3.674 \(\int \cos (x) \cos (\sin (x)) \, dx\)

Optimal. Leaf size=3 \[ \sin (\sin (x)) \]

[Out]

Sin[Sin[x]]

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Rubi [A]  time = 0.0084907, antiderivative size = 3, 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, 2637} \[ \sin (\sin (x)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \cos (x) \cos (\sin (x)) \, dx &=\operatorname{Subst}(\int \cos (x) \, dx,x,\sin (x))\\ &=\sin (\sin (x))\\ \end{align*}

Mathematica [A]  time = 1.48304, size = 3, normalized size = 1. \[ \sin (\sin (x)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sin[Sin[x]]

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Maple [A]  time = 0.007, size = 4, normalized size = 1.3 \begin{align*} \sin \left ( \sin \left ( x \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin(sin(x))

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Maxima [A]  time = 0.969698, size = 4, normalized size = 1.33 \begin{align*} \sin \left (\sin \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin(sin(x))

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Fricas [B]  time = 2.15008, size = 51, normalized size = 17. \begin{align*} \sin \left (\frac{2 \, \tan \left (\frac{1}{2} \, x\right )}{\tan \left (\frac{1}{2} \, x\right )^{2} + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin(2*tan(1/2*x)/(tan(1/2*x)^2 + 1))

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Sympy [A]  time = 0.540292, size = 3, normalized size = 1. \begin{align*} \sin{\left (\sin{\left (x \right )} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin(sin(x))

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Giac [A]  time = 1.07498, size = 4, normalized size = 1.33 \begin{align*} \sin \left (\sin \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin(sin(x))