3.919 \(\int \frac{1}{\sec (x)+\sin (x) \tan (x)} \, dx\)

Optimal. Leaf size=3 \[ \tan ^{-1}(\sin (x)) \]

[Out]

ArcTan[Sin[x]]

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Rubi [A]  time = 0.0301497, antiderivative size = 3, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {4397, 3190, 203} \[ \tan ^{-1}(\sin (x)) \]

Antiderivative was successfully verified.

[In]

Int[(Sec[x] + Sin[x]*Tan[x])^(-1),x]

[Out]

ArcTan[Sin[x]]

Rule 4397

Int[u_, x_Symbol] :> Int[TrigSimplify[u], x] /; TrigSimplifyQ[u]

Rule 3190

Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e +
f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

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

Mathematica [A]  time = 0.0175787, size = 3, normalized size = 1. \[ \tan ^{-1}(\sin (x)) \]

Antiderivative was successfully verified.

[In]

Integrate[(Sec[x] + Sin[x]*Tan[x])^(-1),x]

[Out]

ArcTan[Sin[x]]

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Maple [A]  time = 0.043, size = 4, normalized size = 1.3 \begin{align*} \arctan \left ( \sin \left ( x \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sec(x)+sin(x)*tan(x)),x)

[Out]

arctan(sin(x))

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Maxima [B]  time = 0.971993, size = 61, normalized size = 20.33 \begin{align*} \frac{1}{2} \, \arctan \left (\sin \left (2 \, x\right ) + 2 \, \sin \left (x\right ), \cos \left (2 \, x\right ) + 2 \, \cos \left (x\right ) - 1\right ) - \frac{1}{2} \, \arctan \left (\sin \left (2 \, x\right ) - 2 \, \sin \left (x\right ), \cos \left (2 \, x\right ) - 2 \, \cos \left (x\right ) - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*arctan2(sin(2*x) + 2*sin(x), cos(2*x) + 2*cos(x) - 1) - 1/2*arctan2(sin(2*x) - 2*sin(x), cos(2*x) - 2*cos(
x) - 1)

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Fricas [A]  time = 2.09466, size = 22, normalized size = 7.33 \begin{align*} \arctan \left (\sin \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(sin(x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(sin(x)*tan(x) + sec(x)), x)

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Giac [A]  time = 1.09693, size = 4, normalized size = 1.33 \begin{align*} \arctan \left (\sin \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(sin(x))