3.11.50 \(\int \frac {e^{-\frac {-2 e+x}{x}} (-32 e x^2+32 x^3+e^{2-2 x} (-32 e-32 x^2)+e^{\frac {-2 e+x}{2 x}} (32 e x-32 x^2)+e^{1-x} (-64 e x+32 x^2-32 x^3+e^{\frac {-2 e+x}{2 x}} (32 e+32 x^2)))}{x^2} \, dx\)

Optimal. Leaf size=33 \[ 2+\left (-4+4 e^{-\frac {-e+\frac {x}{2}}{x}} \left (e^{1-x}+x\right )\right )^2 \]

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Rubi [B]  time = 3.91, antiderivative size = 89, normalized size of antiderivative = 2.70, number of steps used = 10, number of rules used = 6, integrand size = 117, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {6688, 12, 6742, 6686, 6706, 2288} \begin {gather*} \frac {32 e^{\frac {2 e}{x}-x} \left (x^2+2 e\right )}{\left (\frac {2 e}{x^2}+1\right ) x}+\frac {16 \left (\sqrt {e}-e^{e/x} x\right )^2}{e}+16 e^{-2 x+\frac {2 e}{x}+1}-32 e^{-x+\frac {e}{x}+\frac {1}{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-32*E*x^2 + 32*x^3 + E^(2 - 2*x)*(-32*E - 32*x^2) + E^((-2*E + x)/(2*x))*(32*E*x - 32*x^2) + E^(1 - x)*(-
64*E*x + 32*x^2 - 32*x^3 + E^((-2*E + x)/(2*x))*(32*E + 32*x^2)))/(E^((-2*E + x)/x)*x^2),x]

[Out]

16*E^(1 + (2*E)/x - 2*x) - 32*E^(1/2 + E/x - x) + (16*(Sqrt[E] - E^(E/x)*x)^2)/E + (32*E^((2*E)/x - x)*(2*E +
x^2))/((1 + (2*E)/x^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 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

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]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {32 e^{-1+\frac {e}{x}-2 x} \left (e^{1+\frac {e}{x}}-e^{\frac {1}{2}+x}+e^{\frac {e}{x}+x} x\right ) \left (-e^2-e^{1+x} x-e x^2+e^x x^2\right )}{x^2} \, dx\\ &=32 \int \frac {e^{-1+\frac {e}{x}-2 x} \left (e^{1+\frac {e}{x}}-e^{\frac {1}{2}+x}+e^{\frac {e}{x}+x} x\right ) \left (-e^2-e^{1+x} x-e x^2+e^x x^2\right )}{x^2} \, dx\\ &=32 \int \left (\frac {e^{-1+\frac {e}{x}} (e-x) \left (\sqrt {e}-e^{e/x} x\right )}{x}-\frac {e^{1+\frac {2 e}{x}-2 x} \left (e+x^2\right )}{x^2}-\frac {e^{\frac {e}{x}-x} \left (-e^{3/2}+2 e^{1+\frac {e}{x}} x-\sqrt {e} x^2-e^{e/x} x^2+e^{e/x} x^3\right )}{x^2}\right ) \, dx\\ &=32 \int \frac {e^{-1+\frac {e}{x}} (e-x) \left (\sqrt {e}-e^{e/x} x\right )}{x} \, dx-32 \int \frac {e^{1+\frac {2 e}{x}-2 x} \left (e+x^2\right )}{x^2} \, dx-32 \int \frac {e^{\frac {e}{x}-x} \left (-e^{3/2}+2 e^{1+\frac {e}{x}} x-\sqrt {e} x^2-e^{e/x} x^2+e^{e/x} x^3\right )}{x^2} \, dx\\ &=16 e^{1+\frac {2 e}{x}-2 x}+\frac {16 \left (\sqrt {e}-e^{e/x} x\right )^2}{e}-32 \int \left (-\frac {e^{\frac {1}{2}+\frac {e}{x}-x} \left (e+x^2\right )}{x^2}+\frac {e^{\frac {2 e}{x}-x} \left (2 e-x+x^2\right )}{x}\right ) \, dx\\ &=16 e^{1+\frac {2 e}{x}-2 x}+\frac {16 \left (\sqrt {e}-e^{e/x} x\right )^2}{e}+32 \int \frac {e^{\frac {1}{2}+\frac {e}{x}-x} \left (e+x^2\right )}{x^2} \, dx-32 \int \frac {e^{\frac {2 e}{x}-x} \left (2 e-x+x^2\right )}{x} \, dx\\ &=16 e^{1+\frac {2 e}{x}-2 x}-32 e^{\frac {1}{2}+\frac {e}{x}-x}+\frac {16 \left (\sqrt {e}-e^{e/x} x\right )^2}{e}+\frac {32 e^{\frac {2 e}{x}-x} \left (2 e+x^2\right )}{\left (1+\frac {2 e}{x^2}\right ) x}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 2.26, size = 81, normalized size = 2.45 \begin {gather*} -32 e^{e/x} \left (\frac {1}{\sqrt {e}}-e^{\frac {e}{x}-x}\right ) x-32 \left (-\frac {1}{2} e^{1+\frac {2 e}{x}-2 x}+e^{\frac {1}{2}+\frac {e}{x}-x}-\frac {1}{2} e^{-1+\frac {2 e}{x}} x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-32*E*x^2 + 32*x^3 + E^(2 - 2*x)*(-32*E - 32*x^2) + E^((-2*E + x)/(2*x))*(32*E*x - 32*x^2) + E^(1 -
x)*(-64*E*x + 32*x^2 - 32*x^3 + E^((-2*E + x)/(2*x))*(32*E + 32*x^2)))/(E^((-2*E + x)/x)*x^2),x]

