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

Optimal. Leaf size=27 \[ -\frac{14}{25} e^{3 x} \sin (4 x)-\frac{23}{25} e^{3 x} \cos (4 x) \]

[Out]

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

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Rubi [A]  time = 0.0801718, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {6742, 4433, 4432} \[ -\frac{14}{25} e^{3 x} \sin (4 x)-\frac{23}{25} e^{3 x} \cos (4 x) \]

Antiderivative was successfully verified.

[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 6742

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

Rule 4433

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 4432

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]

Rubi steps

\begin{align*} \int e^{3 x} (-5 \cos (4 x)+2 \sin (4 x)) \, dx &=\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]  time = 0.0912628, size = 22, normalized size = 0.81 \[ -\frac{1}{25} e^{3 x} (14 \sin (4 x)+23 \cos (4 x)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.017, size = 103, normalized size = 3.8 \begin{align*} -{\frac{ \left ( 24\,\cos \left ( x \right ) +32\,\sin \left ( x \right ) \right ){{\rm e}^{3\,x}} \left ( \cos \left ( x \right ) \right ) ^{3}}{5}}+{\frac{ \left ( 24\,\cos \left ( x \right ) +16\,\sin \left ( x \right ) \right ){{\rm e}^{3\,x}}\cos \left ( x \right ) }{5}}-{\frac{3\, \left ({{\rm e}^{x}} \right ) ^{3}}{5}}-{\frac{8\,{{\rm e}^{3\,x}}\cos \left ( 4\,x \right ) }{25}}+{\frac{6\,{{\rm e}^{3\,x}}\sin \left ( 4\,x \right ) }{25}}-{\frac{8\,{{\rm e}^{3\,x}}\cos \left ( 2\,x \right ) }{13}}+{\frac{12\,{{\rm e}^{3\,x}}\sin \left ( 2\,x \right ) }{13}}-{\frac{4\,{{\rm e}^{3\,x}} \left ( 3\,\sin \left ( 2\,x \right ) -2\,\cos \left ( 2\,x \right ) \right ) }{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8/5*(3*cos(x)+4*sin(x))*exp(3*x)*cos(x)^3+8/5*(3*cos(x)+2*sin(x))*exp(3*x)*cos(x)-3/5*exp(x)^3-8/25*exp(3*x)*
cos(4*x)+6/25*exp(3*x)*sin(4*x)-8/13*exp(3*x)*cos(2*x)+12/13*exp(3*x)*sin(2*x)-4/13*exp(3*x)*(3*sin(2*x)-2*cos
(2*x))

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Maxima [A]  time = 1.10554, size = 53, normalized size = 1.96 \begin{align*} -\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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 0.451213, size = 68, normalized size = 2.52 \begin{align*} -\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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]  time = 0.328428, size = 27, normalized size = 1. \begin{align*} - \frac{14 e^{3 x} \sin{\left (4 x \right )}}{25} - \frac{23 e^{3 x} \cos{\left (4 x \right )}}{25} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Giac [A]  time = 1.14005, size = 53, normalized size = 1.96 \begin{align*} -\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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)