3.21.98 \(\int \frac {2000-2000 x+(-x+x^2)^{\frac {1}{400} (-400 x^2+x^3)} (400 x-801 x^2+2 x^3+(800 x-803 x^2+3 x^3) \log (-x+x^2))}{-2000+2000 x} \, dx\)

Optimal. Leaf size=29 \[ -x+\frac {1}{5} \left (-x+x^2\right )^{x \left (-x+\frac {x^2}{400}\right )} \]

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Rubi [F]  time = 1.69, 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 {2000-2000 x+\left (-x+x^2\right )^{\frac {1}{400} \left (-400 x^2+x^3\right )} \left (400 x-801 x^2+2 x^3+\left (800 x-803 x^2+3 x^3\right ) \log \left (-x+x^2\right )\right )}{-2000+2000 x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(2000 - 2000*x + (-x + x^2)^((-400*x^2 + x^3)/400)*(400*x - 801*x^2 + 2*x^3 + (800*x - 803*x^2 + 3*x^3)*Lo
g[-x + x^2]))/(-2000 + 2000*x),x]

[Out]

-x - (399*Defer[Int][((-1 + x)*x)^((-1 + x/400)*x^2), x])/2000 - (399*Defer[Int][((-1 + x)*x)^((-1 + x/400)*x^
2)/(-1 + x), x])/2000 - (799*Defer[Int][x*((-1 + x)*x)^((-1 + x/400)*x^2), x])/2000 - (2*Log[-((1 - x)*x)]*Def
er[Int][x*((-1 + x)*x)^((-1 + x/400)*x^2), x])/5 + Defer[Int][x^2*((-1 + x)*x)^((-1 + x/400)*x^2), x]/1000 + (
3*Log[-((1 - x)*x)]*Defer[Int][x^2*((-1 + x)*x)^((-1 + x/400)*x^2), x])/2000 + (2*Defer[Int][Defer[Int][x*((-1
+ x)*x)^((-1 + x/400)*x^2), x]/(-1 + x), x])/5 + (2*Defer[Int][Defer[Int][x*((-1 + x)*x)^((-1 + x/400)*x^2),
x]/x, x])/5 - (3*Defer[Int][Defer[Int][x^2*((-1 + x)*x)^((-1 + x/400)*x^2), x]/(-1 + x), x])/2000 - (3*Defer[I
nt][Defer[Int][x^2*((-1 + x)*x)^((-1 + x/400)*x^2), x]/x, x])/2000

