3.725 \(\int \frac{\sec (x) \tan (x)}{1+\sec ^2(x)} \, dx\)

Optimal. Leaf size=5 \[ -\tan ^{-1}(\cos (x)) \]

[Out]

-ArcTan[Cos[x]]

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Rubi [A]  time = 0.0332322, antiderivative size = 5, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {4339, 203} \[ -\tan ^{-1}(\cos (x)) \]

Antiderivative was successfully verified.

[In]

Int[(Sec[x]*Tan[x])/(1 + Sec[x]^2),x]

[Out]

-ArcTan[Cos[x]]

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

Mathematica [A]  time = 0.0222478, size = 5, normalized size = 1. \[ -\tan ^{-1}(\cos (x)) \]

Antiderivative was successfully verified.

[In]

Integrate[(Sec[x]*Tan[x])/(1 + Sec[x]^2),x]

[Out]

-ArcTan[Cos[x]]

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Maple [A]  time = 0.025, size = 4, normalized size = 0.8 \begin{align*} \arctan \left ( \sec \left ( x \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(sec(x))

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Maxima [A]  time = 1.45471, size = 7, normalized size = 1.4 \begin{align*} -\arctan \left (\cos \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arctan(cos(x))

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Fricas [A]  time = 2.49107, size = 23, normalized size = 4.6 \begin{align*} -\arctan \left (\cos \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arctan(cos(x))

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Sympy [A]  time = 0.269607, size = 3, normalized size = 0.6 \begin{align*} \operatorname{atan}{\left (\sec{\left (x \right )} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)*tan(x)/(1+sec(x)**2),x)

[Out]

atan(sec(x))

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Giac [A]  time = 1.08875, size = 7, normalized size = 1.4 \begin{align*} -\arctan \left (\cos \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arctan(cos(x))