\(\int e^{3 x} \cos (5 x) \, dx\) [680]

Optimal result

Integrand size = 10, antiderivative size = 27 \[ \int e^{3 x} \cos (5 x) \, dx=\frac {3}{34} e^{3 x} \cos (5 x)+\frac {5}{34} e^{3 x} \sin (5 x) \]

[Out]

3/34*exp(3*x)*cos(5*x)+5/34*exp(3*x)*sin(5*x)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {4518} \[ \int e^{3 x} \cos (5 x) \, dx=\frac {5}{34} e^{3 x} \sin (5 x)+\frac {3}{34} e^{3 x} \cos (5 x) \]

[In]

Int[E^(3*x)*Cos[5*x],x]

[Out]

(3*E^(3*x)*Cos[5*x])/34 + (5*E^(3*x)*Sin[5*x])/34

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]

Rubi steps \begin{align*} \text {integral}& = \frac {3}{34} e^{3 x} \cos (5 x)+\frac {5}{34} e^{3 x} \sin (5 x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int e^{3 x} \cos (5 x) \, dx=\frac {1}{34} e^{3 x} (3 \cos (5 x)+5 \sin (5 x)) \]

[In]

Integrate[E^(3*x)*Cos[5*x],x]

[Out]

(E^(3*x)*(3*Cos[5*x] + 5*Sin[5*x]))/34

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74

method	result	size
parallelrisch	\(\frac {{\mathrm e}^{3 x} \left (3 \cos \left (5 x \right )+5 \sin \left (5 x \right )\right )}{34}\)	\(20\)
default	\(\frac {3 \,{\mathrm e}^{3 x} \cos \left (5 x \right )}{34}+\frac {5 \,{\mathrm e}^{3 x} \sin \left (5 x \right )}{34}\)	\(22\)
risch	\(\frac {3 \,{\mathrm e}^{\left (3+5 i\right ) x}}{68}-\frac {5 i {\mathrm e}^{\left (3+5 i\right ) x}}{68}+\frac {3 \,{\mathrm e}^{\left (3-5 i\right ) x}}{68}+\frac {5 i {\mathrm e}^{\left (3-5 i\right ) x}}{68}\)	\(36\)
norman	\(\frac {\frac {5 \,{\mathrm e}^{3 x} \tan \left (\frac {5 x}{2}\right )}{17}-\frac {3 \,{\mathrm e}^{3 x} \left (\tan ^{2}\left (\frac {5 x}{2}\right )\right )}{34}+\frac {3 \,{\mathrm e}^{3 x}}{34}}{1+\tan ^{2}\left (\frac {5 x}{2}\right )}\)	\(41\)

[In]

int(exp(3*x)*cos(5*x),x,method=_RETURNVERBOSE)

[Out]

1/34*exp(3*x)*(3*cos(5*x)+5*sin(5*x))

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78 \[ \int e^{3 x} \cos (5 x) \, dx=\frac {3}{34} \, \cos \left (5 \, x\right ) e^{\left (3 \, x\right )} + \frac {5}{34} \, e^{\left (3 \, x\right )} \sin \left (5 \, x\right ) \]

[In]

integrate(exp(3*x)*cos(5*x),x, algorithm="fricas")

[Out]

3/34*cos(5*x)*e^(3*x) + 5/34*e^(3*x)*sin(5*x)

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int e^{3 x} \cos (5 x) \, dx=\frac {5 e^{3 x} \sin {\left (5 x \right )}}{34} + \frac {3 e^{3 x} \cos {\left (5 x \right )}}{34} \]

[In]

integrate(exp(3*x)*cos(5*x),x)

[Out]

5*exp(3*x)*sin(5*x)/34 + 3*exp(3*x)*cos(5*x)/34

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int e^{3 x} \cos (5 x) \, dx=\frac {1}{34} \, {\left (3 \, \cos \left (5 \, x\right ) + 5 \, \sin \left (5 \, x\right )\right )} e^{\left (3 \, x\right )} \]

[In]

integrate(exp(3*x)*cos(5*x),x, algorithm="maxima")

[Out]

1/34*(3*cos(5*x) + 5*sin(5*x))*e^(3*x)

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int e^{3 x} \cos (5 x) \, dx=\frac {1}{34} \, {\left (3 \, \cos \left (5 \, x\right ) + 5 \, \sin \left (5 \, x\right )\right )} e^{\left (3 \, x\right )} \]

[In]

integrate(exp(3*x)*cos(5*x),x, algorithm="giac")

[Out]

1/34*(3*cos(5*x) + 5*sin(5*x))*e^(3*x)

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int e^{3 x} \cos (5 x) \, dx=\frac {{\mathrm {e}}^{3\,x}\,\left (3\,\cos \left (5\,x\right )+5\,\sin \left (5\,x\right )\right )}{34} \]

[In]

int(cos(5*x)*exp(3*x),x)

[Out]

(exp(3*x)*(3*cos(5*x) + 5*sin(5*x)))/34