3.64.62 \(\int \frac {-x-x^2+(x^2+x^3) \log (x)+e^{-2 x} \log ^2(4) (-1+x \log (x))+e^{-x} \log (4) (-1-2 x+(x+2 x^2) \log (x))+(2 x^2 \log (x)+e^{-x} (2+6 x) \log (4) \log (x)+2 e^{-2 x} \log ^2(4) \log (x)) \log (\frac {x+x^2+e^{-x} x \log (4)}{x+e^{-x} \log (4)})}{(x^2+x^3) \log (x)+e^{-x} (x+2 x^2) \log (4) \log (x)+e^{-2 x} x \log ^2(4) \log (x)} \, dx\)

Optimal. Leaf size=26 \[ x+\log ^2\left (x+\frac {x}{x+e^{-x} \log (4)}\right )-\log (\log (x)) \]

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Rubi [F]  time = 9.01, 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 {-x-x^2+\left (x^2+x^3\right ) \log (x)+e^{-2 x} \log ^2(4) (-1+x \log (x))+e^{-x} \log (4) \left (-1-2 x+\left (x+2 x^2\right ) \log (x)\right )+\left (2 x^2 \log (x)+e^{-x} (2+6 x) \log (4) \log (x)+2 e^{-2 x} \log ^2(4) \log (x)\right ) \log \left (\frac {x+x^2+e^{-x} x \log (4)}{x+e^{-x} \log (4)}\right )}{\left (x^2+x^3\right ) \log (x)+e^{-x} \left (x+2 x^2\right ) \log (4) \log (x)+e^{-2 x} x \log ^2(4) \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-x - x^2 + (x^2 + x^3)*Log[x] + (Log[4]^2*(-1 + x*Log[x]))/E^(2*x) + (Log[4]*(-1 - 2*x + (x + 2*x^2)*Log[
x]))/E^x + (2*x^2*Log[x] + ((2 + 6*x)*Log[4]*Log[x])/E^x + (2*Log[4]^2*Log[x])/E^(2*x))*Log[(x + x^2 + (x*Log[
4])/E^x)/(x + Log[4]/E^x)])/((x^2 + x^3)*Log[x] + ((x + 2*x^2)*Log[4]*Log[x])/E^x + (x*Log[4]^2*Log[x])/E^(2*x
)),x]

[Out]

x - Log[Log[x]] + 2*Log[4]*Log[(x*(E^x*(1 + x) + Log[4]))/(E^x*x + Log[4])]*Defer[Int][(E^x*x + Log[4])^(-1),
x] + 2*Log[4]*Log[(x*(E^x*(1 + x) + Log[4]))/(E^x*x + Log[4])]*Defer[Int][1/(x*(E^x*x + Log[4])), x] - 2*Log[4
]*Log[(x*(E^x*(1 + x) + Log[4]))/(E^x*x + Log[4])]*Defer[Int][(E^x + E^x*x + Log[4])^(-1), x] - 2*Log[4]*Log[(
x*(E^x*(1 + x) + Log[4]))/(E^x*x + Log[4])]*Defer[Int][1/((1 + x)*(E^x + E^x*x + Log[4])), x] + 2*Defer[Int][L
og[(x*(E^x*(1 + x) + Log[4]))/(E^x*x + Log[4])]/(1 + x), x] - 2*Log[4]*Defer[Int][Defer[Int][(E^x*x + Log[4])^
(-1), x]/(1 + x), x] - 2*Log[4]^2*Defer[Int][Defer[Int][(E^x*x + Log[4])^(-1), x]/(E^x*x + Log[4]), x] - 2*Log
[4]^2*Defer[Int][Defer[Int][(E^x*x + Log[4])^(-1), x]/(x*(E^x*x + Log[4])), x] + 2*Log[4]^2*Defer[Int][Defer[I
nt][(E^x*x + Log[4])^(-1), x]/(E^x + E^x*x + Log[4]), x] + 2*Log[4]^2*Defer[Int][Defer[Int][(E^x*x + Log[4])^(
-1), x]/((1 + x)*(E^x + E^x*x + Log[4])), x] + 2*Log[4]*Defer[Int][Defer[Int][(E^x*(1 + x) + Log[4])^(-1), x]/
(1 + x), x] + 2*Log[4]^2*Defer[Int][Defer[Int][(E^x*(1 + x) + Log[4])^(-1), x]/(E^x*x + Log[4]), x] + 2*Log[4]
