3.11.3 \(\int \frac {e^4 (-1+17 x-4 x^2-12 x^3+3 x^4)+e^4 (4 x-3 x^3) \log (x)+(e^4 (20-5 x-12 x^2+3 x^3)+e^4 (5-3 x^2) \log (x)) \log (-4+x-\log (x))+(e^4 (-4 x+x^2)-e^4 x \log (x)+(e^4 (-4+x)-e^4 \log (x)) \log (-4+x-\log (x))) \log (x+\log (-4+x-\log (x)))}{4 x-x^2+x \log (x)+(4-x+\log (x)) \log (-4+x-\log (x))} \, dx\)

Optimal. Leaf size=28 \[ e^4 \left (9+x \left (5-x^2-\log (x+\log (-4+x-\log (x)))\right )\right ) \]

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Rubi [F]  time = 2.90, 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^4 \left (-1+17 x-4 x^2-12 x^3+3 x^4\right )+e^4 \left (4 x-3 x^3\right ) \log (x)+\left (e^4 \left (20-5 x-12 x^2+3 x^3\right )+e^4 \left (5-3 x^2\right ) \log (x)\right ) \log (-4+x-\log (x))+\left (e^4 \left (-4 x+x^2\right )-e^4 x \log (x)+\left (e^4 (-4+x)-e^4 \log (x)\right ) \log (-4+x-\log (x))\right ) \log (x+\log (-4+x-\log (x)))}{4 x-x^2+x \log (x)+(4-x+\log (x)) \log (-4+x-\log (x))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^4*(-1 + 17*x - 4*x^2 - 12*x^3 + 3*x^4) + E^4*(4*x - 3*x^3)*Log[x] + (E^4*(20 - 5*x - 12*x^2 + 3*x^3) +
E^4*(5 - 3*x^2)*Log[x])*Log[-4 + x - Log[x]] + (E^4*(-4*x + x^2) - E^4*x*Log[x] + (E^4*(-4 + x) - E^4*Log[x])*
Log[-4 + x - Log[x]])*Log[x + Log[-4 + x - Log[x]]])/(4*x - x^2 + x*Log[x] + (4 - x + Log[x])*Log[-4 + x - Log
[x]]),x]

[Out]

