3.80.88 \(\int \frac {-1-x+e^{3 x} (-3 x-3 x^2)+e^x (1+2 x+x^2+x^3)+(1+3 x+2 x^2+x^3+e^{2 x} (-3 x-3 x^2)) \log (x)+(e^x (x+x^2)+(x+x^2) \log (x)) \log (\frac {x+x^2}{e^x+\log (x)})}{e^{3 x} (-x-x^2)+e^x (x^2+x^3)+(x^2+x^3+e^{2 x} (-x-x^2)) \log (x)+(e^x (x+x^2)+(x+x^2) \log (x)) \log (\frac {x+x^2}{e^x+\log (x)})} \, dx\)

Optimal. Leaf size=28 \[ \log \left (e^x \left (-e^{2 x}+x+\log \left (\frac {x (1+x)}{e^x+\log (x)}\right )\right )\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 180.00, 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*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

$Aborted

Rubi steps

Aborted

________________________________________________________________________________________

Mathematica [A]  time = 0.28, size = 28, normalized size = 1.00 \begin {gather*} x+\log \left (e^{2 x}-x-\log \left (\frac {x (1+x)}{e^x+\log (x)}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.62, size = 25, normalized size = 0.89 \begin {gather*} x + \log \left (x - e^{\left (2 \, x\right )} + \log \left (\frac {x^{2} + x}{e^{x} + \log \relax (x)}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + log(x - e^(2*x) + log((x^2 + x)/(e^x + log(x))))

________________________________________________________________________________________

giac [A]  time = 0.33, size = 25, normalized size = 0.89 \begin {gather*} x + \log \left (x - e^{\left (2 \, x\right )} + \log \left (x + 1\right ) + \log \relax (x) - \log \left (e^{x} + \log \relax (x)\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + log(x - e^(2*x) + log(x + 1) + log(x) - log(e^x + log(x)))

________________________________________________________________________________________

maple [C]  time = 0.19, size = 260, normalized size = 9.29

method	result	size
risch	\(x +\ln \left (\ln \left (\ln \relax (x )+{\mathrm e}^{x}\right )-\frac {i \left (\pi \,\mathrm {csgn}\left (\frac {i}{\ln \relax (x )+{\mathrm e}^{x}}\right ) \mathrm {csgn}\left (\frac {i \left (x +1\right )}{\ln \relax (x )+{\mathrm e}^{x}}\right )^{2}-\pi \,\mathrm {csgn}\left (\frac {i}{\ln \relax (x )+{\mathrm e}^{x}}\right ) \mathrm {csgn}\left (\frac {i \left (x +1\right )}{\ln \relax (x )+{\mathrm e}^{x}}\right ) \mathrm {csgn}\left (i \left (x +1\right )\right )-\pi \mathrm {csgn}\left (\frac {i \left (x +1\right )}{\ln \relax (x )+{\mathrm e}^{x}}\right )^{3}+\pi \mathrm {csgn}\left (\frac {i \left (x +1\right )}{\ln \relax (x )+{\mathrm e}^{x}}\right )^{2} \mathrm {csgn}\left (i \left (x +1\right )\right )+\pi \,\mathrm {csgn}\left (\frac {i \left (x +1\right )}{\ln \relax (x )+{\mathrm e}^{x}}\right ) \mathrm {csgn}\left (\frac {i x \left (x +1\right )}{\ln \relax (x )+{\mathrm e}^{x}}\right )^{2}-\pi \,\mathrm {csgn}\left (\frac {i \left (x +1\right )}{\ln \relax (x )+{\mathrm e}^{x}}\right ) \mathrm {csgn}\left (\frac {i x \left (x +1\right )}{\ln \relax (x )+{\mathrm e}^{x}}\right ) \mathrm {csgn}\left (i x \right )-\pi \mathrm {csgn}\left (\frac {i x \left (x +1\right )}{\ln \relax (x )+{\mathrm e}^{x}}\right )^{3}+\pi \mathrm {csgn}\left (\frac {i x \left (x +1\right )}{\ln \relax (x )+{\mathrm e}^{x}}\right )^{2} \mathrm {csgn}\left (i x \right )+2 i {\mathrm e}^{2 x}-2 i x -2 i \ln \relax (x )-2 i \ln \left (x +1\right )\right )}{2}\right )\)	\(260\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.54, size = 27, normalized size = 0.96 \begin {gather*} x + \log \left (-x + e^{\left (2 \, x\right )} - \log \left (x + 1\right ) - \log \relax (x) + \log \left (e^{x} + \log \relax (x)\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + log(-x + e^(2*x) - log(x + 1) - log(x) + log(e^x + log(x)))

________________________________________________________________________________________

mupad [B]  time = 5.62, size = 24, normalized size = 0.86 \begin {gather*} x+\ln \left (x-{\mathrm {e}}^{2\,x}+\ln \left (\frac {x\,\left (x+1\right )}{{\mathrm {e}}^x+\ln \relax (x)}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x + exp(3*x)*(3*x + 3*x^2) - exp(x)*(2*x + x^2 + x^3 + 1) - log(x)*(3*x - exp(2*x)*(3*x + 3*x^2) + 2*x^2
+ x^3 + 1) - log((x + x^2)/(exp(x) + log(x)))*(exp(x)*(x + x^2) + log(x)*(x + x^2)) + 1)/(exp(x)*(x^2 + x^3)
- exp(3*x)*(x + x^2) + log((x + x^2)/(exp(x) + log(x)))*(exp(x)*(x + x^2) + log(x)*(x + x^2)) + log(x)*(x^2 -
exp(2*x)*(x + x^2) + x^3)),x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 1.98, size = 22, normalized size = 0.79 \begin {gather*} x + \log {\left (x - e^{2 x} + \log {\left (\frac {x^{2} + x}{e^{x} + \log {\relax (x )}} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x**2+x)*ln(x)+(x**2+x)*exp(x))*ln((x**2+x)/(ln(x)+exp(x)))+((-3*x**2-3*x)*exp(x)**2+x**3+2*x**2+3
*x+1)*ln(x)+(-3*x**2-3*x)*exp(x)**3+(x**3+x**2+2*x+1)*exp(x)-x-1)/(((x**2+x)*ln(x)+(x**2+x)*exp(x))*ln((x**2+x
)/(ln(x)+exp(x)))+((-x**2-x)*exp(x)**2+x**3+x**2)*ln(x)+(-x**2-x)*exp(x)**3+(x**3+x**2)*exp(x)),x)

[Out]

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

________________________________________________________________________________________