\(\int \frac {e^{-1-3 x} \log ^2(-x+x^2) ((-9+18 x) \log (x)+(-3+3 x+(9-9 x^2) \log (x)) \log (-x+x^2))}{-x^4+x^5} \, dx\) [1311]

Optimal result

Integrand size = 62, antiderivative size = 26 \[ \int \frac {e^{-1-3 x} \log ^2\left (-x+x^2\right ) \left ((-9+18 x) \log (x)+\left (-3+3 x+\left (9-9 x^2\right ) \log (x)\right ) \log \left (-x+x^2\right )\right )}{-x^4+x^5} \, dx=\frac {3 e^{-1+3 \left (-x+\log \left (\log \left (-x+x^2\right )\right )\right )} \log (x)}{x^3} \]

[Out]

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

Rubi [F]

\[ \int \frac {e^{-1-3 x} \log ^2\left (-x+x^2\right ) \left ((-9+18 x) \log (x)+\left (-3+3 x+\left (9-9 x^2\right ) \log (x)\right ) \log \left (-x+x^2\right )\right )}{-x^4+x^5} \, dx=\int \frac {e^{-1-3 x} \log ^2\left (-x+x^2\right ) \left ((-9+18 x) \log (x)+\left (-3+3 x+\left (9-9 x^2\right ) \log (x)\right ) \log \left (-x+x^2\right )\right )}{-x^4+x^5} \, dx \]

[In]

Int[(E^(-1 - 3*x)*Log[-x + x^2]^2*((-9 + 18*x)*Log[x] + (-3 + 3*x + (9 - 9*x^2)*Log[x])*Log[-x + x^2]))/(-x^4
+ x^5),x]

[Out]

9*Defer[Int][(E^(-1 - 3*x)*Log[x]*Log[(-1 + x)*x]^2)/(-1 + x), x] + 9*Defer[Int][(E^(-1 - 3*x)*Log[x]*Log[(-1
+ x)*x]^2)/x^4, x] - 9*Defer[Int][(E^(-1 - 3*x)*Log[x]*Log[(-1 + x)*x]^2)/x^3, x] - 9*Defer[Int][(E^(-1 - 3*x)
*Log[x]*Log[(-1 + x)*x]^2)/x^2, x] - 9*Defer[Int][(E^(-1 - 3*x)*Log[x]*Log[(-1 + x)*x]^2)/x, x] + 3*Defer[Int]
[(E^(-1 - 3*x)*Log[(-1 + x)*x]^3)/x^4, x] - 9*Defer[Int][(E^(-1 - 3*x)*Log[x]*Log[(-1 + x)*x]^3)/x^4, x] - 9*D
efer[Int][(E^(-1 - 3*x)*Log[x]*Log[(-1 + x)*x]^3)/x^3, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{-1-3 x} \log ^2\left (-x+x^2\right ) \left ((-9+18 x) \log (x)+\left (-3+3 x+\left (9-9 x^2\right ) \log (x)\right ) \log \left (-x+x^2\right )\right )}{(-1+x) x^4} \, dx \\ & = \int \frac {e^{-1-3 x} \log ^2((-1+x) x) \left (-((-9+18 x) \log (x))-\left (-3+3 x+\left (9-9 x^2\right ) \log (x)\right ) \log \left (-x+x^2\right )\right )}{(1-x) x^4} \, dx \\ & = \int \left (\frac {9 e^{-1-3 x} (-1+2 x) \log (x) \log ^2((-1+x) x)}{(-1+x) x^4}-\frac {3 e^{-1-3 x} (-1+3 \log (x)+3 x \log (x)) \log ^3((-1+x) x)}{x^4}\right ) \, dx \\ & = -\left (3 \int \frac {e^{-1-3 x} (-1+3 \log (x)+3 x \log (x)) \log ^3((-1+x) x)}{x^4} \, dx\right )+9 \int \frac {e^{-1-3 x} (-1+2 x) \log (x) \log ^2((-1+x) x)}{(-1+x) x^4} \, dx \\ & = -\left (3 \int \left (-\frac {e^{-1-3 x} \log ^3((-1+x) x)}{x^4}+\frac {3 e^{-1-3 x} \log (x) \log ^3((-1+x) x)}{x^4}+\frac {3 e^{-1-3 x} \log (x) \log ^3((-1+x) x)}{x^3}\right ) \, dx\right )+9 \int \left (\frac {e^{-1-3 x} \log (x) \log ^2((-1+x) x)}{-1+x}+\frac {e^{-1-3 x} \log (x) \log ^2((-1+x) x)}{x^4}-\frac {e^{-1-3 x} \log (x) \log ^2((-1+x) x)}{x^3}-\frac {e^{-1-3 x} \log (x) \log ^2((-1+x) x)}{x^2}-\frac {e^{-1-3 x} \log (x) \log ^2((-1+x) x)}{x}\right ) \, dx \\ & = 3 \int \frac {e^{-1-3 x} \log ^3((-1+x) x)}{x^4} \, dx+9 \int \frac {e^{-1-3 x} \log (x) \log ^2((-1+x) x)}{-1+x} \, dx+9 \int \frac {e^{-1-3 x} \log (x) \log ^2((-1+x) x)}{x^4} \, dx-9 \int \frac {e^{-1-3 x} \log (x) \log ^2((-1+x) x)}{x^3} \, dx-9 \int \frac {e^{-1-3 x} \log (x) \log ^2((-1+x) x)}{x^2} \, dx-9 \int \frac {e^{-1-3 x} \log (x) \log ^2((-1+x) x)}{x} \, dx-9 \int \frac {e^{-1-3 x} \log (x) \log ^3((-1+x) x)}{x^4} \, dx-9 \int \frac {e^{-1-3 x} \log (x) \log ^3((-1+x) x)}{x^3} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.68 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {e^{-1-3 x} \log ^2\left (-x+x^2\right ) \left ((-9+18 x) \log (x)+\left (-3+3 x+\left (9-9 x^2\right ) \log (x)\right ) \log \left (-x+x^2\right )\right )}{-x^4+x^5} \, dx=\frac {3 e^{-1-3 x} \log (x) \log ^3((-1+x) x)}{x^3} \]

