3.651 \(\int e^{4 x^3} x^2 \cos \left (7 x^3\right ) \, dx\)

Optimal. Leaf size=35 \[ \frac{7}{195} e^{4 x^3} \sin \left (7 x^3\right )+\frac{4}{195} e^{4 x^3} \cos \left (7 x^3\right ) \]

[Out]

(4*E^(4*x^3)*Cos[7*x^3])/195 + (7*E^(4*x^3)*Sin[7*x^3])/195

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Rubi [A]  time = 0.29185, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{7}{195} e^{4 x^3} \sin \left (7 x^3\right )+\frac{4}{195} e^{4 x^3} \cos \left (7 x^3\right ) \]

Antiderivative was successfully verified.

[In]  Int[E^(4*x^3)*x^2*Cos[7*x^3],x]

[Out]

(4*E^(4*x^3)*Cos[7*x^3])/195 + (7*E^(4*x^3)*Sin[7*x^3])/195

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Rubi in Sympy [A]  time = 12.4995, size = 32, normalized size = 0.91 \[ \frac{7 e^{4 x^{3}} \sin{\left (7 x^{3} \right )}}{195} + \frac{4 e^{4 x^{3}} \cos{\left (7 x^{3} \right )}}{195} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(4*x**3)*x**2*cos(7*x**3),x)

[Out]

7*exp(4*x**3)*sin(7*x**3)/195 + 4*exp(4*x**3)*cos(7*x**3)/195

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Mathematica [A]  time = 0.0294429, size = 28, normalized size = 0.8 \[ \frac{1}{195} e^{4 x^3} \left (7 \sin \left (7 x^3\right )+4 \cos \left (7 x^3\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[E^(4*x^3)*x^2*Cos[7*x^3],x]

[Out]

(E^(4*x^3)*(4*Cos[7*x^3] + 7*Sin[7*x^3]))/195

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Maple [A]  time = 0.037, size = 53, normalized size = 1.5 \[{1 \left ({\frac{14\,{{\rm e}^{4\,{x}^{3}}}}{195}\tan \left ({\frac{7\,{x}^{3}}{2}} \right ) }-{\frac{4\,{{\rm e}^{4\,{x}^{3}}}}{195} \left ( \tan \left ({\frac{7\,{x}^{3}}{2}} \right ) \right ) ^{2}}+{\frac{4\,{{\rm e}^{4\,{x}^{3}}}}{195}} \right ) \left ( 1+ \left ( \tan \left ({\frac{7\,{x}^{3}}{2}} \right ) \right ) ^{2} \right ) ^{-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(4*x^3)*x^2*cos(7*x^3),x)

[Out]

(14/195*exp(4*x^3)*tan(7/2*x^3)-4/195*exp(4*x^3)*tan(7/2*x^3)^2+4/195*exp(4*x^3)
)/(1+tan(7/2*x^3)^2)

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Maxima [A]  time = 0.815436, size = 39, normalized size = 1.11 \[ \frac{4}{195} \, \cos \left (7 \, x^{3}\right ) e^{\left (4 \, x^{3}\right )} + \frac{7}{195} \, e^{\left (4 \, x^{3}\right )} \sin \left (7 \, x^{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2*cos(7*x^3)*e^(4*x^3),x, algorithm="maxima")

[Out]

4/195*cos(7*x^3)*e^(4*x^3) + 7/195*e^(4*x^3)*sin(7*x^3)

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Fricas [A]  time = 0.271841, size = 39, normalized size = 1.11 \[ \frac{4}{195} \, \cos \left (7 \, x^{3}\right ) e^{\left (4 \, x^{3}\right )} + \frac{7}{195} \, e^{\left (4 \, x^{3}\right )} \sin \left (7 \, x^{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2*cos(7*x^3)*e^(4*x^3),x, algorithm="fricas")

[Out]

4/195*cos(7*x^3)*e^(4*x^3) + 7/195*e^(4*x^3)*sin(7*x^3)

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Sympy [A]  time = 2.54623, size = 32, normalized size = 0.91 \[ \frac{7 e^{4 x^{3}} \sin{\left (7 x^{3} \right )}}{195} + \frac{4 e^{4 x^{3}} \cos{\left (7 x^{3} \right )}}{195} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(exp(4*x**3)*x**2*cos(7*x**3),x)

[Out]

7*exp(4*x**3)*sin(7*x**3)/195 + 4*exp(4*x**3)*cos(7*x**3)/195

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GIAC/XCAS [A]  time = 0.248135, size = 34, normalized size = 0.97 \[ \frac{1}{195} \,{\left (4 \, \cos \left (7 \, x^{3}\right ) + 7 \, \sin \left (7 \, x^{3}\right )\right )} e^{\left (4 \, x^{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2*cos(7*x^3)*e^(4*x^3),x, algorithm="giac")

[Out]

1/195*(4*cos(7*x^3) + 7*sin(7*x^3))*e^(4*x^3)