3.44.13 \(\int \frac {-5+e^{8 e^{2-2 x}+24 e^{1-x} x+18 x^2} (-80 e^{2-2 x}+e^{1-x} (120-120 x)+180 x)}{32+2 e^{16 e^{2-2 x}+48 e^{1-x} x+36 x^2}+e^{8 e^{2-2 x}+24 e^{1-x} x+18 x^2} (-16-4 x)+16 x+2 x^2} \, dx\)

Optimal. Leaf size=30 \[ \frac {5}{2 \left (4-e^{2 \left (x+2 \left (e^{1-x}+x\right )\right )^2}+x\right )} \]

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Rubi [F]  time = 4.21, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-5+e^{8 e^{2-2 x}+24 e^{1-x} x+18 x^2} \left (-80 e^{2-2 x}+e^{1-x} (120-120 x)+180 x\right )}{32+2 e^{16 e^{2-2 x}+48 e^{1-x} x+36 x^2}+e^{8 e^{2-2 x}+24 e^{1-x} x+18 x^2} (-16-4 x)+16 x+2 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-5 + E^(8*E^(2 - 2*x) + 24*E^(1 - x)*x + 18*x^2)*(-80*E^(2 - 2*x) + E^(1 - x)*(120 - 120*x) + 180*x))/(32
+ 2*E^(16*E^(2 - 2*x) + 48*E^(1 - x)*x + 36*x^2) + E^(8*E^(2 - 2*x) + 24*E^(1 - x)*x + 18*x^2)*(-16 - 4*x) +
16*x + 2*x^2),x]

[Out]