[In]

Integrate[(E^(-1 - 3*x)*Log[-x + x^2]^2*((-9 + 18*x)*Log[x] + (-3 + 3*x + (9 - 9*x^2)*Log[x])*Log[-x + x^2]))/
(-x^4 + x^5),x]

[Out]

(3*E^(-1 - 3*x)*Log[x]*Log[(-1 + x)*x]^3)/x^3

Maple [A] (verified)

Time = 7.02 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08

method	result	size
parallelrisch	\(\frac {3 \ln \left (x \right ) {\mathrm e}^{-1} {\mathrm e}^{3 \ln \left (\ln \left (x^{2}-x \right )\right )-3 x}}{x^{3}}\)	\(28\)
risch	\(\text {Expression too large to display}\)	\(1414\)

[In]

int((((-9*x^2+9)*ln(x)+3*x-3)*ln(x^2-x)+(18*x-9)*ln(x))*exp(3*ln(ln(x^2-x))-3*x)/(x^5-x^4)/exp(1)/ln(x^2-x),x,
method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {e^{-1-3 x} \log ^2\left (-x+x^2\right ) \left ((-9+18 x) \log (x)+\left (-3+3 x+\left (9-9 x^2\right ) \log (x)\right ) \log \left (-x+x^2\right )\right )}{-x^4+x^5} \, dx=\frac {3 \, e^{\left (-3 \, x + 3 \, \log \left (\log \left (x^{2} - x\right )\right ) - 1\right )} \log \left (x\right )}{x^{3}} \]

[In]

integrate((((-9*x^2+9)*log(x)+3*x-3)*log(x^2-x)+(18*x-9)*log(x))*exp(3*log(log(x^2-x))-3*x)/(x^5-x^4)/exp(1)/l
og(x^2-x),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {e^{-1-3 x} \log ^2\left (-x+x^2\right ) \left ((-9+18 x) \log (x)+\left (-3+3 x+\left (9-9 x^2\right ) \log (x)\right ) \log \left (-x+x^2\right )\right )}{-x^4+x^5} \, dx=\frac {3 e^{- 3 x} \log {\left (x \right )} \log {\left (x^{2} - x \right )}^{3}}{e x^{3}} \]

[In]

integrate((((-9*x**2+9)*ln(x)+3*x-3)*ln(x**2-x)+(18*x-9)*ln(x))*exp(3*ln(ln(x**2-x))-3*x)/(x**5-x**4)/exp(1)/l
n(x**2-x),x)

[Out]

3*exp(-1)*exp(-3*x)*log(x)*log(x**2 - x)**3/x**3

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (24) = 48\).

