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

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

[Out]

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

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Rubi [A]  time = 0.0183404, 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 = {3183, 3182} \[ \frac{\sqrt{\cos ^2(x)+1} F\left (\left .x+\frac{\pi }{2}\right |-1\right )}{\sqrt{-\cos ^2(x)-1}} \]

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 3183

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]

Rule 3182

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

Mathematica [A]  time = 0.0403404, size = 33, normalized size = 1.03 \[ \frac{\sqrt{\cos (2 x)+3} F\left (x\left |\frac{1}{2}\right .\right )}{\sqrt{2} \sqrt{-\cos (2 x)-3}} \]

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 = 0.943, size = 62, normalized size = 1.9 \begin{align*}{\frac{i{\it EllipticF} \left ( i\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{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/(-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)/(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., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-\cos \left (x\right )^{2} - 1}}\,{d x} \end{align*}

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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{2}{\sqrt{e^{\left (4 i \, x\right )} + 6 \, e^{\left (2 i \, x\right )} + 1}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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

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)