3.50.59 \(\int \frac {4-2 x+e^x (-4+2 x)+(12-18 x+12 x^2+e^x (-12+14 x-10 x^2)) \log (x)+(-8+e^x (8-8 x)+8 x) \log (x) \log ((x-e^x x) \log (x))}{(8 x^3-12 x^4+6 x^5-x^6+e^x (-8 x^3+12 x^4-6 x^5+x^6)) \log (x)} \, dx\)

Optimal. Leaf size=29 \[ \frac {2 \left (-1+2 x+\log \left (\left (1-e^x\right ) x \log (x)\right )\right )}{(2-x)^2 x^2} \]

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Rubi [F]  time = 4.23, 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 {4-2 x+e^x (-4+2 x)+\left (12-18 x+12 x^2+e^x \left (-12+14 x-10 x^2\right )\right ) \log (x)+\left (-8+e^x (8-8 x)+8 x\right ) \log (x) \log \left (\left (x-e^x x\right ) \log (x)\right )}{\left (8 x^3-12 x^4+6 x^5-x^6+e^x \left (-8 x^3+12 x^4-6 x^5+x^6\right )\right ) \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(4 - 2*x + E^x*(-4 + 2*x) + (12 - 18*x + 12*x^2 + E^x*(-12 + 14*x - 10*x^2))*Log[x] + (-8 + E^x*(8 - 8*x)
+ 8*x)*Log[x]*Log[(x - E^x*x)*Log[x]])/((8*x^3 - 12*x^4 + 6*x^5 - x^6 + E^x*(-8*x^3 + 12*x^4 - 6*x^5 + x^6))*L
og[x]),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (-\frac {-2+x}{\log (x)}-\frac {6-9 x+6 x^2+e^x \left (-6+7 x-5 x^2\right )-4 \left (-1+e^x\right ) (-1+x) \log \left (-\left (\left (-1+e^x\right ) x \log (x)\right )\right )}{-1+e^x}\right )}{(2-x)^3 x^3} \, dx\\ &=2 \int \frac {-\frac {-2+x}{\log (x)}-\frac {6-9 x+6 x^2+e^x \left (-6+7 x-5 x^2\right )-4 \left (-1+e^x\right ) (-1+x) \log \left (-\left (\left (-1+e^x\right ) x \log (x)\right )\right )}{-1+e^x}}{(2-x)^3 x^3} \, dx\\ &=2 \int \left (\frac {1}{\left (-1+e^x\right ) (-2+x)^2 x^2}+\frac {-2+x-6 \log (x)+7 x \log (x)-5 x^2 \log (x)+4 \log (x) \log \left (-\left (\left (-1+e^x\right ) x \log (x)\right )\right )-4 x \log (x) \log \left (-\left (\left (-1+e^x\right ) x \log (x)\right )\right )}{(-2+x)^3 x^3 \log (x)}\right ) \, dx\\ &=2 \int \frac {1}{\left (-1+e^x\right ) (-2+x)^2 x^2} \, dx+2 \int \frac {-2+x-6 \log (x)+7 x \log (x)-5 x^2 \log (x)+4 \log (x) \log \left (-\left (\left (-1+e^x\right ) x \log (x)\right )\right )-4 x \log (x) \log \left (-\left (\left (-1+e^x\right ) x \log (x)\right )\right )}{(-2+x)^3 x^3 \log (x)} \, dx\\ &=2 \int \left (\frac {1}{4 \left (-1+e^x\right ) (-2+x)^2}-\frac {1}{4 \left (-1+e^x\right ) (-2+x)}+\frac {1}{4 \left (-1+e^x\right ) x^2}+\frac {1}{4 \left (-1+e^x\right ) x}\right ) \, dx+2 \int \left (-\frac {6}{(-2+x)^3 x^3}+\frac {7}{(-2+x)^3 x^2}-\frac {5}{(-2+x)^3 x}-\frac {2}{(-2+x)^3 x^3 \log (x)}+\frac {1}{(-2+x)^3 x^2 \log (x)}-\frac {4 (-1+x) \log \left (-\left (\left (-1+e^x\right ) x \log (x)\right )\right )}{(-2+x)^3 x^3}\right ) \, dx\\ &=\frac {1}{2} \int \frac {1}{\left (-1+e^x\right ) (-2+x)^2} \, dx-\frac {1}{2} \int \frac {1}{\left (-1+e^x\right ) (-2+x)} \, dx+\frac {1}{2} \int \frac {1}{\left (-1+e^x\right ) x^2} \, dx+\frac {1}{2} \int \frac {1}{\left (-1+e^x\right ) x} \, dx+2 \int \frac {1}{(-2+x)^3 x^2 \log (x)} \, dx-4 \int \frac {1}{(-2+x)^3 x^3 \log (x)} \, dx-8 \int \frac {(-1+x) \log \left (-\left (\left (-1+e^x\right ) x \log (x)\right )\right )}{(-2+x)^3 x^3} \, dx-10 \int \frac {1}{(-2+x)^3 x} \, dx-12 \int \frac {1}{(-2+x)^3 x^3} \, dx+14 \int \frac {1}{(-2+x)^3 x^2} \, dx\\ &=\frac {2 \log \left (\left (1-e^x\right ) x \log (x)\right )}{(2-x)^2 x^2}+\frac {1}{2} \int \frac {1}{\left (-1+e^x\right ) (-2+x)^2} \, dx-\frac {1}{2} \int \frac {1}{\left (-1+e^x\right ) (-2+x)} \, dx+\frac {1}{2} \int \frac {1}{\left (-1+e^x\right ) x^2} \, dx+\frac {1}{2} \int \frac {1}{\left (-1+e^x\right ) x} \, dx+2 \int \frac {1}{(-2+x)^3 x^2 \log (x)} \, dx-4 \int \frac {1}{(-2+x)^3 x^3 \log (x)} \, dx+8 \int \frac {-1+e^x-\log (x)+e^x (1+x) \log (x)}{4 \left (1-e^x\right ) (2-x)^2 x^3 \log (x)} \, dx-10 \int \left (\frac {1}{2 (-2+x)^3}-\frac {1}{4 (-2+x)^2}+\frac {1}{8 (-2+x)}-\frac {1}{8 x}\right ) \, dx-12 \int \left (\frac {1}{8 (-2+x)^3}-\frac {3}{16 (-2+x)^2}+\frac {3}{16 (-2+x)}-\frac {1}{8 x^3}-\frac {3}{16 x^2}-\frac {3}{16 x}\right ) \, dx+14 \int \left (\frac {1}{4 (-2+x)^3}-\frac {1}{4 (-2+x)^2}+\frac {3}{16 (-2+x)}-\frac {1}{8 x^2}-\frac {3}{16 x}\right ) \, dx\\ &=\frac {3}{2 (2-x)^2}+\frac {5}{4 (2-x)}-\frac {3}{4 x^2}-\frac {1}{2 x}-\frac {7}{8} \log (2-x)+\frac {7 \log (x)}{8}+\frac {2 \log \left (\left (1-e^x\right ) x \log (x)\right )}{(2-x)^2 x^2}+\frac {1}{2} \int \frac {1}{\left (-1+e^x\right ) (-2+x)^2} \, dx-\frac {1}{2} \int \frac {1}{\left (-1+e^x\right ) (-2+x)} \, dx+\frac {1}{2} \int \frac {1}{\left (-1+e^x\right ) x^2} \, dx+\frac {1}{2} \int \frac {1}{\left (-1+e^x\right ) x} \, dx+2 \int \frac {1}{(-2+x)^3 x^2 \log (x)} \, dx+2 \int \frac {-1+e^x-\log (x)+e^x (1+x) \log (x)}{\left (1-e^x\right ) (2-x)^2 x^3 \log (x)} \, dx-4 \int \frac {1}{(-2+x)^3 x^3 \log (x)} \, dx\\ &=\frac {3}{2 (2-x)^2}+\frac {5}{4 (2-x)}-\frac {3}{4 x^2}-\frac {1}{2 x}-\frac {7}{8} \log (2-x)+\frac {7 \log (x)}{8}+\frac {2 \log \left (\left (1-e^x\right ) x \log (x)\right )}{(2-x)^2 x^2}+\frac {1}{2} \int \frac {1}{\left (-1+e^x\right ) (-2+x)^2} \, dx-\frac {1}{2} \int \frac {1}{\left (-1+e^x\right ) (-2+x)} \, dx+\frac {1}{2} \int \frac {1}{\left (-1+e^x\right ) x^2} \, dx+\frac {1}{2} \int \frac {1}{\left (-1+e^x\right ) x} \, dx+2 \int \frac {1}{(-2+x)^3 x^2 \log (x)} \, dx+2 \int \left (-\frac {1}{\left (-1+e^x\right ) (-2+x)^2 x^2}+\frac {-1-\log (x)-x \log (x)}{(-2+x)^2 x^3 \log (x)}\right ) \, dx-4 \int \frac {1}{(-2+x)^3 x^3 \log (x)} \, dx\\ &=\frac {3}{2 (2-x)^2}+\frac {5}{4 (2-x)}-\frac {3}{4 x^2}-\frac {1}{2 x}-\frac {7}{8} \log (2-x)+\frac {7 \log (x)}{8}+\frac {2 \log \left (\left (1-e^x\right ) x \log (x)\right )}{(2-x)^2 x^2}+\frac {1}{2} \int \frac {1}{\left (-1+e^x\right ) (-2+x)^2} \, dx-\frac {1}{2} \int \frac {1}{\left (-1+e^x\right ) (-2+x)} \, dx+\frac {1}{2} \int \frac {1}{\left (-1+e^x\right ) x^2} \, dx+\frac {1}{2} \int \frac {1}{\left (-1+e^x\right ) x} \, dx-2 \int \frac {1}{\left (-1+e^x\right ) (-2+x)^2 x^2} \, dx+2 \int \frac {1}{(-2+x)^3 x^2 \log (x)} \, dx+2 \int \frac {-1-\log (x)-x \log (x)}{(-2+x)^2 x^3 \log (x)} \, dx-4 \int \frac {1}{(-2+x)^3 x^3 \log (x)} \, dx\\ &=\frac {3}{2 (2-x)^2}+\frac {5}{4 (2-x)}-\frac {3}{4 x^2}-\frac {1}{2 x}-\frac {7}{8} \log (2-x)+\frac {7 \log (x)}{8}+\frac {2 \log \left (\left (1-e^x\right ) x \log (x)\right )}{(2-x)^2 x^2}+\frac {1}{2} \int \frac {1}{\left (-1+e^x\right ) (-2+x)^2} \, dx-\frac {1}{2} \int \frac {1}{\left (-1+e^x\right ) (-2+x)} \, dx+\frac {1}{2} \int \frac {1}{\left (-1+e^x\right ) x^2} \, dx+\frac {1}{2} \int \frac {1}{\left (-1+e^x\right ) x} \, dx-2 \int \left (\frac {1}{4 \left (-1+e^x\right ) (-2+x)^2}-\frac {1}{4 \left (-1+e^x\right ) (-2+x)}+\frac {1}{4 \left (-1+e^x\right ) x^2}+\frac {1}{4 \left (-1+e^x\right ) x}\right ) \, dx+2 \int \left (\frac {-1-x}{(-2+x)^2 x^3}-\frac {1}{(-2+x)^2 x^3 \log (x)}\right ) \, dx+2 \int \frac {1}{(-2+x)^3 x^2 \log (x)} \, dx-4 \int \frac {1}{(-2+x)^3 x^3 \log (x)} \, dx\\ &=\frac {3}{2 (2-x)^2}+\frac {5}{4 (2-x)}-\frac {3}{4 x^2}-\frac {1}{2 x}-\frac {7}{8} \log (2-x)+\frac {7 \log (x)}{8}+\frac {2 \log \left (\left (1-e^x\right ) x \log (x)\right )}{(2-x)^2 x^2}+2 \int \frac {-1-x}{(-2+x)^2 x^3} \, dx-2 \int \frac {1}{(-2+x)^2 x^3 \log (x)} \, dx+2 \int \frac {1}{(-2+x)^3 x^2 \log (x)} \, dx-4 \int \frac {1}{(-2+x)^3 x^3 \log (x)} \, dx\\ &=\frac {3}{2 (2-x)^2}+\frac {5}{4 (2-x)}-\frac {3}{4 x^2}-\frac {1}{2 x}-\frac {7}{8} \log (2-x)+\frac {7 \log (x)}{8}+\frac {2 \log \left (\left (1-e^x\right ) x \log (x)\right )}{(2-x)^2 x^2}+2 \int \left (-\frac {3}{8 (-2+x)^2}+\frac {7}{16 (-2+x)}-\frac {1}{4 x^3}-\frac {1}{2 x^2}-\frac {7}{16 x}\right ) \, dx-2 \int \frac {1}{(-2+x)^2 x^3 \log (x)} \, dx+2 \int \frac {1}{(-2+x)^3 x^2 \log (x)} \, dx-4 \int \frac {1}{(-2+x)^3 x^3 \log (x)} \, dx\\ &=\frac {3}{2 (2-x)^2}+\frac {1}{2 (2-x)}-\frac {1}{2 x^2}+\frac {1}{2 x}+\frac {2 \log \left (\left (1-e^x\right ) x \log (x)\right )}{(2-x)^2 x^2}-2 \int \frac {1}{(-2+x)^2 x^3 \log (x)} \, dx+2 \int \frac {1}{(-2+x)^3 x^2 \log (x)} \, dx-4 \int \frac {1}{(-2+x)^3 x^3 \log (x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.80, size = 26, normalized size = 0.90 \begin {gather*} \frac {2 \left (-1+2 x+\log \left (-\left (\left (-1+e^x\right ) x \log (x)\right )\right )\right )}{(-2+x)^2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4 - 2*x + E^x*(-4 + 2*x) + (12 - 18*x + 12*x^2 + E^x*(-12 + 14*x - 10*x^2))*Log[x] + (-8 + E^x*(8 -
8*x) + 8*x)*Log[x]*Log[(x - E^x*x)*Log[x]])/((8*x^3 - 12*x^4 + 6*x^5 - x^6 + E^x*(-8*x^3 + 12*x^4 - 6*x^5 + x
^6))*Log[x]),x]

