\(\int e^{3 x} (-5 \cos (4 x)+2 \sin (4 x)) \, dx\) [51]

Optimal result

Integrand size = 19, antiderivative size = 27 \[ \int e^{3 x} (-5 \cos (4 x)+2 \sin (4 x)) \, dx=-\frac {23}{25} e^{3 x} \cos (4 x)-\frac {14}{25} e^{3 x} \sin (4 x) \]

[Out]

-23/25*exp(3*x)*cos(4*x)-14/25*exp(3*x)*sin(4*x)

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {6874, 4518, 4517} \[ \int e^{3 x} (-5 \cos (4 x)+2 \sin (4 x)) \, dx=-\frac {14}{25} e^{3 x} \sin (4 x)-\frac {23}{25} e^{3 x} \cos (4 x) \]

[In]

Int[E^(3*x)*(-5*Cos[4*x] + 2*Sin[4*x]),x]

[Out]

(-23*E^(3*x)*Cos[4*x])/25 - (14*E^(3*x)*Sin[4*x])/25

Rule 4517

Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sin[(d_.) + (e_.)*(x_)], x_Symbol] :> Simp[b*c*Log[F]*F^(c*(a + b*x))*(S
in[d + e*x]/(e^2 + b^2*c^2*Log[F]^2)), x] - Simp[e*F^(c*(a + b*x))*(Cos[d + e*x]/(e^2 + b^2*c^2*Log[F]^2)), x]
/; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2 + b^2*c^2*Log[F]^2, 0]

Rule 4518

Int[Cos[(d_.) + (e_.)*(x_)]*(F_)^((c_.)*((a_.) + (b_.)*(x_))), x_Symbol] :> Simp[b*c*Log[F]*F^(c*(a + b*x))*(C
os[d + e*x]/(e^2 + b^2*c^2*Log[F]^2)), x] + Simp[e*F^(c*(a + b*x))*(Sin[d + e*x]/(e^2 + b^2*c^2*Log[F]^2)), x]
/; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2 + b^2*c^2*Log[F]^2, 0]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = \int \left (-5 e^{3 x} \cos (4 x)+2 e^{3 x} \sin (4 x)\right ) \, dx \\ & = 2 \int e^{3 x} \sin (4 x) \, dx-5 \int e^{3 x} \cos (4 x) \, dx \\ & = -\frac {23}{25} e^{3 x} \cos (4 x)-\frac {14}{25} e^{3 x} \sin (4 x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int e^{3 x} (-5 \cos (4 x)+2 \sin (4 x)) \, dx=-\frac {1}{25} e^{3 x} (23 \cos (4 x)+14 \sin (4 x)) \]

[In]

Integrate[E^(3*x)*(-5*Cos[4*x] + 2*Sin[4*x]),x]

[Out]

-1/25*(E^(3*x)*(23*Cos[4*x] + 14*Sin[4*x]))

Maple [A] (verified)

Time = 0.54 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74

method	result	size
parallelrisch	\(-\frac {{\mathrm e}^{3 x} \left (23 \cos \left (4 x \right )+14 \sin \left (4 x \right )\right )}{25}\)	\(20\)
parts	\(-\frac {23 \,{\mathrm e}^{3 x} \cos \left (4 x \right )}{25}-\frac {14 \,{\mathrm e}^{3 x} \sin \left (4 x \right )}{25}\)	\(22\)
risch	\(-\frac {23 \,{\mathrm e}^{\left (3+4 i\right ) x}}{50}+\frac {7 i {\mathrm e}^{\left (3+4 i\right ) x}}{25}-\frac {23 \,{\mathrm e}^{\left (3-4 i\right ) x}}{50}-\frac {7 i {\mathrm e}^{\left (3-4 i\right ) x}}{25}\)	\(36\)
norman	\(\frac {-\frac {28 \,{\mathrm e}^{3 x} \tan \left (2 x \right )}{25}+\frac {23 \,{\mathrm e}^{3 x} \tan \left (2 x \right )^{2}}{25}-\frac {23 \,{\mathrm e}^{3 x}}{25}}{1+\tan \left (2 x \right )^{2}}\)	\(41\)
default	\(-\frac {8 \left (3 \cos \left (x \right )+4 \sin \left (x \right )\right ) {\mathrm e}^{3 x} \cos \left (x \right )^{3}}{5}+\frac {8 \left (3 \cos \left (x \right )+2 \sin \left (x \right )\right ) {\mathrm e}^{3 x} \cos \left (x \right )}{5}-\frac {3 \,{\mathrm e}^{3 x}}{5}-\frac {8 \,{\mathrm e}^{3 x} \cos \left (4 x \right )}{25}+\frac {6 \,{\mathrm e}^{3 x} \sin \left (4 x \right )}{25}-\frac {8 \,{\mathrm e}^{3 x} \cos \left (2 x \right )}{13}+\frac {12 \,{\mathrm e}^{3 x} \sin \left (2 x \right )}{13}-\frac {4 \,{\mathrm e}^{3 x} \left (3 \sin \left (2 x \right )-2 \cos \left (2 x \right )\right )}{13}\)	\(103\)

