3.1.62 \(\int \frac {1}{\sqrt {-1-\cos ^2(x)}} \, dx\) [62]

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

[Out]

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

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Rubi [A]
time = 0.01, 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 = {3262, 3261} \begin {gather*} \frac {\sqrt {\cos ^2(x)+1} F\left (\left .x+\frac {\pi }{2}\right |-1\right )}{\sqrt {-\cos ^2(x)-1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[-1 - Cos[x]^2],x]

[Out]

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

Rule 3261

Int[1/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Simp[(1/(Sqrt[a]*f))*EllipticF[e + f*x, -b/a]
, x] /; FreeQ[{a, b, e, f}, x] && GtQ[a, 0]

Rule 3262

Int[1/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Dist[Sqrt[1 + b*(Sin[e + f*x]^2/a)]/Sqrt[a +
b*Sin[e + f*x]^2], Int[1/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 \frac {1}{\sqrt {-1-\cos ^2(x)}} \, dx &=\frac {\sqrt {1+\cos ^2(x)} \int \frac {1}{\sqrt {1+\cos ^2(x)}} \, dx}{\sqrt {-1-\cos ^2(x)}}\\ &=\frac {\sqrt {1+\cos ^2(x)} F\left (\left .\frac {\pi }{2}+x\right |-1\right )}{\sqrt {-1-\cos ^2(x)}}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 33, normalized size = 1.03 \begin {gather*} \frac {\sqrt {3+\cos (2 x)} F\left (x\left |\frac {1}{2}\right .\right )}{\sqrt {2} \sqrt {-3-\cos (2 x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[-1 - Cos[x]^2],x]

[Out]

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

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Maple [A]
time = 0.28, size = 62, normalized size = 1.94

method	result	size
default	\(\frac {i \sqrt {-\left (1+\cos ^{2}\left (x \right )\right ) \left (\sin ^{2}\left (x \right )\right )}\, \sqrt {1+\cos ^{2}\left (x \right )}\, \sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, \EllipticF \left (i \cos \left (x \right ), i\right )}{\sqrt {\cos ^{4}\left (x \right )-1}\, \sin \left (x \right ) \sqrt {-1-\left (\cos ^{2}\left (x \right )\right )}}\)	\(62\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

[Out]

integrate(1/sqrt(-cos(x)^2 - 1), x)

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Fricas [A]
time = 0.08, size = 41, normalized size = 1.28 \begin {gather*} 2 \, {\left (2 \, \sqrt {2} + 3\right )} \sqrt {2 \, \sqrt {2} - 3} F(\arcsin \left (\sqrt {2 \, \sqrt {2} - 3} e^{\left (i \, x\right )}\right )\,|\,12 \, \sqrt {2} + 17) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(2*sqrt(2) + 3)*sqrt(2*sqrt(2) - 3)*elliptic_f(arcsin(sqrt(2*sqrt(2) - 3)*e^(I*x)), 12*sqrt(2) + 17)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/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/(-1-cos(x)^2)^(1/2),x, algorithm="giac")

[Out]

integrate(1/sqrt(-cos(x)^2 - 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(- cos(x)^2 - 1)^(1/2),x)

[Out]

int(1/(- cos(x)^2 - 1)^(1/2), x)

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