(-5*Defer[Int][(-4 + E^((2*(2*E + 3*E^x*x)^2)/E^(2*x)) - x)^(-2), x])/2 + 60*Defer[Int][E^(1 + 8*E^(2 - 2*x) -
x + 24*E^(1 - x)*x + 18*x^2)/(-4 + E^((2*(2*E + 3*E^x*x)^2)/E^(2*x)) - x)^2, x] - 40*Defer[Int][E^(2 + 8*E^(2
- 2*x) + 24*E^(1 - x)*x + 2*x*(-1 + 9*x))/(-4 + E^((2*(2*E + 3*E^x*x)^2)/E^(2*x)) - x)^2, x] + 90*Defer[Int][
(E^((2*(2*E + 3*E^x*x)^2)/E^(2*x))*x)/(-4 + E^((2*(2*E + 3*E^x*x)^2)/E^(2*x)) - x)^2, x] - 60*Defer[Int][(E^(1
+ 8*E^(2 - 2*x) - x + 24*E^(1 - x)*x + 18*x^2)*x)/(-4 + E^((2*(2*E + 3*E^x*x)^2)/E^(2*x)) - x)^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-5+20 \exp \left (8 e^{2-2 x}+24 e^{1-x} x+2 x (-1+9 x)\right ) \left (-2 e+3 e^x\right ) \left (2 e+3 e^x x\right )}{2 \left (4-e^{2 e^{-2 x} \left (2 e+3 e^x x\right )^2}+x\right )^2} \, dx\\ &=\frac {1}{2} \int \frac {-5+20 \exp \left (8 e^{2-2 x}+24 e^{1-x} x+2 x (-1+9 x)\right ) \left (-2 e+3 e^x\right ) \left (2 e+3 e^x x\right )}{\left (4-e^{2 e^{-2 x} \left (2 e+3 e^x x\right )^2}+x\right )^2} \, dx\\ &=\frac {1}{2} \int \left (-\frac {5}{\left (-4+e^{2 e^{-2 x} \left (2 e+3 e^x x\right )^2}-x\right )^2}+\frac {120 e^{1+8 e^{2-2 x}-x+24 e^{1-x} x+18 x^2}}{\left (-4+e^{2 e^{-2 x} \left (2 e+3 e^x x\right )^2}-x\right )^2}-\frac {80 \exp \left (2+8 e^{2-2 x}+24 e^{1-x} x+2 x (-1+9 x)\right )}{\left (-4+e^{2 e^{-2 x} \left (2 e+3 e^x x\right )^2}-x\right )^2}+\frac {180 e^{2 e^{-2 x} \left (2 e+3 e^x x\right )^2} x}{\left (-4+e^{2 e^{-2 x} \left (2 e+3 e^x x\right )^2}-x\right )^2}-\frac {120 e^{1+8 e^{2-2 x}-x+24 e^{1-x} x+18 x^2} x}{\left (-4+e^{2 e^{-2 x} \left (2 e+3 e^x x\right )^2}-x\right )^2}\right ) \, dx\\ &=-\left (\frac {5}{2} \int \frac {1}{\left (-4+e^{2 e^{-2 x} \left (2 e+3 e^x x\right )^2}-x\right )^2} \, dx\right )-40 \int \frac {\exp \left (2+8 e^{2-2 x}+24 e^{1-x} x+2 x (-1+9 x)\right )}{\left (-4+e^{2 e^{-2 x} \left (2 e+3 e^x x\right )^2}-x\right )^2} \, dx+60 \int \frac {e^{1+8 e^{2-2 x}-x+24 e^{1-x} x+18 x^2}}{\left (-4+e^{2 e^{-2 x} \left (2 e+3 e^x x\right )^2}-x\right )^2} \, dx-60 \int \frac {e^{1+8 e^{2-2 x}-x+24 e^{1-x} x+18 x^2} x}{\left (-4+e^{2 e^{-2 x} \left (2 e+3 e^x x\right )^2}-x\right )^2} \, dx+90 \int \frac {e^{2 e^{-2 x} \left (2 e+3 e^x x\right )^2} x}{\left (-4+e^{2 e^{-2 x} \left (2 e+3 e^x x\right )^2}-x\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.40, size = 32, normalized size = 1.07 \begin {gather*} -\frac {5}{2 \left (-4+e^{2 e^{-2 x} \left (2 e+3 e^x x\right )^2}-x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-5 + E^(8*E^(2 - 2*x) + 24*E^(1 - x)*x + 18*x^2)*(-80*E^(2 - 2*x) + E^(1 - x)*(120 - 120*x) + 180*x
))/(32 + 2*E^(16*E^(2 - 2*x) + 48*E^(1 - x)*x + 36*x^2) + E^(8*E^(2 - 2*x) + 24*E^(1 - x)*x + 18*x^2)*(-16 - 4
*x) + 16*x + 2*x^2),x]

[Out]

-5/(2*(-4 + E^((2*(2*E + 3*E^x*x)^2)/E^(2*x)) - x))

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fricas [A]  time = 0.73, size = 33, normalized size = 1.10 \begin {gather*} \frac {5}{2 \, {\left (x - e^{\left (18 \, x^{2} + 24 \, x e^{\left (-x + 1\right )} + 8 \, e^{\left (-2 \, x + 2\right )}\right )} + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-80*exp(-x+1)^2+(-120*x+120)*exp(-x+1)+180*x)*exp(8*exp(-x+1)^2+24*x*exp(-x+1)+18*x^2)-5)/(2*exp(8
*exp(-x+1)^2+24*x*exp(-x+1)+18*x^2)^2+(-16-4*x)*exp(8*exp(-x+1)^2+24*x*exp(-x+1)+18*x^2)+2*x^2+16*x+32),x, alg
orithm="fricas")

[Out]

5/2/(x - e^(18*x^2 + 24*x*e^(-x + 1) + 8*e^(-2*x + 2)) + 4)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-80*exp(-x+1)^2+(-120*x+120)*exp(-x+1)+180*x)*exp(8*exp(-x+1)^2+24*x*exp(-x+1)+18*x^2)-5)/(2*exp(8
*exp(-x+1)^2+24*x*exp(-x+1)+18*x^2)^2+(-16-4*x)*exp(8*exp(-x+1)^2+24*x*exp(-x+1)+18*x^2)+2*x^2+16*x+32),x, alg
orithm="giac")

