3.68 \(\int e^{-1+3 x} \cos (e^{-1+3 x}) \sin (1+e^{-1+3 x}) \, dx\)

Optimal. Leaf size=30 \[ \frac{1}{6} e^{3 x-1} \sin (1)-\frac{1}{12} \cos \left (2 e^{3 x-1}+1\right ) \]

[Out]

-Cos[1 + 2*E^(-1 + 3*x)]/12 + (E^(-1 + 3*x)*Sin[1])/6

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Rubi [A]  time = 0.0355204, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {2282, 4574, 2638} \[ \frac{1}{6} e^{3 x-1} \sin (1)-\frac{1}{12} \cos \left (2 e^{3 x-1}+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[E^(-1 + 3*x)*Cos[E^(-1 + 3*x)]*Sin[1 + E^(-1 + 3*x)],x]

[Out]

-Cos[1 + 2*E^(-1 + 3*x)]/12 + (E^(-1 + 3*x)*Sin[1])/6

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 4574

Int[Cos[w_]^(q_.)*Sin[v_]^(p_.), x_Symbol] :> Int[ExpandTrigReduce[Sin[v]^p*Cos[w]^q, x], x] /; IGtQ[p, 0] &&
IGtQ[q, 0] && ((PolynomialQ[v, x] && PolynomialQ[w, x]) || (BinomialQ[{v, w}, x] && IndependentQ[Cancel[v/w],
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^{-1+3 x} \cos \left (e^{-1+3 x}\right ) \sin \left (1+e^{-1+3 x}\right ) \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \cos (x) \sin (1+x) \, dx,x,e^{-1+3 x}\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{\sin (1)}{2}+\frac{1}{2} \sin (1+2 x)\right ) \, dx,x,e^{-1+3 x}\right )\\ &=\frac{1}{6} e^{-1+3 x} \sin (1)+\frac{1}{6} \operatorname{Subst}\left (\int \sin (1+2 x) \, dx,x,e^{-1+3 x}\right )\\ &=-\frac{1}{12} \cos \left (1+2 e^{-1+3 x}\right )+\frac{1}{6} e^{-1+3 x} \sin (1)\\ \end{align*}

Mathematica [A]  time = 0.0608572, size = 30, normalized size = 1. \[ \frac{1}{6} e^{3 x-1} \sin (1)-\frac{1}{12} \cos \left (2 e^{3 x-1}+1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[E^(-1 + 3*x)*Cos[E^(-1 + 3*x)]*Sin[1 + E^(-1 + 3*x)],x]

[Out]

-Cos[1 + 2*E^(-1 + 3*x)]/12 + (E^(-1 + 3*x)*Sin[1])/6

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Maple [A]  time = 0.051, size = 25, normalized size = 0.8 \begin{align*} -{\frac{\cos \left ( 1+2\,{{\rm e}^{-1+3\,x}} \right ) }{12}}+{\frac{{{\rm e}^{-1+3\,x}}\sin \left ( 1 \right ) }{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-1+3*x)*cos(exp(-1+3*x))*sin(1+exp(-1+3*x)),x)

[Out]

-1/12*cos(1+2*exp(-1+3*x))+1/6*exp(-1+3*x)*sin(1)

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Maxima [A]  time = 0.986635, size = 32, normalized size = 1.07 \begin{align*} \frac{1}{6} \, e^{\left (3 \, x - 1\right )} \sin \left (1\right ) - \frac{1}{12} \, \cos \left (2 \, e^{\left (3 \, x - 1\right )} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(-1+3*x)*cos(exp(-1+3*x))*sin(1+exp(-1+3*x)),x, algorithm="maxima")

[Out]

1/6*e^(3*x - 1)*sin(1) - 1/12*cos(2*e^(3*x - 1) + 1)

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Fricas [A]  time = 0.483324, size = 140, normalized size = 4.67 \begin{align*} -\frac{1}{6} \, \cos \left (1\right ) \cos \left (e^{\left (3 \, x - 1\right )}\right )^{2} + \frac{1}{6} \, \cos \left (e^{\left (3 \, x - 1\right )}\right ) \sin \left (1\right ) \sin \left (e^{\left (3 \, x - 1\right )}\right ) + \frac{1}{6} \, e^{\left (3 \, x - 1\right )} \sin \left (1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(-1+3*x)*cos(exp(-1+3*x))*sin(1+exp(-1+3*x)),x, algorithm="fricas")

[Out]

-1/6*cos(1)*cos(e^(3*x - 1))^2 + 1/6*cos(e^(3*x - 1))*sin(1)*sin(e^(3*x - 1)) + 1/6*e^(3*x - 1)*sin(1)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(-1+3*x)*cos(exp(-1+3*x))*sin(1+exp(-1+3*x)),x)

[Out]

Timed out

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Giac [A]  time = 1.12193, size = 32, normalized size = 1.07 \begin{align*} \frac{1}{6} \, e^{\left (3 \, x - 1\right )} \sin \left (1\right ) - \frac{1}{12} \, \cos \left (2 \, e^{\left (3 \, x - 1\right )} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(-1+3*x)*cos(exp(-1+3*x))*sin(1+exp(-1+3*x)),x, algorithm="giac")

[Out]

1/6*e^(3*x - 1)*sin(1) - 1/12*cos(2*e^(3*x - 1) + 1)