^2*Defer[Int][Defer[Int][(E^x*(1 + x) + Log[4])^(-1), x]/(x*(E^x*x + Log[4])), x] - 2*Log[4]^2*Defer[Int][Defe
r[Int][(E^x*(1 + x) + Log[4])^(-1), x]/(E^x + E^x*x + Log[4]), x] - 2*Log[4]^2*Defer[Int][Defer[Int][(E^x*(1 +
x) + Log[4])^(-1), x]/((1 + x)*(E^x + E^x*x + Log[4])), x] + 2*Log[4]*Defer[Int][Defer[Int][1/((1 + x)*(E^x*(
1 + x) + Log[4])), x]/(1 + x), x] + 2*Log[4]^2*Defer[Int][Defer[Int][1/((1 + x)*(E^x*(1 + x) + Log[4])), x]/(E
^x*x + Log[4]), x] + 2*Log[4]^2*Defer[Int][Defer[Int][1/((1 + x)*(E^x*(1 + x) + Log[4])), x]/(x*(E^x*x + Log[4
])), x] - 2*Log[4]^2*Defer[Int][Defer[Int][1/((1 + x)*(E^x*(1 + x) + Log[4])), x]/(E^x + E^x*x + Log[4]), x] -
2*Log[4]^2*Defer[Int][Defer[Int][1/((1 + x)*(E^x*(1 + x) + Log[4])), x]/((1 + x)*(E^x + E^x*x + Log[4])), x]
- 2*Log[4]*Defer[Int][Defer[Int][(E^x*x^2 + x*Log[4])^(-1), x]/(1 + x), x] - 2*Log[4]^2*Defer[Int][Defer[Int][
(E^x*x^2 + x*Log[4])^(-1), x]/(E^x*x + Log[4]), x] - 2*Log[4]^2*Defer[Int][Defer[Int][(E^x*x^2 + x*Log[4])^(-1
), x]/(x*(E^x*x + Log[4])), x] + 2*Log[4]^2*Defer[Int][Defer[Int][(E^x*x^2 + x*Log[4])^(-1), x]/(E^x + E^x*x +
Log[4]), x] + 2*Log[4]^2*Defer[Int][Defer[Int][(E^x*x^2 + x*Log[4])^(-1), x]/((1 + x)*(E^x + E^x*x + Log[4]))
, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {2 (1+x) \log (4) \log \left (\frac {x \left (e^x (1+x)+\log (4)\right )}{e^x x+\log (4)}\right )}{x \left (e^x x+\log (4)\right )}-\frac {2 (2+x) \log (4) \log \left (\frac {x \left (e^x (1+x)+\log (4)\right )}{e^x x+\log (4)}\right )}{(1+x) \left (e^x+e^x x+\log (4)\right )}+\frac {-1-x+x \log (x)+x^2 \log (x)+2 x \log (x) \log \left (\frac {x \left (e^x (1+x)+\log (4)\right )}{e^x x+\log (4)}\right )}{x (1+x) \log (x)}\right ) \, dx\\ &=(2 \log (4)) \int \frac {(1+x) \log \left (\frac {x \left (e^x (1+x)+\log (4)\right )}{e^x x+\log (4)}\right )}{x \left (e^x x+\log (4)\right )} \, dx-(2 \log (4)) \int \frac {(2+x) \log \left (\frac {x \left (e^x (1+x)+\log (4)\right )}{e^x x+\log (4)}\right )}{(1+x) \left (e^x+e^x x+\log (4)\right )} \, dx+\int \frac {-1-x+x \log (x)+x^2 \log (x)+2 x \log (x) \log \left (\frac {x \left (e^x (1+x)+\log (4)\right )}{e^x x+\log (4)}\right )}{x (1+x) \log (x)} \, dx\\ &=(2 \log (4)) \int \frac {\left (e^{2 x} x^2+e^x (1+3 x) \log (4)+\log ^2(4)\right ) \left (\int \frac {1}{e^x (1+x)+\log (4)} \, dx+\int \frac {1}{(1+x) \left (e^x (1+x)+\log (4)\right )} \, dx\right )}{x \left (e^x x+\log (4)\right ) \left (e^x (1+x)+\log (4)\right )} \, dx-(2 \log (4)) \int \frac {\left (e^{2 x} x^2+e^x (1+3 x) \log (4)+\log ^2(4)\right ) \left (\int \frac {1}{e^x x+\log (4)} \, dx+\int \frac {1}{e^x x^2+x \log (4)} \, dx\right )}{x \left (e^x x+\log (4)\right ) \left (e^x (1+x)+\log (4)\right )} \, dx+\left (2 \log (4) \log \left (\frac {x \left (e^x (1+x)+\log (4)\right )}{e^x x+\log (4)}\right )\right ) \int \frac {1}{e^x x+\log (4)} \, dx+\left (2 \log (4) \log \left (\frac {x \left (e^x (1+x)+\log (4)\right )}{e^x x+\log (4)}\right )\right ) \int \frac {1}{x \left (e^x x+\log (4)\right )} \, dx-\left (2 \log (4) \log \left (\frac {x \left (e^x (1+x)+\log (4)\right )}{e^x x+\log (4)}\right )\right ) \int \frac {1}{e^x+e^x x+\log (4)} \, dx-\left (2 \log (4) \log \left (\frac {x \left (e^x (1+x)+\log (4)\right )}{e^x x+\log (4)}\right )\right ) \int \frac {1}{(1+x) \left (e^x+e^x x+\log (4)\right )} \, dx+\int \left (\frac {-1+x \log (x)}{x \log (x)}+\frac {2 \log \left (\frac {x \left (e^x (1+x)+\log (4)\right )}{e^x x+\log (4)}\right )}{1+x}\right ) \, dx\\ &=2 \int \frac {\log \left (\frac {x \left (e^x (1+x)+\log (4)\right )}{e^x x+\log (4)}\right )}{1+x} \, dx+(2 \log (4)) \int \left (\frac {\int \frac {1}{e^x (1+x)+\log (4)} \, dx+\int \frac {1}{(1+x) \left (e^x (1+x)+\log (4)\right )} \, dx}{1+x}+\frac {(1+x) \log (4) \left (\int \frac {1}{e^x (1+x)+\log (4)} \, dx+\int \frac {1}{(1+x) \left (e^x (1+x)+\log (4)\right )} \, dx\right )}{x \left (e^x x+\log (4)\right )}-\frac {(2+x) \log (4) \left (\int \frac {1}{e^x (1+x)+\log (4)} \, dx+\int \frac {1}{(1+x) \left (e^x (1+x)+\log (4)\right )} \, dx\right )}{(1+x) \left (e^x+e^x x+\log (4)\right )}\right ) \, dx-(2 \log (4)) \int \left (\frac {\int \frac {1}{e^x x+\log (4)} \, dx+\int \frac {1}{e^x x^2+x \log (4)} \, dx}{1+x}+\frac {(1+x) \log (4) \left (\int \frac {1}{e^x x+\log (4)} \, dx+\int \frac {1}{e^x x^2+x \log (4)} \, dx\right )}{x \left (e^x x+\log (4)\right )}-\frac {(2+x) \log (4) \left (\int \frac {1}{e^x x+\log (4)} \, dx+\int \frac {1}{e^x x^2+x \log (4)} \, dx\right )}{(1+x) \left (e^x+e^x x+\log (4)\right )}\right ) \, dx+\left (2 \log (4) \log \left (\frac {x \left (e^x (1+x)+\log (4)\right )}{e^x x+\log (4)}\right )\right ) \int \frac {1}{e^x x+\log (4)} \, dx+\left (2 \log (4) \log \left (\frac {x \left (e^x (1+x)+\log (4)\right )}{e^x x+\log (4)}\right )\right ) \int \frac {1}{x \left (e^x x+\log (4)\right )} \, dx-\left (2 \log (4) \log \left (\frac {x \left (e^x (1+x)+\log (4)\right )}{e^x x+\log (4)}\right )\right ) \int \frac {1}{e^x+e^x x+\log (4)} \, dx-\left (2 \log (4) \log \left (\frac {x \left (e^x (1+x)+\log (4)\right )}{e^x x+\log (4)}\right )\right ) \int \frac {1}{(1+x) \left (e^x+e^x x+\log (4)\right )} \, dx+\int \frac {-1+x \log (x)}{x \log (x)} \, dx\\ &=2 \int \frac {\log \left (\frac {x \left (e^x (1+x)+\log (4)\right )}{e^x x+\log (4)}\right )}{1+x} \, dx+(2 \log (4)) \int \frac {\int \frac {1}{e^x (1+x)+\log (4)} \, dx+\int \frac {1}{(1+x) \left (e^x (1+x)+\log (4)\right )} \, dx}{1+x} \, dx-(2 \log (4)) \int \frac {\int \frac {1}{e^x x+\log (4)} \, dx+\int \frac {1}{e^x x^2+x \log (4)} \, dx}{1+x} \, dx+\left (2 \log ^2(4)\right ) \int \frac {(1+x) \left (\int \frac {1}{e^x (1+x)+\log (4)} \, dx+\int \frac {1}{(1+x) \left (e^x (1+x)+\log (4)\right )} \, dx\right )}{x \left (e^x x+\log (4)\right )} \, dx-\left (2 \log ^2(4)\right ) \int \frac {(2+x) \left (\int \frac {1}{e^x (1+x)+\log (4)} \, dx+\int \frac {1}{(1+x) \left (e^x (1+x)+\log (4)\right )} \, dx\right )}{(1+x) \left (e^x+e^x x+\log (4)\right )} \, dx-\left (2 \log ^2(4)\right ) \int \frac {(1+x) \left (\int \frac {1}{e^x x+\log (4)} \, dx+\int \frac {1}{e^x x^2+x \log (4)} \, dx\right )}{x \left (e^x x+\log (4)\right )} \, dx+\left (2 \log ^2(4)\right ) \int \frac {(2+x) \left (\int \frac {1}{e^x x+\log (4)} \, dx+\int \frac {1}{e^x x^2+x \log (4)} \, dx\right )}{(1+x) \left (e^x+e^x x+\log (4)\right )} \, dx+\left (2 \log (4) \log \left (\frac {x \left (e^x (1+x)+\log (4)\right )}{e^x x+\log (4)}\right )\right ) \int \frac {1}{e^x x+\log (4)} \, dx+\left (2 \log (4) \log \left (\frac {x \left (e^x (1+x)+\log (4)\right )}{e^x x+\log (4)}\right )\right ) \int \frac {1}{x \left (e^x x+\log (4)\right )} \, dx-\left (2 \log (4) \log \left (\frac {x \left (e^x (1+x)+\log (4)\right )}{e^x x+\log (4)}\right )\right ) \int \frac {1}{e^x+e^x x+\log (4)} \, dx-\left (2 \log (4) \log \left (\frac {x \left (e^x (1+x)+\log (4)\right )}{e^x x+\log (4)}\right )\right ) \int \frac {1}{(1+x) \left (e^x+e^x x+\log (4)\right )} \, dx+\int \left (1-\frac {1}{x \log (x)}\right ) \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [F]  time = 0.84, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-x-x^2+\left (x^2+x^3\right ) \log (x)+e^{-2 x} \log ^2(4) (-1+x \log (x))+e^{-x} \log (4) \left (-1-2 x+\left (x+2 x^2\right ) \log (x)\right )+\left (2 x^2 \log (x)+e^{-x} (2+6 x) \log (4) \log (x)+2 e^{-2 x} \log ^2(4) \log (x)\right ) \log \left (\frac {x+x^2+e^{-x} x \log (4)}{x+e^{-x} \log (4)}\right )}{\left (x^2+x^3\right ) \log (x)+e^{-x} \left (x+2 x^2\right ) \log (4) \log (x)+e^{-2 x} x \log ^2(4) \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(-x - x^2 + (x^2 + x^3)*Log[x] + (Log[4]^2*(-1 + x*Log[x]))/E^(2*x) + (Log[4]*(-1 - 2*x + (x + 2*x^2