[In]

int(exp(3*x)*(-5*cos(4*x)+2*sin(4*x)),x,method=_RETURNVERBOSE)

[Out]

-1/25*exp(3*x)*(23*cos(4*x)+14*sin(4*x))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78 \[ \int e^{3 x} (-5 \cos (4 x)+2 \sin (4 x)) \, dx=-\frac {23}{25} \, \cos \left (4 \, x\right ) e^{\left (3 \, x\right )} - \frac {14}{25} \, e^{\left (3 \, x\right )} \sin \left (4 \, x\right ) \]

[In]

integrate(exp(3*x)*(-5*cos(4*x)+2*sin(4*x)),x, algorithm="fricas")

[Out]

-23/25*cos(4*x)*e^(3*x) - 14/25*e^(3*x)*sin(4*x)

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int e^{3 x} (-5 \cos (4 x)+2 \sin (4 x)) \, dx=- \frac {14 e^{3 x} \sin {\left (4 x \right )}}{25} - \frac {23 e^{3 x} \cos {\left (4 x \right )}}{25} \]

[In]

integrate(exp(3*x)*(-5*cos(4*x)+2*sin(4*x)),x)

[Out]

-14*exp(3*x)*sin(4*x)/25 - 23*exp(3*x)*cos(4*x)/25

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.44 \[ \int e^{3 x} (-5 \cos (4 x)+2 \sin (4 x)) \, dx=-\frac {2}{25} \, {\left (4 \, \cos \left (4 \, x\right ) - 3 \, \sin \left (4 \, x\right )\right )} e^{\left (3 \, x\right )} - \frac {1}{5} \, {\left (3 \, \cos \left (4 \, x\right ) + 4 \, \sin \left (4 \, x\right )\right )} e^{\left (3 \, x\right )} \]

[In]

integrate(exp(3*x)*(-5*cos(4*x)+2*sin(4*x)),x, algorithm="maxima")

[Out]

-2/25*(4*cos(4*x) - 3*sin(4*x))*e^(3*x) - 1/5*(3*cos(4*x) + 4*sin(4*x))*e^(3*x)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.44 \[ \int e^{3 x} (-5 \cos (4 x)+2 \sin (4 x)) \, dx=-\frac {2}{25} \, {\left (4 \, \cos \left (4 \, x\right ) - 3 \, \sin \left (4 \, x\right )\right )} e^{\left (3 \, x\right )} - \frac {1}{5} \, {\left (3 \, \cos \left (4 \, x\right ) + 4 \, \sin \left (4 \, x\right )\right )} e^{\left (3 \, x\right )} \]

[In]

integrate(exp(3*x)*(-5*cos(4*x)+2*sin(4*x)),x, algorithm="giac")

[Out]

-2/25*(4*cos(4*x) - 3*sin(4*x))*e^(3*x) - 1/5*(3*cos(4*x) + 4*sin(4*x))*e^(3*x)

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int e^{3 x} (-5 \cos (4 x)+2 \sin (4 x)) \, dx=-\frac {{\mathrm {e}}^{3\,x}\,\left (23\,\cos \left (4\,x\right )+14\,\sin \left (4\,x\right )\right )}{25} \]

[In]

int(-exp(3*x)*(5*cos(4*x) - 2*sin(4*x)),x)

[Out]

-(exp(3*x)*(23*cos(4*x) + 14*sin(4*x)))/25