[Out]

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

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fricas [A]  time = 0.68, size = 36, normalized size = 1.24 \begin {gather*} \frac {2 \, {\left (2 \, x + \log \left (-{\left (x e^{x} - x\right )} \log \relax (x)\right ) - 1\right )}}{x^{4} - 4 \, x^{3} + 4 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-8*x+8)*exp(x)+8*x-8)*log(x)*log((x-exp(x)*x)*log(x))+((-10*x^2+14*x-12)*exp(x)+12*x^2-18*x+12)*l
og(x)+(2*x-4)*exp(x)+4-2*x)/((x^6-6*x^5+12*x^4-8*x^3)*exp(x)-x^6+6*x^5-12*x^4+8*x^3)/log(x),x, algorithm="fric
as")

[Out]

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

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-8*x+8)*exp(x)+8*x-8)*log(x)*log((x-exp(x)*x)*log(x))+((-10*x^2+14*x-12)*exp(x)+12*x^2-18*x+12)*l
og(x)+(2*x-4)*exp(x)+4-2*x)/((x^6-6*x^5+12*x^4-8*x^3)*exp(x)-x^6+6*x^5-12*x^4+8*x^3)/log(x),x, algorithm="giac
")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Simplification assuming t_nostep near 0Simplification assuming t_nostep near 0Simplification assuming t_nos
tep near 0S

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maple [C]  time = 0.20, size = 257, normalized size = 8.86

method	result	size
risch	\(\frac {2 \ln \left ({\mathrm e}^{x}-1\right )}{\left (x^{2}-4 x +4\right ) x^{2}}+\frac {i \pi \mathrm {csgn}\left (i x \ln \relax (x ) \left ({\mathrm e}^{x}-1\right )\right )^{3}-i \pi \mathrm {csgn}\left (i \ln \relax (x ) \left ({\mathrm e}^{x}-1\right )\right )^{3}+i \pi \,\mathrm {csgn}\left (i \ln \relax (x ) \left ({\mathrm e}^{x}-1\right )\right ) \mathrm {csgn}\left (i x \ln \relax (x ) \left ({\mathrm e}^{x}-1\right )\right )^{2}+i \pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{x}-1\right )\right ) \mathrm {csgn}\left (i \ln \relax (x ) \left ({\mathrm e}^{x}-1\right )\right )^{2}+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \ln \relax (x ) \left ({\mathrm e}^{x}-1\right )\right )^{2}-2 i \pi \mathrm {csgn}\left (i x \ln \relax (x ) \left ({\mathrm e}^{x}-1\right )\right )^{2}+i \pi \,\mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i \ln \relax (x ) \left ({\mathrm e}^{x}-1\right )\right )^{2}-i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \relax (x ) \left ({\mathrm e}^{x}-1\right )\right ) \mathrm {csgn}\left (i x \ln \relax (x ) \left ({\mathrm e}^{x}-1\right )\right )-i \pi \,\mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-1\right )\right ) \mathrm {csgn}\left (i \ln \relax (x ) \left ({\mathrm e}^{x}-1\right )\right )+2 i \pi -2+4 x +2 \ln \relax (x )+2 \ln \left (\ln \relax (x )\right )}{\left (x^{2}-4 x +4\right ) x^{2}}\)	\(257\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-8*x+8)*exp(x)+8*x-8)*ln(x)*ln((x-exp(x)*x)*ln(x))+((-10*x^2+14*x-12)*exp(x)+12*x^2-18*x+12)*ln(x)+(2*x
-4)*exp(x)+4-2*x)/((x^6-6*x^5+12*x^4-8*x^3)*exp(x)-x^6+6*x^5-12*x^4+8*x^3)/ln(x),x,method=_RETURNVERBOSE)