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-1+\frac {x ((-1+x) x)^{\left (-1+\frac {x}{400}\right ) x^2} \left (-400+801 x-2 x^2-800 \log ((-1+x) x)+803 x \log ((-1+x) x)-3 x^2 \log ((-1+x) x)\right )}{2000 (1-x)}\right ) \, dx\\ &=-x+\frac {\int \frac {x ((-1+x) x)^{\left (-1+\frac {x}{400}\right ) x^2} \left (-400+801 x-2 x^2-800 \log ((-1+x) x)+803 x \log ((-1+x) x)-3 x^2 \log ((-1+x) x)\right )}{1-x} \, dx}{2000}\\ &=-x+\frac {\int \left (\frac {x ((-1+x) x)^{\left (-1+\frac {x}{400}\right ) x^2} \left (400-801 x+2 x^2\right )}{-1+x}+x ((-1+x) x)^{\left (-1+\frac {x}{400}\right ) x^2} (-800+3 x) \log ((-1+x) x)\right ) \, dx}{2000}\\ &=-x+\frac {\int \frac {x ((-1+x) x)^{\left (-1+\frac {x}{400}\right ) x^2} \left (400-801 x+2 x^2\right )}{-1+x} \, dx}{2000}+\frac {\int x ((-1+x) x)^{\left (-1+\frac {x}{400}\right ) x^2} (-800+3 x) \log ((-1+x) x) \, dx}{2000}\\ &=-x+\frac {\int \left (-399 ((-1+x) x)^{\left (-1+\frac {x}{400}\right ) x^2}-\frac {399 ((-1+x) x)^{\left (-1+\frac {x}{400}\right ) x^2}}{-1+x}-799 x ((-1+x) x)^{\left (-1+\frac {x}{400}\right ) x^2}+2 x^2 ((-1+x) x)^{\left (-1+\frac {x}{400}\right ) x^2}\right ) \, dx}{2000}-\frac {\int \frac {(-1+2 x) \left (-800 \int x ((-1+x) x)^{\left (-1+\frac {x}{400}\right ) x^2} \, dx+3 \int x^2 ((-1+x) x)^{\left (-1+\frac {x}{400}\right ) x^2} \, dx\right )}{(-1+x) x} \, dx}{2000}+\frac {(3 \log ((-1+x) x)) \int x^2 ((-1+x) x)^{\left (-1+\frac {x}{400}\right ) x^2} \, dx}{2000}-\frac {1}{5} (2 \log ((-1+x) x)) \int x ((-1+x) x)^{\left (-1+\frac {x}{400}\right ) x^2} \, dx\\ &=-x-\frac {\int \left (-\frac {800 (-1+2 x) \int x ((-1+x) x)^{\left (-1+\frac {x}{400}\right ) x^2} \, dx}{(-1+x) x}+\frac {3 (-1+2 x) \int x^2 ((-1+x) x)^{\left (-1+\frac {x}{400}\right ) x^2} \, dx}{(-1+x) x}\right ) \, dx}{2000}+\frac {\int x^2 ((-1+x) x)^{\left (-1+\frac {x}{400}\right ) x^2} \, dx}{1000}-\frac {399 \int ((-1+x) x)^{\left (-1+\frac {x}{400}\right ) x^2} \, dx}{2000}-\frac {399 \int \frac {((-1+x) x)^{\left (-1+\frac {x}{400}\right ) x^2}}{-1+x} \, dx}{2000}-\frac {799 \int x ((-1+x) x)^{\left (-1+\frac {x}{400}\right ) x^2} \, dx}{2000}+\frac {(3 \log ((-1+x) x)) \int x^2 ((-1+x) x)^{\left (-1+\frac {x}{400}\right ) x^2} \, dx}{2000}-\frac {1}{5} (2 \log ((-1+x) x)) \int x ((-1+x) x)^{\left (-1+\frac {x}{400}\right ) x^2} \, dx\\ &=-x+\frac {\int x^2 ((-1+x) x)^{\left (-1+\frac {x}{400}\right ) x^2} \, dx}{1000}-\frac {3 \int \frac {(-1+2 x) \int x^2 ((-1+x) x)^{\left (-1+\frac {x}{400}\right ) x^2} \, dx}{(-1+x) x} \, dx}{2000}-\frac {399 \int ((-1+x) x)^{\left (-1+\frac {x}{400}\right ) x^2} \, dx}{2000}-\frac {399 \int \frac {((-1+x) x)^{\left (-1+\frac {x}{400}\right ) x^2}}{-1+x} \, dx}{2000}-\frac {799 \int x ((-1+x) x)^{\left (-1+\frac {x}{400}\right ) x^2} \, dx}{2000}+\frac {2}{5} \int \frac {(-1+2 x) \int x ((-1+x) x)^{\left (-1+\frac {x}{400}\right ) x^2} \, dx}{(-1+x) x} \, dx+\frac {(3 \log ((-1+x) x)) \int x^2 ((-1+x) x)^{\left (-1+\frac {x}{400}\right ) x^2} \, dx}{2000}-\frac {1}{5} (2 \log ((-1+x) x)) \int x ((-1+x) x)^{\left (-1+\frac {x}{400}\right ) x^2} \, dx\\ &=-x+\frac {\int x^2 ((-1+x) x)^{\left (-1+\frac {x}{400}\right ) x^2} \, dx}{1000}-\frac {3 \int \left (\frac {\int x^2 ((-1+x) x)^{\left (-1+\frac {x}{400}\right ) x^2} \, dx}{-1+x}+\frac {\int x^2 ((-1+x) x)^{\left (-1+\frac {x}{400}\right ) x^2} \, dx}{x}\right ) \, dx}{2000}-\frac {399 \int ((-1+x) x)^{\left (-1+\frac {x}{400}\right ) x^2} \, dx}{2000}-\frac {399 \int \frac {((-1+x) x)^{\left (-1+\frac {x}{400}\right ) x^2}}{-1+x} \, dx}{2000}-\frac {799 \int x ((-1+x) x)^{\left (-1+\frac {x}{400}\right ) x^2} \, dx}{2000}+\frac {2}{5} \int \left (\frac {\int x ((-1+x) x)^{\left (-1+\frac {x}{400}\right ) x^2} \, dx}{-1+x}+\frac {\int x ((-1+x) x)^{\left (-1+\frac {x}{400}\right ) x^2} \, dx}{x}\right ) \, dx+\frac {(3 \log ((-1+x) x)) \int x^2 ((-1+x) x)^{\left (-1+\frac {x}{400}\right ) x^2} \, dx}{2000}-\frac {1}{5} (2 \log ((-1+x) x)) \int x ((-1+x) x)^{\left (-1+\frac {x}{400}\right ) x^2} \, dx\\ &=-x+\frac {\int x^2 ((-1+x) x)^{\left (-1+\frac {x}{400}\right ) x^2} \, dx}{1000}-\frac {3 \int \frac {\int x^2 ((-1+x) x)^{\left (-1+\frac {x}{400}\right ) x^2} \, dx}{-1+x} \, dx}{2000}-\frac {3 \int \frac {\int x^2 ((-1+x) x)^{\left (-1+\frac {x}{400}\right ) x^2} \, dx}{x} \, dx}{2000}-\frac {399 \int ((-1+x) x)^{\left (-1+\frac {x}{400}\right ) x^2} \, dx}{2000}-\frac {399 \int \frac {((-1+x) x)^{\left (-1+\frac {x}{400}\right ) x^2}}{-1+x} \, dx}{2000}-\frac {799 \int x ((-1+x) x)^{\left (-1+\frac {x}{400}\right ) x^2} \, dx}{2000}+\frac {2}{5} \int \frac {\int x ((-1+x) x)^{\left (-1+\frac {x}{400}\right ) x^2} \, dx}{-1+x} \, dx+\frac {2}{5} \int \frac {\int x ((-1+x) x)^{\left (-1+\frac {x}{400}\right ) x^2} \, dx}{x} \, dx+\frac {(3 \log ((-1+x) x)) \int x^2 ((-1+x) x)^{\left (-1+\frac {x}{400}\right ) x^2} \, dx}{2000}-\frac {1}{5} (2 \log ((-1+x) x)) \int x ((-1+x) x)^{\left (-1+\frac {x}{400}\right ) x^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.30, size = 26, normalized size = 0.90 \begin {gather*} \frac {-2000 x+400 ((-1+x) x)^{\frac {1}{400} (-400+x) x^2}}{2000} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2000 - 2000*x + (-x + x^2)^((-400*x^2 + x^3)/400)*(400*x - 801*x^2 + 2*x^3 + (800*x - 803*x^2 + 3*x
^3)*Log[-x + x^2]))/(-2000 + 2000*x),x]

