\(\int e^x \cos (e^x) \, dx\) [64]

Optimal result

Integrand size = 8, antiderivative size = 4 \[ \int e^x \cos \left (e^x\right ) \, dx=\sin \left (e^x\right ) \]

[Out]

sin(exp(x))

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 4, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2320, 2717} \[ \int e^x \cos \left (e^x\right ) \, dx=\sin \left (e^x\right ) \]

[In]

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

[Out]

Sin[E^x]

Rule 2320

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 2717

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

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 4, normalized size of antiderivative = 1.00 \[ \int e^x \cos \left (e^x\right ) \, dx=\sin \left (e^x\right ) \]

[In]

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

[Out]

Sin[E^x]

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 4, normalized size of antiderivative = 1.00

method	result	size
derivativedivides	\(\sin \left ({\mathrm e}^{x}\right )\)	\(4\)
default	\(\sin \left ({\mathrm e}^{x}\right )\)	\(4\)
risch	\(\sin \left ({\mathrm e}^{x}\right )\)	\(4\)
parallelrisch	\(\sin \left ({\mathrm e}^{x}\right )\)	\(4\)
norman	\(\frac {2 \tan \left (\frac {{\mathrm e}^{x}}{2}\right )}{1+\tan \left (\frac {{\mathrm e}^{x}}{2}\right )^{2}}\)	\(19\)

[In]

int(exp(x)*cos(exp(x)),x,method=_RETURNVERBOSE)

[Out]

sin(exp(x))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.75 \[ \int e^x \cos \left (e^x\right ) \, dx=\sin \left (e^{x}\right ) \]

[In]

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

[Out]

sin(e^x)

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.75 \[ \int e^x \cos \left (e^x\right ) \, dx=\sin {\left (e^{x} \right )} \]

[In]

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

[Out]

sin(exp(x))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.75 \[ \int e^x \cos \left (e^x\right ) \, dx=\sin \left (e^{x}\right ) \]

[In]

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

[Out]

sin(e^x)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.75 \[ \int e^x \cos \left (e^x\right ) \, dx=\sin \left (e^{x}\right ) \]

[In]

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

[Out]

sin(e^x)

Mupad [B] (verification not implemented)

Time = 27.70 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.75 \[ \int e^x \cos \left (e^x\right ) \, dx=\sin \left ({\mathrm {e}}^x\right ) \]

[In]

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

[Out]

sin(exp(x))