\(\int e^x \cos (4+3 x) \, dx\) [683]

Optimal result

Integrand size = 10, antiderivative size = 27 \[ \int e^x \cos (4+3 x) \, dx=\frac {1}{10} e^x \cos (4+3 x)+\frac {3}{10} e^x \sin (4+3 x) \]

[Out]

1/10*exp(x)*cos(4+3*x)+3/10*exp(x)*sin(4+3*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} \[ \int e^x \cos (4+3 x) \, dx=\frac {3}{10} e^x \sin (3 x+4)+\frac {1}{10} e^x \cos (3 x+4) \]

[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 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 {1}{10} e^x \cos (4+3 x)+\frac {3}{10} e^x \sin (4+3 x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int e^x \cos (4+3 x) \, dx=\frac {1}{10} e^x (\cos (4+3 x)+3 \sin (4+3 x)) \]

[In]

Integrate[E^x*Cos[4 + 3*x],x]

[Out]

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

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74

method	result	size
parallelrisch	\(\frac {{\mathrm e}^{x} \left (\cos \left (4+3 x \right )+3 \sin \left (4+3 x \right )\right )}{10}\)	\(20\)
default	\(\frac {{\mathrm e}^{x} \cos \left (4+3 x \right )}{10}+\frac {3 \,{\mathrm e}^{x} \sin \left (4+3 x \right )}{10}\)	\(22\)
risch	\(\left (\frac {1}{20}-\frac {3 i}{20}\right ) {\mathrm e}^{x} {\mathrm e}^{3 i x} {\mathrm e}^{4 i}+\left (\frac {1}{20}+\frac {3 i}{20}\right ) {\mathrm e}^{x} {\mathrm e}^{-3 i x} {\mathrm e}^{-4 i}\)	\(30\)
norman	\(\frac {\frac {3 \,{\mathrm e}^{x} \tan \left (2+\frac {3 x}{2}\right )}{5}-\frac {{\mathrm e}^{x} \left (\tan ^{2}\left (2+\frac {3 x}{2}\right )\right )}{10}+\frac {{\mathrm e}^{x}}{10}}{1+\tan ^{2}\left (2+\frac {3 x}{2}\right )}\)	\(41\)

[In]

int(exp(x)*cos(4+3*x),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78 \[ \int e^x \cos (4+3 x) \, dx=\frac {1}{10} \, \cos \left (3 \, x + 4\right ) e^{x} + \frac {3}{10} \, e^{x} \sin \left (3 \, x + 4\right ) \]

[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)

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int e^x \cos (4+3 x) \, dx=\frac {3 e^{x} \sin {\left (3 x + 4 \right )}}{10} + \frac {e^{x} \cos {\left (3 x + 4 \right )}}{10} \]

[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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int e^x \cos (4+3 x) \, dx=\frac {1}{10} \, {\left (\cos \left (3 \, x + 4\right ) + 3 \, \sin \left (3 \, x + 4\right )\right )} e^{x} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int e^x \cos (4+3 x) \, dx=\frac {1}{10} \, {\left (\cos \left (3 \, x + 4\right ) + 3 \, \sin \left (3 \, x + 4\right )\right )} e^{x} \]

[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

Mupad [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int e^x \cos (4+3 x) \, dx=\frac {{\mathrm {e}}^x\,\left (\cos \left (3\,x+4\right )+3\,\sin \left (3\,x+4\right )\right )}{10} \]

[In]

int(exp(x)*cos(3*x + 4),x)

[Out]

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