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[Tan[x]/Sqrt[3]]

________________________________________________________________________________________

Rubi [A]  time = 0.0467953, antiderivative size = 9, normalized size of antiderivative = 1., 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 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]

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]

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*}

Mathematica [B]  time = 0.0437544, 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]

________________________________________________________________________________________

Maple [C]  time = 0.164, size = 103, normalized size = 11.4 \begin{align*} -{\frac{\sqrt{3}\sqrt{2}\sqrt{6} \left ( \sin \left ( x \right ) \right ) ^{2}}{9\,\cos \left ( x \right ) \left ( -1+\cos \left ( x \right ) \right ) }\sqrt{{\frac{2\,\cos \left ( x \right ) -1}{1+\cos \left ( x \right ) }}}\sqrt{{\frac{1+2\,\cos \left ( x \right ) }{1+\cos \left ( x \right ) }}} \left ({\it EllipticF} \left ({\frac{\sqrt{3} \left ( -1+\cos \left ( x \right ) \right ) }{\sin \left ( x \right ) }},{\frac{1}{3}} \right ) -2\,{\it EllipticPi} \left ({\frac{\sqrt{3} \left ( -1+\cos \left ( x \right ) \right ) }{\sin \left ( x \right ) }},1/3,1/3 \right ) \right ){\frac{1}{\sqrt{{\frac{4\, \left ( \cos \left ( x \right ) \right ) ^{2}-1}{ \left ( \cos \left ( x \right ) \right ) ^{2}}}}}}} \end{align*}

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*3^(1/2)*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+cos(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)/cos(x)/(-1+cos(x))

________________________________________________________________________________________

Maxima [A]  time = 1.41891, size = 11, normalized size = 1.22 \begin{align*} \arcsin \left (\frac{1}{3} \, \sqrt{3} \tan \left (x\right )\right ) \end{align*}

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))

________________________________________________________________________________________

Fricas [B]  time = 2.11589, size = 76, normalized size = 8.44 \begin{align*} -\arctan \left (\frac{\sqrt{\frac{4 \, \cos \left (x\right )^{2} - 1}{\cos \left (x\right )^{2}}} \cos \left (x\right )}{\sin \left (x\right )}\right ) \end{align*}

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))

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec \left (x\right )^{2}}{\sqrt{-\sec \left (x\right )^{2} + 4}}\,{d x} \end{align*}

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)