\(\int \frac {x^{11}-2 x^6 \log (x)+x \log ^2(x)+e^{\frac {-256 x^5-81 x^6+81 \log (2)+(256+81 x) \log (x)}{-81 x^5+81 \log (x)}} (-x^{11}+(1-5 x^5) \log (2)+2 x^6 \log (x)-x \log ^2(x))}{x^{11}-2 x^6 \log (x)+x \log ^2(x)} \, dx\) [7007]

Optimal result

Integrand size = 106, antiderivative size = 24 \[ \int \frac {x^{11}-2 x^6 \log (x)+x \log ^2(x)+e^{\frac {-256 x^5-81 x^6+81 \log (2)+(256+81 x) \log (x)}{-81 x^5+81 \log (x)}} \left (-x^{11}+\left (1-5 x^5\right ) \log (2)+2 x^6 \log (x)-x \log ^2(x)\right )}{x^{11}-2 x^6 \log (x)+x \log ^2(x)} \, dx=-e^{\frac {256}{81}+x+\frac {\log (2)}{-x^5+\log (x)}}+x \]

[Out]

x-exp(256/81+x+ln(2)/(ln(x)-x^5))

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(95\) vs. \(2(24)=48\).

Time = 4.46 (sec) , antiderivative size = 95, normalized size of antiderivative = 3.96, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {6873, 6874, 2326} \[ \int \frac {x^{11}-2 x^6 \log (x)+x \log ^2(x)+e^{\frac {-256 x^5-81 x^6+81 \log (2)+(256+81 x) \log (x)}{-81 x^5+81 \log (x)}} \left (-x^{11}+\left (1-5 x^5\right ) \log (2)+2 x^6 \log (x)-x \log ^2(x)\right )}{x^{11}-2 x^6 \log (x)+x \log ^2(x)} \, dx=x-\frac {2^{\frac {1}{\log (x)-x^5}} e^{\frac {x^5 (81 x+256)}{81 \left (x^5-\log (x)\right )}} x^{-\frac {81 x^5+81 x-81 \log (x)+256}{81 \left (x^5-\log (x)\right )}} \left (\log (2)-x^5 \log (32)\right )}{\left (\frac {1}{x}-5 x^4\right ) \log (2)} \]

[In]

Int[(x^11 - 2*x^6*Log[x] + x*Log[x]^2 + E^((-256*x^5 - 81*x^6 + 81*Log[2] + (256 + 81*x)*Log[x])/(-81*x^5 + 81
*Log[x]))*(-x^11 + (1 - 5*x^5)*Log[2] + 2*x^6*Log[x] - x*Log[x]^2))/(x^11 - 2*x^6*Log[x] + x*Log[x]^2),x]

[Out]

x - (2^(-x^5 + Log[x])^(-1)*E^((x^5*(256 + 81*x))/(81*(x^5 - Log[x])))*(Log[2] - x^5*Log[32]))/(x^((256 + 81*x
+ 81*x^5 - 81*Log[x])/(81*(x^5 - Log[x])))*(x^(-1) - 5*x^4)*Log[2])

Rule 2326

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, x]))}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rule 6873

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = \int \frac {x^{11}-2 x^6 \log (x)+x \log ^2(x)+\exp \left (\frac {-256 x^5-81 x^6+81 \log (2)+(256+81 x) \log (x)}{-81 x^5+81 \log (x)}\right ) \left (-x^{11}+\left (1-5 x^5\right ) \log (2)+2 x^6 \log (x)-x \log ^2(x)\right )}{x \left (x^5-\log (x)\right )^2} \, dx \\ & = \int \left (1+\frac {2^{\frac {1}{-x^5+\log (x)}} e^{\frac {x^5 (256+81 x)}{81 \left (x^5-\log (x)\right )}} x^{\frac {-256-81 x-81 x^5+81 \log (x)}{81 \left (x^5-\log (x)\right )}} \left (-x^{11}+\log (2)-x^5 \log (32)+2 x^6 \log (x)-x \log ^2(x)\right )}{\left (x^5-\log (x)\right )^2}\right ) \, dx \\ & = x+\int \frac {2^{\frac {1}{-x^5+\log (x)}} e^{\frac {x^5 (256+81 x)}{81 \left (x^5-\log (x)\right )}} x^{\frac {-256-81 x-81 x^5+81 \log (x)}{81 \left (x^5-\log (x)\right )}} \left (-x^{11}+\log (2)-x^5 \log (32)+2 x^6 \log (x)-x \log ^2(x)\right )}{\left (x^5-\log (x)\right )^2} \, dx \\ & = x-\frac {2^{\frac {1}{-x^5+\log (x)}} e^{\frac {x^5 (256+81 x)}{81 \left (x^5-\log (x)\right )}} x^{-\frac {256+81 x+81 x^5-81 \log (x)}{81 \left (x^5-\log (x)\right )}} \left (\log (2)-x^5 \log (32)\right )}{\left (\frac {1}{x}-5 x^4\right ) \log (2)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96 \[ \int \frac {x^{11}-2 x^6 \log (x)+x \log ^2(x)+e^{\frac {-256 x^5-81 x^6+81 \log (2)+(256+81 x) \log (x)}{-81 x^5+81 \log (x)}} \left (-x^{11}+\left (1-5 x^5\right ) \log (2)+2 x^6 \log (x)-x \log ^2(x)\right )}{x^{11}-2 x^6 \log (x)+x \log ^2(x)} \, dx=-2^{\frac {1}{-x^5+\log (x)}} e^{\frac {256}{81}+x}+x \]

