3.791 \(\int \cos (x) \cos (2 \sin (x)) \, dx\)

Optimal. Leaf size=9 \[ \frac{1}{2} \sin (2 \sin (x)) \]

[Out]

Sin[2*Sin[x]]/2

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Rubi [A]  time = 0.0104997, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4334, 2637} \[ \frac{1}{2} \sin (2 \sin (x)) \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]*Cos[2*Sin[x]],x]

[Out]

Sin[2*Sin[x]]/2

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 (2 \sin (x)) \, dx &=\operatorname{Subst}(\int \cos (2 x) \, dx,x,\sin (x))\\ &=\frac{1}{2} \sin (2 \sin (x))\\ \end{align*}

Mathematica [A]  time = 1.37057, size = 9, normalized size = 1. \[ \frac{1}{2} \sin (2 \sin (x)) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]*Cos[2*Sin[x]],x]

[Out]

Sin[2*Sin[x]]/2

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Maple [A]  time = 0.013, size = 8, normalized size = 0.9 \begin{align*}{\frac{\sin \left ( 2\,\sin \left ( x \right ) \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)*cos(2*sin(x)),x)

[Out]

1/2*sin(2*sin(x))

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Maxima [A]  time = 0.964211, size = 9, normalized size = 1. \begin{align*} \frac{1}{2} \, \sin \left (2 \, \sin \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*cos(2*sin(x)),x, algorithm="maxima")

[Out]

1/2*sin(2*sin(x))

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Fricas [B]  time = 2.03069, size = 57, normalized size = 6.33 \begin{align*} \frac{1}{2} \, \sin \left (\frac{4 \, \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(2*sin(x)),x, algorithm="fricas")

[Out]

1/2*sin(4*tan(1/2*x)/(tan(1/2*x)^2 + 1))

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Sympy [A]  time = 0.515596, size = 7, normalized size = 0.78 \begin{align*} \frac{\sin{\left (2 \sin{\left (x \right )} \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*cos(2*sin(x)),x)

[Out]

sin(2*sin(x))/2

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Giac [A]  time = 1.09994, size = 9, normalized size = 1. \begin{align*} \frac{1}{2} \, \sin \left (2 \, \sin \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*cos(2*sin(x)),x, algorithm="giac")

[Out]

1/2*sin(2*sin(x))