3.42.69 \(\int \frac {-1+e^{1250-1000 x+300 x^2-40 x^3+2 x^4} (2000-2200 x+840 x^2-136 x^3+8 x^4)}{-36+e^{1250-1000 x+300 x^2-40 x^3+2 x^4} (24-12 x)+18 x+e^{2500-2000 x+600 x^2-80 x^3+4 x^4} (-4+2 x)+(-24+e^{1250-1000 x+300 x^2-40 x^3+2 x^4} (8-4 x)+12 x) \log (-2+x)+(-4+2 x) \log ^2(-2+x)} \, dx\)

Optimal. Leaf size=25 \[ \frac {1}{2 \left (3-e^{2 (5-x)^4}+\log (-2+x)\right )} \]

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Rubi [F]  time = 22.84, 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 {-1+e^{1250-1000 x+300 x^2-40 x^3+2 x^4} \left (2000-2200 x+840 x^2-136 x^3+8 x^4\right )}{-36+e^{1250-1000 x+300 x^2-40 x^3+2 x^4} (24-12 x)+18 x+e^{2500-2000 x+600 x^2-80 x^3+4 x^4} (-4+2 x)+\left (-24+e^{1250-1000 x+300 x^2-40 x^3+2 x^4} (8-4 x)+12 x\right ) \log (-2+x)+(-4+2 x) \log ^2(-2+x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-1 + E^(1250 - 1000*x + 300*x^2 - 40*x^3 + 2*x^4)*(2000 - 2200*x + 840*x^2 - 136*x^3 + 8*x^4))/(-36 + E^(
1250 - 1000*x + 300*x^2 - 40*x^3 + 2*x^4)*(24 - 12*x) + 18*x + E^(2500 - 2000*x + 600*x^2 - 80*x^3 + 4*x^4)*(-
4 + 2*x) + (-24 + E^(1250 - 1000*x + 300*x^2 - 40*x^3 + 2*x^4)*(8 - 4*x) + 12*x)*Log[-2 + x] + (-4 + 2*x)*Log[
-2 + x]^2),x]

[Out]

