3.37.81 \(\int \frac {2+e^{1+e^{e^3} (-1+6 e^x-9 e^{2 x})+x-6 e^x x+9 e^{2 x} x} (3 x^2+e^{e^3} (18 e^x x^2-54 e^{2 x} x^2)+e^x (-18 x^2-18 x^3)+e^{2 x} (27 x^2+54 x^3))}{3 x^2} \, dx\) [3681]

Optimal. Leaf size=31 \[ e^{1+\left (-1+3 e^x\right )^2 \left (-e^{e^3}+x\right )}-\frac {2}{3 x} \]

[Out]

exp((x-exp(exp(3)))*(3*exp(x)-1)^2+1)-2/3/x

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Rubi [A]
time = 1.08, antiderivative size = 43, normalized size of antiderivative = 1.39, number of steps used = 4, number of rules used = 3, integrand size = 112, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.027, Rules used = {12, 14, 6838} \begin {gather*} \exp \left (-e^{e^3} \left (1-3 e^x\right )^2-6 e^x x+9 e^{2 x} x+x+1\right )-\frac {2}{3 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + E^(1 + E^E^3*(-1 + 6*E^x - 9*E^(2*x)) + x - 6*E^x*x + 9*E^(2*x)*x)*(3*x^2 + E^E^3*(18*E^x*x^2 - 54*E^
(2*x)*x^2) + E^x*(-18*x^2 - 18*x^3) + E^(2*x)*(27*x^2 + 54*x^3)))/(3*x^2),x]

[Out]

E^(1 - E^E^3*(1 - 3*E^x)^2 + x - 6*E^x*x + 9*E^(2*x)*x) - 2/(3*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 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 6838

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}{3} \int \frac {2+\exp \left (1+e^{e^3} \left (-1+6 e^x-9 e^{2 x}\right )+x-6 e^x x+9 e^{2 x} x\right ) \left (3 x^2+e^{e^3} \left (18 e^x x^2-54 e^{2 x} x^2\right )+e^x \left (-18 x^2-18 x^3\right )+e^{2 x} \left (27 x^2+54 x^3\right )\right )}{x^2} \, dx\\ &=\frac {1}{3} \int \left (\frac {2}{x^2}+3 \exp \left (1-e^{e^3} \left (1-3 e^x\right )^2+x-6 e^x x+9 e^{2 x} x\right ) \left (1-3 e^x\right ) \left (1-3 e^x \left (1-2 e^{e^3}\right )-6 e^x x\right )\right ) \, dx\\ &=-\frac {2}{3 x}+\int \exp \left (1-e^{e^3} \left (1-3 e^x\right )^2+x-6 e^x x+9 e^{2 x} x\right ) \left (1-3 e^x\right ) \left (1-3 e^x \left (1-2 e^{e^3}\right )-6 e^x x\right ) \, dx\\ &=\exp \left (1-e^{e^3} \left (1-3 e^x\right )^2+x-6 e^x x+9 e^{2 x} x\right )-\frac {2}{3 x}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.36, size = 50, normalized size = 1.61 \begin {gather*} e^{1-e^{e^3}+6 e^x \left (e^{e^3}-x\right )-9 e^{2 x} \left (e^{e^3}-x\right )+x}-\frac {2}{3 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + E^(1 + E^E^3*(-1 + 6*E^x - 9*E^(2*x)) + x - 6*E^x*x + 9*E^(2*x)*x)*(3*x^2 + E^E^3*(18*E^x*x^2 -
54*E^(2*x)*x^2) + E^x*(-18*x^2 - 18*x^3) + E^(2*x)*(27*x^2 + 54*x^3)))/(3*x^2),x]

[Out]

E^(1 - E^E^3 + 6*E^x*(E^E^3 - x) - 9*E^(2*x)*(E^E^3 - x) + x) - 2/(3*x)

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Maple [A]
time = 0.74, size = 41, normalized size = 1.32

method	result	size
norman	\(\frac {-\frac {2}{3}+x \,{\mathrm e}^{\left (-9 \,{\mathrm e}^{2 x}+6 \,{\mathrm e}^{x}-1\right ) {\mathrm e}^{{\mathrm e}^{3}}+9 x \,{\mathrm e}^{2 x}-6 \,{\mathrm e}^{x} x +x +1}}{x}\)	\(41\)
risch	\(-\frac {2}{3 x}+{\mathrm e}^{6 \,{\mathrm e}^{{\mathrm e}^{3}+x}-6 \,{\mathrm e}^{x} x -9 \,{\mathrm e}^{2 x +{\mathrm e}^{3}}+9 x \,{\mathrm e}^{2 x}-{\mathrm e}^{{\mathrm e}^{3}}+x +1}\)	\(44\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*(((-54*exp(x)^2*x^2+18*exp(x)*x^2)*exp(exp(3))+(54*x^3+27*x^2)*exp(x)^2+(-18*x^3-18*x^2)*exp(x)+3*x^2)
*exp((-9*exp(x)^2+6*exp(x)-1)*exp(exp(3))+9*x*exp(x)^2-6*exp(x)*x+x+1)+2)/x^2,x,method=_RETURNVERBOSE)

[Out]

