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

Optimal. Leaf size=24 \[ -e^{\frac {256}{81}+x+\frac {\log (2)}{-x^5+\log (x)}}+x \]

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Rubi [B]  time = 6.83, antiderivative size = 95, normalized size of antiderivative = 3.96, number of steps used = 4, number of rules used = 3, integrand size = 106, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {6741, 6742, 2288} \begin {gather*} 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)} \end {gather*}

Warning: Unable to verify antiderivative.

[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 2288

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 6741

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

Rule 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.19, size = 23, normalized size = 0.96 \begin {gather*} -2^{\frac {1}{-x^5+\log (x)}} e^{\frac {256}{81}+x}+x \end {gather*}

Antiderivative was successfully verified.

[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

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fricas [A]  time = 0.61, size = 41, normalized size = 1.71 \begin {gather*} x - e^{\left (\frac {81 \, x^{6} + 256 \, x^{5} - {\left (81 \, x + 256\right )} \log \relax (x) - 81 \, \log \relax (2)}{81 \, {\left (x^{5} - \log \relax (x)\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)))

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giac [A]  time = 0.75, size = 41, normalized size = 1.71 \begin {gather*} x - e^{\left (\frac {81 \, x^{6} + 256 \, x^{5} - 81 \, x \log \relax (x) - 81 \, \log \relax (2) - 256 \, \log \relax (x)}{81 \, {\left (x^{5} - \log \relax (x)\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)))

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maple [A]  time = 0.08, size = 42, normalized size = 1.75

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

Verification of antiderivative is not currently implemented for this CAS.

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

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maxima [A]  time = 0.55, size = 22, normalized size = 0.92 \begin {gather*} x - e^{\left (x - \frac {\log \relax (2)}{x^{5} - \log \relax (x)} + \frac {256}{81}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 4.35, size = 89, normalized size = 3.71 \begin {gather*} x-2^{\frac {81}{81\,\ln \relax (x)-81\,x^5}}\,x^{\frac {256}{81\,\ln \relax (x)-81\,x^5}}\,x^{\frac {81\,x}{81\,\ln \relax (x)-81\,x^5}}\,{\mathrm {e}}^{-\frac {81\,x^6}{81\,\ln \relax (x)-81\,x^5}}\,{\mathrm {e}}^{-\frac {256\,x^5}{81\,\ln \relax (x)-81\,x^5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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sympy [B]  time = 0.51, size = 36, normalized size = 1.50 \begin {gather*} x - e^{\frac {- 81 x^{6} - 256 x^{5} + \left (81 x + 256\right ) \log {\relax (x )} + 81 \log {\relax (2 )}}{- 81 x^{5} + 81 \log {\relax (x )}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)))

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