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

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 = 0.17855, 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, Rules used = {6715, 4433} \[ \frac{7}{195} e^{4 x^3} \sin \left (7 x^3\right )+\frac{4}{195} e^{4 x^3} \cos \left (7 x^3\right ) \]

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

Rule 6715

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

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^{4 x^3} x^2 \cos \left (7 x^3\right ) \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int e^{4 x} \cos (7 x) \, dx,x,x^3\right )\\ &=\frac{4}{195} e^{4 x^3} \cos \left (7 x^3\right )+\frac{7}{195} e^{4 x^3} \sin \left (7 x^3\right )\\ \end{align*}

Mathematica [A]  time = 0.0534506, 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 = 0.052, size = 53, normalized size = 1.5 \begin{align*}{ \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}} \end{align*}

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 = 0.968881, size = 39, normalized size = 1.11 \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(4*x^3)*x^2*cos(7*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 = 0.868176, size = 77, normalized size = 2.2 \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(4*x^3)*x^2*cos(7*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 = 2.24314, size = 32, normalized size = 0.91 \begin{align*} \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} \end{align*}

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 [A]  time = 1.23861, size = 34, normalized size = 0.97 \begin{align*} \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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(4*x^3)*x^2*cos(7*x^3),x, algorithm="giac")

[Out]

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