[Out]

-32*E^(E/x)*(1/Sqrt[E] - E^(E/x - x))*x - 32*(-1/2*E^(1 + (2*E)/x - 2*x) + E^(1/2 + E/x - x) - (E^(-1 + (2*E)/
x)*x^2)/2)

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fricas [B]  time = 0.80, size = 63, normalized size = 1.91 \begin {gather*} 16 \, {\left (x^{2} + 2 \, {\left (x - e^{\left (\frac {x - 2 \, e}{2 \, x}\right )}\right )} e^{\left (-x + 1\right )} - 2 \, x e^{\left (\frac {x - 2 \, e}{2 \, x}\right )} + e^{\left (-2 \, x + 2\right )}\right )} e^{\left (-\frac {x - 2 \, e}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-32*exp(1)-32*x^2)*exp(-x+1)^2+((32*exp(1)+32*x^2)*exp(1/2*(-2*exp(1)+x)/x)-64*x*exp(1)-32*x^3+32*
x^2)*exp(-x+1)+(32*x*exp(1)-32*x^2)*exp(1/2*(-2*exp(1)+x)/x)-32*x^2*exp(1)+32*x^3)/x^2/exp(1/2*(-2*exp(1)+x)/x
)^2,x, algorithm="fricas")

[Out]

16*(x^2 + 2*(x - e^(1/2*(x - 2*e)/x))*e^(-x + 1) - 2*x*e^(1/2*(x - 2*e)/x) + e^(-2*x + 2))*e^(-(x - 2*e)/x)

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giac [B]  time = 0.59, size = 131, normalized size = 3.97 \begin {gather*} 16 \, x^{2} e^{\left (\frac {2 \, e}{x} - 1\right )} + 16 \, {\left (2 \, x e^{\left (-\frac {2 \, x^{2} - x - 2 \, e}{2 \, x} - \frac {x^{2} - 2 \, e}{x}\right )} - 2 \, x e^{\left (-\frac {x^{2} - 2 \, e}{x}\right )} - 2 \, e^{\left (-\frac {2 \, x^{2} - x - 2 \, e}{x}\right )} + e^{\left (-\frac {3 \, {\left (2 \, x^{2} - x - 2 \, e\right )}}{2 \, x}\right )}\right )} e^{\left (\frac {2 \, x^{2} - x - 2 \, e}{2 \, x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-32*exp(1)-32*x^2)*exp(-x+1)^2+((32*exp(1)+32*x^2)*exp(1/2*(-2*exp(1)+x)/x)-64*x*exp(1)-32*x^3+32*
x^2)*exp(-x+1)+(32*x*exp(1)-32*x^2)*exp(1/2*(-2*exp(1)+x)/x)-32*x^2*exp(1)+32*x^3)/x^2/exp(1/2*(-2*exp(1)+x)/x
)^2,x, algorithm="giac")

[Out]

16*x^2*e^(2*e/x - 1) + 16*(2*x*e^(-1/2*(2*x^2 - x - 2*e)/x - (x^2 - 2*e)/x) - 2*x*e^(-(x^2 - 2*e)/x) - 2*e^(-(
2*x^2 - x - 2*e)/x) + e^(-3/2*(2*x^2 - x - 2*e)/x))*e^(1/2*(2*x^2 - x - 2*e)/x)

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maple [B]  time = 0.10, size = 66, normalized size = 2.00

method	result	size
risch	\(\left (-32 x -32 \,{\mathrm e}^{1-x}\right ) {\mathrm e}^{\frac {2 \,{\mathrm e}-x}{2 x}}+\left (16 x^{2}+32 x \,{\mathrm e}^{1-x}+16 \,{\mathrm e}^{-2 x +2}\right ) {\mathrm e}^{\frac {2 \,{\mathrm e}-x}{x}}\)	\(66\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-32*exp(1)-32*x^2)*exp(1-x)^2+((32*exp(1)+32*x^2)*exp(1/2*(-2*exp(1)+x)/x)-64*x*exp(1)-32*x^3+32*x^2)*ex
p(1-x)+(32*x*exp(1)-32*x^2)*exp(1/2*(-2*exp(1)+x)/x)-32*x^2*exp(1)+32*x^3)/x^2/exp(1/2*(-2*exp(1)+x)/x)^2,x,me
thod=_RETURNVERBOSE)

