3.41.82 \(\int \frac {e (1-x)+e^6 (-x+2 x^2)-e x \log (x)+(e^6 (2+x-x^2)+e (-1+x) \log (x)) \log (\frac {e^5 (-2-x+x^2)+(1-x) \log (x)}{e^5})+(e^{10} (2+x-x^2)+e^5 (-6+x+5 x^2-2 x^3)+(3+e^5 (-1+x)-5 x+2 x^2) \log (x)) \log ^2(\frac {e^5 (-2-x+x^2)+(1-x) \log (x)}{e^5})}{(e^5 (2+x-x^2)+(-1+x) \log (x)) \log ^2(\frac {e^5 (-2-x+x^2)+(1-x) \log (x)}{e^5})} \, dx\)

Optimal. Leaf size=31 \[ x \left (-3+e^5+x+\frac {e}{\log \left (-2+(1-x) \left (-x+\frac {\log (x)}{e^5}\right )\right )}\right ) \]

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Rubi [F]  time = 2.32, 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 {e (1-x)+e^6 \left (-x+2 x^2\right )-e x \log (x)+\left (e^6 \left (2+x-x^2\right )+e (-1+x) \log (x)\right ) \log \left (\frac {e^5 \left (-2-x+x^2\right )+(1-x) \log (x)}{e^5}\right )+\left (e^{10} \left (2+x-x^2\right )+e^5 \left (-6+x+5 x^2-2 x^3\right )+\left (3+e^5 (-1+x)-5 x+2 x^2\right ) \log (x)\right ) \log ^2\left (\frac {e^5 \left (-2-x+x^2\right )+(1-x) \log (x)}{e^5}\right )}{\left (e^5 \left (2+x-x^2\right )+(-1+x) \log (x)\right ) \log ^2\left (\frac {e^5 \left (-2-x+x^2\right )+(1-x) \log (x)}{e^5}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E*(1 - x) + E^6*(-x + 2*x^2) - E*x*Log[x] + (E^6*(2 + x - x^2) + E*(-1 + x)*Log[x])*Log[(E^5*(-2 - x + x^
2) + (1 - x)*Log[x])/E^5] + (E^10*(2 + x - x^2) + E^5*(-6 + x + 5*x^2 - 2*x^3) + (3 + E^5*(-1 + x) - 5*x + 2*x
^2)*Log[x])*Log[(E^5*(-2 - x + x^2) + (1 - x)*Log[x])/E^5]^2)/((E^5*(2 + x - x^2) + (-1 + x)*Log[x])*Log[(E^5*
(-2 - x + x^2) + (1 - x)*Log[x])/E^5]^2),x]

[Out]