[Out]

2/(x^2-4*x+4)/x^2*ln(exp(x)-1)+(I*Pi*csgn(I*x*ln(x)*(exp(x)-1))^3-I*Pi*csgn(I*ln(x)*(exp(x)-1))^3+I*Pi*csgn(I*
ln(x)*(exp(x)-1))*csgn(I*x*ln(x)*(exp(x)-1))^2+I*Pi*csgn(I*(exp(x)-1))*csgn(I*ln(x)*(exp(x)-1))^2+I*Pi*csgn(I*
x)*csgn(I*x*ln(x)*(exp(x)-1))^2-2*I*Pi*csgn(I*x*ln(x)*(exp(x)-1))^2+I*Pi*csgn(I*ln(x))*csgn(I*ln(x)*(exp(x)-1)
)^2-I*Pi*csgn(I*x)*csgn(I*ln(x)*(exp(x)-1))*csgn(I*x*ln(x)*(exp(x)-1))-I*Pi*csgn(I*ln(x))*csgn(I*(exp(x)-1))*c
sgn(I*ln(x)*(exp(x)-1))+2*I*Pi-2+4*x+2*ln(x)+2*ln(ln(x)))/(x^2-4*x+4)/x^2

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maxima [A]  time = 0.41, size = 35, normalized size = 1.21 \begin {gather*} \frac {2 \, {\left (2 \, x + \log \relax (x) + \log \left (-e^{x} + 1\right ) + \log \left (\log \relax (x)\right ) - 1\right )}}{x^{4} - 4 \, x^{3} + 4 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-8*x+8)*exp(x)+8*x-8)*log(x)*log((x-exp(x)*x)*log(x))+((-10*x^2+14*x-12)*exp(x)+12*x^2-18*x+12)*l
og(x)+(2*x-4)*exp(x)+4-2*x)/((x^6-6*x^5+12*x^4-8*x^3)*exp(x)-x^6+6*x^5-12*x^4+8*x^3)/log(x),x, algorithm="maxi
ma")

[Out]

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

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mupad [B]  time = 3.52, size = 26, normalized size = 0.90 \begin {gather*} \frac {2\,\left (2\,x+\ln \left (\ln \relax (x)\,\left (x-x\,{\mathrm {e}}^x\right )\right )-1\right )}{x^2\,{\left (x-2\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x + log(x)*(18*x + exp(x)*(10*x^2 - 14*x + 12) - 12*x^2 - 12) - exp(x)*(2*x - 4) + log(log(x)*(x - x*ex
p(x)))*log(x)*(exp(x)*(8*x - 8) - 8*x + 8) - 4)/(log(x)*(exp(x)*(8*x^3 - 12*x^4 + 6*x^5 - x^6) - 8*x^3 + 12*x^
4 - 6*x^5 + x^6)),x)

[Out]

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

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sympy [A]  time = 0.78, size = 44, normalized size = 1.52 \begin {gather*} - \frac {2 - 4 x}{x^{4} - 4 x^{3} + 4 x^{2}} + \frac {2 \log {\left (\left (- x e^{x} + x\right ) \log {\relax (x )} \right )}}{x^{4} - 4 x^{3} + 4 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-8*x+8)*exp(x)+8*x-8)*ln(x)*ln((x-exp(x)*x)*ln(x))+((-10*x**2+14*x-12)*exp(x)+12*x**2-18*x+12)*ln
(x)+(2*x-4)*exp(x)+4-2*x)/((x**6-6*x**5+12*x**4-8*x**3)*exp(x)-x**6+6*x**5-12*x**4+8*x**3)/ln(x),x)

[Out]

-(2 - 4*x)/(x**4 - 4*x**3 + 4*x**2) + 2*log((-x*exp(x) + x)*log(x))/(x**4 - 4*x**3 + 4*x**2)

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