3.86.51 \(\int \frac {e^{-\frac {19-40 x+20 x^2+e^{16} (-4+4 x)+e^{32} (-4+8 x-4 x^2)}{4-8 x+4 x^2}} (-1+e^{16} (-2+2 x)+e^{\frac {19-40 x+20 x^2+e^{16} (-4+4 x)+e^{32} (-4+8 x-4 x^2)}{4-8 x+4 x^2}} (2-6 x+6 x^2-2 x^3))}{-2+6 x-6 x^2+2 x^3} \, dx\) [8551]

Optimal. Leaf size=23 \[ e^{-5+\left (e^{16}-\frac {1}{2 (-1+x)}\right )^2}-x \]

[Out]

1/exp(5-(exp(16)-1/2/(-1+x))^2)-x

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Rubi [F]
time = 6.95, 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 {\exp \left (-\frac {19-40 x+20 x^2+e^{16} (-4+4 x)+e^{32} \left (-4+8 x-4 x^2\right )}{4-8 x+4 x^2}\right ) \left (-1+e^{16} (-2+2 x)+\exp \left (\frac {19-40 x+20 x^2+e^{16} (-4+4 x)+e^{32} \left (-4+8 x-4 x^2\right )}{4-8 x+4 x^2}\right ) \left (2-6 x+6 x^2-2 x^3\right )\right )}{-2+6 x-6 x^2+2 x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-1 + E^16*(-2 + 2*x) + E^((19 - 40*x + 20*x^2 + E^16*(-4 + 4*x) + E^32*(-4 + 8*x - 4*x^2))/(4 - 8*x + 4*x
^2))*(2 - 6*x + 6*x^2 - 2*x^3))/(E^((19 - 40*x + 20*x^2 + E^16*(-4 + 4*x) + E^32*(-4 + 8*x - 4*x^2))/(4 - 8*x
+ 4*x^2))*(-2 + 6*x - 6*x^2 + 2*x^3)),x]

[Out]

