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*exp(1/2*sin(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 = 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 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 \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*}

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Mathematica [A]  time = 0.01, size = 10, normalized size = 1.00 \[ 2 e^{\frac {\sin (x)}{2}} \]

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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fricas [A]  time = 0.72, size = 12, normalized size = 1.20 \[ 2 \, e^{\left (\cos \left (\frac {1}{2} \, x\right ) \sin \left (\frac {1}{2} \, x\right )\right )} \]

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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giac [B]  time = 0.14, size = 18, normalized size = 1.80 \[ 2 \, e^{\left (\frac {\tan \left (\frac {1}{2} \, x\right )}{\tan \left (\frac {1}{2} \, x\right )^{2} + 1}\right )} \]

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))

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maple [A]  time = 0.08, size = 13, normalized size = 1.30 \[ 2 \,{\mathrm e}^{\cos \left (\frac {x}{2}\right ) \sin \left (\frac {x}{2}\right )} \]

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 = 0.33, size = 7, normalized size = 0.70 \[ 2 \, e^{\left (\frac {1}{2} \, \sin \relax (x)\right )} \]

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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mupad [B]  time = 2.95, size = 7, normalized size = 0.70 \[ 2\,{\mathrm {e}}^{\frac {\sin \relax (x)}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*exp(sin(x)/2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{\sin {\left (\frac {x}{2} \right )} \cos {\left (\frac {x}{2} \right )}} \cos {\relax (x )}\, dx \]

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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