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

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

[Out]

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

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Rubi [A]  time = 0.0117849, antiderivative size = 10, normalized size of antiderivative = 1., 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 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])

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]

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*}

Mathematica [A]  time = 0.0213795, size = 7, normalized size = 0.7 \[ e^{\sin (x) \cos (x)} \]

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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Maple [A]  time = 0.015, size = 7, normalized size = 0.7 \begin{align*}{{\rm e}^{\cos \left ( x \right ) \sin \left ( x \right ) }} \end{align*}

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 = 3.1773, size = 9, normalized size = 0.9 \begin{align*} e^{\left (\frac{1}{2} \, \sin \left (2 \, x\right )\right )} \end{align*}

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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Fricas [A]  time = 2.15711, size = 26, normalized size = 2.6 \begin{align*} e^{\left (\cos \left (x\right ) \sin \left (x\right )\right )} \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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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Giac [A]  time = 1.09902, size = 16, normalized size = 1.6 \begin{align*} e^{\left (\frac{\tan \left (x\right )}{\tan \left (x\right )^{2} + 1}\right )} \end{align*}

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))