3.62 \(\int e^x \sin (e^x) \, dx\)

Optimal. Leaf size=6 \[ -\cos \left (e^x\right ) \]

[Out]

-cos(exp(x))

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Rubi [A]  time = 0.01, antiderivative size = 6, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2282, 2638} \[ -\cos \left (e^x\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.01, size = 6, normalized size = 1.00 \[ -\cos \left (e^x\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Cos[E^x]

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fricas [A]  time = 2.29, size = 5, normalized size = 0.83 \[ -\cos \left (e^{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-cos(e^x)

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giac [A]  time = 0.14, size = 5, normalized size = 0.83 \[ -\cos \left (e^{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-cos(e^x)

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maple [A]  time = 0.00, size = 6, normalized size = 1.00 \[ -\cos \left ({\mathrm e}^{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-cos(exp(x))

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maxima [A]  time = 0.34, size = 5, normalized size = 0.83 \[ -\cos \left (e^{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-cos(e^x)

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mupad [B]  time = 2.21, size = 5, normalized size = 0.83 \[ -\cos \left ({\mathrm {e}}^x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-cos(exp(x))

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sympy [A]  time = 0.23, size = 5, normalized size = 0.83 \[ - \cos {\left (e^{x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-cos(exp(x))

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