5*E^4*x - E^4*x^3 - 5*E^4*Defer[Int][x/(x + Log[-4 + x - Log[x]]), x] + 3*E^4*Defer[Int][x^3/(x + Log[-4 + x -
Log[x]]), x] + E^4*Defer[Int][1/((-4 + x - Log[x])*(x + Log[-4 + x - Log[x]])), x] - 17*E^4*Defer[Int][x/((-4
+ x - Log[x])*(x + Log[-4 + x - Log[x]])), x] + 4*E^4*Defer[Int][x^2/((-4 + x - Log[x])*(x + Log[-4 + x - Log
[x]])), x] + 12*E^4*Defer[Int][x^3/((-4 + x - Log[x])*(x + Log[-4 + x - Log[x]])), x] - 3*E^4*Defer[Int][x^4/(
(-4 + x - Log[x])*(x + Log[-4 + x - Log[x]])), x] - 4*E^4*Defer[Int][(x*Log[x])/((-4 + x - Log[x])*(x + Log[-4
+ x - Log[x]])), x] + 3*E^4*Defer[Int][(x^3*Log[x])/((-4 + x - Log[x])*(x + Log[-4 + x - Log[x]])), x] - E^4*
Defer[Int][Log[x + Log[-4 + x - Log[x]]], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^4 \left (-1+17 x-4 x^2-12 x^3+3 x^4\right )+e^4 \left (4 x-3 x^3\right ) \log (x)+\left (e^4 \left (20-5 x-12 x^2+3 x^3\right )+e^4 \left (5-3 x^2\right ) \log (x)\right ) \log (-4+x-\log (x))+\left (e^4 \left (-4 x+x^2\right )-e^4 x \log (x)+\left (e^4 (-4+x)-e^4 \log (x)\right ) \log (-4+x-\log (x))\right ) \log (x+\log (-4+x-\log (x)))}{(4-x+\log (x)) (x+\log (-4+x-\log (x)))} \, dx\\ &=\int \frac {e^4 \left (-1+17 x-4 x^2-12 x^3+3 x^4+\left (4 x-3 x^3\right ) \log (x)+\left (-5+3 x^2\right ) (-4+x-\log (x)) \log (-4+x-\log (x))+(-4+x-\log (x)) (x+\log (-4+x-\log (x))) \log (x+\log (-4+x-\log (x)))\right )}{(4-x+\log (x)) (x+\log (-4+x-\log (x)))} \, dx\\ &=e^4 \int \frac {-1+17 x-4 x^2-12 x^3+3 x^4+\left (4 x-3 x^3\right ) \log (x)+\left (-5+3 x^2\right ) (-4+x-\log (x)) \log (-4+x-\log (x))+(-4+x-\log (x)) (x+\log (-4+x-\log (x))) \log (x+\log (-4+x-\log (x)))}{(4-x+\log (x)) (x+\log (-4+x-\log (x)))} \, dx\\ &=e^4 \int \left (\frac {1}{(-4+x-\log (x)) (x+\log (-4+x-\log (x)))}-\frac {17 x}{(-4+x-\log (x)) (x+\log (-4+x-\log (x)))}+\frac {4 x^2}{(-4+x-\log (x)) (x+\log (-4+x-\log (x)))}+\frac {12 x^3}{(-4+x-\log (x)) (x+\log (-4+x-\log (x)))}-\frac {3 x^4}{(-4+x-\log (x)) (x+\log (-4+x-\log (x)))}+\frac {x \left (-4+3 x^2\right ) \log (x)}{(-4+x-\log (x)) (x+\log (-4+x-\log (x)))}-\frac {\left (-5+3 x^2\right ) \log (-4+x-\log (x))}{x+\log (-4+x-\log (x))}-\log (x+\log (-4+x-\log (x)))\right ) \, dx\\ &=e^4 \int \frac {1}{(-4+x-\log (x)) (x+\log (-4+x-\log (x)))} \, dx+e^4 \int \frac {x \left (-4+3 x^2\right ) \log (x)}{(-4+x-\log (x)) (x+\log (-4+x-\log (x)))} \, dx-e^4 \int \frac {\left (-5+3 x^2\right ) \log (-4+x-\log (x))}{x+\log (-4+x-\log (x))} \, dx-e^4 \int \log (x+\log (-4+x-\log (x))) \, dx-\left (3 e^4\right ) \int \frac {x^4}{(-4+x-\log (x)) (x+\log (-4+x-\log (x)))} \, dx+\left (4 e^4\right ) \int \frac {x^2}{(-4+x-\log (x)) (x+\log (-4+x-\log (x)))} \, dx+\left (12 e^4\right ) \int \frac {x^3}{(-4+x-\log (x)) (x+\log (-4+x-\log (x)))} \, dx-\left (17 e^4\right ) \int \frac {x}{(-4+x-\log (x)) (x+\log (-4+x-\log (x)))} \, dx\\ &=e^4 \int \frac {1}{(-4+x-\log (x)) (x+\log (-4+x-\log (x)))} \, dx-e^4 \int \left (-5+3 x^2+\frac {x \left (5-3 x^2\right )}{x+\log (-4+x-\log (x))}\right ) \, dx+e^4 \int \left (-\frac {4 x \log (x)}{(-4+x-\log (x)) (x+\log (-4+x-\log (x)))}+\frac {3 x^3 \log (x)}{(-4+x-\log (x)) (x+\log (-4+x-\log (x)))}\right ) \, dx-e^4 \int \log (x+\log (-4+x-\log (x))) \, dx-\left (3 e^4\right ) \int \frac {x^4}{(-4+x-\log (x)) (x+\log (-4+x-\log (x)))} \, dx+\left (4 e^4\right ) \int \frac {x^2}{(-4+x-\log (x)) (x+\log (-4+x-\log (x)))} \, dx+\left (12 e^4\right ) \int \frac {x^3}{(-4+x-\log (x)) (x+\log (-4+x-\log (x)))} \, dx-\left (17 e^4\right ) \int \frac {x}{(-4+x-\log (x)) (x+\log (-4+x-\log (x)))} \, dx\\ &=5 e^4 x-e^4 x^3-e^4 \int \frac {x \left (5-3 x^2\right )}{x+\log (-4+x-\log (x))} \, dx+e^4 \int \frac {1}{(-4+x-\log (x)) (x+\log (-4+x-\log (x)))} \, dx-e^4 \int \log (x+\log (-4+x-\log (x))) \, dx-\left (3 e^4\right ) \int \frac {x^4}{(-4+x-\log (x)) (x+\log (-4+x-\log (x)))} \, dx+\left (3 e^4\right ) \int \frac {x^3 \log (x)}{(-4+x-\log (x)) (x+\log (-4+x-\log (x)))} \, dx+\left (4 e^4\right ) \int \frac {x^2}{(-4+x-\log (x)) (x+\log (-4+x-\log (x)))} \, dx-\left (4 e^4\right ) \int \frac {x \log (x)}{(-4+x-\log (x)) (x+\log (-4+x-\log (x)))} \, dx+\left (12 e^4\right ) \int \frac {x^3}{(-4+x-\log (x)) (x+\log (-4+x-\log (x)))} \, dx-\left (17 e^4\right ) \int \frac {x}{(-4+x-\log (x)) (x+\log (-4+x-\log (x)))} \, dx\\ &=5 e^4 x-e^4 x^3+e^4 \int \frac {1}{(-4+x-\log (x)) (x+\log (-4+x-\log (x)))} \, dx-e^4 \int \left (\frac {5 x}{x+\log (-4+x-\log (x))}-\frac {3 x^3}{x+\log (-4+x-\log (x))}\right ) \, dx-e^4 \int \log (x+\log (-4+x-\log (x))) \, dx-\left (3 e^4\right ) \int \frac {x^4}{(-4+x-\log (x)) (x+\log (-4+x-\log (x)))} \, dx+\left (3 e^4\right ) \int \frac {x^3 \log (x)}{(-4+x-\log (x)) (x+\log (-4+x-\log (x)))} \, dx+\left (4 e^4\right ) \int \frac {x^2}{(-4+x-\log (x)) (x+\log (-4+x-\log (x)))} \, dx-\left (4 e^4\right ) \int \frac {x \log (x)}{(-4+x-\log (x)) (x+\log (-4+x-\log (x)))} \, dx+\left (12 e^4\right ) \int \frac {x^3}{(-4+x-\log (x)) (x+\log (-4+x-\log (x)))} \, dx-\left (17 e^4\right ) \int \frac {x}{(-4+x-\log (x)) (x+\log (-4+x-\log (x)))} \, dx\\ &=5 e^4 x-e^4 x^3+e^4 \int \frac {1}{(-4+x-\log (x)) (x+\log (-4+x-\log (x)))} \, dx-e^4 \int \log (x+\log (-4+x-\log (x))) \, dx+\left (3 e^4\right ) \int \frac {x^3}{x+\log (-4+x-\log (x))} \, dx-\left (3 e^4\right ) \int \frac {x^4}{(-4+x-\log (x)) (x+\log (-4+x-\log (x)))} \, dx+\left (3 e^4\right ) \int \frac {x^3 \log (x)}{(-4+x-\log (x)) (x+\log (-4+x-\log (x)))} \, dx+\left (4 e^4\right ) \int \frac {x^2}{(-4+x-\log (x)) (x+\log (-4+x-\log (x)))} \, dx-\left (4 e^4\right ) \int \frac {x \log (x)}{(-4+x-\log (x)) (x+\log (-4+x-\log (x)))} \, dx-\left (5 e^4\right ) \int \frac {x}{x+\log (-4+x-\log (x))} \, dx+\left (12 e^4\right ) \int \frac {x^3}{(-4+x-\log (x)) (x+\log (-4+x-\log (x)))} \, dx-\left (17 e^4\right ) \int \frac {x}{(-4+x-\log (x)) (x+\log (-4+x-\log (x)))} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.13, size = 27, normalized size = 0.96 \begin {gather*} e^4 \left (5 x-x^3-x \log (x+\log (-4+x-\log (x)))\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^4*(-1 + 17*x - 4*x^2 - 12*x^3 + 3*x^4) + E^4*(4*x - 3*x^3)*Log[x] + (E^4*(20 - 5*x - 12*x^2 + 3*x
^3) + E^4*(5 - 3*x^2)*Log[x])*Log[-4 + x - Log[x]] + (E^4*(-4*x + x^2) - E^4*x*Log[x] + (E^4*(-4 + x) - E^4*Lo
g[x])*Log[-4 + x - Log[x]])*Log[x + Log[-4 + x - Log[x]]])/(4*x - x^2 + x*Log[x] + (4 - x + Log[x])*Log[-4 + x
- Log[x]]),x]

