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]

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

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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 = {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 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 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])

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*}

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Mathematica [B]  time = 0.06, 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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fricas [B]  time = 0.83, size = 45, normalized size = 5.00 \[ -\frac {1}{4} \, \arctan \left (\frac {{\left (9 \, \cos \relax (x)^{3} - 8 \, \cos \relax (x)\right )} \sqrt {\frac {5 \, \cos \relax (x)^{2} - 4}{\cos \relax (x)^{2}}}}{4 \, {\left (5 \, \cos \relax (x)^{2} - 4\right )} \sin \relax (x)}\right ) \]

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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giac [A]  time = 0.17, size = 7, normalized size = 0.78 \[ \frac {1}{2} \, \arcsin \left (2 \, \tan \relax (x)\right ) \]

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))

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maple [C]  time = 0.69, size = 171, normalized size = 19.00 \[ -\frac {\sqrt {2}\, \sqrt {\frac {2 \cos \relax (x ) \sqrt {5}+5 \cos \relax (x )-2 \sqrt {5}-4}{1+\cos \relax (x )}}\, \sqrt {-\frac {2 \left (2 \cos \relax (x ) \sqrt {5}-5 \cos \relax (x )-2 \sqrt {5}+4\right )}{1+\cos \relax (x )}}\, \left (\EllipticF \left (\frac {\left (-1+\cos \relax (x )\right ) \left (\sqrt {5}+2\right )}{\sin \relax (x )}, 9-4 \sqrt {5}\right )-2 \EllipticPi \left (\frac {\sqrt {9+4 \sqrt {5}}\, \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}, \frac {1}{9+4 \sqrt {5}}, \frac {\sqrt {9-4 \sqrt {5}}}{\sqrt {9+4 \sqrt {5}}}\right )\right ) \left (\sin ^{2}\relax (x )\right )}{\sqrt {\frac {5 \left (\cos ^{2}\relax (x )\right )-4}{\cos \relax (x )^{2}}}\, \cos \relax (x ) \left (-1+\cos \relax (x )\right ) \sqrt {9+4 \sqrt {5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.42, size = 7, normalized size = 0.78 \[ \frac {1}{2} \, \arcsin \left (2 \, \tan \relax (x)\right ) \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec ^{2}{\relax (x )}}{\sqrt {- \left (2 \tan {\relax (x )} - 1\right ) \left (2 \tan {\relax (x )} + 1\right )}}\, dx \]

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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