-500*Defer[Int][E^(1250 + 1000*x + 300*x^2 + 40*x^3 + 2*x^4)/(3*E^(40*x*(25 + x^2)) - E^(1250 + 300*x^2 + 2*x^
4) + E^(40*x*(25 + x^2))*Log[-2 + x])^2, x] - Defer[Int][E^(80*x*(25 + x^2))/((-2 + x)*(3*E^(40*x*(25 + x^2))
- E^(1250 + 300*x^2 + 2*x^4) + E^(40*x*(25 + x^2))*Log[-2 + x])^2), x]/2 + 300*Defer[Int][(E^(1250 + 1000*x +
300*x^2 + 40*x^3 + 2*x^4)*x)/(3*E^(40*x*(25 + x^2)) - E^(1250 + 300*x^2 + 2*x^4) + E^(40*x*(25 + x^2))*Log[-2
+ x])^2, x] - 60*Defer[Int][(E^(1250 + 1000*x + 300*x^2 + 40*x^3 + 2*x^4)*x^2)/(3*E^(40*x*(25 + x^2)) - E^(125
0 + 300*x^2 + 2*x^4) + E^(40*x*(25 + x^2))*Log[-2 + x])^2, x] + 4*Defer[Int][(E^(1250 + 1000*x + 300*x^2 + 40*
x^3 + 2*x^4)*x^3)/(3*E^(40*x*(25 + x^2)) - E^(1250 + 300*x^2 + 2*x^4) + E^(40*x*(25 + x^2))*Log[-2 + x])^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{80 x \left (25+x^2\right )} \left (1-8 e^{2 (-5+x)^4} (-5+x)^3 (-2+x)\right )}{2 (2-x) \left (3 e^{40 x \left (25+x^2\right )}-e^{2 \left (625+150 x^2+x^4\right )}+e^{40 x \left (25+x^2\right )} \log (-2+x)\right )^2} \, dx\\ &=\frac {1}{2} \int \frac {e^{80 x \left (25+x^2\right )} \left (1-8 e^{2 (-5+x)^4} (-5+x)^3 (-2+x)\right )}{(2-x) \left (3 e^{40 x \left (25+x^2\right )}-e^{2 \left (625+150 x^2+x^4\right )}+e^{40 x \left (25+x^2\right )} \log (-2+x)\right )^2} \, dx\\ &=\frac {1}{2} \int \left (-\frac {e^{80 x \left (25+x^2\right )}}{(-2+x) \left (3 e^{40 x \left (25+x^2\right )}-e^{1250+300 x^2+2 x^4}+e^{40 x \left (25+x^2\right )} \log (-2+x)\right )^2}+\frac {2000 e^{2 (-5+x)^4+80 x \left (25+x^2\right )}}{(-2+x) \left (3 e^{40 x \left (25+x^2\right )}-e^{1250+300 x^2+2 x^4}+e^{40 x \left (25+x^2\right )} \log (-2+x)\right )^2}-\frac {2200 e^{2 (-5+x)^4+80 x \left (25+x^2\right )} x}{(-2+x) \left (3 e^{40 x \left (25+x^2\right )}-e^{1250+300 x^2+2 x^4}+e^{40 x \left (25+x^2\right )} \log (-2+x)\right )^2}+\frac {840 e^{2 (-5+x)^4+80 x \left (25+x^2\right )} x^2}{(-2+x) \left (3 e^{40 x \left (25+x^2\right )}-e^{1250+300 x^2+2 x^4}+e^{40 x \left (25+x^2\right )} \log (-2+x)\right )^2}-\frac {136 e^{2 (-5+x)^4+80 x \left (25+x^2\right )} x^3}{(-2+x) \left (3 e^{40 x \left (25+x^2\right )}-e^{1250+300 x^2+2 x^4}+e^{40 x \left (25+x^2\right )} \log (-2+x)\right )^2}+\frac {8 e^{2 (-5+x)^4+80 x \left (25+x^2\right )} x^4}{(-2+x) \left (3 e^{40 x \left (25+x^2\right )}-e^{1250+300 x^2+2 x^4}+e^{40 x \left (25+x^2\right )} \log (-2+x)\right )^2}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {e^{80 x \left (25+x^2\right )}}{(-2+x) \left (3 e^{40 x \left (25+x^2\right )}-e^{1250+300 x^2+2 x^4}+e^{40 x \left (25+x^2\right )} \log (-2+x)\right )^2} \, dx\right )+4 \int \frac {e^{2 (-5+x)^4+80 x \left (25+x^2\right )} x^4}{(-2+x) \left (3 e^{40 x \left (25+x^2\right )}-e^{1250+300 x^2+2 x^4}+e^{40 x \left (25+x^2\right )} \log (-2+x)\right )^2} \, dx-68 \int \frac {e^{2 (-5+x)^4+80 x \left (25+x^2\right )} x^3}{(-2+x) \left (3 e^{40 x \left (25+x^2\right )}-e^{1250+300 x^2+2 x^4}+e^{40 x \left (25+x^2\right )} \log (-2+x)\right )^2} \, dx+420 \int \frac {e^{2 (-5+x)^4+80 x \left (25+x^2\right )} x^2}{(-2+x) \left (3 e^{40 x \left (25+x^2\right )}-e^{1250+300 x^2+2 x^4}+e^{40 x \left (25+x^2\right )} \log (-2+x)\right )^2} \, dx+1000 \int \frac {e^{2 (-5+x)^4+80 x \left (25+x^2\right )}}{(-2+x) \left (3 e^{40 x \left (25+x^2\right )}-e^{1250+300 x^2+2 x^4}+e^{40 x \left (25+x^2\right )} \log (-2+x)\right )^2} \, dx-1100 \int \frac {e^{2 (-5+x)^4+80 x \left (25+x^2\right )} x}{(-2+x) \left (3 e^{40 x \left (25+x^2\right )}-e^{1250+300 x^2+2 x^4}+e^{40 x \left (25+x^2\right )} \log (-2+x)\right )^2} \, dx\\ &=-\left (\frac {1}{2} \int \frac {e^{80 x \left (25+x^2\right )}}{(-2+x) \left (3 e^{40 x \left (25+x^2\right )}-e^{1250+300 x^2+2 x^4}+e^{40 x \left (25+x^2\right )} \log (-2+x)\right )^2} \, dx\right )+4 \int \frac {e^{1250+1000 x+300 x^2+40 x^3+2 x^4} x^4}{(-2+x) \left (3 e^{40 x \left (25+x^2\right )}-e^{1250+300 x^2+2 x^4}+e^{40 x \left (25+x^2\right )} \log (-2+x)\right )^2} \, dx-68 \int \frac {e^{1250+1000 x+300 x^2+40 x^3+2 x^4} x^3}{(-2+x) \left (3 e^{40 x \left (25+x^2\right )}-e^{1250+300 x^2+2 x^4}+e^{40 x \left (25+x^2\right )} \log (-2+x)\right )^2} \, dx+420 \int \frac {e^{1250+1000 x+300 x^2+40 x^3+2 x^4} x^2}{(-2+x) \left (3 e^{40 x \left (25+x^2\right )}-e^{1250+300 x^2+2 x^4}+e^{40 x \left (25+x^2\right )} \log (-2+x)\right )^2} \, dx+1000 \int \frac {e^{1250+1000 x+300 x^2+40 x^3+2 x^4}}{(-2+x) \left (3 e^{40 x \left (25+x^2\right )}-e^{1250+300 x^2+2 x^4}+e^{40 x \left (25+x^2\right )} \log (-2+x)\right )^2} \, dx-1100 \int \frac {e^{1250+1000 x+300 x^2+40 x^3+2 x^4} x}{(-2+x) \left (3 e^{40 x \left (25+x^2\right )}-e^{1250+300 x^2+2 x^4}+e^{40 x \left (25+x^2\right )} \log (-2+x)\right )^2} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.60, size = 60, normalized size = 2.40 \begin {gather*} \frac {e^{40 x \left (25+x^2\right )}}{2 \left (3 e^{40 x \left (25+x^2\right )}-e^{2 \left (625+150 x^2+x^4\right )}+e^{40 x \left (25+x^2\right )} \log (-2+x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + E^(1250 - 1000*x + 300*x^2 - 40*x^3 + 2*x^4)*(2000 - 2200*x + 840*x^2 - 136*x^3 + 8*x^4))/(-36
+ E^(1250 - 1000*x + 300*x^2 - 40*x^3 + 2*x^4)*(24 - 12*x) + 18*x + E^(2500 - 2000*x + 600*x^2 - 80*x^3 + 4*x
^4)*(-4 + 2*x) + (-24 + E^(1250 - 1000*x + 300*x^2 - 40*x^3 + 2*x^4)*(8 - 4*x) + 12*x)*Log[-2 + x] + (-4 + 2*x
)*Log[-2 + x]^2),x]

