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

Optimal. Leaf size=10 \[ e^{\frac {1}{2} \sin (2 x)} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4356, 2194} \[ e^{\frac {1}{2} \sin (2 x)} \]

Antiderivative was successfully verified.

[In]

Int[E^(Cos[x]*Sin[x])*Cos[2*x],x]

[Out]

E^(Sin[2*x]/2)

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 4356

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]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Cos] || EqQ[F, cos])

Rubi steps

\begin {align*} \int e^{\cos (x) \sin (x)} \cos (2 x) \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int e^{x/2} \, dx,x,\sin (2 x)\right )\\ &=e^{\frac {1}{2} \sin (2 x)}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 7, normalized size = 0.70 \[ e^{\sin (x) \cos (x)} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(Cos[x]*Sin[x])*Cos[2*x],x]

[Out]

E^(Cos[x]*Sin[x])

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fricas [A]  time = 0.79, size = 6, normalized size = 0.60 \[ e^{\left (\cos \relax (x) \sin \relax (x)\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(cos(x)*sin(x))

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giac [A]  time = 0.15, size = 12, normalized size = 1.20 \[ e^{\left (\frac {\tan \relax (x)}{\tan \relax (x)^{2} + 1}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(tan(x)/(tan(x)^2 + 1))

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maple [A]  time = 0.08, size = 7, normalized size = 0.70 \[ {\mathrm e}^{\cos \relax (x ) \sin \relax (x )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(cos(x)*sin(x))

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maxima [A]  time = 0.95, size = 7, normalized size = 0.70 \[ e^{\left (\frac {1}{2} \, \sin \left (2 \, x\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 2.99, size = 7, normalized size = 0.70 \[ {\mathrm {e}}^{\frac {\sin \left (2\,x\right )}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(sin(2*x)/2)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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