\(\int \frac {e^{\frac {1}{4} (-9+24 x-16 x^2-(-12+16 x) \log (4 x-x^2)-4 \log ^2(4 x-x^2))} (-36-6 x+90 x^2-24 x^3+(24+36 x-12 x^2) \log (4 x-x^2))}{-4 x+x^2} \, dx\) [7500]

Optimal result

Integrand size = 93, antiderivative size = 27 \[ \int \frac {e^{\frac {1}{4} \left (-9+24 x-16 x^2-(-12+16 x) \log \left (4 x-x^2\right )-4 \log ^2\left (4 x-x^2\right )\right )} \left (-36-6 x+90 x^2-24 x^3+\left (24+36 x-12 x^2\right ) \log \left (4 x-x^2\right )\right )}{-4 x+x^2} \, dx=1+3 e^{-\left (-\frac {3}{2}+2 x+\log \left (4 x-x^2\right )\right )^2} \]

[Out]

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

Rubi [F]

\[ \int \frac {e^{\frac {1}{4} \left (-9+24 x-16 x^2-(-12+16 x) \log \left (4 x-x^2\right )-4 \log ^2\left (4 x-x^2\right )\right )} \left (-36-6 x+90 x^2-24 x^3+\left (24+36 x-12 x^2\right ) \log \left (4 x-x^2\right )\right )}{-4 x+x^2} \, dx=\int \frac {\exp \left (\frac {1}{4} \left (-9+24 x-16 x^2-(-12+16 x) \log \left (4 x-x^2\right )-4 \log ^2\left (4 x-x^2\right )\right )\right ) \left (-36-6 x+90 x^2-24 x^3+\left (24+36 x-12 x^2\right ) \log \left (4 x-x^2\right )\right )}{-4 x+x^2} \, dx \]

[In]

Int[(E^((-9 + 24*x - 16*x^2 - (-12 + 16*x)*Log[4*x - x^2] - 4*Log[4*x - x^2]^2)/4)*(-36 - 6*x + 90*x^2 - 24*x^
3 + (24 + 36*x - 12*x^2)*Log[4*x - x^2]))/(-4*x + x^2),x]

[Out]