[Out]

E^(40*x*(25 + x^2))/(2*(3*E^(40*x*(25 + x^2)) - E^(2*(625 + 150*x^2 + x^4)) + E^(40*x*(25 + x^2))*Log[-2 + x])
)

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fricas [A]  time = 0.58, size = 33, normalized size = 1.32 \begin {gather*} -\frac {1}{2 \, {\left (e^{\left (2 \, x^{4} - 40 \, x^{3} + 300 \, x^{2} - 1000 \, x + 1250\right )} - \log \left (x - 2\right ) - 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^4-136*x^3+840*x^2-2200*x+2000)*exp(x^4-20*x^3+150*x^2-500*x+625)^2-1)/((2*x-4)*log(x-2)^2+((-4
*x+8)*exp(x^4-20*x^3+150*x^2-500*x+625)^2+12*x-24)*log(x-2)+(2*x-4)*exp(x^4-20*x^3+150*x^2-500*x+625)^4+(-12*x
+24)*exp(x^4-20*x^3+150*x^2-500*x+625)^2+18*x-36),x, algorithm="fricas")

[Out]

-1/2/(e^(2*x^4 - 40*x^3 + 300*x^2 - 1000*x + 1250) - log(x - 2) - 3)

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giac [A]  time = 1.06, size = 33, normalized size = 1.32 \begin {gather*} -\frac {1}{2 \, {\left (e^{\left (2 \, x^{4} - 40 \, x^{3} + 300 \, x^{2} - 1000 \, x + 1250\right )} - \log \left (x - 2\right ) - 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^4-136*x^3+840*x^2-2200*x+2000)*exp(x^4-20*x^3+150*x^2-500*x+625)^2-1)/((2*x-4)*log(x-2)^2+((-4
*x+8)*exp(x^4-20*x^3+150*x^2-500*x+625)^2+12*x-24)*log(x-2)+(2*x-4)*exp(x^4-20*x^3+150*x^2-500*x+625)^4+(-12*x
+24)*exp(x^4-20*x^3+150*x^2-500*x+625)^2+18*x-36),x, algorithm="giac")

[Out]

-1/2/(e^(2*x^4 - 40*x^3 + 300*x^2 - 1000*x + 1250) - log(x - 2) - 3)

