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

Optimal. Leaf size=13 \[ e^x \sin \left (e^x\right )+\cos \left (e^x\right ) \]

[Out]

Cos[E^x] + E^x*Sin[E^x]

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Rubi [A]  time = 0.0165981, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {2282, 3296, 2638} \[ e^x \sin \left (e^x\right )+\cos \left (e^x\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Cos[E^x] + E^x*Sin[E^x]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

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

Mathematica [A]  time = 0.0166958, size = 13, normalized size = 1. \[ e^x \sin \left (e^x\right )+\cos \left (e^x\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Cos[E^x] + E^x*Sin[E^x]

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Maple [B]  time = 0.02, size = 24, normalized size = 1.9 \begin{align*}{ \left ( 2\,{{\rm e}^{x}}\tan \left ( 1/2\,{{\rm e}^{x}} \right ) +2 \right ) \left ( 1+ \left ( \tan \left ({\frac{{{\rm e}^{x}}}{2}} \right ) \right ) ^{2} \right ) ^{-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*exp(x)*tan(1/2*exp(x))+2)/(1+tan(1/2*exp(x))^2)

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Maxima [A]  time = 1.00955, size = 14, normalized size = 1.08 \begin{align*} e^{x} \sin \left (e^{x}\right ) + \cos \left (e^{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^x*sin(e^x) + cos(e^x)

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Fricas [A]  time = 0.47621, size = 34, normalized size = 2.62 \begin{align*} e^{x} \sin \left (e^{x}\right ) + \cos \left (e^{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^x*sin(e^x) + cos(e^x)

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Sympy [A]  time = 11.9689, size = 12, normalized size = 0.92 \begin{align*} e^{x} \sin{\left (e^{x} \right )} + \cos{\left (e^{x} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(x)*sin(exp(x)) + cos(exp(x))

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Giac [A]  time = 1.12902, size = 14, normalized size = 1.08 \begin{align*} e^{x} \sin \left (e^{x}\right ) + \cos \left (e^{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^x*sin(e^x) + cos(e^x)