[In]

Integrate[(x^11 - 2*x^6*Log[x] + x*Log[x]^2 + E^((-256*x^5 - 81*x^6 + 81*Log[2] + (256 + 81*x)*Log[x])/(-81*x^
5 + 81*Log[x]))*(-x^11 + (1 - 5*x^5)*Log[2] + 2*x^6*Log[x] - x*Log[x]^2))/(x^11 - 2*x^6*Log[x] + x*Log[x]^2),x
]

[Out]

-(2^(-x^5 + Log[x])^(-1)*E^(256/81 + x)) + x

Maple [A] (verified)

Time = 227.15 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.71

method	result	size
parallelrisch	\(x -{\mathrm e}^{\frac {\left (81 x +256\right ) \ln \left (x \right )+81 \ln \left (2\right )-81 x^{6}-256 x^{5}}{81 \ln \left (x \right )-81 x^{5}}}\)	\(41\)
risch	\(x -{\mathrm e}^{\frac {-81 x^{6}-256 x^{5}+81 x \ln \left (x \right )+256 \ln \left (x \right )+81 \ln \left (2\right )}{81 \ln \left (x \right )-81 x^{5}}}\)	\(42\)

[In]

int(((-x*ln(x)^2+2*x^6*ln(x)+(-5*x^5+1)*ln(2)-x^11)*exp(((81*x+256)*ln(x)+81*ln(2)-81*x^6-256*x^5)/(81*ln(x)-8
1*x^5))+x*ln(x)^2-2*x^6*ln(x)+x^11)/(x*ln(x)^2-2*x^6*ln(x)+x^11),x,method=_RETURNVERBOSE)

[Out]

x-exp(1/81/(ln(x)-x^5)*((81*x+256)*ln(x)+81*ln(2)-81*x^6-256*x^5))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.71 \[ \int \frac {x^{11}-2 x^6 \log (x)+x \log ^2(x)+e^{\frac {-256 x^5-81 x^6+81 \log (2)+(256+81 x) \log (x)}{-81 x^5+81 \log (x)}} \left (-x^{11}+\left (1-5 x^5\right ) \log (2)+2 x^6 \log (x)-x \log ^2(x)\right )}{x^{11}-2 x^6 \log (x)+x \log ^2(x)} \, dx=x - e^{\left (\frac {81 \, x^{6} + 256 \, x^{5} - {\left (81 \, x + 256\right )} \log \left (x\right ) - 81 \, \log \left (2\right )}{81 \, {\left (x^{5} - \log \left (x\right )\right )}}\right )} \]

[In]

integrate(((-x*log(x)^2+2*x^6*log(x)+(-5*x^5+1)*log(2)-x^11)*exp(((81*x+256)*log(x)+81*log(2)-81*x^6-256*x^5)/
(81*log(x)-81*x^5))+x*log(x)^2-2*x^6*log(x)+x^11)/(x*log(x)^2-2*x^6*log(x)+x^11),x, algorithm="fricas")

[Out]

x - e^(1/81*(81*x^6 + 256*x^5 - (81*x + 256)*log(x) - 81*log(2))/(x^5 - log(x)))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (17) = 34\).

Time = 0.20 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.50 \[ \int \frac {x^{11}-2 x^6 \log (x)+x \log ^2(x)+e^{\frac {-256 x^5-81 x^6+81 \log (2)+(256+81 x) \log (x)}{-81 x^5+81 \log (x)}} \left (-x^{11}+\left (1-5 x^5\right ) \log (2)+2 x^6 \log (x)-x \log ^2(x)\right )}{x^{11}-2 x^6 \log (x)+x \log ^2(x)} \, dx=x - e^{\frac {- 81 x^{6} - 256 x^{5} + \left (81 x + 256\right ) \log {\left (x \right )} + 81 \log {\left (2 \right )}}{- 81 x^{5} + 81 \log {\left (x \right )}}} \]

