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

Optimal. Leaf size=21 \[ 2 e^{\cos (x)} \cos \left (\frac {x}{2}+\sin (x)\right ) \sin \left (\frac {x}{2}\right ) \]

[Out]

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

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Rubi [F]
time = 0.03, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int e^{\cos (x)} \cos (2 x+\sin (x)) \, dx \end {gather*}

Verification is not applicable to the result.

[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 {gather*} \begin {aligned} \text {Integral} &=\int e^{\cos (x)} \cos (2 x+\sin (x)) \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.05, size = 21, normalized size = 1.00 \begin {gather*} 2 e^{\cos (x)} \cos \left (\frac {x}{2}+\sin (x)\right ) \sin \left (\frac {x}{2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2*E^Cos[x]*Cos[x/2 + Sin[x]]*Sin[x/2]

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Maple [C] Result contains complex when optimal does not.
time = 1.80, size = 52, normalized size = 2.48

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (16) = 32\).
time = 0.62, size = 84, normalized size = 4.00 \begin {gather*} {\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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int e^{\cos {\left (x \right )}} \cos {\left (2 x + \sin {\left (x \right )} \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (16) = 32\).
time = 0.55, size = 136, normalized size = 6.48 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Mupad [B]
time = 0.36, size = 15, normalized size = 0.71 \begin {gather*} {\mathrm {e}}^{\cos \left (x\right )}\,\left (\sin \left (x+\sin \left (x\right )\right )-\sin \left (\sin \left (x\right )\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(cos(x))*cos(2*x + sin(x)),x)

[Out]

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

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Chatgpt [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {not solved} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

not solved

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