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 = 0.01, antiderivative size = 3, normalized size of antiderivative = 1.00, 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 2637

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

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*}

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Mathematica [A]  time = 1.47, size = 3, normalized size = 1.00 \[ \sin (\sin (x)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sin[Sin[x]]

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fricas [B]  time = 1.64, size = 17, normalized size = 5.67 \[ \sin \left (\frac {2 \, \tan \left (\frac {1}{2} \, x\right )}{\tan \left (\frac {1}{2} \, x\right )^{2} + 1}\right ) \]

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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giac [A]  time = 0.14, size = 3, normalized size = 1.00 \[ \sin \left (\sin \relax (x)\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin(sin(x))

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maple [A]  time = 0.01, size = 4, normalized size = 1.33 \[ \sin \left (\sin \relax (x )\right ) \]

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 = 0.32, size = 3, normalized size = 1.00 \[ \sin \left (\sin \relax (x)\right ) \]

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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mupad [B]  time = 2.95, size = 3, normalized size = 1.00 \[ \sin \left (\sin \relax (x)\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin(sin(x))

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sympy [A]  time = 0.41, size = 3, normalized size = 1.00 \[ \sin {\left (\sin {\relax (x )} \right )} \]

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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