Time = 0.27 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.31 \[ \int \frac {e^{-1-3 x} \log ^2\left (-x+x^2\right ) \left ((-9+18 x) \log (x)+\left (-3+3 x+\left (9-9 x^2\right ) \log (x)\right ) \log \left (-x+x^2\right )\right )}{-x^4+x^5} \, dx=\frac {3 \, {\left (e^{\left (-3 \, x\right )} \log \left (x - 1\right )^{3} \log \left (x\right ) + 3 \, e^{\left (-3 \, x\right )} \log \left (x - 1\right )^{2} \log \left (x\right )^{2} + 3 \, e^{\left (-3 \, x\right )} \log \left (x - 1\right ) \log \left (x\right )^{3} + e^{\left (-3 \, x\right )} \log \left (x\right )^{4}\right )} e^{\left (-1\right )}}{x^{3}} \]

[In]

integrate((((-9*x^2+9)*log(x)+3*x-3)*log(x^2-x)+(18*x-9)*log(x))*exp(3*log(log(x^2-x))-3*x)/(x^5-x^4)/exp(1)/l
og(x^2-x),x, algorithm="maxima")

[Out]

3*(e^(-3*x)*log(x - 1)^3*log(x) + 3*e^(-3*x)*log(x - 1)^2*log(x)^2 + 3*e^(-3*x)*log(x - 1)*log(x)^3 + e^(-3*x)
*log(x)^4)*e^(-1)/x^3

Giac [F]

\[ \int \frac {e^{-1-3 x} \log ^2\left (-x+x^2\right ) \left ((-9+18 x) \log (x)+\left (-3+3 x+\left (9-9 x^2\right ) \log (x)\right ) \log \left (-x+x^2\right )\right )}{-x^4+x^5} \, dx=\int { -\frac {3 \, {\left ({\left (3 \, {\left (x^{2} - 1\right )} \log \left (x\right ) - x + 1\right )} \log \left (x^{2} - x\right ) - 3 \, {\left (2 \, x - 1\right )} \log \left (x\right )\right )} e^{\left (-3 \, x + 3 \, \log \left (\log \left (x^{2} - x\right )\right ) - 1\right )}}{{\left (x^{5} - x^{4}\right )} \log \left (x^{2} - x\right )} \,d x } \]

[In]

integrate((((-9*x^2+9)*log(x)+3*x-3)*log(x^2-x)+(18*x-9)*log(x))*exp(3*log(log(x^2-x))-3*x)/(x^5-x^4)/exp(1)/l
og(x^2-x),x, algorithm="giac")

[Out]

integrate(-3*((3*(x^2 - 1)*log(x) - x + 1)*log(x^2 - x) - 3*(2*x - 1)*log(x))*e^(-3*x + 3*log(log(x^2 - x)) -
1)/((x^5 - x^4)*log(x^2 - x)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{-1-3 x} \log ^2\left (-x+x^2\right ) \left ((-9+18 x) \log (x)+\left (-3+3 x+\left (9-9 x^2\right ) \log (x)\right ) \log \left (-x+x^2\right )\right )}{-x^4+x^5} \, dx=\int -\frac {{\mathrm {e}}^{-1}\,{\mathrm {e}}^{3\,\ln \left (\ln \left (x^2-x\right )\right )-3\,x}\,\left (\ln \left (x\right )\,\left (18\,x-9\right )-\ln \left (x^2-x\right )\,\left (\ln \left (x\right )\,\left (9\,x^2-9\right )-3\,x+3\right )\right )}{\ln \left (x^2-x\right )\,\left (x^4-x^5\right )} \,d x \]

[In]

int(-(exp(-1)*exp(3*log(log(x^2 - x)) - 3*x)*(log(x)*(18*x - 9) - log(x^2 - x)*(log(x)*(9*x^2 - 9) - 3*x + 3))
)/(log(x^2 - x)*(x^4 - x^5)),x)

[Out]

int(-(exp(-1)*exp(3*log(log(x^2 - x)) - 3*x)*(log(x)*(18*x - 9) - log(x^2 - x)*(log(x)*(9*x^2 - 9) - 3*x + 3))
)/(log(x^2 - x)*(x^4 - x^5)), x)