[Out]

E^4*(5*x - x^3 - x*Log[x + Log[-4 + x - Log[x]]])

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fricas [A]  time = 0.75, size = 28, normalized size = 1.00 \begin {gather*} -x e^{4} \log \left (x + \log \left (x - \log \relax (x) - 4\right )\right ) - {\left (x^{3} - 5 \, x\right )} e^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-exp(4)*log(x)+(x-4)*exp(4))*log(-log(x)+x-4)-x*exp(4)*log(x)+(x^2-4*x)*exp(4))*log(log(-log(x)+x
-4)+x)+((-3*x^2+5)*exp(4)*log(x)+(3*x^3-12*x^2-5*x+20)*exp(4))*log(-log(x)+x-4)+(-3*x^3+4*x)*exp(4)*log(x)+(3*
x^4-12*x^3-4*x^2+17*x-1)*exp(4))/((log(x)-x+4)*log(-log(x)+x-4)+x*log(x)-x^2+4*x),x, algorithm="fricas")

[Out]

-x*e^4*log(x + log(x - log(x) - 4)) - (x^3 - 5*x)*e^4

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giac [A]  time = 0.61, size = 29, normalized size = 1.04 \begin {gather*} -x^{3} e^{4} - x e^{4} \log \left (x + \log \left (x - \log \relax (x) - 4\right )\right ) + 5 \, x e^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-exp(4)*log(x)+(x-4)*exp(4))*log(-log(x)+x-4)-x*exp(4)*log(x)+(x^2-4*x)*exp(4))*log(log(-log(x)+x
-4)+x)+((-3*x^2+5)*exp(4)*log(x)+(3*x^3-12*x^2-5*x+20)*exp(4))*log(-log(x)+x-4)+(-3*x^3+4*x)*exp(4)*log(x)+(3*
x^4-12*x^3-4*x^2+17*x-1)*exp(4))/((log(x)-x+4)*log(-log(x)+x-4)+x*log(x)-x^2+4*x),x, algorithm="giac")

[Out]