[Out]

Timed out

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maple [A]  time = 0.11, size = 34, normalized size = 1.13

method	result	size
risch	\(\frac {5}{2 \left (x -{\mathrm e}^{8 \,{\mathrm e}^{-2 x +2}+24 x \,{\mathrm e}^{1-x}+18 x^{2}}+4\right )}\)	\(34\)
norman	\(\frac {5}{2 \left (x -{\mathrm e}^{8 \,{\mathrm e}^{-2 x +2}+24 x \,{\mathrm e}^{1-x}+18 x^{2}}+4\right )}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-80*exp(1-x)^2+(-120*x+120)*exp(1-x)+180*x)*exp(8*exp(1-x)^2+24*x*exp(1-x)+18*x^2)-5)/(2*exp(8*exp(1-x)^
2+24*x*exp(1-x)+18*x^2)^2+(-16-4*x)*exp(8*exp(1-x)^2+24*x*exp(1-x)+18*x^2)+2*x^2+16*x+32),x,method=_RETURNVERB
OSE)

[Out]

5/2/(x-exp(8*exp(-2*x+2)+24*x*exp(1-x)+18*x^2)+4)

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maxima [A]  time = 0.42, size = 33, normalized size = 1.10 \begin {gather*} \frac {5}{2 \, {\left (x - e^{\left (18 \, x^{2} + 24 \, x e^{\left (-x + 1\right )} + 8 \, e^{\left (-2 \, x + 2\right )}\right )} + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-80*exp(-x+1)^2+(-120*x+120)*exp(-x+1)+180*x)*exp(8*exp(-x+1)^2+24*x*exp(-x+1)+18*x^2)-5)/(2*exp(8
*exp(-x+1)^2+24*x*exp(-x+1)+18*x^2)^2+(-16-4*x)*exp(8*exp(-x+1)^2+24*x*exp(-x+1)+18*x^2)+2*x^2+16*x+32),x, alg
orithm="maxima")

[Out]

5/2/(x - e^(18*x^2 + 24*x*e^(-x + 1) + 8*e^(-2*x + 2)) + 4)

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mupad [B]  time = 3.41, size = 36, normalized size = 1.20 \begin {gather*} \frac {5}{2\,\left (x-{\mathrm {e}}^{8\,{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^2}\,{\mathrm {e}}^{18\,x^2}\,{\mathrm {e}}^{24\,x\,{\mathrm {e}}^{-x}\,\mathrm {e}}+4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(8*exp(2 - 2*x) + 24*x*exp(1 - x) + 18*x^2)*(80*exp(2 - 2*x) - 180*x + exp(1 - x)*(120*x - 120)) + 5)
/(16*x + 2*exp(16*exp(2 - 2*x) + 48*x*exp(1 - x) + 36*x^2) - exp(8*exp(2 - 2*x) + 24*x*exp(1 - x) + 18*x^2)*(4
*x + 16) + 2*x^2 + 32),x)

[Out]

5/(2*(x - exp(8*exp(-2*x)*exp(2))*exp(18*x^2)*exp(24*x*exp(-x)*exp(1)) + 4))

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sympy [A]  time = 0.25, size = 32, normalized size = 1.07 \begin {gather*} - \frac {5}{- 2 x + 2 e^{18 x^{2} + 24 x e^{1 - x} + 8 e^{2 - 2 x}} - 8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-80*exp(-x+1)**2+(-120*x+120)*exp(-x+1)+180*x)*exp(8*exp(-x+1)**2+24*x*exp(-x+1)+18*x**2)-5)/(2*ex
p(8*exp(-x+1)**2+24*x*exp(-x+1)+18*x**2)**2+(-16-4*x)*exp(8*exp(-x+1)**2+24*x*exp(-x+1)+18*x**2)+2*x**2+16*x+3
2),x)

[Out]

-5/(-2*x + 2*exp(18*x**2 + 24*x*exp(1 - x) + 8*exp(2 - 2*x)) - 8)

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