3.710 \(\int \frac{\sec ^2(x)}{\sqrt{1-4 \tan ^2(x)}} \, dx\)

Optimal. Leaf size=9 \[ \frac{1}{2} \sin ^{-1}(2 \tan (x)) \]

[Out]

ArcSin[2*Tan[x]]/2

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Rubi [A]  time = 0.0468873, 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 = {3675, 216} \[ \frac{1}{2} \sin ^{-1}(2 \tan (x)) \]

Antiderivative was successfully verified.

[In]

Int[Sec[x]^2/Sqrt[1 - 4*Tan[x]^2],x]

[Out]

ArcSin[2*Tan[x]]/2

Rule 3675

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> With[
{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff/(c^(m - 1)*f), Subst[Int[(c^2 + ff^2*x^2)^(m/2 - 1)*(a + b*(ff*x)
^n)^p, x], x, (c*Tan[e + f*x])/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[m/2] && (IntegersQ[n, p
] || IGtQ[m, 0] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])

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

Mathematica [B]  time = 0.0622612, size = 47, normalized size = 5.22 \[ \frac{\sqrt{5 \cos (2 x)-3} \sec (x) \tan ^{-1}\left (\frac{2 \sin (x)}{\sqrt{1-5 \sin ^2(x)}}\right )}{2 \sqrt{2-8 \tan ^2(x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[x]^2/Sqrt[1 - 4*Tan[x]^2],x]

[Out]

(ArcTan[(2*Sin[x])/Sqrt[1 - 5*Sin[x]^2]]*Sqrt[-3 + 5*Cos[2*x]]*Sec[x])/(2*Sqrt[2 - 8*Tan[x]^2])

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Maple [C]  time = 0.361, size = 172, normalized size = 19.1 \begin{align*}{\frac{\sqrt{2} \left ( \sin \left ( x \right ) \right ) ^{2}}{\sqrt{9+4\,\sqrt{5}}\cos \left ( x \right ) \left ( -1+\cos \left ( x \right ) \right ) }\sqrt{{\frac{2\,\cos \left ( x \right ) \sqrt{5}-2\,\sqrt{5}+5\,\cos \left ( x \right ) -4}{1+\cos \left ( x \right ) }}}\sqrt{-2\,{\frac{2\,\cos \left ( x \right ) \sqrt{5}-2\,\sqrt{5}-5\,\cos \left ( x \right ) +4}{1+\cos \left ( x \right ) }}} \left ( 2\,{\it EllipticPi} \left ({\frac{\sqrt{9+4\,\sqrt{5}} \left ( -1+\cos \left ( x \right ) \right ) }{\sin \left ( x \right ) }}, \left ( 9+4\,\sqrt{5} \right ) ^{-1},{\frac{\sqrt{9-4\,\sqrt{5}}}{\sqrt{9+4\,\sqrt{5}}}} \right ) -{\it EllipticF} \left ({\frac{ \left ( -1+\cos \left ( x \right ) \right ) \left ( \sqrt{5}+2 \right ) }{\sin \left ( x \right ) }},9-4\,\sqrt{5} \right ) \right ){\frac{1}{\sqrt{{\frac{5\, \left ( \cos \left ( x \right ) \right ) ^{2}-4}{ \left ( \cos \left ( x \right ) \right ) ^{2}}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(9+4*5^(1/2))^(1/2)*2^(1/2)*((2*cos(x)*5^(1/2)-2*5^(1/2)+5*cos(x)-4)/(1+cos(x)))^(1/2)*(-2*(2*cos(x)*5^(1/2)
-2*5^(1/2)-5*cos(x)+4)/(1+cos(x)))^(1/2)*(2*EllipticPi((9+4*5^(1/2))^(1/2)*(-1+cos(x))/sin(x),1/(9+4*5^(1/2)),
(9-4*5^(1/2))^(1/2)/(9+4*5^(1/2))^(1/2))-EllipticF((-1+cos(x))*(5^(1/2)+2)/sin(x),9-4*5^(1/2)))*sin(x)^2/((5*c
os(x)^2-4)/cos(x)^2)^(1/2)/cos(x)/(-1+cos(x))

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Maxima [A]  time = 1.48008, size = 9, normalized size = 1. \begin{align*} \frac{1}{2} \, \arcsin \left (2 \, \tan \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.24892, size = 135, normalized size = 15. \begin{align*} -\frac{1}{4} \, \arctan \left (\frac{{\left (9 \, \cos \left (x\right )^{3} - 8 \, \cos \left (x\right )\right )} \sqrt{\frac{5 \, \cos \left (x\right )^{2} - 4}{\cos \left (x\right )^{2}}}}{4 \,{\left (5 \, \cos \left (x\right )^{2} - 4\right )} \sin \left (x\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*arctan(1/4*(9*cos(x)^3 - 8*cos(x))*sqrt((5*cos(x)^2 - 4)/cos(x)^2)/((5*cos(x)^2 - 4)*sin(x)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sec(x)**2/sqrt(-(2*tan(x) - 1)*(2*tan(x) + 1)), x)

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Giac [A]  time = 1.15862, size = 9, normalized size = 1. \begin{align*} \frac{1}{2} \, \arcsin \left (2 \, \tan \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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