3.52.43 \(\int \frac {1}{2} e^{2+e^{\frac {1}{2} (-60 x^2+15 x^3)}-x} (-2+e^{\frac {1}{2} (-60 x^2+15 x^3)} (-120 x+45 x^2)) \, dx\)

Optimal. Leaf size=19 \[ e^{2+e^{\frac {15}{2} (-4+x) x^2}-x} \]

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Rubi [A]  time = 0.15, antiderivative size = 24, normalized size of antiderivative = 1.26, number of steps used = 2, number of rules used = 2, integrand size = 57, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.035, Rules used = {12, 6706} \begin {gather*} e^{e^{-\frac {15}{2} \left (4 x^2-x^3\right )}-x+2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(2 + E^((-60*x^2 + 15*x^3)/2) - x)*(-2 + E^((-60*x^2 + 15*x^3)/2)*(-120*x + 45*x^2)))/2,x]

[Out]

E^(2 + E^((-15*(4*x^2 - x^3))/2) - x)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int e^{2+e^{\frac {1}{2} \left (-60 x^2+15 x^3\right )}-x} \left (-2+e^{\frac {1}{2} \left (-60 x^2+15 x^3\right )} \left (-120 x+45 x^2\right )\right ) \, dx\\ &=e^{2+e^{-\frac {15}{2} \left (4 x^2-x^3\right )}-x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.35, size = 19, normalized size = 1.00 \begin {gather*} e^{2+e^{\frac {15}{2} (-4+x) x^2}-x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(2 + E^((-60*x^2 + 15*x^3)/2) - x)*(-2 + E^((-60*x^2 + 15*x^3)/2)*(-120*x + 45*x^2)))/2,x]

[Out]

E^(2 + E^((15*(-4 + x)*x^2)/2) - x)

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fricas [A]  time = 0.60, size = 18, normalized size = 0.95 \begin {gather*} e^{\left (-x + e^{\left (\frac {15}{2} \, x^{3} - 30 \, x^{2}\right )} + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((45*x^2-120*x)*exp(15/2*x^3-30*x^2)-2)*exp(exp(15/2*x^3-30*x^2)+2-x),x, algorithm="fricas")

[Out]

e^(-x + e^(15/2*x^3 - 30*x^2) + 2)

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giac [A]  time = 0.65, size = 18, normalized size = 0.95 \begin {gather*} e^{\left (-x + e^{\left (\frac {15}{2} \, x^{3} - 30 \, x^{2}\right )} + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((45*x^2-120*x)*exp(15/2*x^3-30*x^2)-2)*exp(exp(15/2*x^3-30*x^2)+2-x),x, algorithm="giac")

[Out]

e^(-x + e^(15/2*x^3 - 30*x^2) + 2)

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maple [A]  time = 0.10, size = 16, normalized size = 0.84

method	result	size
risch	\({\mathrm e}^{{\mathrm e}^{\frac {15 \left (x -4\right ) x^{2}}{2}}-x +2}\)	\(16\)
norman	\({\mathrm e}^{{\mathrm e}^{\frac {15}{2} x^{3}-30 x^{2}}+2-x}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*((45*x^2-120*x)*exp(15/2*x^3-30*x^2)-2)*exp(exp(15/2*x^3-30*x^2)+2-x),x,method=_RETURNVERBOSE)

[Out]

exp(exp(15/2*(x-4)*x^2)-x+2)

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maxima [A]  time = 0.67, size = 18, normalized size = 0.95 \begin {gather*} e^{\left (-x + e^{\left (\frac {15}{2} \, x^{3} - 30 \, x^{2}\right )} + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((45*x^2-120*x)*exp(15/2*x^3-30*x^2)-2)*exp(exp(15/2*x^3-30*x^2)+2-x),x, algorithm="maxima")

[Out]

e^(-x + e^(15/2*x^3 - 30*x^2) + 2)

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mupad [B]  time = 0.10, size = 21, normalized size = 1.11 \begin {gather*} {\mathrm {e}}^{-x}\,{\mathrm {e}}^2\,{\mathrm {e}}^{{\mathrm {e}}^{\frac {15\,x^3}{2}}\,{\mathrm {e}}^{-30\,x^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(exp((15*x^3)/2 - 30*x^2) - x + 2)*(exp((15*x^3)/2 - 30*x^2)*(120*x - 45*x^2) + 2))/2,x)

[Out]

exp(-x)*exp(2)*exp(exp((15*x^3)/2)*exp(-30*x^2))

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sympy [A]  time = 0.38, size = 17, normalized size = 0.89 \begin {gather*} e^{- x + e^{\frac {15 x^{3}}{2} - 30 x^{2}} + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((45*x**2-120*x)*exp(15/2*x**3-30*x**2)-2)*exp(exp(15/2*x**3-30*x**2)+2-x),x)

[Out]

exp(-x + exp(15*x**3/2 - 30*x**2) + 2)

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