\(\int e^{\cos (x)} \cos (2 x+\sin (x)) \, dx\) [313]

Optimal result

Integrand size = 12, antiderivative size = 21 \[ \int e^{\cos (x)} \cos (2 x+\sin (x)) \, dx=2 e^{\cos (x)} \cos \left (\frac {x}{2}+\sin (x)\right ) \sin \left (\frac {x}{2}\right ) \]

[Out]

2*exp(cos(x))*cos(1/2*x+sin(x))*sin(1/2*x)

Rubi [F]

\[ \int e^{\cos (x)} \cos (2 x+\sin (x)) \, dx=\int e^{\cos (x)} \cos (2 x+\sin (x)) \, dx \]

[In]

Int[E^Cos[x]*Cos[2*x + Sin[x]],x]

[Out]

Defer[Int][E^Cos[x]*Cos[2*x + Sin[x]], x]

Rubi steps \begin{align*} \text {integral}& = \int e^{\cos (x)} \cos (2 x+\sin (x)) \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int e^{\cos (x)} \cos (2 x+\sin (x)) \, dx=2 e^{\cos (x)} \cos \left (\frac {x}{2}+\sin (x)\right ) \sin \left (\frac {x}{2}\right ) \]

[In]

Integrate[E^Cos[x]*Cos[2*x + Sin[x]],x]

[Out]

2*E^Cos[x]*Cos[x/2 + Sin[x]]*Sin[x/2]

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 2.17 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.48

method	result	size
risch	\(-\frac {i {\mathrm e}^{i x} {\mathrm e}^{{\mathrm e}^{i x}}}{2}+\frac {i {\mathrm e}^{{\mathrm e}^{i x}}}{2}+\frac {i {\mathrm e}^{-i x} {\mathrm e}^{{\mathrm e}^{-i x}}}{2}-\frac {i {\mathrm e}^{{\mathrm e}^{-i x}}}{2}\)	\(52\)

[In]

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

[Out]

-1/2*I*exp(I*x)*exp(exp(I*x))+1/2*I*exp(exp(I*x))+1/2*I*exp(1/exp(I*x))*exp(-I*x)-1/2*I*exp(1/exp(I*x))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (16) = 32\).

Time = 0.26 (sec) , antiderivative size = 84, normalized size of antiderivative = 4.00 \[ \int e^{\cos (x)} \cos (2 x+\sin (x)) \, dx={\left (2 \, \cos \left (x\right ) - 1\right )} \cos \left (\frac {2 \, {\left (x \tan \left (\frac {1}{2} \, x\right )^{2} + x + \tan \left (\frac {1}{2} \, x\right )\right )}}{\tan \left (\frac {1}{2} \, x\right )^{2} + 1}\right ) e^{\cos \left (x\right )} \sin \left (x\right ) - {\left (2 \, \cos \left (x\right )^{2} - \cos \left (x\right ) - 1\right )} e^{\cos \left (x\right )} \sin \left (\frac {2 \, {\left (x \tan \left (\frac {1}{2} \, x\right )^{2} + x + \tan \left (\frac {1}{2} \, x\right )\right )}}{\tan \left (\frac {1}{2} \, x\right )^{2} + 1}\right ) \]

[In]

integrate(exp(cos(x))*cos(2*x+sin(x)),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int e^{\cos (x)} \cos (2 x+\sin (x)) \, dx=\int e^{\cos {\left (x \right )}} \cos {\left (2 x + \sin {\left (x \right )} \right )}\, dx \]

[In]

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

[Out]

Integral(exp(cos(x))*cos(2*x + sin(x)), x)

Maxima [F]

\[ \int e^{\cos (x)} \cos (2 x+\sin (x)) \, dx=\int { \cos \left (2 \, x + \sin \left (x\right )\right ) e^{\cos \left (x\right )} \,d x } \]

[In]

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

[Out]

1/4*e^cos(x)*sin(2*x + sin(x)) + 1/2*e^cos(x)*sin(x + sin(x)) - 1/2*e^cos(x)*sin(sin(x)) - 1/4*integrate(cos(3
*x + sin(x))*e^cos(x), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (16) = 32\).

Time = 0.28 (sec) , antiderivative size = 136, normalized size of antiderivative = 6.48 \[ \int e^{\cos (x)} \cos (2 x+\sin (x)) \, dx=\frac {\cos \left (x\right )^{3} e^{\cos \left (x\right )} \sin \left (2 \, x + \sin \left (x\right )\right ) - \cos \left (2 \, x + \sin \left (x\right )\right ) \cos \left (x\right )^{2} e^{\cos \left (x\right )} \sin \left (x\right ) + \cos \left (x\right ) e^{\cos \left (x\right )} \sin \left (2 \, x + \sin \left (x\right )\right ) \sin \left (x\right )^{2} - \cos \left (2 \, x + \sin \left (x\right )\right ) e^{\cos \left (x\right )} \sin \left (x\right )^{3} - \cos \left (x\right )^{2} e^{\cos \left (x\right )} \sin \left (2 \, x + \sin \left (x\right )\right ) + 2 \, \cos \left (2 \, x + \sin \left (x\right )\right ) \cos \left (x\right ) e^{\cos \left (x\right )} \sin \left (x\right ) + e^{\cos \left (x\right )} \sin \left (2 \, x + \sin \left (x\right )\right ) \sin \left (x\right )^{2}}{\cos \left (x\right )^{4} + 2 \, \cos \left (x\right )^{2} \sin \left (x\right )^{2} + \sin \left (x\right )^{4}} \]

[In]

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

[Out]

(cos(x)^3*e^cos(x)*sin(2*x + sin(x)) - cos(2*x + sin(x))*cos(x)^2*e^cos(x)*sin(x) + cos(x)*e^cos(x)*sin(2*x +
sin(x))*sin(x)^2 - cos(2*x + sin(x))*e^cos(x)*sin(x)^3 - cos(x)^2*e^cos(x)*sin(2*x + sin(x)) + 2*cos(2*x + sin
(x))*cos(x)*e^cos(x)*sin(x) + e^cos(x)*sin(2*x + sin(x))*sin(x)^2)/(cos(x)^4 + 2*cos(x)^2*sin(x)^2 + sin(x)^4)

Mupad [B] (verification not implemented)

Time = 17.19 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.71 \[ \int e^{\cos (x)} \cos (2 x+\sin (x)) \, dx={\mathrm {e}}^{\cos \left (x\right )}\,\left (\sin \left (x+\sin \left (x\right )\right )-\sin \left (\sin \left (x\right )\right )\right ) \]

[In]

int(exp(cos(x))*cos(2*x + sin(x)),x)

[Out]

exp(cos(x))*(sin(x + sin(x)) - sin(sin(x)))