\(\int e^x \sin (e^x) \, dx\) [62]

Optimal result

Integrand size = 8, antiderivative size = 6 \[ \int e^x \sin \left (e^x\right ) \, dx=-\cos \left (e^x\right ) \]

[Out]

-cos(exp(x))

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 6, 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, 2718} \[ \int e^x \sin \left (e^x\right ) \, dx=-\cos \left (e^x\right ) \]

[In]

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

[Out]

-Cos[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 2718

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

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00 \[ \int e^x \sin \left (e^x\right ) \, dx=-\cos \left (e^x\right ) \]

[In]

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

[Out]

-Cos[E^x]

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00

method	result	size
derivativedivides	\(-\cos \left ({\mathrm e}^{x}\right )\)	\(6\)
default	\(-\cos \left ({\mathrm e}^{x}\right )\)	\(6\)
risch	\(-\cos \left ({\mathrm e}^{x}\right )\)	\(6\)
parallelrisch	\(-\cos \left ({\mathrm e}^{x}\right )-1\)	\(8\)
norman	\(-\frac {2}{1+\tan \left (\frac {{\mathrm e}^{x}}{2}\right )^{2}}\)	\(14\)

[In]

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

[Out]

-cos(exp(x))

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.83 \[ \int e^x \sin \left (e^x\right ) \, dx=-\cos \left (e^{x}\right ) \]

[In]

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

[Out]

-cos(e^x)

Sympy [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.83 \[ \int e^x \sin \left (e^x\right ) \, dx=- \cos {\left (e^{x} \right )} \]

[In]

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

[Out]

-cos(exp(x))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.83 \[ \int e^x \sin \left (e^x\right ) \, dx=-\cos \left (e^{x}\right ) \]

[In]

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

[Out]

-cos(e^x)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.83 \[ \int e^x \sin \left (e^x\right ) \, dx=-\cos \left (e^{x}\right ) \]

[In]

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

[Out]

-cos(e^x)

Mupad [B] (verification not implemented)

Time = 27.65 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.83 \[ \int e^x \sin \left (e^x\right ) \, dx=-\cos \left ({\mathrm {e}}^x\right ) \]

[In]

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

[Out]

-cos(exp(x))