-((3 - E^5)*x) + x^2 - E*Defer[Int][1/((-2*E^5 - E^5*x + E^5*x^2 + Log[x] - x*Log[x])*Log[-2 - x + x^2 - ((-1
+ x)*Log[x])/E^5]^2), x] + E*(1 + E^5)*Defer[Int][x/((-2*E^5 - E^5*x + E^5*x^2 + Log[x] - x*Log[x])*Log[-2 - x
+ x^2 - ((-1 + x)*Log[x])/E^5]^2), x] - 2*E^6*Defer[Int][x^2/((-2*E^5 - E^5*x + E^5*x^2 + Log[x] - x*Log[x])*
Log[-2 - x + x^2 - ((-1 + x)*Log[x])/E^5]^2), x] + E*Defer[Int][(x*Log[x])/((-2*E^5 - E^5*x + E^5*x^2 + Log[x]
- x*Log[x])*Log[-2 - x + x^2 - ((-1 + x)*Log[x])/E^5]^2), x] + E*Defer[Int][Log[-2 - x + x^2 - ((-1 + x)*Log[
x])/E^5]^(-1), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e-e x+e^6 x (-1+2 x)-e x \log (x)+\left (e^6 \left (2+x-x^2\right )+e (-1+x) \log (x)\right ) \log \left (-2-x+x^2-\frac {(-1+x) \log (x)}{e^5}\right )-\left (-3+e^5+2 x\right ) \left (e^5 \left (-2-x+x^2\right )-(-1+x) \log (x)\right ) \log ^2\left (-2-x+x^2-\frac {(-1+x) \log (x)}{e^5}\right )}{\left (e^5 \left (2+x-x^2\right )+(-1+x) \log (x)\right ) \log ^2\left (-2-x+x^2-\frac {(-1+x) \log (x)}{e^5}\right )} \, dx\\ &=\int \left (-3 \left (1-\frac {e^5}{3}\right )+2 x+\frac {e \left (1-\left (1+e^5\right ) x+2 e^5 x^2-x \log (x)\right )}{\left (2 e^5+e^5 x-e^5 x^2-\log (x)+x \log (x)\right ) \log ^2\left (-2-x+x^2-\frac {(-1+x) \log (x)}{e^5}\right )}+\frac {e}{\log \left (-2-x+x^2-\frac {(-1+x) \log (x)}{e^5}\right )}\right ) \, dx\\ &=-\left (\left (3-e^5\right ) x\right )+x^2+e \int \frac {1-\left (1+e^5\right ) x+2 e^5 x^2-x \log (x)}{\left (2 e^5+e^5 x-e^5 x^2-\log (x)+x \log (x)\right ) \log ^2\left (-2-x+x^2-\frac {(-1+x) \log (x)}{e^5}\right )} \, dx+e \int \frac {1}{\log \left (-2-x+x^2-\frac {(-1+x) \log (x)}{e^5}\right )} \, dx\\ &=-\left (\left (3-e^5\right ) x\right )+x^2+e \int \left (-\frac {1}{\left (-2 e^5-e^5 x+e^5 x^2+\log (x)-x \log (x)\right ) \log ^2\left (-2-x+x^2-\frac {(-1+x) \log (x)}{e^5}\right )}+\frac {\left (1+e^5\right ) x}{\left (-2 e^5-e^5 x+e^5 x^2+\log (x)-x \log (x)\right ) \log ^2\left (-2-x+x^2-\frac {(-1+x) \log (x)}{e^5}\right )}-\frac {2 e^5 x^2}{\left (-2 e^5-e^5 x+e^5 x^2+\log (x)-x \log (x)\right ) \log ^2\left (-2-x+x^2-\frac {(-1+x) \log (x)}{e^5}\right )}+\frac {x \log (x)}{\left (-2 e^5-e^5 x+e^5 x^2+\log (x)-x \log (x)\right ) \log ^2\left (-2-x+x^2-\frac {(-1+x) \log (x)}{e^5}\right )}\right ) \, dx+e \int \frac {1}{\log \left (-2-x+x^2-\frac {(-1+x) \log (x)}{e^5}\right )} \, dx\\ &=-\left (\left (3-e^5\right ) x\right )+x^2-e \int \frac {1}{\left (-2 e^5-e^5 x+e^5 x^2+\log (x)-x \log (x)\right ) \log ^2\left (-2-x+x^2-\frac {(-1+x) \log (x)}{e^5}\right )} \, dx+e \int \frac {x \log (x)}{\left (-2 e^5-e^5 x+e^5 x^2+\log (x)-x \log (x)\right ) \log ^2\left (-2-x+x^2-\frac {(-1+x) \log (x)}{e^5}\right )} \, dx+e \int \frac {1}{\log \left (-2-x+x^2-\frac {(-1+x) \log (x)}{e^5}\right )} \, dx-\left (2 e^6\right ) \int \frac {x^2}{\left (-2 e^5-e^5 x+e^5 x^2+\log (x)-x \log (x)\right ) \log ^2\left (-2-x+x^2-\frac {(-1+x) \log (x)}{e^5}\right )} \, dx+\left (e \left (1+e^5\right )\right ) \int \frac {x}{\left (-2 e^5-e^5 x+e^5 x^2+\log (x)-x \log (x)\right ) \log ^2\left (-2-x+x^2-\frac {(-1+x) \log (x)}{e^5}\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.12, size = 31, normalized size = 1.00 \begin {gather*} x \left (-3+e^5+x+\frac {e}{\log \left (-2-x+x^2-\frac {(-1+x) \log (x)}{e^5}\right )}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E*(1 - x) + E^6*(-x + 2*x^2) - E*x*Log[x] + (E^6*(2 + x - x^2) + E*(-1 + x)*Log[x])*Log[(E^5*(-2 -
x + x^2) + (1 - x)*Log[x])/E^5] + (E^10*(2 + x - x^2) + E^5*(-6 + x + 5*x^2 - 2*x^3) + (3 + E^5*(-1 + x) - 5*x
+ 2*x^2)*Log[x])*Log[(E^5*(-2 - x + x^2) + (1 - x)*Log[x])/E^5]^2)/((E^5*(2 + x - x^2) + (-1 + x)*Log[x])*Log
[(E^5*(-2 - x + x^2) + (1 - x)*Log[x])/E^5]^2),x]

