3.700 \(\int \frac{\sec ^2(x)}{11-5 \tan (x)+5 \tan ^2(x)} \, dx\)

Optimal. Leaf size=53 \[ \frac{2 x}{\sqrt{195}}-\frac{2 \tan ^{-1}\left (\frac{10 \cos ^2(x)+12 \sin (x) \cos (x)-5}{12 \cos ^2(x)-10 \sin (x) \cos (x)+\sqrt{195}+10}\right )}{\sqrt{195}} \]

[Out]

(2*x)/Sqrt[195] - (2*ArcTan[(-5 + 10*Cos[x]^2 + 12*Cos[x]*Sin[x])/(10 + Sqrt[195] + 12*Cos[x]^2 - 10*Cos[x]*Si
n[x])])/Sqrt[195]

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Rubi [A]  time = 0.0656439, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {4342, 618, 204} \[ \frac{2 x}{\sqrt{195}}-\frac{2 \tan ^{-1}\left (\frac{10 \cos ^2(x)+12 \sin (x) \cos (x)-5}{12 \cos ^2(x)-10 \sin (x) \cos (x)+\sqrt{195}+10}\right )}{\sqrt{195}} \]

Antiderivative was successfully verified.

[In]

Int[Sec[x]^2/(11 - 5*Tan[x] + 5*Tan[x]^2),x]

[Out]

(2*x)/Sqrt[195] - (2*ArcTan[(-5 + 10*Cos[x]^2 + 12*Cos[x]*Sin[x])/(10 + Sqrt[195] + 12*Cos[x]^2 - 10*Cos[x]*Si
n[x])])/Sqrt[195]

Rule 4342

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))]^2, x_Symbol] :> With[{d = FreeFactors[Tan[c*(a + b*x)], x]}, Dist[d/
(b*c), Subst[Int[SubstFor[1, Tan[c*(a + b*x)]/d, u, x], x], x, Tan[c*(a + b*x)]/d], x] /; FunctionOfQ[Tan[c*(a
+ b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x] && NonsumQ[u] && (EqQ[F, Sec] || EqQ[F, sec])

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\sec ^2(x)}{11-5 \tan (x)+5 \tan ^2(x)} \, dx &=\operatorname{Subst}\left (\int \frac{1}{11-5 x+5 x^2} \, dx,x,\tan (x)\right )\\ &=-\left (2 \operatorname{Subst}\left (\int \frac{1}{-195-x^2} \, dx,x,-5+10 \tan (x)\right )\right )\\ &=\frac{2 x}{\sqrt{195}}+\frac{2 \tan ^{-1}\left (\frac{5-10 \cos ^2(x)-12 \cos (x) \sin (x)}{10+\sqrt{195}+12 \cos ^2(x)-10 \cos (x) \sin (x)}\right )}{\sqrt{195}}\\ \end{align*}

Mathematica [A]  time = 0.0549315, size = 22, normalized size = 0.42 \[ -\frac{2 \tan ^{-1}\left (\sqrt{\frac{5}{39}} (1-2 \tan (x))\right )}{\sqrt{195}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[x]^2/(11 - 5*Tan[x] + 5*Tan[x]^2),x]

[Out]

(-2*ArcTan[Sqrt[5/39]*(1 - 2*Tan[x])])/Sqrt[195]

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Maple [A]  time = 0.064, size = 18, normalized size = 0.3 \begin{align*}{\frac{2\,\sqrt{195}}{195}\arctan \left ({\frac{ \left ( 10\,\tan \left ( x \right ) -5 \right ) \sqrt{195}}{195}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(x)^2/(11-5*tan(x)+5*tan(x)^2),x)

[Out]

2/195*195^(1/2)*arctan(1/195*(10*tan(x)-5)*195^(1/2))

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Maxima [A]  time = 1.46819, size = 23, normalized size = 0.43 \begin{align*} \frac{2}{195} \, \sqrt{195} \arctan \left (\frac{1}{39} \, \sqrt{195}{\left (2 \, \tan \left (x\right ) - 1\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^2/(11-5*tan(x)+5*tan(x)^2),x, algorithm="maxima")

[Out]

2/195*sqrt(195)*arctan(1/39*sqrt(195)*(2*tan(x) - 1))

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Fricas [A]  time = 2.10278, size = 188, normalized size = 3.55 \begin{align*} \frac{1}{195} \, \sqrt{195} \arctan \left (-\frac{192 \, \sqrt{195} \cos \left (x\right )^{2} - 160 \, \sqrt{195} \cos \left (x\right ) \sin \left (x\right ) - 35 \, \sqrt{195}}{195 \,{\left (10 \, \cos \left (x\right )^{2} + 12 \, \cos \left (x\right ) \sin \left (x\right ) - 5\right )}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^2/(11-5*tan(x)+5*tan(x)^2),x, algorithm="fricas")

[Out]

1/195*sqrt(195)*arctan(-1/195*(192*sqrt(195)*cos(x)^2 - 160*sqrt(195)*cos(x)*sin(x) - 35*sqrt(195))/(10*cos(x)
^2 + 12*cos(x)*sin(x) - 5))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec ^{2}{\left (x \right )}}{5 \tan ^{2}{\left (x \right )} - 5 \tan{\left (x \right )} + 11}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)**2/(11-5*tan(x)+5*tan(x)**2),x)

[Out]

Integral(sec(x)**2/(5*tan(x)**2 - 5*tan(x) + 11), x)

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Giac [A]  time = 1.12237, size = 23, normalized size = 0.43 \begin{align*} \frac{2}{195} \, \sqrt{195} \arctan \left (\frac{1}{39} \, \sqrt{195}{\left (2 \, \tan \left (x\right ) - 1\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^2/(11-5*tan(x)+5*tan(x)^2),x, algorithm="giac")

[Out]

2/195*sqrt(195)*arctan(1/39*sqrt(195)*(2*tan(x) - 1))