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maple [A]  time = 0.03, size = 21, normalized size = 0.84

method	result	size
risch	\(-\frac {1}{2 \left ({\mathrm e}^{2 \left (x -5\right )^{4}}-\ln \left (x -2\right )-3\right )}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((8*x^4-136*x^3+840*x^2-2200*x+2000)*exp(x^4-20*x^3+150*x^2-500*x+625)^2-1)/((2*x-4)*ln(x-2)^2+((-4*x+8)*e
xp(x^4-20*x^3+150*x^2-500*x+625)^2+12*x-24)*ln(x-2)+(2*x-4)*exp(x^4-20*x^3+150*x^2-500*x+625)^4+(-12*x+24)*exp
(x^4-20*x^3+150*x^2-500*x+625)^2+18*x-36),x,method=_RETURNVERBOSE)

[Out]

-1/2/(exp(2*(x-5)^4)-ln(x-2)-3)

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maxima [B]  time = 0.50, size = 53, normalized size = 2.12 \begin {gather*} \frac {e^{\left (40 \, x^{3} + 1000 \, x\right )}}{2 \, {\left ({\left (e^{\left (1000 \, x\right )} \log \left (x - 2\right ) + 3 \, e^{\left (1000 \, x\right )}\right )} e^{\left (40 \, x^{3}\right )} - e^{\left (2 \, x^{4} + 300 \, x^{2} + 1250\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^4-136*x^3+840*x^2-2200*x+2000)*exp(x^4-20*x^3+150*x^2-500*x+625)^2-1)/((2*x-4)*log(x-2)^2+((-4
*x+8)*exp(x^4-20*x^3+150*x^2-500*x+625)^2+12*x-24)*log(x-2)+(2*x-4)*exp(x^4-20*x^3+150*x^2-500*x+625)^4+(-12*x
+24)*exp(x^4-20*x^3+150*x^2-500*x+625)^2+18*x-36),x, algorithm="maxima")

[Out]

1/2*e^(40*x^3 + 1000*x)/((e^(1000*x)*log(x - 2) + 3*e^(1000*x))*e^(40*x^3) - e^(2*x^4 + 300*x^2 + 1250))

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mupad [B]  time = 3.07, size = 36, normalized size = 1.44 \begin {gather*} \frac {1}{2\,\left (\ln \left (x-2\right )-{\mathrm {e}}^{-1000\,x}\,{\mathrm {e}}^{1250}\,{\mathrm {e}}^{2\,x^4}\,{\mathrm {e}}^{-40\,x^3}\,{\mathrm {e}}^{300\,x^2}+3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(300*x^2 - 1000*x - 40*x^3 + 2*x^4 + 1250)*(840*x^2 - 2200*x - 136*x^3 + 8*x^4 + 2000) - 1)/(18*x + lo
g(x - 2)^2*(2*x - 4) - log(x - 2)*(exp(300*x^2 - 1000*x - 40*x^3 + 2*x^4 + 1250)*(4*x - 8) - 12*x + 24) - exp(
300*x^2 - 1000*x - 40*x^3 + 2*x^4 + 1250)*(12*x - 24) + exp(600*x^2 - 2000*x - 80*x^3 + 4*x^4 + 2500)*(2*x - 4
) - 36),x)

[Out]

1/(2*(log(x - 2) - exp(-1000*x)*exp(1250)*exp(2*x^4)*exp(-40*x^3)*exp(300*x^2) + 3))

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sympy [A]  time = 0.51, size = 34, normalized size = 1.36 \begin {gather*} - \frac {1}{2 e^{2 x^{4} - 40 x^{3} + 300 x^{2} - 1000 x + 1250} - 2 \log {\left (x - 2 \right )} - 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x**4-136*x**3+840*x**2-2200*x+2000)*exp(x**4-20*x**3+150*x**2-500*x+625)**2-1)/((2*x-4)*ln(x-2)*
*2+((-4*x+8)*exp(x**4-20*x**3+150*x**2-500*x+625)**2+12*x-24)*ln(x-2)+(2*x-4)*exp(x**4-20*x**3+150*x**2-500*x+
625)**4+(-12*x+24)*exp(x**4-20*x**3+150*x**2-500*x+625)**2+18*x-36),x)

[Out]

-1/(2*exp(2*x**4 - 40*x**3 + 300*x**2 - 1000*x + 1250) - 2*log(x - 2) - 6)

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