3.709 \(\int \frac {\sec ^2(x)}{\sqrt {4-\sec ^2(x)}} \, dx\)

Optimal. Leaf size=9 \[ \sin ^{-1}\left (\frac {\tan (x)}{\sqrt {3}}\right ) \]

[Out]

arcsin(1/3*tan(x)*3^(1/2))

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Rubi [A]  time = 0.05, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {4146, 216} \[ \sin ^{-1}\left (\frac {\tan (x)}{\sqrt {3}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sec[x]^2/Sqrt[4 - Sec[x]^2],x]

[Out]

ArcSin[Tan[x]/Sqrt[3]]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 4146

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = Fre
eFactors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[(1 + ff^2*x^2)^(m/2 - 1)*ExpandToSum[a + b*(1 + ff^2*x^2)^(n/
2), x]^p, x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2] && IntegerQ[n/2]

Rubi steps

\begin {align*} \int \frac {\sec ^2(x)}{\sqrt {4-\sec ^2(x)}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{\sqrt {3-x^2}} \, dx,x,\tan (x)\right )\\ &=\sin ^{-1}\left (\frac {\tan (x)}{\sqrt {3}}\right )\\ \end {align*}

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Mathematica [B]  time = 0.04, size = 43, normalized size = 4.78 \[ \frac {\sqrt {2 \cos (2 x)+1} \sec (x) \tan ^{-1}\left (\frac {\sin (x)}{\sqrt {3-4 \sin ^2(x)}}\right )}{\sqrt {4-\sec ^2(x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[x]^2/Sqrt[4 - Sec[x]^2],x]

[Out]

(ArcTan[Sin[x]/Sqrt[3 - 4*Sin[x]^2]]*Sqrt[1 + 2*Cos[2*x]]*Sec[x])/Sqrt[4 - Sec[x]^2]

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fricas [B]  time = 0.83, size = 25, normalized size = 2.78 \[ -\arctan \left (\frac {\sqrt {\frac {4 \, \cos \relax (x)^{2} - 1}{\cos \relax (x)^{2}}} \cos \relax (x)}{\sin \relax (x)}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^2/(4-sec(x)^2)^(1/2),x, algorithm="fricas")

[Out]

-arctan(sqrt((4*cos(x)^2 - 1)/cos(x)^2)*cos(x)/sin(x))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec \relax (x)^{2}}{\sqrt {-\sec \relax (x)^{2} + 4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^2/(4-sec(x)^2)^(1/2),x, algorithm="giac")

[Out]

integrate(sec(x)^2/sqrt(-sec(x)^2 + 4), x)

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maple [C]  time = 0.28, size = 103, normalized size = 11.44 \[ -\frac {\sqrt {2}\, \sqrt {\frac {2 \cos \relax (x )-1}{1+\cos \relax (x )}}\, \sqrt {6}\, \sqrt {\frac {1+2 \cos \relax (x )}{1+\cos \relax (x )}}\, \left (\EllipticF \left (\frac {\sqrt {3}\, \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}, \frac {1}{3}\right )-2 \EllipticPi \left (\frac {\sqrt {3}\, \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}, \frac {1}{3}, \frac {1}{3}\right )\right ) \left (\sin ^{2}\relax (x )\right ) \sqrt {3}}{9 \sqrt {\frac {4 \left (\cos ^{2}\relax (x )\right )-1}{\cos \relax (x )^{2}}}\, \cos \relax (x ) \left (-1+\cos \relax (x )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(x)^2/(4-sec(x)^2)^(1/2),x)

[Out]

-1/9*2^(1/2)*((2*cos(x)-1)/(1+cos(x)))^(1/2)*6^(1/2)*((1+2*cos(x))/(1+cos(x)))^(1/2)*(EllipticF(3^(1/2)*(-1+co
s(x))/sin(x),1/3)-2*EllipticPi(3^(1/2)*(-1+cos(x))/sin(x),1/3,1/3))*sin(x)^2/((4*cos(x)^2-1)/cos(x)^2)^(1/2)/c
os(x)/(-1+cos(x))*3^(1/2)

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maxima [A]  time = 0.43, size = 8, normalized size = 0.89 \[ \arcsin \left (\frac {1}{3} \, \sqrt {3} \tan \relax (x)\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^2/(4-sec(x)^2)^(1/2),x, algorithm="maxima")

[Out]

arcsin(1/3*sqrt(3)*tan(x))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.11 \[ \int \frac {1}{{\cos \relax (x)}^2\,\sqrt {4-\frac {1}{{\cos \relax (x)}^2}}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec ^{2}{\relax (x )}}{\sqrt {- \left (\sec {\relax (x )} - 2\right ) \left (\sec {\relax (x )} + 2\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)**2/(4-sec(x)**2)**(1/2),x)

[Out]

Integral(sec(x)**2/sqrt(-(sec(x) - 2)*(sec(x) + 2)), x)

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