-x + Defer[Int][E^((45 + 4*E^16 + 4*E^32 - 4*(22 + E^16 + 2*E^32)*x + 4*(11 + E^32)*x^2)/(4*(-1 + x)^2))/(1 -
x)^2, x] - Defer[Int][1/(E^((19 - 4*E^16 - 4*E^32 - 4*(10 - E^16 - 2*E^32)*x + 4*(5 - E^32)*x^2)/(4 - 8*x + 4*
x^2))*(-1 + x)^3), x]/2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (-\frac {19-4 e^{16}-4 e^{32}-4 \left (10-e^{16}-2 e^{32}\right ) x+4 \left (5-e^{32}\right ) x^2}{4-8 x+4 x^2}\right ) \left (1-e^{16} (-2+2 x)-\exp \left (\frac {19-40 x+20 x^2+e^{16} (-4+4 x)+e^{32} \left (-4+8 x-4 x^2\right )}{4-8 x+4 x^2}\right ) \left (2-6 x+6 x^2-2 x^3\right )\right )}{2-6 x+6 x^2-2 x^3} \, dx\\ &=\int \left (-\exp \left (-e^{32}+\frac {19}{4 (-1+x)^2}+\frac {e^{16}}{-1+x}-\frac {10 x}{(-1+x)^2}+\frac {5 x^2}{(-1+x)^2}-\frac {19-4 e^{16}-4 e^{32}-4 \left (10-e^{16}-2 e^{32}\right ) x+4 \left (5-e^{32}\right ) x^2}{4-8 x+4 x^2}\right )+\frac {\exp \left (-\frac {19-4 e^{16}-4 e^{32}-4 \left (10-e^{16}-2 e^{32}\right ) x+4 \left (5-e^{32}\right ) x^2}{4-8 x+4 x^2}\right ) \left (-1-2 e^{16}+2 e^{16} x\right )}{2 (-1+x)^3}\right ) \, dx\\ &=\frac {1}{2} \int \frac {\exp \left (-\frac {19-4 e^{16}-4 e^{32}-4 \left (10-e^{16}-2 e^{32}\right ) x+4 \left (5-e^{32}\right ) x^2}{4-8 x+4 x^2}\right ) \left (-1-2 e^{16}+2 e^{16} x\right )}{(-1+x)^3} \, dx-\int \exp \left (-e^{32}+\frac {19}{4 (-1+x)^2}+\frac {e^{16}}{-1+x}-\frac {10 x}{(-1+x)^2}+\frac {5 x^2}{(-1+x)^2}-\frac {19-4 e^{16}-4 e^{32}-4 \left (10-e^{16}-2 e^{32}\right ) x+4 \left (5-e^{32}\right ) x^2}{4-8 x+4 x^2}\right ) \, dx\\ &=\frac {1}{2} \int \left (-\frac {\exp \left (-\frac {19-4 e^{16}-4 e^{32}-4 \left (10-e^{16}-2 e^{32}\right ) x+4 \left (5-e^{32}\right ) x^2}{4-8 x+4 x^2}\right )}{(-1+x)^3}+\frac {2 \exp \left (16-\frac {19-4 e^{16}-4 e^{32}-4 \left (10-e^{16}-2 e^{32}\right ) x+4 \left (5-e^{32}\right ) x^2}{4-8 x+4 x^2}\right )}{(-1+x)^2}\right ) \, dx-\int 1 \, dx\\ &=-x-\frac {1}{2} \int \frac {\exp \left (-\frac {19-4 e^{16}-4 e^{32}-4 \left (10-e^{16}-2 e^{32}\right ) x+4 \left (5-e^{32}\right ) x^2}{4-8 x+4 x^2}\right )}{(-1+x)^3} \, dx+\int \frac {\exp \left (16-\frac {19-4 e^{16}-4 e^{32}-4 \left (10-e^{16}-2 e^{32}\right ) x+4 \left (5-e^{32}\right ) x^2}{4-8 x+4 x^2}\right )}{(-1+x)^2} \, dx\\ &=-x-\frac {1}{2} \int \frac {\exp \left (-\frac {19-4 e^{16}-4 e^{32}-4 \left (10-e^{16}-2 e^{32}\right ) x+4 \left (5-e^{32}\right ) x^2}{4-8 x+4 x^2}\right )}{(-1+x)^3} \, dx+\int \frac {\exp \left (\frac {45+4 e^{16}+4 e^{32}-4 \left (22+e^{16}+2 e^{32}\right ) x+4 \left (11+e^{32}\right ) x^2}{4 (-1+x)^2}\right )}{(1-x)^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 1.34, size = 30, normalized size = 1.30 \begin {gather*} e^{-5+e^{32}+\frac {1}{4 (-1+x)^2}-\frac {e^{16}}{-1+x}}-x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + E^16*(-2 + 2*x) + E^((19 - 40*x + 20*x^2 + E^16*(-4 + 4*x) + E^32*(-4 + 8*x - 4*x^2))/(4 - 8*x
+ 4*x^2))*(2 - 6*x + 6*x^2 - 2*x^3))/(E^((19 - 40*x + 20*x^2 + E^16*(-4 + 4*x) + E^32*(-4 + 8*x - 4*x^2))/(4
- 8*x + 4*x^2))*(-2 + 6*x - 6*x^2 + 2*x^3)),x]

[Out]

E^(-5 + E^32 + 1/(4*(-1 + x)^2) - E^16/(-1 + x)) - x

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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 1.54, size = 93, normalized size = 4.04

method	result	size
risch	\(-x +{\mathrm e}^{-\frac {-4 \,{\mathrm e}^{32} x^{2}+8 \,{\mathrm e}^{32} x +4 x \,{\mathrm e}^{16}+20 x^{2}-4 \,{\mathrm e}^{32}-4 \,{\mathrm e}^{16}-40 x +19}{4 \left (x -1\right )^{2}}}\)	\(48\)
derivativedivides	\(-x +1+{\mathrm e}^{{\mathrm e}^{32}-\frac {{\mathrm e}^{16}}{x -1}+\frac {1}{4 \left (x -1\right )^{2}}-5}-i {\mathrm e}^{16} \sqrt {\pi }\, {\mathrm e}^{-5} \erf \left (\frac {i}{2 x -2}-i {\mathrm e}^{16}\right )+i \sqrt {\pi }\, {\mathrm e}^{11} \erf \left (\frac {i}{2 x -2}-i {\mathrm e}^{16}\right )\)	\(93\)
default	\(-x +1+{\mathrm e}^{{\mathrm e}^{32}-\frac {{\mathrm e}^{16}}{x -1}+\frac {1}{4 \left (x -1\right )^{2}}-5}-i {\mathrm e}^{16} \sqrt {\pi }\, {\mathrm e}^{-5} \erf \left (\frac {i}{2 x -2}-i {\mathrm e}^{16}\right )+i \sqrt {\pi }\, {\mathrm e}^{11} \erf \left (\frac {i}{2 x -2}-i {\mathrm e}^{16}\right )\)	\(93\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x^3+6*x^2-6*x+2)*exp(((-4*x^2+8*x-4)*exp(16)^2+(4*x-4)*exp(16)+20*x^2-40*x+19)/(4*x^2-8*x+4))+(2*x-2)
*exp(16)-1)/(2*x^3-6*x^2+6*x-2)/exp(((-4*x^2+8*x-4)*exp(16)^2+(4*x-4)*exp(16)+20*x^2-40*x+19)/(4*x^2-8*x+4)),x
,method=_RETURNVERBOSE)