)*Log[x]))/E^x + (2*x^2*Log[x] + ((2 + 6*x)*Log[4]*Log[x])/E^x + (2*Log[4]^2*Log[x])/E^(2*x))*Log[(x + x^2 + (
x*Log[4])/E^x)/(x + Log[4]/E^x)])/((x^2 + x^3)*Log[x] + ((x + 2*x^2)*Log[4]*Log[x])/E^x + (x*Log[4]^2*Log[x])/
E^(2*x)),x]

[Out]

Integrate[(-x - x^2 + (x^2 + x^3)*Log[x] + (Log[4]^2*(-1 + x*Log[x]))/E^(2*x) + (Log[4]*(-1 - 2*x + (x + 2*x^2
)*Log[x]))/E^x + (2*x^2*Log[x] + ((2 + 6*x)*Log[4]*Log[x])/E^x + (2*Log[4]^2*Log[x])/E^(2*x))*Log[(x + x^2 + (
x*Log[4])/E^x)/(x + Log[4]/E^x)])/((x^2 + x^3)*Log[x] + ((x + 2*x^2)*Log[4]*Log[x])/E^x + (x*Log[4]^2*Log[x])/
E^(2*x)), x]

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fricas [A]  time = 0.61, size = 42, normalized size = 1.62 \begin {gather*} \log \left (\frac {x^{2} + x e^{\left (-x + \log \left (2 \, \log \relax (2)\right )\right )} + x}{x + e^{\left (-x + \log \left (2 \, \log \relax (2)\right )\right )}}\right )^{2} + x - \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{0} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

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maple [C]  time = 0.65, size = 1475, normalized size = 56.73

method	result	size
risch	\(x -2 \ln \left (\ln \relax (2)\right )^{2}-2 \ln \relax (2)^{2}-\ln \left (\ln \relax (x )\right )+\ln \relax (x )^{2}-i \pi \ln \left (2 \ln \relax (2) {\mathrm e}^{-x}+x \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i x \left (2 \ln \relax (2) {\mathrm e}^{-x}+x +1\right )}{2 \ln \relax (2) {\mathrm e}^{-x}+x}\right )^{2}-i \pi \ln \left (2 \ln \relax (2) {\mathrm e}^{-x}+x \right ) \mathrm {csgn}\left (\frac {i \left (2 \ln \relax (2) {\mathrm e}^{-x}+x +1\right )}{2 \ln \relax (2) {\mathrm e}^{-x}+x}\right ) \mathrm {csgn}\left (\frac {i x \left (2 \ln \relax (2) {\mathrm e}^{-x}+x +1\right )}{2 \ln \relax (2) {\mathrm e}^{-x}+x}\right )^{2}+i \pi \ln \relax (x ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i x \left (2 \ln \relax (2) {\mathrm e}^{-x}+x +1\right )}{2 \ln \relax (2) {\mathrm e}^{-x}+x}\right )^{2}-i \pi \ln \left (2 \ln \relax (2) {\mathrm e}^{-x}+x \right ) \mathrm {csgn}\left (\frac {i}{2 \ln \relax (2) {\mathrm e}^{-x}+x}\right ) \mathrm {csgn}\left (\frac {i \left (2 \ln \relax (2) {\mathrm e}^{-x}+x +1\right )}{2 \ln \relax (2) {\mathrm e}^{-x}+x}\right )^{2}-i \pi \ln \left (2 \ln \relax (2) {\mathrm e}^{-x}+x \right ) \mathrm {csgn}\left (i \left (2 \ln \relax (2) {\mathrm