36*Defer[Int][E^((-9 + 24*x - 16*x^2 - 4*Log[-((-4 + x)*x)]^2)/4)*((4 - x)*x)^(2 - 4*x), x] + 6*Defer[Int][E^(
(-9 + 24*x - 16*x^2 - 4*Log[-((-4 + x)*x)]^2)/4)*x*((4 - x)*x)^(2 - 4*x), x] - 90*Defer[Int][E^((-9 + 24*x - 1
6*x^2 - 4*Log[-((-4 + x)*x)]^2)/4)*x^2*((4 - x)*x)^(2 - 4*x), x] + 24*Defer[Int][E^((-9 + 24*x - 16*x^2 - 4*Lo
g[-((-4 + x)*x)]^2)/4)*x^3*((4 - x)*x)^(2 - 4*x), x] - 24*Defer[Int][E^((-9 + 24*x - 16*x^2 - 4*Log[-((-4 + x)
*x)]^2)/4)*((4 - x)*x)^(2 - 4*x)*Log[(4 - x)*x], x] - 36*Defer[Int][E^((-9 + 24*x - 16*x^2 - 4*Log[-((-4 + x)*
x)]^2)/4)*x*((4 - x)*x)^(2 - 4*x)*Log[(4 - x)*x], x] + 12*Defer[Int][E^((-9 + 24*x - 16*x^2 - 4*Log[-((-4 + x)
*x)]^2)/4)*x^2*((4 - x)*x)^(2 - 4*x)*Log[(4 - x)*x], x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\exp \left (\frac {1}{4} \left (-9+24 x-16 x^2-(-12+16 x) \log \left (4 x-x^2\right )-4 \log ^2\left (4 x-x^2\right )\right )\right ) \left (-36-6 x+90 x^2-24 x^3+\left (24+36 x-12 x^2\right ) \log \left (4 x-x^2\right )\right )}{(-4+x) x} \, dx \\ & = \int 6 \exp \left (\frac {1}{4} \left (-9+24 x-16 x^2-4 \log ^2(-((-4+x) x))\right )\right ) ((4-x) x)^{2-4 x} \left (2+3 x-x^2\right ) (3-4 x-2 \log (-((-4+x) x))) \, dx \\ & = 6 \int \exp \left (\frac {1}{4} \left (-9+24 x-16 x^2-4 \log ^2(-((-4+x) x))\right )\right ) ((4-x) x)^{2-4 x} \left (2+3 x-x^2\right ) (3-4 x-2 \log (-((-4+x) x))) \, dx \\ & = 6 \int \left (6 \exp \left (\frac {1}{4} \left (-9+24 x-16 x^2-4 \log ^2(-((-4+x) x))\right )\right ) ((4-x) x)^{2-4 x}+\exp \left (\frac {1}{4} \left (-9+24 x-16 x^2-4 \log ^2(-((-4+x) x))\right )\right ) x ((4-x) x)^{2-4 x}-15 \exp \left (\frac {1}{4} \left (-9+24 x-16 x^2-4 \log ^2(-((-4+x) x))\right )\right ) x^2 ((4-x) x)^{2-4 x}+4 \exp \left (\frac {1}{4} \left (-9+24 x-16 x^2-4 \log ^2(-((-4+x) x))\right )\right ) x^3 ((4-x) x)^{2-4 x}+2 \exp \left (\frac {1}{4} \left (-9+24 x-16 x^2-4 \log ^2(-((-4+x) x))\right )\right ) ((4-x) x)^{2-4 x} \left (-2-3 x+x^2\right ) \log ((4-x) x)\right ) \, dx \\ & = 6 \int \exp \left (\frac {1}{4} \left (-9+24 x-16 x^2-4 \log ^2(-((-4+x) x))\right )\right ) x ((4-x) x)^{2-4 x} \, dx+12 \int \exp \left (\frac {1}{4} \left (-9+24 x-16 x^2-4 \log ^2(-((-4+x) x))\right )\right ) ((4-x) x)^{2-4 x} \left (-2-3 x+x^2\right ) \log ((4-x) x) \, dx+24 \int \exp \left (\frac {1}{4} \left (-9+24 x-16 x^2-4 \log ^2(-((-4+x) x))\right )\right ) x^3 ((4-x) x)^{2-4 x} \, dx+36 \int \exp \left (\frac {1}{4} \left (-9+24 x-16 x^2-4 \log ^2(-((-4+x) x))\right )\right ) ((4-x) x)^{2-4 x} \, dx-90 \int \exp \left (\frac {1}{4} \left (-9+24 x-16 x^2-4 \log ^2(-((-4+x) x))\right )\right ) x^2 ((4-x) x)^{2-4 x} \, dx \\ & = 6 \int \exp \left (\frac {1}{4} \left (-9+24 x-16 x^2-4 \log ^2(-((-4+x) x))\right )\right ) x ((4-x) x)^{2-4 x} \, dx+12 \int \left (-2 \exp \left (\frac {1}{4} \left (-9+24 x-16 x^2-4 \log ^2(-((-4+x) x))\right )\right ) ((4-x) x)^{2-4 x} \log ((4-x) x)-3 \exp \left (\frac {1}{4} \left (-9+24 x-16 x^2-4 \log ^2(-((-4+x) x))\right )\right ) x ((4-x) x)^{2-4 x} \log ((4-x) x)+\exp \left (\frac {1}{4} \left (-9+24 x-16 x^2-4 \log ^2(-((-4+x) x))\right )\right ) x^2 ((4-x) x)^{2-4 x} \log ((4-x) x)\right ) \, dx+24 \int \exp \left (\frac {1}{4} \left (-9+24 x-16 x^2-4 \log ^2(-((-4+x) x))\right )\right ) x^3 ((4-x) x)^{2-4 x} \, dx+36 \int \exp \left (\frac {1}{4} \left (-9+24 x-16 x^2-4 \log ^2(-((-4+x) x))\right )\right ) ((4-x) x)^{2-4 x} \, dx-90 \int \exp \left (\frac {1}{4} \left (-9+24 x-16 x^2-4 \log ^2(-((-4+x) x))\right )\right ) x^2 ((4-x) x)^{2-4 x} \, dx \\ & = 6 \int \exp \left (\frac {1}{4} \left (-9+24 x-16 x^2-4 \log ^2(-((-4+x) x))\right )\right ) x ((4-x) x)^{2-4 x} \, dx+12 \int \exp \left (\frac {1}{4} \left (-9+24 x-16 x^2-4 \log ^2(-((-4+x) x))\right )\right ) x^2 ((4-x) x)^{2-4 x} \log ((4-x) x) \, dx+24 \int \exp \left (\frac {1}{4} \left (-9+24 x-16 x^2-4 \log ^2(-((-4+x) x))\right )\right ) x^3 ((4-x) x)^{2-4 x} \, dx-24 \int \exp \left (\frac {1}{4} \left (-9+24 x-16 x^2-4 \log ^2(-((-4+x) x))\right )\right ) ((4-x) x)^{2-4 x} \log ((4-x) x) \, dx+36 \int \exp \left (\frac {1}{4} \left (-9+24 x-16 x^2-4 \log ^2(-((-4+x) x))\right )\right ) ((4-x) x)^{2-4 x} \, dx-36 \int \exp \left (\frac {1}{4} \left (-9+24 x-16 x^2-4 \log ^2(-((-4+x) x))\right )\right ) x ((4-x) x)^{2-4 x} \log ((4-x) x) \, dx-90 \int \exp \left (\frac {1}{4} \left (-9+24 x-16 x^2-4 \log ^2(-((-4+x) x))\right )\right ) x^2 ((4-x) x)^{2-4 x} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 1.96 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.67 \[ \int \frac {e^{\frac {1}{4} \left (-9+24 x-16 x^2-(-12+16 x) \log \left (4 x-x^2\right )-4 \log ^2\left (4 x-x^2\right )\right )} \left (-36-6 x+90 x^2-24 x^3+\left (24+36 x-12 x^2\right ) \log \left (4 x-x^2\right )\right )}{-4 x+x^2} \, dx=-3 e^{-\frac {9}{4}+6 x-4 x^2-\log ^2(-((-4+x) x))} (-4+x)^3 x^3 (-((-4+x) x))^{-4 x} \]

