/2)*EllipticE(cos(x),I)/(1-cos(x)^4)^(1/2)/sin(x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\cos \relax (x)^{2} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(cos(x)^2 + 1), x)

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mupad [B]  time = 0.01, size = 7, normalized size = 0.78 \[ \sqrt {2}\,\mathrm {E}\left (x\middle |\frac {1}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(x)^2 + 1)^(1/2),x)

[Out]

2^(1/2)*ellipticE(x, 1/2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\cos ^{2}{\relax (x )} + 1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(cos(x)**2 + 1), x)

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