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

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

[Out]

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

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Rubi [A]  time = 0.0109039, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {4356, 2194} \[ 2 e^{\frac{\sin (x)}{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0086436, size = 10, normalized size = 1. \[ 2 e^{\frac{\sin (x)}{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.017, size = 13, normalized size = 1.3 \begin{align*} 2\,{{\rm e}^{\cos \left ( x/2 \right ) \sin \left ( x/2 \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(cos(1/2*x)*sin(1/2*x))*cos(x),x)

[Out]

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

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Maxima [A]  time = 0.996401, size = 9, normalized size = 0.9 \begin{align*} 2 \, e^{\left (\frac{1}{2} \, \sin \left (x\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(cos(1/2*x)*sin(1/2*x))*cos(x),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.05714, size = 39, normalized size = 3.9 \begin{align*} 2 \, e^{\left (\cos \left (\frac{1}{2} \, x\right ) \sin \left (\frac{1}{2} \, x\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(cos(1/2*x)*sin(1/2*x))*cos(x),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(cos(1/2*x)*sin(1/2*x))*cos(x),x)

[Out]

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

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Giac [B]  time = 1.10507, size = 24, normalized size = 2.4 \begin{align*} 2 \, e^{\left (\frac{\tan \left (\frac{1}{2} \, x\right )}{\tan \left (\frac{1}{2} \, x\right )^{2} + 1}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(cos(1/2*x)*sin(1/2*x))*cos(x),x, algorithm="giac")

[Out]

2*e^(tan(1/2*x)/(tan(1/2*x)^2 + 1))