\(\int \cos (x) \cos (2 \sin (x)) \, dx\) [791]

Optimal result

Integrand size = 8, antiderivative size = 9 \[ \int \cos (x) \cos (2 \sin (x)) \, dx=\frac {1}{2} \sin (2 \sin (x)) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4419, 2717} \[ \int \cos (x) \cos (2 \sin (x)) \, dx=\frac {1}{2} \sin (2 \sin (x)) \]

[In]

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

[Out]

Sin[2*Sin[x]]/2

Rule 2717

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

Rule 4419

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

Mathematica [A] (verified)

Time = 1.16 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00 \[ \int \cos (x) \cos (2 \sin (x)) \, dx=\frac {1}{2} \sin (2 \sin (x)) \]

[In]

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

[Out]

Sin[2*Sin[x]]/2

Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.89

method	result	size
derivativedivides	\(\frac {\sin \left (2 \sin \left (x \right )\right )}{2}\)	\(8\)
default	\(\frac {\sin \left (2 \sin \left (x \right )\right )}{2}\)	\(8\)
risch	\(\frac {\sin \left (2 \sin \left (x \right )\right )}{2}\)	\(8\)
parallelrisch	\(\frac {\sin \left (2 \sin \left (x \right )\right )}{2}\)	\(8\)
norman	\(\frac {\tan \left (\frac {x}{2}\right )^{2} \tan \left (\frac {2 \tan \left (\frac {x}{2}\right )}{1+\tan \left (\frac {x}{2}\right )^{2}}\right )+\tan \left (\frac {2 \tan \left (\frac {x}{2}\right )}{1+\tan \left (\frac {x}{2}\right )^{2}}\right )}{\left (1+\tan \left (\frac {x}{2}\right )^{2}\right ) \left (1+\tan \left (\frac {2 \tan \left (\frac {x}{2}\right )}{1+\tan \left (\frac {x}{2}\right )^{2}}\right )^{2}\right )}\)	\(77\)

[In]

int(cos(x)*cos(2*sin(x)),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 19 vs. \(2 (7) = 14\).

Time = 0.26 (sec) , antiderivative size = 19, normalized size of antiderivative = 2.11 \[ \int \cos (x) \cos (2 \sin (x)) \, dx=\frac {1}{2} \, \sin \left (\frac {4 \, \tan \left (\frac {1}{2} \, x\right )}{\tan \left (\frac {1}{2} \, x\right )^{2} + 1}\right ) \]

[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))

Sympy [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.78 \[ \int \cos (x) \cos (2 \sin (x)) \, dx=\frac {\sin {\left (2 \sin {\left (x \right )} \right )}}{2} \]

[In]

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

[Out]

sin(2*sin(x))/2

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.78 \[ \int \cos (x) \cos (2 \sin (x)) \, dx=\frac {1}{2} \, \sin \left (2 \, \sin \left (x\right )\right ) \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.78 \[ \int \cos (x) \cos (2 \sin (x)) \, dx=\frac {1}{2} \, \sin \left (2 \, \sin \left (x\right )\right ) \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.78 \[ \int \cos (x) \cos (2 \sin (x)) \, dx=\frac {\sin \left (2\,\sin \left (x\right )\right )}{2} \]

[In]

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

[Out]

sin(2*sin(x))/2