[Out]

x*(-3 + E^5 + x + E/Log[-2 - x + x^2 - ((-1 + x)*Log[x])/E^5])

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fricas [B]  time = 0.62, size = 66, normalized size = 2.13 \begin {gather*} \frac {x e + {\left (x^{2} + x e^{5} - 3 \, x\right )} \log \left ({\left ({\left (x^{2} - x - 2\right )} e^{5} - {\left (x - 1\right )} \log \relax (x)\right )} e^{\left (-5\right )}\right )}{\log \left ({\left ({\left (x^{2} - x - 2\right )} e^{5} - {\left (x - 1\right )} \log \relax (x)\right )} e^{\left (-5\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((x-1)*exp(5)+2*x^2-5*x+3)*log(x)+(-x^2+x+2)*exp(5)^2+(-2*x^3+5*x^2+x-6)*exp(5))*log(((-x+1)*log(x
)+(x^2-x-2)*exp(5))/exp(5))^2+((x-1)*exp(1)*log(x)+(-x^2+x+2)*exp(1)*exp(5))*log(((-x+1)*log(x)+(x^2-x-2)*exp(
5))/exp(5))-x*exp(1)*log(x)+(2*x^2-x)*exp(1)*exp(5)+(-x+1)*exp(1))/((x-1)*log(x)+(-x^2+x+2)*exp(5))/log(((-x+1
)*log(x)+(x^2-x-2)*exp(5))/exp(5))^2,x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.82, size = 130, normalized size = 4.19 \begin {gather*} \frac {x^{2} \log \left (x^{2} e^{5} - x e^{5} - x \log \relax (x) - 2 \, e^{5} + \log \relax (x)\right ) + x e^{5} \log \left (x^{2} e^{5} - x e^{5} - x \log \relax (x) - 2 \, e^{5} + \log \relax (x)\right ) - 5 \, x^{2} - 5 \, x e^{5} + x e - 3 \, x \log \left (x^{2} e^{5} - x e^{5} - x \log \relax (x) - 2 \, e^{5} + \log \relax (x)\right ) + 15 \, x}{\log \left (x^{2} e^{5} - x e^{5} - x \log \relax (x) - 2 \, e^{5} + \log \relax (x)\right ) - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((x-1)*exp(5)+2*x^2-5*x+3)*log(x)+(-x^2+x+2)*exp(5)^2+(-2*x^3+5*x^2+x-6)*exp(5))*log(((-x+1)*log(x
)+(x^2-x-2)*exp(5))/exp(5))^2+((x-1)*exp(1)*log(x)+(-x^2+x+2)*exp(1)*exp(5))*log(((-x+1)*log(x)+(x^2-x-2)*exp(
5))/exp(5))-x*exp(1)*log(x)+(2*x^2-x)*exp(1)*exp(5)+(-x+1)*exp(1))/((x-1)*log(x)+(-x^2+x+2)*exp(5))/log(((-x+1
)*log(x)+(x^2-x-2)*exp(5))/exp(5))^2,x, algorithm="giac")