-x^3*e^4 - x*e^4*log(x + log(x - log(x) - 4)) + 5*x*e^4

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maple [A]  time = 0.08, size = 28, normalized size = 1.00

method	result	size
risch	\(-x \,{\mathrm e}^{4} \ln \left (\ln \left (-\ln \relax (x )+x -4\right )+x \right )-{\mathrm e}^{4} x \left (x^{2}-5\right )\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-exp(4)*ln(x)+(x-4)*exp(4))*ln(-ln(x)+x-4)-x*exp(4)*ln(x)+(x^2-4*x)*exp(4))*ln(ln(-ln(x)+x-4)+x)+((-3*x
^2+5)*exp(4)*ln(x)+(3*x^3-12*x^2-5*x+20)*exp(4))*ln(-ln(x)+x-4)+(-3*x^3+4*x)*exp(4)*ln(x)+(3*x^4-12*x^3-4*x^2+
17*x-1)*exp(4))/((ln(x)-x+4)*ln(-ln(x)+x-4)+x*ln(x)-x^2+4*x),x,method=_RETURNVERBOSE)

[Out]

-x*exp(4)*ln(ln(-ln(x)+x-4)+x)-exp(4)*x*(x^2-5)

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maxima [A]  time = 0.39, size = 29, normalized size = 1.04 \begin {gather*} -x^{3} e^{4} - x e^{4} \log \left (x + \log \left (x - \log \relax (x) - 4\right )\right ) + 5 \, x e^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-exp(4)*log(x)+(x-4)*exp(4))*log(-log(x)+x-4)-x*exp(4)*log(x)+(x^2-4*x)*exp(4))*log(log(-log(x)+x
-4)+x)+((-3*x^2+5)*exp(4)*log(x)+(3*x^3-12*x^2-5*x+20)*exp(4))*log(-log(x)+x-4)+(-3*x^3+4*x)*exp(4)*log(x)+(3*
x^4-12*x^3-4*x^2+17*x-1)*exp(4))/((log(x)-x+4)*log(-log(x)+x-4)+x*log(x)-x^2+4*x),x, algorithm="maxima")

[Out]

-x^3*e^4 - x*e^4*log(x + log(x - log(x) - 4)) + 5*x*e^4

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mupad [B]  time = 1.15, size = 21, normalized size = 0.75 \begin {gather*} -x\,{\mathrm {e}}^4\,\left (\ln \left (x+\ln \left (x-\ln \relax (x)-4\right )\right )+x^2-5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x + log(x - log(x) - 4))*(exp(4)*(4*x - x^2) - log(x - log(x) - 4)*(exp(4)*(x - 4) - exp(4)*log(x))
+ x*exp(4)*log(x)) + exp(4)*(4*x^2 - 17*x + 12*x^3 - 3*x^4 + 1) + log(x - log(x) - 4)*(exp(4)*(5*x + 12*x^2 -
3*x^3 - 20) + exp(4)*log(x)*(3*x^2 - 5)) - exp(4)*log(x)*(4*x - 3*x^3))/(4*x + log(x - log(x) - 4)*(log(x) - x
+ 4) + x*log(x) - x^2),x)

[Out]

-x*exp(4)*(log(x + log(x - log(x) - 4)) + x^2 - 5)

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sympy [A]  time = 8.64, size = 29, normalized size = 1.04 \begin {gather*} - x^{3} e^{4} - x e^{4} \log {\left (x + \log {\left (x - \log {\relax (x )} - 4 \right )} \right )} + 5 x e^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-exp(4)*ln(x)+(x-4)*exp(4))*ln(-ln(x)+x-4)-x*exp(4)*ln(x)+(x**2-4*x)*exp(4))*ln(ln(-ln(x)+x-4)+x)
+((-3*x**2+5)*exp(4)*ln(x)+(3*x**3-12*x**2-5*x+20)*exp(4))*ln(-ln(x)+x-4)+(-3*x**3+4*x)*exp(4)*ln(x)+(3*x**4-1
2*x**3-4*x**2+17*x-1)*exp(4))/((ln(x)-x+4)*ln(-ln(x)+x-4)+x*ln(x)-x**2+4*x),x)

[Out]

-x**3*exp(4) - x*exp(4)*log(x + log(x - log(x) - 4)) + 5*x*exp(4)

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