[Out]

(-2000*x + 400*((-1 + x)*x)^(((-400 + x)*x^2)/400))/2000

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fricas [A]  time = 0.58, size = 25, normalized size = 0.86 \begin {gather*} \frac {1}{5} \, {\left (x^{2} - x\right )}^{\frac {1}{400} \, x^{3} - x^{2}} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*x^3-803*x^2+800*x)*log(x^2-x)+2*x^3-801*x^2+400*x)*exp(1/400*(x^3-400*x^2)*log(x^2-x))-2000*x+2
000)/(2000*x-2000),x, algorithm="fricas")

[Out]

1/5*(x^2 - x)^(1/400*x^3 - x^2) - x

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, x^{3} - 801 \, x^{2} + {\left (3 \, x^{3} - 803 \, x^{2} + 800 \, x\right )} \log \left (x^{2} - x\right ) + 400 \, x\right )} {\left (x^{2} - x\right )}^{\frac {1}{400} \, x^{3} - x^{2}} - 2000 \, x + 2000}{2000 \, {\left (x - 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*x^3-803*x^2+800*x)*log(x^2-x)+2*x^3-801*x^2+400*x)*exp(1/400*(x^3-400*x^2)*log(x^2-x))-2000*x+2
000)/(2000*x-2000),x, algorithm="giac")

[Out]

integrate(1/2000*((2*x^3 - 801*x^2 + (3*x^3 - 803*x^2 + 800*x)*log(x^2 - x) + 400*x)*(x^2 - x)^(1/400*x^3 - x^
2) - 2000*x + 2000)/(x - 1), x)

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maple [A]  time = 0.34, size = 23, normalized size = 0.79

method	result	size
risch	\(-x +\frac {\left (x^{2}-x \right )^{\frac {x^{2} \left (x -400\right )}{400}}}{5}\)	\(23\)
default	\(-x +\frac {{\mathrm e}^{\frac {\left (x^{3}-400 x^{2}\right ) \ln \left (x^{2}-x \right )}{400}}}{5}\)	\(27\)
norman	\(-x +\frac {{\mathrm e}^{\frac {\left (x^{3}-400 x^{2}\right ) \ln \left (x^{2}-x \right )}{400}}}{5}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((3*x^3-803*x^2+800*x)*ln(x^2-x)+2*x^3-801*x^2+400*x)*exp(1/400*(x^3-400*x^2)*ln(x^2-x))-2000*x+2000)/(20
00*x-2000),x,method=_RETURNVERBOSE)

[Out]

-x+1/5*(x^2-x)^(1/400*x^2*(x-400))

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maxima [A]  time = 0.42, size = 40, normalized size = 1.38 \begin {gather*} -x + \frac {1}{5} \, e^{\left (\frac {1}{400} \, x^{3} \log \left (x - 1\right ) + \frac {1}{400} \, x^{3} \log \relax (x) - x^{2} \log \left (x - 1\right ) - x^{2} \log \relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*x^3-803*x^2+800*x)*log(x^2-x)+2*x^3-801*x^2+400*x)*exp(1/400*(x^3-400*x^2)*log(x^2-x))-2000*x+2
000)/(2000*x-2000),x, algorithm="maxima")

[Out]

-x + 1/5*e^(1/400*x^3*log(x - 1) + 1/400*x^3*log(x) - x^2*log(x - 1) - x^2*log(x))

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mupad [B]  time = 1.50, size = 33, normalized size = 1.14 \begin {gather*} \frac {{\mathrm {e}}^{\frac {x^3\,\ln \left (x^2-x\right )}{400}}}{5\,{\left (x^2-x\right )}^{x^2}}-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-(log(x^2 - x)*(400*x^2 - x^3))/400)*(400*x + log(x^2 - x)*(800*x - 803*x^2 + 3*x^3) - 801*x^2 + 2*x^
3) - 2000*x + 2000)/(2000*x - 2000),x)

[Out]

exp((x^3*log(x^2 - x))/400)/(5*(x^2 - x)^(x^2)) - x

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sympy [A]  time = 0.52, size = 19, normalized size = 0.66 \begin {gather*} - x + \frac {e^{\left (\frac {x^{3}}{400} - x^{2}\right ) \log {\left (x^{2} - x \right )}}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*x**3-803*x**2+800*x)*ln(x**2-x)+2*x**3-801*x**2+400*x)*exp(1/400*(x**3-400*x**2)*ln(x**2-x))-20
00*x+2000)/(2000*x-2000),x)

[Out]

-x + exp((x**3/400 - x**2)*log(x**2 - x))/5

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