3.676 \(\int \cos (x) \sec (\sin (x)) \, dx\)

Optimal. Leaf size=4 \[ \tanh ^{-1}(\sin (\sin (x))) \]

[Out]

ArcTanh[Sin[Sin[x]]]

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Rubi [A]  time = 0.0069563, antiderivative size = 4, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4334, 3770} \[ \tanh ^{-1}(\sin (\sin (x))) \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]*Sec[Sin[x]],x]

[Out]

ArcTanh[Sin[Sin[x]]]

Rule 4334

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

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \cos (x) \sec (\sin (x)) \, dx &=\operatorname{Subst}(\int \sec (x) \, dx,x,\sin (x))\\ &=\tanh ^{-1}(\sin (\sin (x)))\\ \end{align*}

Mathematica [A]  time = 0.0045549, size = 4, normalized size = 1. \[ \tanh ^{-1}(\sin (\sin (x))) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]*Sec[Sin[x]],x]

[Out]

ArcTanh[Sin[Sin[x]]]

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Maple [A]  time = 0.009, size = 9, normalized size = 2.3 \begin{align*} \ln \left ( \sec \left ( \sin \left ( x \right ) \right ) +\tan \left ( \sin \left ( x \right ) \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)*sec(sin(x)),x)

[Out]

ln(sec(sin(x))+tan(sin(x)))

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Maxima [B]  time = 0.950847, size = 11, normalized size = 2.75 \begin{align*} \log \left (\sec \left (\sin \left (x\right )\right ) + \tan \left (\sin \left (x\right )\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*sec(sin(x)),x, algorithm="maxima")

[Out]

log(sec(sin(x)) + tan(sin(x)))

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Fricas [B]  time = 2.08479, size = 140, normalized size = 35. \begin{align*} \frac{1}{2} \, \log \left (\sin \left (\frac{2 \, \tan \left (\frac{1}{2} \, x\right )}{\tan \left (\frac{1}{2} \, x\right )^{2} + 1}\right ) + 1\right ) - \frac{1}{2} \, \log \left (-\sin \left (\frac{2 \, \tan \left (\frac{1}{2} \, x\right )}{\tan \left (\frac{1}{2} \, x\right )^{2} + 1}\right ) + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*sec(sin(x)),x, algorithm="fricas")

[Out]

1/2*log(sin(2*tan(1/2*x)/(tan(1/2*x)^2 + 1)) + 1) - 1/2*log(-sin(2*tan(1/2*x)/(tan(1/2*x)^2 + 1)) + 1)

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Sympy [A]  time = 1.72252, size = 10, normalized size = 2.5 \begin{align*} \log{\left (\tan{\left (\sin{\left (x \right )} \right )} + \sec{\left (\sin{\left (x \right )} \right )} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*sec(sin(x)),x)

[Out]

log(tan(sin(x)) + sec(sin(x)))

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Giac [B]  time = 1.08891, size = 39, normalized size = 9.75 \begin{align*} \frac{1}{4} \, \log \left ({\left | \frac{1}{\sin \left (\sin \left (x\right )\right )} + \sin \left (\sin \left (x\right )\right ) + 2 \right |}\right ) - \frac{1}{4} \, \log \left ({\left | \frac{1}{\sin \left (\sin \left (x\right )\right )} + \sin \left (\sin \left (x\right )\right ) - 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*sec(sin(x)),x, algorithm="giac")

[Out]

1/4*log(abs(1/sin(sin(x)) + sin(sin(x)) + 2)) - 1/4*log(abs(1/sin(sin(x)) + sin(sin(x)) - 2))