3.56 \(\int \sqrt{-1-\cos ^2(x)} \, dx\)

Optimal. Leaf size=32 \[ \frac{\sqrt{-\cos ^2(x)-1} E\left (\left .x+\frac{\pi }{2}\right |-1\right )}{\sqrt{\cos ^2(x)+1}} \]

[Out]

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

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Rubi [A]  time = 0.0222627, antiderivative size = 32, 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} \[ \frac{\sqrt{-\cos ^2(x)-1} E\left (\left .x+\frac{\pi }{2}\right |-1\right )}{\sqrt{\cos ^2(x)+1}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[-1 - Cos[x]^2],x]

[Out]

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

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

Mathematica [A]  time = 0.0375725, size = 34, normalized size = 1.06 \[ -\frac{\sqrt{2} \sqrt{\cos (2 x)+3} E\left (x\left |\frac{1}{2}\right .\right )}{\sqrt{-\cos (2 x)-3}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-1 - Cos[x]^2],x]

[Out]

-((Sqrt[2]*Sqrt[3 + Cos[2*x]]*EllipticE[x, 1/2])/Sqrt[-3 - Cos[2*x]])

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Maple [B]  time = 1.302, size = 75, normalized size = 2.3 \begin{align*}{\frac{-i \left ( 2\,{\it EllipticF} \left ( i\cos \left ( x \right ) ,i \right ) -{\it EllipticE} \left ( i\cos \left ( x \right ) ,i \right ) \right ) }{\sin \left ( x \right ) }\sqrt{- \left ( 1+ \left ( \cos \left ( x \right ) \right ) ^{2} \right ) \left ( \sin \left ( x \right ) \right ) ^{2}}\sqrt{1+ \left ( \cos \left ( x \right ) \right ) ^{2}}\sqrt{ \left ( \sin \left ( x \right ) \right ) ^{2}}{\frac{1}{\sqrt{ \left ( \cos \left ( x \right ) \right ) ^{4}-1}}}{\frac{1}{\sqrt{-1- \left ( \cos \left ( x \right ) \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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*} \frac{2 \,{\left (e^{\left (2 i \, x\right )} - e^{\left (i \, x\right )}\right )}{\rm integral}\left (\frac{4 \, \sqrt{e^{\left (4 i \, x\right )} + 6 \, e^{\left (2 i \, x\right )} + 1}{\left (e^{\left (2 i \, x\right )} + 1\right )}}{e^{\left (6 i \, x\right )} - 2 \, e^{\left (5 i \, x\right )} + 7 \, e^{\left (4 i \, x\right )} - 12 \, e^{\left (3 i \, x\right )} + 7 \, e^{\left (2 i \, x\right )} - 2 \, e^{\left (i \, x\right )} + 1}, x\right ) + \sqrt{e^{\left (4 i \, x\right )} + 6 \, e^{\left (2 i \, x\right )} + 1}{\left (e^{\left (i \, x\right )} + 1\right )}}{2 \,{\left (e^{\left (2 i \, x\right )} - e^{\left (i \, x\right )}\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]

1/2*(2*(e^(2*I*x) - e^(I*x))*integral(4*sqrt(e^(4*I*x) + 6*e^(2*I*x) + 1)*(e^(2*I*x) + 1)/(e^(6*I*x) - 2*e^(5*
I*x) + 7*e^(4*I*x) - 12*e^(3*I*x) + 7*e^(2*I*x) - 2*e^(I*x) + 1), x) + sqrt(e^(4*I*x) + 6*e^(2*I*x) + 1)*(e^(I
*x) + 1))/(e^(2*I*x) - e^(I*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)