3.1.68 \(\int e^{-1+3 x} \cos (e^{-1+3 x}) \sin (1+e^{-1+3 x}) \, dx\) [68]

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

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 30, normalized size of antiderivative = 1.00, 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 = {2320, 4670, 2718} \begin {gather*} \frac {1}{6} e^{3 x-1} \sin (1)-\frac {1}{12} \cos \left (2 e^{3 x-1}+1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

Rule 4670

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]))

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} \text {Subst}\left (\int \cos (x) \sin (1+x) \, dx,x,e^{-1+3 x}\right )\\ &=\frac {1}{3} \text {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} \text {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*}

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Mathematica [A]
time = 0.09, size = 30, normalized size = 1.00 \begin {gather*} -\frac {1}{12} \cos \left (1+2 e^{-1+3 x}\right )+\frac {1}{6} e^{-1+3 x} \sin (1) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.34, size = 25, normalized size = 0.83

method	result	size
derivativedivides	\(-\frac {\cos \left (1+2 \,{\mathrm e}^{-1+3 x}\right )}{12}+\frac {{\mathrm e}^{-1+3 x} \sin \left (1\right )}{6}\)	\(25\)
default	\(-\frac {\cos \left (1+2 \,{\mathrm e}^{-1+3 x}\right )}{12}+\frac {{\mathrm e}^{-1+3 x} \sin \left (1\right )}{6}\)	\(25\)
risch	\(-\frac {\cos \left (1+2 \,{\mathrm e}^{-1+3 x}\right )}{12}+\frac {{\mathrm e}^{-1+3 x} \sin \left (1\right )}{6}\)	\(25\)
norman	\(\frac {\frac {2 \tan \left (\frac {{\mathrm e}^{-1+3 x}}{2}\right ) \tan \left (\frac {1}{2}+\frac {{\mathrm e}^{-1+3 x}}{2}\right )}{3}-\frac {{\mathrm e}^{-1+3 x} \tan \left (\frac {{\mathrm e}^{-1+3 x}}{2}\right )}{3}+\frac {{\mathrm e}^{-1+3 x} \tan \left (\frac {1}{2}+\frac {{\mathrm e}^{-1+3 x}}{2}\right )}{3}+\frac {{\mathrm e}^{-1+3 x} \tan \left (\frac {{\mathrm e}^{-1+3 x}}{2}\right ) \left (\tan ^{2}\left (\frac {1}{2}+\frac {{\mathrm e}^{-1+3 x}}{2}\right )\right )}{3}-\frac {{\mathrm e}^{-1+3 x} \left (\tan ^{2}\left (\frac {{\mathrm e}^{-1+3 x}}{2}\right )\right ) \tan \left (\frac {1}{2}+\frac {{\mathrm e}^{-1+3 x}}{2}\right )}{3}}{\left (1+\tan ^{2}\left (\frac {{\mathrm e}^{-1+3 x}}{2}\right )\right ) \left (1+\tan ^{2}\left (\frac {1}{2}+\frac {{\mathrm e}^{-1+3 x}}{2}\right )\right )}\)	\(153\)

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,method=_RETURNVERBOSE)

[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 = 0.28, size = 24, normalized size = 0.80 \begin {gather*} \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 {gather*}

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 = 1.48, size = 42, normalized size = 1.40 \begin {gather*} -\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 {gather*}

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)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 = 0.44, size = 24, normalized size = 0.80 \begin {gather*} \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 {gather*}

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)

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Mupad [B]
time = 0.31, size = 24, normalized size = 0.80 \begin {gather*} \frac {{\mathrm {e}}^{3\,x-1}\,\sin \left (1\right )}{6}-\frac {\cos \left (2\,{\mathrm {e}}^{3\,x-1}+1\right )}{12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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