[Out]

(-32*x-32*exp(1-x))*exp(1/2*(2*exp(1)-x)/x)+(16*x^2+32*x*exp(1-x)+16*exp(-2*x+2))*exp((2*exp(1)-x)/x)

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maxima [C]  time = 0.50, size = 87, normalized size = 2.64 \begin {gather*} -32 \, {\rm Ei}\left (\frac {e}{x}\right ) e^{\frac {1}{2}} + 16 \, {\left ({\left (2 \, x e^{x} + e\right )} e^{\left (\frac {2 \, e}{x}\right )} - 2 \, e^{\left (x + \frac {e}{x} + \frac {1}{2}\right )}\right )} e^{\left (-2 \, x\right )} + 32 \, e^{\frac {1}{2}} \Gamma \left (-1, -\frac {e}{x}\right ) + 64 \, e \Gamma \left (-1, -\frac {2 \, e}{x}\right ) + 128 \, e \Gamma \left (-2, -\frac {2 \, e}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-32*exp(1)-32*x^2)*exp(-x+1)^2+((32*exp(1)+32*x^2)*exp(1/2*(-2*exp(1)+x)/x)-64*x*exp(1)-32*x^3+32*
x^2)*exp(-x+1)+(32*x*exp(1)-32*x^2)*exp(1/2*(-2*exp(1)+x)/x)-32*x^2*exp(1)+32*x^3)/x^2/exp(1/2*(-2*exp(1)+x)/x
)^2,x, algorithm="maxima")

[Out]

-32*Ei(e/x)*e^(1/2) + 16*((2*x*e^x + e)*e^(2*e/x) - 2*e^(x + e/x + 1/2))*e^(-2*x) + 32*e^(1/2)*gamma(-1, -e/x)
+ 64*e*gamma(-1, -2*e/x) + 128*e*gamma(-2, -2*e/x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {{\mathrm {e}}^{-\frac {2\,\left (\frac {x}{2}-\mathrm {e}\right )}{x}}\,\left ({\mathrm {e}}^{2-2\,x}\,\left (32\,x^2+32\,\mathrm {e}\right )+{\mathrm {e}}^{1-x}\,\left (64\,x\,\mathrm {e}-{\mathrm {e}}^{\frac {\frac {x}{2}-\mathrm {e}}{x}}\,\left (32\,x^2+32\,\mathrm {e}\right )-32\,x^2+32\,x^3\right )+32\,x^2\,\mathrm {e}-{\mathrm {e}}^{\frac {\frac {x}{2}-\mathrm {e}}{x}}\,\left (32\,x\,\mathrm {e}-32\,x^2\right )-32\,x^3\right )}{x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-(2*(x/2 - exp(1)))/x)*(exp(2 - 2*x)*(32*exp(1) + 32*x^2) + exp(1 - x)*(64*x*exp(1) - exp((x/2 - exp
(1))/x)*(32*exp(1) + 32*x^2) - 32*x^2 + 32*x^3) + 32*x^2*exp(1) - exp((x/2 - exp(1))/x)*(32*x*exp(1) - 32*x^2)
- 32*x^3))/x^2,x)

[Out]

int(-(exp(-(2*(x/2 - exp(1)))/x)*(exp(2 - 2*x)*(32*exp(1) + 32*x^2) + exp(1 - x)*(64*x*exp(1) - exp((x/2 - exp
(1))/x)*(32*exp(1) + 32*x^2) - 32*x^2 + 32*x^3) + 32*x^2*exp(1) - exp((x/2 - exp(1))/x)*(32*x*exp(1) - 32*x^2)
- 32*x^3))/x^2, x)

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sympy [B]  time = 82.39, size = 76, normalized size = 2.30 \begin {gather*} 16 x^{2} e^{- \frac {2 \left (\frac {x}{2} - e\right )}{x}} - 32 x e^{- \frac {\frac {x}{2} - e}{x}} + \left (32 x e^{- \frac {2 \left (\frac {x}{2} - e\right )}{x}} - 32 e^{- \frac {\frac {x}{2} - e}{x}}\right ) e^{1 - x} + 16 e^{- \frac {2 \left (\frac {x}{2} - e\right )}{x}} e^{2 - 2 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-32*exp(1)-32*x**2)*exp(-x+1)**2+((32*exp(1)+32*x**2)*exp(1/2*(-2*exp(1)+x)/x)-64*x*exp(1)-32*x**3
+32*x**2)*exp(-x+1)+(32*x*exp(1)-32*x**2)*exp(1/2*(-2*exp(1)+x)/x)-32*x**2*exp(1)+32*x**3)/x**2/exp(1/2*(-2*ex
p(1)+x)/x)**2,x)

[Out]

16*x**2*exp(-2*(x/2 - E)/x) - 32*x*exp(-(x/2 - E)/x) + (32*x*exp(-2*(x/2 - E)/x) - 32*exp(-(x/2 - E)/x))*exp(1
- x) + 16*exp(-2*(x/2 - E)/x)*exp(2 - 2*x)

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