[In]

Integrate[(E^((-9 + 24*x - 16*x^2 - (-12 + 16*x)*Log[4*x - x^2] - 4*Log[4*x - x^2]^2)/4)*(-36 - 6*x + 90*x^2 -
24*x^3 + (24 + 36*x - 12*x^2)*Log[4*x - x^2]))/(-4*x + x^2),x]

[Out]

(-3*E^(-9/4 + 6*x - 4*x^2 - Log[-((-4 + x)*x)]^2)*(-4 + x)^3*x^3)/(-((-4 + x)*x))^(4*x)

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.67

method	result	size
norman	\(3 \,{\mathrm e}^{-\ln \left (-x^{2}+4 x \right )^{2}+\left (3-4 x \right ) \ln \left (-x^{2}+4 x \right )-4 x^{2}+6 x -\frac {9}{4}}\)	\(45\)
risch	\(3 \left (-x^{2}+4 x \right )^{3-4 x} {\mathrm e}^{-\ln \left (-x^{2}+4 x \right )^{2}-\frac {9}{4}-4 x^{2}+6 x}\)	\(45\)
parallelrisch	\(3 \,{\mathrm e}^{-\ln \left (-x^{2}+4 x \right )^{2}+\left (3-4 x \right ) \ln \left (-x^{2}+4 x \right )-4 x^{2}+6 x -\frac {9}{4}}\)	\(45\)

