3.64 \(\int e^x \cos (e^x) \, dx\)

Optimal. Leaf size=4 \[ \sin \left (e^x\right ) \]

[Out]

Sin[E^x]

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Rubi [A]  time = 0.0084126, antiderivative size = 4, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2282, 2637} \[ \sin \left (e^x\right ) \]

Antiderivative was successfully verified.

[In]

Int[E^x*Cos[E^x],x]

[Out]

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 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

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

Mathematica [A]  time = 0.0094151, size = 4, normalized size = 1. \[ \sin \left (e^x\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[E^x*Cos[E^x],x]

[Out]

Sin[E^x]

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Maple [A]  time = 0.007, size = 4, normalized size = 1. \begin{align*} \sin \left ({{\rm e}^{x}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin(exp(x))

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Maxima [A]  time = 0.983681, size = 4, normalized size = 1. \begin{align*} \sin \left (e^{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin(e^x)

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Fricas [A]  time = 0.457546, size = 14, normalized size = 3.5 \begin{align*} \sin \left (e^{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin(e^x)

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Sympy [A]  time = 0.495608, size = 3, normalized size = 0.75 \begin{align*} \sin{\left (e^{x} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin(exp(x))

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Giac [A]  time = 1.13345, size = 4, normalized size = 1. \begin{align*} \sin \left (e^{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin(e^x)