3.364 \(\int e^x \cos (4+3 x) \, dx\)

Optimal. Leaf size=27 \[ \frac{3}{10} e^x \sin (3 x+4)+\frac{1}{10} e^x \cos (3 x+4) \]

[Out]

(E^x*Cos[4 + 3*x])/10 + (3*E^x*Sin[4 + 3*x])/10

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Rubi [A]  time = 0.0089086, 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} \[ \frac{3}{10} e^x \sin (3 x+4)+\frac{1}{10} e^x \cos (3 x+4) \]

Antiderivative was successfully verified.

[In]

Int[E^x*Cos[4 + 3*x],x]

[Out]

(E^x*Cos[4 + 3*x])/10 + (3*E^x*Sin[4 + 3*x])/10

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^x \cos (4+3 x) \, dx &=\frac{1}{10} e^x \cos (4+3 x)+\frac{3}{10} e^x \sin (4+3 x)\\ \end{align*}

Mathematica [A]  time = 0.0421079, size = 22, normalized size = 0.81 \[ \frac{1}{10} e^x (3 \sin (3 x+4)+\cos (3 x+4)) \]

Antiderivative was successfully verified.

[In]

Integrate[E^x*Cos[4 + 3*x],x]

[Out]

(E^x*(Cos[4 + 3*x] + 3*Sin[4 + 3*x]))/10

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Maple [A]  time = 0.007, size = 22, normalized size = 0.8 \begin{align*}{\frac{{{\rm e}^{x}}\cos \left ( 3\,x+4 \right ) }{10}}+{\frac{3\,{{\rm e}^{x}}\sin \left ( 3\,x+4 \right ) }{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)*cos(3*x+4),x)

[Out]

1/10*exp(x)*cos(3*x+4)+3/10*exp(x)*sin(3*x+4)

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Maxima [A]  time = 0.943301, size = 26, normalized size = 0.96 \begin{align*} \frac{1}{10} \,{\left (\cos \left (3 \, x + 4\right ) + 3 \, \sin \left (3 \, x + 4\right )\right )} e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*cos(4+3*x),x, algorithm="maxima")

[Out]

1/10*(cos(3*x + 4) + 3*sin(3*x + 4))*e^x

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Fricas [A]  time = 1.91943, size = 63, normalized size = 2.33 \begin{align*} \frac{1}{10} \, \cos \left (3 \, x + 4\right ) e^{x} + \frac{3}{10} \, e^{x} \sin \left (3 \, x + 4\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*cos(4+3*x),x, algorithm="fricas")

[Out]

1/10*cos(3*x + 4)*e^x + 3/10*e^x*sin(3*x + 4)

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Sympy [A]  time = 0.287545, size = 24, normalized size = 0.89 \begin{align*} \frac{3 e^{x} \sin{\left (3 x + 4 \right )}}{10} + \frac{e^{x} \cos{\left (3 x + 4 \right )}}{10} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*cos(4+3*x),x)

[Out]

3*exp(x)*sin(3*x + 4)/10 + exp(x)*cos(3*x + 4)/10

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Giac [A]  time = 1.05335, size = 26, normalized size = 0.96 \begin{align*} \frac{1}{10} \,{\left (\cos \left (3 \, x + 4\right ) + 3 \, \sin \left (3 \, x + 4\right )\right )} e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*cos(4+3*x),x, algorithm="giac")

[Out]

1/10*(cos(3*x + 4) + 3*sin(3*x + 4))*e^x