[Out]

(x^2*log(x^2*e^5 - x*e^5 - x*log(x) - 2*e^5 + log(x)) + x*e^5*log(x^2*e^5 - x*e^5 - x*log(x) - 2*e^5 + log(x))
- 5*x^2 - 5*x*e^5 + x*e - 3*x*log(x^2*e^5 - x*e^5 - x*log(x) - 2*e^5 + log(x)) + 15*x)/(log(x^2*e^5 - x*e^5 -
x*log(x) - 2*e^5 + log(x)) - 5)

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maple [A]  time = 0.05, size = 42, normalized size = 1.35

method	result	size
risch	\(x \,{\mathrm e}^{5}+x^{2}-3 x +\frac {x \,{\mathrm e}}{\ln \left (\left (\left (1-x \right ) \ln \relax (x )+\left (x^{2}-x -2\right ) {\mathrm e}^{5}\right ) {\mathrm e}^{-5}\right )}\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((((x-1)*exp(5)+2*x^2-5*x+3)*ln(x)+(-x^2+x+2)*exp(5)^2+(-2*x^3+5*x^2+x-6)*exp(5))*ln(((1-x)*ln(x)+(x^2-x-2
)*exp(5))/exp(5))^2+((x-1)*exp(1)*ln(x)+(-x^2+x+2)*exp(1)*exp(5))*ln(((1-x)*ln(x)+(x^2-x-2)*exp(5))/exp(5))-x*
exp(1)*ln(x)+(2*x^2-x)*exp(1)*exp(5)+(1-x)*exp(1))/((x-1)*ln(x)+(-x^2+x+2)*exp(5))/ln(((1-x)*ln(x)+(x^2-x-2)*e
xp(5))/exp(5))^2,x,method=_RETURNVERBOSE)

[Out]

x*exp(5)+x^2-3*x+x*exp(1)/ln(((1-x)*ln(x)+(x^2-x-2)*exp(5))*exp(-5))

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maxima [B]  time = 0.43, size = 84, normalized size = 2.71 \begin {gather*} -\frac {5 \, x^{2} + x {\left (5 \, e^{5} - e - 15\right )} - {\left (x^{2} + x {\left (e^{5} - 3\right )}\right )} \log \left (x^{2} e^{5} - x e^{5} - {\left (x - 1\right )} \log \relax (x) - 2 \, e^{5}\right )}{\log \left (x^{2} e^{5} - x e^{5} - {\left (x - 1\right )} \log \relax (x) - 2 \, e^{5}\right ) - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((x-1)*exp(5)+2*x^2-5*x+3)*log(x)+(-x^2+x+2)*exp(5)^2+(-2*x^3+5*x^2+x-6)*exp(5))*log(((-x+1)*log(x
)+(x^2-x-2)*exp(5))/exp(5))^2+((x-1)*exp(1)*log(x)+(-x^2+x+2)*exp(1)*exp(5))*log(((-x+1)*log(x)+(x^2-x-2)*exp(
5))/exp(5))-x*exp(1)*log(x)+(2*x^2-x)*exp(1)*exp(5)+(-x+1)*exp(1))/((x-1)*log(x)+(-x^2+x+2)*exp(5))/log(((-x+1
)*log(x)+(x^2-x-2)*exp(5))/exp(5))^2,x, algorithm="maxima")

[Out]

-(5*x^2 + x*(5*e^5 - e - 15) - (x^2 + x*(e^5 - 3))*log(x^2*e^5 - x*e^5 - (x - 1)*log(x) - 2*e^5))/(log(x^2*e^5
- x*e^5 - (x - 1)*log(x) - 2*e^5) - 5)