[Out]

-x+1+exp(exp(32)-exp(16)/(x-1)+1/4/(x-1)^2-5)-I*exp(16)*Pi^(1/2)*exp(exp(32)-5-exp(16)^2)*erf(1/2*I/(x-1)-I*ex
p(16))+I*Pi^(1/2)*exp(exp(32)+11-exp(16)^2)*erf(1/2*I/(x-1)-I*exp(16))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 93 vs. \(2 (21) = 42\).
time = 1.42, size = 93, normalized size = 4.04 \begin {gather*} -x + \frac {6 \, x - 5}{2 \, {\left (x^{2} - 2 \, x + 1\right )}} - \frac {3 \, {\left (4 \, x - 3\right )}}{2 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {3 \, {\left (2 \, x - 1\right )}}{2 \, {\left (x^{2} - 2 \, x + 1\right )}} - \frac {1}{2 \, {\left (x^{2} - 2 \, x + 1\right )}} + e^{\left (-\frac {e^{16}}{x - 1} + \frac {1}{4 \, {\left (x^{2} - 2 \, x + 1\right )}} + e^{32} - 5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^3+6*x^2-6*x+2)*exp(((-4*x^2+8*x-4)*exp(16)^2+(-4+4*x)*exp(16)+20*x^2-40*x+19)/(4*x^2-8*x+4))+
(-2+2*x)*exp(16)-1)/(2*x^3-6*x^2+6*x-2)/exp(((-4*x^2+8*x-4)*exp(16)^2+(-4+4*x)*exp(16)+20*x^2-40*x+19)/(4*x^2-
8*x+4)),x, algorithm="maxima")

[Out]

-x + 1/2*(6*x - 5)/(x^2 - 2*x + 1) - 3/2*(4*x - 3)/(x^2 - 2*x + 1) + 3/2*(2*x - 1)/(x^2 - 2*x + 1) - 1/2/(x^2
- 2*x + 1) + e^(-e^16/(x - 1) + 1/4/(x^2 - 2*x + 1) + e^32 - 5)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (21) = 42\).
time = 0.40, size = 46, normalized size = 2.00 \begin {gather*} -x + e^{\left (-\frac {20 \, x^{2} - 4 \, {\left (x^{2} - 2 \, x + 1\right )} e^{32} + 4 \, {\left (x - 1\right )} e^{16} - 40 \, x + 19}{4 \, {\left (x^{2} - 2 \, x + 1\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^3+6*x^2-6*x+2)*exp(((-4*x^2+8*x-4)*exp(16)^2+(-4+4*x)*exp(16)+20*x^2-40*x+19)/(4*x^2-8*x+4))+
(-2+2*x)*exp(16)-1)/(2*x^3-6*x^2+6*x-2)/exp(((-4*x^2+8*x-4)*exp(16)^2+(-4+4*x)*exp(16)+20*x^2-40*x+19)/(4*x^2-
8*x+4)),x, algorithm="fricas")

[Out]

-x + e^(-1/4*(20*x^2 - 4*(x^2 - 2*x + 1)*e^32 + 4*(x - 1)*e^16 - 40*x + 19)/(x^2 - 2*x + 1))

