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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0089844, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {4334, 2194} \[ \frac{1}{2} e^{2 \sin (x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4334

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

Mathematica [A]  time = 0.0065793, size = 10, normalized size = 1. \[ \frac{1}{2} e^{2 \sin (x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 8, normalized size = 0.8 \begin{align*}{\frac{{{\rm e}^{2\,\sin \left ( x \right ) }}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.935649, size = 9, normalized size = 0.9 \begin{align*} \frac{1}{2} \, e^{\left (2 \, \sin \left (x\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 0.510337, size = 24, normalized size = 2.4 \begin{align*} \frac{1}{2} \, e^{\left (2 \, \sin \left (x\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.293039, size = 7, normalized size = 0.7 \begin{align*} \frac{e^{2 \sin{\left (x \right )}}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(2*sin(x))/2

________________________________________________________________________________________

Giac [A]  time = 1.06855, size = 9, normalized size = 0.9 \begin{align*} \frac{1}{2} \, e^{\left (2 \, \sin \left (x\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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