(-2/3+x*exp((-9*exp(x)^2+6*exp(x)-1)*exp(exp(3))+9*x*exp(x)^2-6*exp(x)*x+x+1))/x

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Maxima [A]
time = 0.42, size = 43, normalized size = 1.39 \begin {gather*} -\frac {2}{3 \, x} + e^{\left (9 \, x e^{\left (2 \, x\right )} - 6 \, x e^{x} + x - 9 \, e^{\left (2 \, x + e^{3}\right )} + 6 \, e^{\left (x + e^{3}\right )} - e^{\left (e^{3}\right )} + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(((-54*exp(x)^2*x^2+18*exp(x)*x^2)*exp(exp(3))+(54*x^3+27*x^2)*exp(x)^2+(-18*x^3-18*x^2)*exp(x)+
3*x^2)*exp((-9*exp(x)^2+6*exp(x)-1)*exp(exp(3))+9*x*exp(x)^2-6*exp(x)*x+x+1)+2)/x^2,x, algorithm="maxima")

[Out]

-2/3/x + e^(9*x*e^(2*x) - 6*x*e^x + x - 9*e^(2*x + e^3) + 6*e^(x + e^3) - e^(e^3) + 1)

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Fricas [A]
time = 0.37, size = 43, normalized size = 1.39 \begin {gather*} \frac {3 \, x e^{\left (9 \, x e^{\left (2 \, x\right )} - 6 \, x e^{x} - {\left (9 \, e^{\left (2 \, x\right )} - 6 \, e^{x} + 1\right )} e^{\left (e^{3}\right )} + x + 1\right )} - 2}{3 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(((-54*exp(x)^2*x^2+18*exp(x)*x^2)*exp(exp(3))+(54*x^3+27*x^2)*exp(x)^2+(-18*x^3-18*x^2)*exp(x)+
3*x^2)*exp((-9*exp(x)^2+6*exp(x)-1)*exp(exp(3))+9*x*exp(x)^2-6*exp(x)*x+x+1)+2)/x^2,x, algorithm="fricas")

[Out]

1/3*(3*x*e^(9*x*e^(2*x) - 6*x*e^x - (9*e^(2*x) - 6*e^x + 1)*e^(e^3) + x + 1) - 2)/x

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Sympy [A]
time = 0.19, size = 42, normalized size = 1.35 \begin {gather*} e^{9 x e^{2 x} - 6 x e^{x} + x + \left (- 9 e^{2 x} + 6 e^{x} - 1\right ) e^{e^{3}} + 1} - \frac {2}{3 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(((-54*exp(x)**2*x**2+18*exp(x)*x**2)*exp(exp(3))+(54*x**3+27*x**2)*exp(x)**2+(-18*x**3-18*x**2)
*exp(x)+3*x**2)*exp((-9*exp(x)**2+6*exp(x)-1)*exp(exp(3))+9*x*exp(x)**2-6*exp(x)*x+x+1)+2)/x**2,x)

[Out]

exp(9*x*exp(2*x) - 6*x*exp(x) + x + (-9*exp(2*x) + 6*exp(x) - 1)*exp(exp(3)) + 1) - 2/(3*x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(((-54*exp(x)^2*x^2+18*exp(x)*x^2)*exp(exp(3))+(54*x^3+27*x^2)*exp(x)^2+(-18*x^3-18*x^2)*exp(x)+
3*x^2)*exp((-9*exp(x)^2+6*exp(x)-1)*exp(exp(3))+9*x*exp(x)^2-6*exp(x)*x+x+1)+2)/x^2,x, algorithm="giac")

[Out]

integrate(1/3*(3*(x^2 + 9*(2*x^3 + x^2)*e^(2*x) - 6*(x^3 + x^2)*e^x - 6*(3*x^2*e^(2*x) - x^2*e^x)*e^(e^3))*e^(
9*x*e^(2*x) - 6*x*e^x - (9*e^(2*x) - 6*e^x + 1)*e^(e^3) + x + 1) + 2)/x^2, x)

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Mupad [B]
time = 2.43, size = 49, normalized size = 1.58 \begin {gather*} {\mathrm {e}}^{-6\,x\,{\mathrm {e}}^x}\,\mathrm {e}\,{\mathrm {e}}^{6\,{\mathrm {e}}^{{\mathrm {e}}^3}\,{\mathrm {e}}^x}\,{\mathrm {e}}^{9\,x\,{\mathrm {e}}^{2\,x}}\,{\mathrm {e}}^{-{\mathrm {e}}^{{\mathrm {e}}^3}}\,{\mathrm {e}}^{-9\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{{\mathrm {e}}^3}}\,{\mathrm {e}}^x-\frac {2}{3\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((exp(x + 9*x*exp(2*x) - exp(exp(3))*(9*exp(2*x) - 6*exp(x) + 1) - 6*x*exp(x) + 1)*(exp(exp(3))*(18*x^2*ex
p(x) - 54*x^2*exp(2*x)) - exp(x)*(18*x^2 + 18*x^3) + exp(2*x)*(27*x^2 + 54*x^3) + 3*x^2))/3 + 2/3)/x^2,x)

[Out]

exp(-6*x*exp(x))*exp(1)*exp(6*exp(exp(3))*exp(x))*exp(9*x*exp(2*x))*exp(-exp(exp(3)))*exp(-9*exp(2*x)*exp(exp(
3)))*exp(x) - 2/(3*x)

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