3.368 \(\int e^{\sin (x)} \sin (2 x) \, dx\)

Optimal. Leaf size=15 \[ 2 e^{\sin (x)} \sin (x)-2 e^{\sin (x)} \]

[Out]

-2*E^Sin[x] + 2*E^Sin[x]*Sin[x]

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Rubi [A]  time = 0.0311459, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ 2 e^{\sin (x)} \sin (x)-2 e^{\sin (x)} \]

Antiderivative was successfully verified.

[In]  Int[E^Sin[x]*Sin[2*x],x]

[Out]

-2*E^Sin[x] + 2*E^Sin[x]*Sin[x]

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(sin(x))*sin(2*x),x)

[Out]

Timed out

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Mathematica [A]  time = 0.0170362, size = 11, normalized size = 0.73 \[ e^{\sin (x)} (2 \sin (x)-2) \]

Antiderivative was successfully verified.

[In]  Integrate[E^Sin[x]*Sin[2*x],x]

[Out]

E^Sin[x]*(-2 + 2*Sin[x])

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Maple [A]  time = 0.013, size = 14, normalized size = 0.9 \[ -2\,{{\rm e}^{\sin \left ( x \right ) }}+2\,{{\rm e}^{\sin \left ( x \right ) }}\sin \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(sin(x))*sin(2*x),x)

[Out]

-2*exp(sin(x))+2*exp(sin(x))*sin(x)

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Maxima [A]  time = 1.42882, size = 12, normalized size = 0.8 \[ 2 \,{\left (\sin \left (x\right ) - 1\right )} e^{\sin \left (x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^sin(x)*sin(2*x),x, algorithm="maxima")

[Out]

2*(sin(x) - 1)*e^sin(x)

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Fricas [A]  time = 0.250407, size = 12, normalized size = 0.8 \[ 2 \,{\left (\sin \left (x\right ) - 1\right )} e^{\sin \left (x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^sin(x)*sin(2*x),x, algorithm="fricas")

[Out]

2*(sin(x) - 1)*e^sin(x)

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Sympy [A]  time = 3.6733, size = 15, normalized size = 1. \[ 2 e^{\sin{\left (x \right )}} \sin{\left (x \right )} - 2 e^{\sin{\left (x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(exp(sin(x))*sin(2*x),x)

[Out]

2*exp(sin(x))*sin(x) - 2*exp(sin(x))

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GIAC/XCAS [A]  time = 0.217663, size = 721, normalized size = 48.07 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^sin(x)*sin(2*x),x, algorithm="giac")

[Out]

2*(e^(2*tan(1/2*x)/(tan(1/2*x)^2 + 1))*tan(1/2*x)^6*tan(x)^2 - 2*e^(2*tan(1/2*x)
/(tan(1/2*x)^2 + 1))*tan(1/2*x)^5*tan(x)^2 - e^(2*tan(1/2*x)/(tan(1/2*x)^2 + 1))
*tan(1/2*x)^6 + 8*e^(2*tan(1/2*x)/(tan(1/2*x)^2 + 1))*tan(1/2*x)^5*tan(x) - 5*e^
(2*tan(1/2*x)/(tan(1/2*x)^2 + 1))*tan(1/2*x)^4*tan(x)^2 + 2*e^(2*tan(1/2*x)/(tan
(1/2*x)^2 + 1))*tan(1/2*x)^5 - 16*e^(2*tan(1/2*x)/(tan(1/2*x)^2 + 1))*tan(1/2*x)
^4*tan(x) + 12*e^(2*tan(1/2*x)/(tan(1/2*x)^2 + 1))*tan(1/2*x)^3*tan(x)^2 + 5*e^(
2*tan(1/2*x)/(tan(1/2*x)^2 + 1))*tan(1/2*x)^4 - 5*e^(2*tan(1/2*x)/(tan(1/2*x)^2
+ 1))*tan(1/2*x)^2*tan(x)^2 - 12*e^(2*tan(1/2*x)/(tan(1/2*x)^2 + 1))*tan(1/2*x)^
3 + 16*e^(2*tan(1/2*x)/(tan(1/2*x)^2 + 1))*tan(1/2*x)^2*tan(x) - 2*e^(2*tan(1/2*
x)/(tan(1/2*x)^2 + 1))*tan(1/2*x)*tan(x)^2 + 5*e^(2*tan(1/2*x)/(tan(1/2*x)^2 + 1
))*tan(1/2*x)^2 - 8*e^(2*tan(1/2*x)/(tan(1/2*x)^2 + 1))*tan(1/2*x)*tan(x) + e^(2
*tan(1/2*x)/(tan(1/2*x)^2 + 1))*tan(x)^2 + 2*e^(2*tan(1/2*x)/(tan(1/2*x)^2 + 1))
*tan(1/2*x) - e^(2*tan(1/2*x)/(tan(1/2*x)^2 + 1)))/(tan(1/2*x)^6*tan(x)^2 + tan(
1/2*x)^6 + 3*tan(1/2*x)^4*tan(x)^2 + 3*tan(1/2*x)^4 + 3*tan(1/2*x)^2*tan(x)^2 +
3*tan(1/2*x)^2 + tan(x)^2 + 1)