[In]

int(((-12*x^2+36*x+24)*ln(-x^2+4*x)-24*x^3+90*x^2-6*x-36)/(x^2-4*x)/exp(ln(-x^2+4*x)^2+1/4*(16*x-12)*ln(-x^2+4
*x)+4*x^2-6*x+9/4),x,method=_RETURNVERBOSE)

[Out]

3/exp(ln(-x^2+4*x)^2+1/4*(16*x-12)*ln(-x^2+4*x)+4*x^2-6*x+9/4)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.63 \[ \int \frac {e^{\frac {1}{4} \left (-9+24 x-16 x^2-(-12+16 x) \log \left (4 x-x^2\right )-4 \log ^2\left (4 x-x^2\right )\right )} \left (-36-6 x+90 x^2-24 x^3+\left (24+36 x-12 x^2\right ) \log \left (4 x-x^2\right )\right )}{-4 x+x^2} \, dx=3 \, e^{\left (-4 \, x^{2} - {\left (4 \, x - 3\right )} \log \left (-x^{2} + 4 \, x\right ) - \log \left (-x^{2} + 4 \, x\right )^{2} + 6 \, x - \frac {9}{4}\right )} \]

[In]

integrate(((-12*x^2+36*x+24)*log(-x^2+4*x)-24*x^3+90*x^2-6*x-36)/(x^2-4*x)/exp(log(-x^2+4*x)^2+1/4*(16*x-12)*l
og(-x^2+4*x)+4*x^2-6*x+9/4),x, algorithm="fricas")

[Out]

3*e^(-4*x^2 - (4*x - 3)*log(-x^2 + 4*x) - log(-x^2 + 4*x)^2 + 6*x - 9/4)

Sympy [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.37 \[ \int \frac {e^{\frac {1}{4} \left (-9+24 x-16 x^2-(-12+16 x) \log \left (4 x-x^2\right )-4 \log ^2\left (4 x-x^2\right )\right )} \left (-36-6 x+90 x^2-24 x^3+\left (24+36 x-12 x^2\right ) \log \left (4 x-x^2\right )\right )}{-4 x+x^2} \, dx=3 e^{- 4 x^{2} + 6 x - \left (4 x - 3\right ) \log {\left (- x^{2} + 4 x \right )} - \log {\left (- x^{2} + 4 x \right )}^{2} - \frac {9}{4}} \]

[In]

integrate(((-12*x**2+36*x+24)*ln(-x**2+4*x)-24*x**3+90*x**2-6*x-36)/(x**2-4*x)/exp(ln(-x**2+4*x)**2+1/4*(16*x-
12)*ln(-x**2+4*x)+4*x**2-6*x+9/4),x)

[Out]

3*exp(-4*x**2 + 6*x - (4*x - 3)*log(-x**2 + 4*x) - log(-x**2 + 4*x)**2 - 9/4)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (26) = 52\).

Time = 0.34 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.67 \[ \int \frac {e^{\frac {1}{4} \left (-9+24 x-16 x^2-(-12+16 x) \log \left (4 x-x^2\right )-4 \log ^2\left (4 x-x^2\right )\right )} \left (-36-6 x+90 x^2-24 x^3+\left (24+36 x-12 x^2\right ) \log \left (4 x-x^2\right )\right )}{-4 x+x^2} \, dx=-3 \, {\left (x^{6} - 12 \, x^{5} + 48 \, x^{4} - 64 \, x^{3}\right )} e^{\left (-4 \, x^{2} - 4 \, x \log \left (x\right ) - \log \left (x\right )^{2} - 4 \, x \log \left (-x + 4\right ) - 2 \, \log \left (x\right ) \log \left (-x + 4\right ) - \log \left (-x + 4\right )^{2} + 6 \, x - \frac {9}{4}\right )} \]

[In]

integrate(((-12*x^2+36*x+24)*log(-x^2+4*x)-24*x^3+90*x^2-6*x-36)/(x^2-4*x)/exp(log(-x^2+4*x)^2+1/4*(16*x-12)*l
og(-x^2+4*x)+4*x^2-6*x+9/4),x, algorithm="maxima")

[Out]