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (15) = 30\).
time = 0.19, size = 46, normalized size = 2.00 \begin {gather*} - x + e^{- \frac {20 x^{2} - 40 x + \left (4 x - 4\right ) e^{16} + \left (- 4 x^{2} + 8 x - 4\right ) e^{32} + 19}{4 x^{2} - 8 x + 4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x**3+6*x**2-6*x+2)*exp(((-4*x**2+8*x-4)*exp(16)**2+(-4+4*x)*exp(16)+20*x**2-40*x+19)/(4*x**2-8*
x+4))+(-2+2*x)*exp(16)-1)/(2*x**3-6*x**2+6*x-2)/exp(((-4*x**2+8*x-4)*exp(16)**2+(-4+4*x)*exp(16)+20*x**2-40*x+
19)/(4*x**2-8*x+4)),x)

[Out]

-x + exp(-(20*x**2 - 40*x + (4*x - 4)*exp(16) + (-4*x**2 + 8*x - 4)*exp(32) + 19)/(4*x**2 - 8*x + 4))

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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(((-2*x^3+6*x^2-6*x+2)*exp(((-4*x^2+8*x-4)*exp(16)^2+(-4+4*x)*exp(16)+20*x^2-40*x+19)/(4*x^2-8*x+4))+
(-2+2*x)*exp(16)-1)/(2*x^3-6*x^2+6*x-2)/exp(((-4*x^2+8*x-4)*exp(16)^2+(-4+4*x)*exp(16)+20*x^2-40*x+19)/(4*x^2-
8*x+4)),x, algorithm="giac")

[Out]

integrate(1/2*(2*(x - 1)*e^16 - 2*(x^3 - 3*x^2 + 3*x - 1)*e^(1/4*(20*x^2 - 4*(x^2 - 2*x + 1)*e^32 + 4*(x - 1)*
e^16 - 40*x + 19)/(x^2 - 2*x + 1)) - 1)*e^(-1/4*(20*x^2 - 4*(x^2 - 2*x + 1)*e^32 + 4*(x - 1)*e^16 - 40*x + 19)
/(x^2 - 2*x + 1))/(x^3 - 3*x^2 + 3*x - 1), x)

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Mupad [B]
time = 5.54, size = 144, normalized size = 6.26 \begin {gather*} {\mathrm {e}}^{\frac {40\,x}{4\,x^2-8\,x+4}}\,{\mathrm {e}}^{-\frac {20\,x^2}{4\,x^2-8\,x+4}}\,{\mathrm {e}}^{-\frac {4\,x\,{\mathrm {e}}^{16}}{4\,x^2-8\,x+4}}\,{\mathrm {e}}^{-\frac {8\,x\,{\mathrm {e}}^{32}}{4\,x^2-8\,x+4}}\,{\mathrm {e}}^{-\frac {19}{4\,x^2-8\,x+4}}\,{\mathrm {e}}^{\frac {4\,x^2\,{\mathrm {e}}^{32}}{4\,x^2-8\,x+4}}\,{\mathrm {e}}^{\frac {4\,{\mathrm {e}}^{16}}{4\,x^2-8\,x+4}}\,{\mathrm {e}}^{\frac {4\,{\mathrm {e}}^{32}}{4\,x^2-8\,x+4}}-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-(20*x^2 - exp(32)*(4*x^2 - 8*x + 4) - 40*x + exp(16)*(4*x - 4) + 19)/(4*x^2 - 8*x + 4))*(exp((20*x^
2 - exp(32)*(4*x^2 - 8*x + 4) - 40*x + exp(16)*(4*x - 4) + 19)/(4*x^2 - 8*x + 4))*(6*x - 6*x^2 + 2*x^3 - 2) -
exp(16)*(2*x - 2) + 1))/(6*x - 6*x^2 + 2*x^3 - 2),x)

[Out]

exp((40*x)/(4*x^2 - 8*x + 4))*exp(-(20*x^2)/(4*x^2 - 8*x + 4))*exp(-(4*x*exp(16))/(4*x^2 - 8*x + 4))*exp(-(8*x
*exp(32))/(4*x^2 - 8*x + 4))*exp(-19/(4*x^2 - 8*x + 4))*exp((4*x^2*exp(32))/(4*x^2 - 8*x + 4))*exp((4*exp(16))
/(4*x^2 - 8*x + 4))*exp((4*exp(32))/(4*x^2 - 8*x + 4)) - x

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