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

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

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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*}

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Mathematica [A]  time = 0.04, 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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fricas [A]  time = 0.40, size = 21, normalized size = 0.78 \[ \frac {1}{10} \, \cos \left (3 \, x + 4\right ) e^{x} + \frac {3}{10} \, e^{x} \sin \left (3 \, x + 4\right ) \]

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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giac [A]  time = 0.18, size = 19, normalized size = 0.70 \[ \frac {1}{10} \, {\left (\cos \left (3 \, x + 4\right ) + 3 \, \sin \left (3 \, x + 4\right )\right )} e^{x} \]

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

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maple [A]  time = 0.12, size = 22, normalized size = 0.81 \[ \frac {\cos \left (3 x +4\right ) {\mathrm e}^{x}}{10}+\frac {3 \,{\mathrm e}^{x} \sin \left (3 x +4\right )}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.86, size = 19, normalized size = 0.70 \[ \frac {1}{10} \, {\left (\cos \left (3 \, x + 4\right ) + 3 \, \sin \left (3 \, x + 4\right )\right )} e^{x} \]

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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mupad [B]  time = 3.53, size = 19, normalized size = 0.70 \[ \frac {{\mathrm {e}}^x\,\left (\cos \left (3\,x+4\right )+3\,\sin \left (3\,x+4\right )\right )}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)*cos(3*x + 4),x)

[Out]

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

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sympy [A]  time = 0.29, size = 24, normalized size = 0.89 \[ \frac {3 e^{x} \sin {\left (3 x + 4 \right )}}{10} + \frac {e^{x} \cos {\left (3 x + 4 \right )}}{10} \]

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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