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

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 = 0.0192777, 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 \[ \frac{3}{10} e^x \sin (3 x+4)+\frac{1}{10} e^x \cos (3 x+4) \]

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

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Rubi in Sympy [A]  time = 2.52161, 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]  rubi_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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Mathematica [A]  time = 0.0260844, 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 \[{\frac{{{\rm e}^{x}}\cos \left ( 4+3\,x \right ) }{10}}+{\frac{3\,{{\rm e}^{x}}\sin \left ( 4+3\,x \right ) }{10}} \]

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 = 0.772715, size = 26, normalized size = 0.96 \[ \frac{1}{10} \,{\left (\cos \left (3 \, x + 4\right ) + 3 \, \sin \left (3 \, x + 4\right )\right )} e^{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(3*x + 4)*e^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 = 0.240848, size = 28, normalized size = 1.04 \[ \frac{1}{10} \, \cos \left (3 \, x + 4\right ) e^{x} + \frac{3}{10} \, e^{x} \sin \left (3 \, x + 4\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(3*x + 4)*e^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 = 0.319667, 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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GIAC/XCAS [A]  time = 0.440208, size = 26, normalized size = 0.96 \[ \frac{1}{10} \,{\left (\cos \left (3 \, x + 4\right ) + 3 \, \sin \left (3 \, x + 4\right )\right )} e^{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(3*x + 4)*e^x,x, algorithm="giac")

[Out]

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