3.2.74 \(\int e^{2 x} \cos (3 x) \, dx\) [174]

Optimal. Leaf size=27 \[ \frac {2}{13} e^{2 x} \cos (3 x)+\frac {3}{13} e^{2 x} \sin (3 x) \]

[Out]

2/13*exp(2*x)*cos(3*x)+3/13*exp(2*x)*sin(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 = {4518} \begin {gather*} \frac {3}{13} e^{2 x} \sin (3 x)+\frac {2}{13} e^{2 x} \cos (3 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(2*x)*Cos[3*x],x]

[Out]

(2*E^(2*x)*Cos[3*x])/13 + (3*E^(2*x)*Sin[3*x])/13

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 {gather*} \begin {aligned} \text {Integral} &=\frac {2}{13} e^{2 x} \cos (3 x)+\frac {3}{13} e^{2 x} \sin (3 x)\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.02, size = 22, normalized size = 0.81 \begin {gather*} \frac {1}{13} e^{2 x} (2 \cos (3 x)+3 \sin (3 x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(2*x)*Cos[3*x],x]

[Out]

(E^(2*x)*(2*Cos[3*x] + 3*Sin[3*x]))/13

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Maple [A]
time = 0.06, size = 22, normalized size = 0.81

method	result	size
parallelrisch	\(\frac {{\mathrm e}^{2 x} \left (2 \cos \left (3 x \right )+3 \sin \left (3 x \right )\right )}{13}\)	\(20\)
default	\(\frac {2 \,{\mathrm e}^{2 x} \cos \left (3 x \right )}{13}+\frac {3 \,{\mathrm e}^{2 x} \sin \left (3 x \right )}{13}\)	\(22\)
risch	\(\frac {{\mathrm e}^{\left (2+3 i\right ) x}}{13}-\frac {3 i {\mathrm e}^{\left (2+3 i\right ) x}}{26}+\frac {{\mathrm e}^{\left (2-3 i\right ) x}}{13}+\frac {3 i {\mathrm e}^{\left (2-3 i\right ) x}}{26}\)	\(36\)
norman	\(\frac {\frac {6 \,{\mathrm e}^{2 x} \tan \left (\frac {3 x}{2}\right )}{13}-\frac {2 \,{\mathrm e}^{2 x} \left (\tan ^{2}\left (\frac {3 x}{2}\right )\right )}{13}+\frac {2 \,{\mathrm e}^{2 x}}{13}}{1+\tan ^{2}\left (\frac {3 x}{2}\right )}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(2*x)*cos(3*x),x,method=_RETURNVERBOSE)

[Out]

2/13*exp(2*x)*cos(3*x)+3/13*exp(2*x)*sin(3*x)

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Maxima [A]
time = 0.33, size = 19, normalized size = 0.70 \begin {gather*} \frac {1}{13} \, {\left (2 \, \cos \left (3 \, x\right ) + 3 \, \sin \left (3 \, x\right )\right )} e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(2*x)*cos(3*x),x, algorithm="maxima")

[Out]

1/13*(2*cos(3*x) + 3*sin(3*x))*e^(2*x)

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Fricas [A]
time = 0.58, size = 21, normalized size = 0.78 \begin {gather*} \frac {2}{13} \, \cos \left (3 \, x\right ) e^{\left (2 \, x\right )} + \frac {3}{13} \, e^{\left (2 \, x\right )} \sin \left (3 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(2*x)*cos(3*x),x, algorithm="fricas")

[Out]

2/13*cos(3*x)*e^(2*x) + 3/13*e^(2*x)*sin(3*x)

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Sympy [A]
time = 0.10, size = 26, normalized size = 0.96 \begin {gather*} \frac {3 e^{2 x} \sin {\left (3 x \right )}}{13} + \frac {2 e^{2 x} \cos {\left (3 x \right )}}{13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(2*x)*cos(3*x),x)

[Out]

3*exp(2*x)*sin(3*x)/13 + 2*exp(2*x)*cos(3*x)/13

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Giac [A]
time = 0.45, size = 19, normalized size = 0.70 \begin {gather*} \frac {1}{13} \, {\left (2 \, \cos \left (3 \, x\right ) + 3 \, \sin \left (3 \, x\right )\right )} e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(2*x)*cos(3*x),x, algorithm="giac")

[Out]

1/13*(2*cos(3*x) + 3*sin(3*x))*e^(2*x)

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Mupad [B]
time = 0.03, size = 19, normalized size = 0.70 \begin {gather*} \frac {{\mathrm {e}}^{2\,x}\,\left (2\,\cos \left (3\,x\right )+3\,\sin \left (3\,x\right )\right )}{13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(3*x)*exp(2*x),x)

[Out]

(exp(2*x)*(2*cos(3*x) + 3*sin(3*x)))/13

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Chatgpt [F] Failed to verify
time = 1.00, size = 19, normalized size = 0.70 \begin {gather*} \frac {{\mathrm e}^{2 x} \left (13 \cos \left (3 x \right )-6 \sin \left (3 x \right )\right )}{13} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

int(exp(2*x)*cos(3*x),x)

[Out]

1/13*exp(2*x)*(13*cos(3*x)-6*sin(3*x))

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