3.1.55 \(\int \sqrt {1+\cos ^2(x)} \, dx\) [55]

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

[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 = {3256} \begin {gather*} E\left (\left .x+\frac {\pi }{2}\right |-1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + Cos[x]^2],x]

[Out]

EllipticE[Pi/2 + x, -1]

Rule 3256

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Simp[(Sqrt[a]/f)*EllipticE[e + f*x, -b/a], 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 = 0.03, size = 11, normalized size = 1.22 \begin {gather*} \sqrt {2} E\left (x\left |\frac {1}{2}\right .\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 + Cos[x]^2],x]

[Out]

Sqrt[2]*EllipticE[x, 1/2]

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 40 vs. \(2 (17 ) = 34\).
time = 0.32, size = 41, normalized size = 4.56

method	result	size
default	\(-\frac {\sqrt {\left (1+\cos ^{2}\left (x \right )\right ) \left (\sin ^{2}\left (x \right )\right )}\, \sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, \EllipticE \left (\cos \left (x \right ), i\right )}{\sqrt {1-\left (\cos ^{4}\left (x \right )\right )}\, \sin \left (x \right )}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+cos(x)^2)^(1/2),x,method=_RETURNVERBOSE)

[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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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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Fricas [F]
time = 0.08, size = 10, normalized size = 1.11 \begin {gather*} {\rm integral}\left (\sqrt {\cos \left (x\right )^{2} + 1}, x\right ) \end {gather*}

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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {\cos ^{2}{\left (x \right )} + 1}\, dx \end {gather*}

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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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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