3.37.87 \(\int \frac {-2 x^2-e^x x^2+e^{\frac {-3-2 x+3 e^4 x+e^x x^2}{x}} (-3+e^x (-x^2-x^3))}{x^2} \, dx\)

Optimal. Leaf size=31 \[ 1-e^x-e^{-2+3 e^4-\frac {3}{x}+e^x x}-2 x \]

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Rubi [A]  time = 0.62, antiderivative size = 30, normalized size of antiderivative = 0.97, number of steps used = 5, number of rules used = 4, integrand size = 60, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {14, 2194, 6688, 6706} \begin {gather*} -2 x-e^x-e^{e^x x-\frac {3}{x}+3 e^4-2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-2*x^2 - E^x*x^2 + E^((-3 - 2*x + 3*E^4*x + E^x*x^2)/x)*(-3 + E^x*(-x^2 - x^3)))/x^2,x]

[Out]

-E^x - E^(-2 + 3*E^4 - 3/x + E^x*x) - 2*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 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, 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} &=\int \left (-2-e^x+\frac {e^{-2 \left (1-\frac {3 e^4}{2}\right )-\frac {3}{x}+e^x x} \left (-3-e^x x^2-e^x x^3\right )}{x^2}\right ) \, dx\\ &=-2 x-\int e^x \, dx+\int \frac {e^{-2 \left (1-\frac {3 e^4}{2}\right )-\frac {3}{x}+e^x x} \left (-3-e^x x^2-e^x x^3\right )}{x^2} \, dx\\ &=-e^x-2 x+\int \frac {e^{-2 \left (1-\frac {3 e^4}{2}\right )-\frac {3}{x}+e^x x} \left (-3-e^x x^2 (1+x)\right )}{x^2} \, dx\\ &=-e^x-e^{-2+3 e^4-\frac {3}{x}+e^x x}-2 x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.31, size = 30, normalized size = 0.97 \begin {gather*} -e^x-e^{-2+3 e^4-\frac {3}{x}+e^x x}-2 x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2*x^2 - E^x*x^2 + E^((-3 - 2*x + 3*E^4*x + E^x*x^2)/x)*(-3 + E^x*(-x^2 - x^3)))/x^2,x]

[Out]

-E^x - E^(-2 + 3*E^4 - 3/x + E^x*x) - 2*x

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fricas [A]  time = 0.61, size = 31, normalized size = 1.00 \begin {gather*} -2 \, x - e^{x} - e^{\left (\frac {x^{2} e^{x} + 3 \, x e^{4} - 2 \, x - 3}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^3-x^2)*exp(x)-3)*exp((exp(x)*x^2+3*x*exp(4)-2*x-3)/x)-exp(x)*x^2-2*x^2)/x^2,x, algorithm="fric
as")

[Out]

-2*x - e^x - e^((x^2*e^x + 3*x*e^4 - 2*x - 3)/x)

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giac [A]  time = 0.18, size = 26, normalized size = 0.84 \begin {gather*} -2 \, x - e^{\left (x e^{x} - \frac {3}{x} + 3 \, e^{4} - 2\right )} - e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^3-x^2)*exp(x)-3)*exp((exp(x)*x^2+3*x*exp(4)-2*x-3)/x)-exp(x)*x^2-2*x^2)/x^2,x, algorithm="giac
")

[Out]

-2*x - e^(x*e^x - 3/x + 3*e^4 - 2) - e^x

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maple [A]  time = 0.23, size = 32, normalized size = 1.03

method	result	size
risch	\(-2 x -{\mathrm e}^{x}-{\mathrm e}^{\frac {{\mathrm e}^{x} x^{2}+3 x \,{\mathrm e}^{4}-2 x -3}{x}}\)	\(32\)
norman	\(\frac {-2 x^{2}-{\mathrm e}^{x} x -{\mathrm e}^{\frac {{\mathrm e}^{x} x^{2}+3 x \,{\mathrm e}^{4}-2 x -3}{x}} x}{x}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-x^3-x^2)*exp(x)-3)*exp((exp(x)*x^2+3*x*exp(4)-2*x-3)/x)-exp(x)*x^2-2*x^2)/x^2,x,method=_RETURNVERBOSE)

[Out]

-2*x-exp(x)-exp((exp(x)*x^2+3*x*exp(4)-2*x-3)/x)

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maxima [A]  time = 0.56, size = 26, normalized size = 0.84 \begin {gather*} -2 \, x - e^{\left (x e^{x} - \frac {3}{x} + 3 \, e^{4} - 2\right )} - e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^3-x^2)*exp(x)-3)*exp((exp(x)*x^2+3*x*exp(4)-2*x-3)/x)-exp(x)*x^2-2*x^2)/x^2,x, algorithm="maxi
ma")

[Out]

-2*x - e^(x*e^x - 3/x + 3*e^4 - 2) - e^x

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mupad [B]  time = 2.34, size = 28, normalized size = 0.90 \begin {gather*} -2\,x-{\mathrm {e}}^x-{\mathrm {e}}^{x\,{\mathrm {e}}^x}\,{\mathrm {e}}^{3\,{\mathrm {e}}^4}\,{\mathrm {e}}^{-2}\,{\mathrm {e}}^{-\frac {3}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^2*exp(x) + exp(-(2*x - x^2*exp(x) - 3*x*exp(4) + 3)/x)*(exp(x)*(x^2 + x^3) + 3) + 2*x^2)/x^2,x)

[Out]

- 2*x - exp(x) - exp(x*exp(x))*exp(3*exp(4))*exp(-2)*exp(-3/x)

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sympy [A]  time = 0.24, size = 29, normalized size = 0.94 \begin {gather*} - 2 x - e^{x} - e^{\frac {x^{2} e^{x} - 2 x + 3 x e^{4} - 3}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x**3-x**2)*exp(x)-3)*exp((exp(x)*x**2+3*x*exp(4)-2*x-3)/x)-exp(x)*x**2-2*x**2)/x**2,x)

[Out]

-2*x - exp(x) - exp((x**2*exp(x) - 2*x + 3*x*exp(4) - 3)/x)

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