[In]

integrate(((-x*ln(x)**2+2*x**6*ln(x)+(-5*x**5+1)*ln(2)-x**11)*exp(((81*x+256)*ln(x)+81*ln(2)-81*x**6-256*x**5)
/(81*ln(x)-81*x**5))+x*ln(x)**2-2*x**6*ln(x)+x**11)/(x*ln(x)**2-2*x**6*ln(x)+x**11),x)

[Out]

x - exp((-81*x**6 - 256*x**5 + (81*x + 256)*log(x) + 81*log(2))/(-81*x**5 + 81*log(x)))

Maxima [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {x^{11}-2 x^6 \log (x)+x \log ^2(x)+e^{\frac {-256 x^5-81 x^6+81 \log (2)+(256+81 x) \log (x)}{-81 x^5+81 \log (x)}} \left (-x^{11}+\left (1-5 x^5\right ) \log (2)+2 x^6 \log (x)-x \log ^2(x)\right )}{x^{11}-2 x^6 \log (x)+x \log ^2(x)} \, dx=x - e^{\left (x - \frac {\log \left (2\right )}{x^{5} - \log \left (x\right )} + \frac {256}{81}\right )} \]

[In]

integrate(((-x*log(x)^2+2*x^6*log(x)+(-5*x^5+1)*log(2)-x^11)*exp(((81*x+256)*log(x)+81*log(2)-81*x^6-256*x^5)/
(81*log(x)-81*x^5))+x*log(x)^2-2*x^6*log(x)+x^11)/(x*log(x)^2-2*x^6*log(x)+x^11),x, algorithm="maxima")

[Out]

x - e^(x - log(2)/(x^5 - log(x)) + 256/81)

Giac [A] (verification not implemented)

none

Time = 0.59 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.71 \[ \int \frac {x^{11}-2 x^6 \log (x)+x \log ^2(x)+e^{\frac {-256 x^5-81 x^6+81 \log (2)+(256+81 x) \log (x)}{-81 x^5+81 \log (x)}} \left (-x^{11}+\left (1-5 x^5\right ) \log (2)+2 x^6 \log (x)-x \log ^2(x)\right )}{x^{11}-2 x^6 \log (x)+x \log ^2(x)} \, dx=x - e^{\left (\frac {81 \, x^{6} + 256 \, x^{5} - 81 \, x \log \left (x\right ) - 81 \, \log \left (2\right ) - 256 \, \log \left (x\right )}{81 \, {\left (x^{5} - \log \left (x\right )\right )}}\right )} \]

[In]

integrate(((-x*log(x)^2+2*x^6*log(x)+(-5*x^5+1)*log(2)-x^11)*exp(((81*x+256)*log(x)+81*log(2)-81*x^6-256*x^5)/
(81*log(x)-81*x^5))+x*log(x)^2-2*x^6*log(x)+x^11)/(x*log(x)^2-2*x^6*log(x)+x^11),x, algorithm="giac")

[Out]

x - e^(1/81*(81*x^6 + 256*x^5 - 81*x*log(x) - 81*log(2) - 256*log(x))/(x^5 - log(x)))

Mupad [B] (verification not implemented)

Time = 18.10 (sec) , antiderivative size = 89, normalized size of antiderivative = 3.71 \[ \int \frac {x^{11}-2 x^6 \log (x)+x \log ^2(x)+e^{\frac {-256 x^5-81 x^6+81 \log (2)+(256+81 x) \log (x)}{-81 x^5+81 \log (x)}} \left (-x^{11}+\left (1-5 x^5\right ) \log (2)+2 x^6 \log (x)-x \log ^2(x)\right )}{x^{11}-2 x^6 \log (x)+x \log ^2(x)} \, dx=x-2^{\frac {81}{81\,\ln \left (x\right )-81\,x^5}}\,x^{\frac {256}{81\,\ln \left (x\right )-81\,x^5}}\,x^{\frac {81\,x}{81\,\ln \left (x\right )-81\,x^5}}\,{\mathrm {e}}^{-\frac {81\,x^6}{81\,\ln \left (x\right )-81\,x^5}}\,{\mathrm {e}}^{-\frac {256\,x^5}{81\,\ln \left (x\right )-81\,x^5}} \]

[In]

int((x*log(x)^2 - 2*x^6*log(x) - exp((81*log(2) + log(x)*(81*x + 256) - 256*x^5 - 81*x^6)/(81*log(x) - 81*x^5)
)*(x*log(x)^2 - 2*x^6*log(x) + log(2)*(5*x^5 - 1) + x^11) + x^11)/(x*log(x)^2 - 2*x^6*log(x) + x^11),x)

[Out]

x - 2^(81/(81*log(x) - 81*x^5))*x^(256/(81*log(x) - 81*x^5))*x^((81*x)/(81*log(x) - 81*x^5))*exp(-(81*x^6)/(81
*log(x) - 81*x^5))*exp(-(256*x^5)/(81*log(x) - 81*x^5))