-3*(x^6 - 12*x^5 + 48*x^4 - 64*x^3)*e^(-4*x^2 - 4*x*log(x) - log(x)^2 - 4*x*log(-x + 4) - 2*log(x)*log(-x + 4)
- log(-x + 4)^2 + 6*x - 9/4)

Giac [A] (verification not implemented)

none

Time = 0.69 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.93 \[ \int \frac {e^{\frac {1}{4} \left (-9+24 x-16 x^2-(-12+16 x) \log \left (4 x-x^2\right )-4 \log ^2\left (4 x-x^2\right )\right )} \left (-36-6 x+90 x^2-24 x^3+\left (24+36 x-12 x^2\right ) \log \left (4 x-x^2\right )\right )}{-4 x+x^2} \, dx=3 \, e^{\left (-4 \, x^{2} - 4 \, x \log \left (-x^{2} + 4 \, x\right ) - \log \left (-x^{2} + 4 \, x\right )^{2} + 6 \, x + 3 \, \log \left (-x^{2} + 4 \, x\right ) - \frac {9}{4}\right )} \]

[In]

integrate(((-12*x^2+36*x+24)*log(-x^2+4*x)-24*x^3+90*x^2-6*x-36)/(x^2-4*x)/exp(log(-x^2+4*x)^2+1/4*(16*x-12)*l
og(-x^2+4*x)+4*x^2-6*x+9/4),x, algorithm="giac")

[Out]

3*e^(-4*x^2 - 4*x*log(-x^2 + 4*x) - log(-x^2 + 4*x)^2 + 6*x + 3*log(-x^2 + 4*x) - 9/4)

Mupad [B] (verification not implemented)

Time = 13.59 (sec) , antiderivative size = 181, normalized size of antiderivative = 6.70 \[ \int \frac {e^{\frac {1}{4} \left (-9+24 x-16 x^2-(-12+16 x) \log \left (4 x-x^2\right )-4 \log ^2\left (4 x-x^2\right )\right )} \left (-36-6 x+90 x^2-24 x^3+\left (24+36 x-12 x^2\right ) \log \left (4 x-x^2\right )\right )}{-4 x+x^2} \, dx=\frac {192\,x^3\,{\mathrm {e}}^{-4\,x^2+6\,x-{\ln \left (4\,x-x^2\right )}^2-\frac {9}{4}}}{{\left (4\,x-x^2\right )}^{4\,x}}-\frac {144\,x^4\,{\mathrm {e}}^{-4\,x^2+6\,x-{\ln \left (4\,x-x^2\right )}^2-\frac {9}{4}}}{{\left (4\,x-x^2\right )}^{4\,x}}+\frac {36\,x^5\,{\mathrm {e}}^{-4\,x^2+6\,x-{\ln \left (4\,x-x^2\right )}^2-\frac {9}{4}}}{{\left (4\,x-x^2\right )}^{4\,x}}-\frac {3\,x^6\,{\mathrm {e}}^{-4\,x^2+6\,x-{\ln \left (4\,x-x^2\right )}^2-\frac {9}{4}}}{{\left (4\,x-x^2\right )}^{4\,x}} \]

[In]

int((exp(6*x - log(4*x - x^2)^2 - (log(4*x - x^2)*(16*x - 12))/4 - 4*x^2 - 9/4)*(6*x - log(4*x - x^2)*(36*x -
12*x^2 + 24) - 90*x^2 + 24*x^3 + 36))/(4*x - x^2),x)

[Out]

(192*x^3*exp(6*x - log(4*x - x^2)^2 - 4*x^2 - 9/4))/(4*x - x^2)^(4*x) - (144*x^4*exp(6*x - log(4*x - x^2)^2 -
4*x^2 - 9/4))/(4*x - x^2)^(4*x) + (36*x^5*exp(6*x - log(4*x - x^2)^2 - 4*x^2 - 9/4))/(4*x - x^2)^(4*x) - (3*x^
6*exp(6*x - log(4*x - x^2)^2 - 4*x^2 - 9/4))/(4*x - x^2)^(4*x)