3.1.56 \(\int \sqrt {-1-\cos ^2(x)} \, dx\) [56]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.02, antiderivative size = 32, normalized size of antiderivative = 1.00, 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 = {3257, 3256} \begin {gather*} \frac {\sqrt {-\cos ^2(x)-1} E\left (\left .x+\frac {\pi }{2}\right |-1\right )}{\sqrt {\cos ^2(x)+1}} \end {gather*}

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 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]

Rule 3257

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]

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 = 0.05, size = 34, normalized size = 1.06 \begin {gather*} -\frac {\sqrt {2} \sqrt {3+\cos (2 x)} E\left (x\left |\frac {1}{2}\right .\right )}{\sqrt {-3-\cos (2 x)}} \end {gather*}

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]])

________________________________________________________________________________________

Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (35 ) = 70\).
time = 0.39, size = 75, normalized size = 2.34

method	result	size
default	\(-\frac {i \sqrt {-\left (1+\cos ^{2}\left (x \right )\right ) \left (\sin ^{2}\left (x \right )\right )}\, \sqrt {1+\cos ^{2}\left (x \right )}\, \sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, \left (2 \EllipticF \left (i \cos \left (x \right ), i\right )-\EllipticE \left (i \cos \left (x \right ), i\right )\right )}{\sqrt {\cos ^{4}\left (x \right )-1}\, \sin \left (x \right ) \sqrt {-1-\left (\cos ^{2}\left (x \right )\right )}}\)	\(75\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)

________________________________________________________________________________________

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)

________________________________________________________________________________________

Fricas [F]
time = 0.08, size = 112, normalized size = 3.50 \begin {gather*} \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 {gather*}

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))

________________________________________________________________________________________

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)

________________________________________________________________________________________

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)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \sqrt {-{\cos \left (x\right )}^2-1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((- cos(x)^2 - 1)^(1/2),x)

[Out]

int((- cos(x)^2 - 1)^(1/2), x)

________________________________________________________________________________________