3.726 \(\int \frac{\sec (x) \tan (x)}{9+4 \sec ^2(x)} \, dx\)

Optimal. Leaf size=11 \[ -\frac{1}{6} \tan ^{-1}\left (\frac{3 \cos (x)}{2}\right ) \]

[Out]

-ArcTan[(3*Cos[x])/2]/6

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Rubi [A]  time = 0.0347168, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {4339, 203} \[ -\frac{1}{6} \tan ^{-1}\left (\frac{3 \cos (x)}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(Sec[x]*Tan[x])/(9 + 4*Sec[x]^2),x]

[Out]

-ArcTan[(3*Cos[x])/2]/6

Rule 4339

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

Rule 203

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

Rubi steps

\begin{align*} \int \frac{\sec (x) \tan (x)}{9+4 \sec ^2(x)} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{4+9 x^2} \, dx,x,\cos (x)\right )\\ &=-\frac{1}{6} \tan ^{-1}\left (\frac{3 \cos (x)}{2}\right )\\ \end{align*}

Mathematica [A]  time = 0.0268567, size = 11, normalized size = 1. \[ -\frac{1}{6} \tan ^{-1}\left (\frac{3 \cos (x)}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(Sec[x]*Tan[x])/(9 + 4*Sec[x]^2),x]

[Out]

-ArcTan[(3*Cos[x])/2]/6

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Maple [A]  time = 0.019, size = 8, normalized size = 0.7 \begin{align*}{\frac{1}{6}\arctan \left ({\frac{2\,\sec \left ( x \right ) }{3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(x)*tan(x)/(9+4*sec(x)^2),x)

[Out]

1/6*arctan(2/3*sec(x))

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Maxima [A]  time = 1.45431, size = 9, normalized size = 0.82 \begin{align*} -\frac{1}{6} \, \arctan \left (\frac{3}{2} \, \cos \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)*tan(x)/(9+4*sec(x)^2),x, algorithm="maxima")

[Out]

-1/6*arctan(3/2*cos(x))

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Fricas [A]  time = 2.41866, size = 34, normalized size = 3.09 \begin{align*} -\frac{1}{6} \, \arctan \left (\frac{3}{2} \, \cos \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)*tan(x)/(9+4*sec(x)^2),x, algorithm="fricas")

[Out]

-1/6*arctan(3/2*cos(x))

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Sympy [A]  time = 0.308155, size = 8, normalized size = 0.73 \begin{align*} \frac{\operatorname{atan}{\left (\frac{2 \sec{\left (x \right )}}{3} \right )}}{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)*tan(x)/(9+4*sec(x)**2),x)

[Out]

atan(2*sec(x)/3)/6

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Giac [A]  time = 1.10702, size = 9, normalized size = 0.82 \begin{align*} -\frac{1}{6} \, \arctan \left (\frac{3}{2} \, \cos \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)*tan(x)/(9+4*sec(x)^2),x, algorithm="giac")

[Out]

-1/6*arctan(3/2*cos(x))