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mupad [B]  time = 3.82, size = 165, normalized size = 5.32 \begin {gather*} \frac {2\,{\mathrm {e}}^6+x\,{\mathrm {e}}^6-\ln \relax (x)\,\left (\mathrm {e}-x\,\mathrm {e}\right )-x^2\,{\mathrm {e}}^6}{\ln \relax (x)+\frac {x+x\,{\mathrm {e}}^5-2\,x^2\,{\mathrm {e}}^5-1}{x}}+\frac {x\,\mathrm {e}-\frac {x\,\mathrm {e}\,\ln \left (-{\mathrm {e}}^{-5}\,\left (\ln \relax (x)\,\left (x-1\right )+{\mathrm {e}}^5\,\left (-x^2+x+2\right )\right )\right )\,\left (2\,{\mathrm {e}}^5-\ln \relax (x)+x\,{\mathrm {e}}^5-x^2\,{\mathrm {e}}^5+x\,\ln \relax (x)\right )}{x+x\,{\mathrm {e}}^5-2\,x^2\,{\mathrm {e}}^5+x\,\ln \relax (x)-1}}{\ln \left (-{\mathrm {e}}^{-5}\,\left (\ln \relax (x)\,\left (x-1\right )+{\mathrm {e}}^5\,\left (-x^2+x+2\right )\right )\right )}+x\,\left ({\mathrm {e}}^5-3\right )+x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(1)*(x - 1) - log(-exp(-5)*(log(x)*(x - 1) + exp(5)*(x - x^2 + 2)))*(exp(6)*(x - x^2 + 2) + exp(1)*lo
g(x)*(x - 1)) - log(-exp(-5)*(log(x)*(x - 1) + exp(5)*(x - x^2 + 2)))^2*(exp(5)*(x + 5*x^2 - 2*x^3 - 6) + log(
x)*(exp(5)*(x - 1) - 5*x + 2*x^2 + 3) + exp(10)*(x - x^2 + 2)) + exp(6)*(x - 2*x^2) + x*exp(1)*log(x))/(log(-e
xp(-5)*(log(x)*(x - 1) + exp(5)*(x - x^2 + 2)))^2*(log(x)*(x - 1) + exp(5)*(x - x^2 + 2))),x)

[Out]

(2*exp(6) + x*exp(6) - log(x)*(exp(1) - x*exp(1)) - x^2*exp(6))/(log(x) + (x + x*exp(5) - 2*x^2*exp(5) - 1)/x)
+ (x*exp(1) - (x*exp(1)*log(-exp(-5)*(log(x)*(x - 1) + exp(5)*(x - x^2 + 2)))*(2*exp(5) - log(x) + x*exp(5) -
x^2*exp(5) + x*log(x)))/(x + x*exp(5) - 2*x^2*exp(5) + x*log(x) - 1))/log(-exp(-5)*(log(x)*(x - 1) + exp(5)*(
x - x^2 + 2))) + x*(exp(5) - 3) + x^2

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sympy [A]  time = 0.67, size = 36, normalized size = 1.16 \begin {gather*} x^{2} + x \left (-3 + e^{5}\right ) + \frac {e x}{\log {\left (\frac {\left (1 - x\right ) \log {\relax (x )} + \left (x^{2} - x - 2\right ) e^{5}}{e^{5}} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((x-1)*exp(5)+2*x**2-5*x+3)*ln(x)+(-x**2+x+2)*exp(5)**2+(-2*x**3+5*x**2+x-6)*exp(5))*ln(((-x+1)*ln
(x)+(x**2-x-2)*exp(5))/exp(5))**2+((x-1)*exp(1)*ln(x)+(-x**2+x+2)*exp(1)*exp(5))*ln(((-x+1)*ln(x)+(x**2-x-2)*e
xp(5))/exp(5))-x*exp(1)*ln(x)+(2*x**2-x)*exp(1)*exp(5)+(-x+1)*exp(1))/((x-1)*ln(x)+(-x**2+x+2)*exp(5))/ln(((-x
+1)*ln(x)+(x**2-x-2)*exp(5))/exp(5))**2,x)

[Out]

x**2 + x*(-3 + exp(5)) + E*x/log(((1 - x)*log(x) + (x**2 - x - 2)*exp(5))*exp(-5))

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