3.282 \(\int e^{3 x} \cos (5 x) \, dx\)

Optimal. Leaf size=27 \[ \frac{5}{34} e^{3 x} \sin (5 x)+\frac{3}{34} e^{3 x} \cos (5 x) \]

[Out]

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

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Rubi [A]  time = 0.0096329, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {4433} \[ \frac{5}{34} e^{3 x} \sin (5 x)+\frac{3}{34} e^{3 x} \cos (5 x) \]

Antiderivative was successfully verified.

[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 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]

Rubi steps

\begin{align*} \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)\\ \end{align*}

Mathematica [A]  time = 0.0270942, size = 22, normalized size = 0.81 \[ \frac{1}{34} e^{3 x} (5 \sin (5 x)+3 \cos (5 x)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.007, size = 22, normalized size = 0.8 \begin{align*}{\frac{3\,{{\rm e}^{3\,x}}\cos \left ( 5\,x \right ) }{34}}+{\frac{5\,{{\rm e}^{3\,x}}\sin \left ( 5\,x \right ) }{34}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.953807, size = 26, normalized size = 0.96 \begin{align*} \frac{1}{34} \,{\left (3 \, \cos \left (5 \, x\right ) + 5 \, \sin \left (5 \, 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)*cos(5*x),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.96981, size = 63, normalized size = 2.33 \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]  time = 0.296948, size = 26, normalized size = 0.96 \begin{align*} \frac{5 e^{3 x} \sin{\left (5 x \right )}}{34} + \frac{3 e^{3 x} \cos{\left (5 x \right )}}{34} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Giac [A]  time = 1.05933, size = 26, normalized size = 0.96 \begin{align*} \frac{1}{34} \,{\left (3 \, \cos \left (5 \, x\right ) + 5 \, \sin \left (5 \, 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)*cos(5*x),x, algorithm="giac")

[Out]

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