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

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

[Out]

EllipticE[Pi/2 + x, -1]

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Rubi [A]  time = 0.0080177, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, 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*}

Mathematica [A]  time = 0.0211922, 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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Maple [B]  time = 0.747, size = 41, normalized size = 4.6 \begin{align*} -{\frac{{\it EllipticE} \left ( \cos \left ( x \right ) ,i \right ) }{\sin \left ( x \right ) }\sqrt{ \left ( 1+ \left ( \cos \left ( x \right ) \right ) ^{2} \right ) \left ( \sin \left ( x \right ) \right ) ^{2}}\sqrt{ \left ( \sin \left ( x \right ) \right ) ^{2}}{\frac{1}{\sqrt{1- \left ( \cos \left ( x \right ) \right ) ^{4}}}}} \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\cos \left (x\right )^{2} + 1}\,{d x} \end{align*}

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., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{\cos \left (x\right )^{2} + 1}, x\right ) \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\cos ^{2}{\left (x \right )} + 1}\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\cos \left (x\right )^{2} + 1}\,{d x} \end{align*}

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)