e}^{-x}+x +1\right )\right ) \mathrm {csgn}\left (\frac {i \left (2 \ln \relax (2) {\mathrm e}^{-x}+x +1\right )}{2 \ln \relax (2) {\mathrm e}^{-x}+x}\right )^{2}+i \pi \ln \left (2 \ln \relax (2) {\mathrm e}^{-x}+x +1\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i x \left (2 \ln \relax (2) {\mathrm e}^{-x}+x +1\right )}{2 \ln \relax (2) {\mathrm e}^{-x}+x}\right )^{2}+i \pi \ln \left (2 \ln \relax (2) {\mathrm e}^{-x}+x +1\right ) \mathrm {csgn}\left (\frac {i}{2 \ln \relax (2) {\mathrm e}^{-x}+x}\right ) \mathrm {csgn}\left (\frac {i \left (2 \ln \relax (2) {\mathrm e}^{-x}+x +1\right )}{2 \ln \relax (2) {\mathrm e}^{-x}+x}\right )^{2}+i \pi \ln \left (2 \ln \relax (2) {\mathrm e}^{-x}+x +1\right ) \mathrm {csgn}\left (i \left (2 \ln \relax (2) {\mathrm e}^{-x}+x +1\right )\right ) \mathrm {csgn}\left (\frac {i \left (2 \ln \relax (2) {\mathrm e}^{-x}+x +1\right )}{2 \ln \relax (2) {\mathrm e}^{-x}+x}\right )^{2}+i \pi \ln \left (2 \ln \relax (2) {\mathrm e}^{-x}+x +1\right ) \mathrm {csgn}\left (\frac {i \left (2 \ln \relax (2) {\mathrm e}^{-x}+x +1\right )}{2 \ln \relax (2) {\mathrm e}^{-x}+x}\right ) \mathrm {csgn}\left (\frac {i x \left (2 \ln \relax (2) {\mathrm e}^{-x}+x +1\right )}{2 \ln \relax (2) {\mathrm e}^{-x}+x}\right )^{2}-i \pi \ln \relax (x ) \mathrm {csgn}\left (\frac {i \left (2 \ln \relax (2) {\mathrm e}^{-x}+x +1\right )}{2 \ln \relax (2) {\mathrm e}^{-x}+x}\right )^{3}-i \pi \ln \relax (x ) \mathrm {csgn}\left (\frac {i x \left (2 \ln \relax (2) {\mathrm e}^{-x}+x +1\right )}{2 \ln \relax (2) {\mathrm e}^{-x}+x}\right )^{3}+i \pi \ln \left (2 \ln \relax (2) {\mathrm e}^{-x}+x \right ) \mathrm {csgn}\left (\frac {i x \left (2 \ln \relax (2) {\mathrm e}^{-x}+x +1\right )}{2 \ln \relax (2) {\mathrm e}^{-x}+x}\right )^{3}+i \pi \ln \left (2 \ln \relax (2) {\mathrm e}^{-x}+x \right ) \mathrm {csgn}\left (\frac {i \left (2 \ln \relax (2) {\mathrm e}^{-x}+x +1\right )}{2 \ln \relax (2) {\mathrm e}^{-x}+x}\right )^{3}-i \pi \ln \left (2 \ln \relax (2) {\mathrm e}^{-x}+x +1\right ) \mathrm {csgn}\left (\frac {i \left (2 \ln \relax (2) {\mathrm e}^{-x}+x +1\right )}{2 \ln \relax (2) {\mathrm e}^{-x}+x}\right )^{3}-i \pi \ln \left (2 \ln \relax (2) {\mathrm e}^{-x}+x +1\right ) \mathrm {csgn}\left (\frac {i x \left (2 \ln \relax (2) {\mathrm e}^{-x}+x +1\right )}{2 \ln \relax (2) {\mathrm e}^{-x}+x}\right )^{3}+\ln \left (2 \ln \relax (2) {\mathrm e}^{-x}+x \right )^{2}-4 \ln \relax (2) \ln \left (\ln \relax (2)\right )+\ln \left (2 \ln \relax (2) {\mathrm e}^{-x}+x +1\right )^{2}+i \pi \ln \relax (x ) \mathrm {csgn}\left (\frac {i}{2 \ln \relax (2) {\mathrm e}^{-x}+x}\right ) \mathrm {csgn}\left (\frac {i \left (2 \ln \relax (2) {\mathrm e}^{-x}+x +1\right )}{2 \ln \relax (2) {\mathrm e}^{-x}+x}\right )^{2}-2 \ln \relax (x ) \ln \left (2 \ln \relax (2) {\mathrm e}^{-x}+x \right )+\left (2 \ln \relax (x )-2 \ln \left (2 \ln \relax (2) {\mathrm e}^{-x}+x \right )\right ) \ln \left (2 \ln \relax (2) {\mathrm e}^{-x}+x +1\right )+i \pi \ln \relax (x ) \mathrm {csgn}\left (i \left (2 \ln \relax (2) {\mathrm e}^{-x}+x +1\right )\right ) \mathrm {csgn}\left (\frac {i \left (2 \ln \relax (2) {\mathrm e}^{-x}+x +1\right )}{2 \ln \relax (2) {\mathrm e}^{-x}+x}\right )^{2}+i \pi \ln \relax (x ) \mathrm {csgn}\left (\frac {i \left (2 \ln \relax (2) {\mathrm e}^{-x}+x +1\right )}{2 \ln \relax (2) {\mathrm e}^{-x}+x}\right ) \mathrm {csgn}\left (\frac {i x \left (2 \ln \relax (2) {\mathrm e}^{-x}+x +1\right )}{2 \ln \relax (2) {\mathrm e}^{-x}+x}\right )^{2}+i \pi \ln \left (2 \ln \relax (2) {\mathrm e}^{-x}+x \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i \left (2 \ln \relax (2) {\mathrm e}^{-x}+x +1\right )}{2 \ln \relax (2) {\mathrm e}^{-x}+x}\right ) \mathrm {csgn}\left (\frac {i x \left (2 \ln \relax (2) {\mathrm e}^{-x}+x +1\right )}{2 \ln \relax (2) {\mathrm e}^{-x}+x}\right )-i \pi \ln \left (2 \ln \relax (2) {\mathrm e}^{-x}+x +1\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i \left (2 \ln \relax (2) {\mathrm e}^{-x}+x +1\right )}{2 \ln \relax (2) {\mathrm e}^{-x}+x}\right ) \mathrm {csgn}\left (\frac {i x \left (2 \ln \relax (2) {\mathrm e}^{-x}+x +1\right )}{2 \ln \relax (2) {\mathrm e}^{-x}+x}\right )-i \pi \ln \left (2 \ln \relax (2) {\mathrm e}^{-x}+x +1\right ) \mathrm {csgn}\left (\frac {i}{2 \ln \relax (2) {\mathrm e}^{-x}+x}\right ) \mathrm {csgn}\left (i \left (2 \ln \relax (2) {\mathrm e}^{-x}+x +1\right )\right ) \mathrm {csgn}\left (\frac {i \left (2 \ln \relax (2) {\mathrm e}^{-x}+x +1\right )}{2 \ln \relax (2) {\mathrm e}^{-x}+x}\right )+i \pi \ln \left (2 \ln \relax (2) {\mathrm e}^{-x}+x \right ) \mathrm {csgn}\left (\frac {i}{2 \ln \relax (2) {\mathrm e}^{-x}+x}\right ) \mathrm {csgn}\left (i \left (2 \ln \relax (2) {\mathrm e}^{-x}+x +1\right )\right ) \mathrm {csgn}\left (\frac {i \left (2 \ln \relax (2) {\mathrm e}^{-x}+x +1\right )}{2 \ln \relax (2) {\mathrm e}^{-x}+x}\right )-i \pi \ln \relax (x ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i \left (2 \ln \relax (2) {\mathrm e}^{-x}+x +1\right )}{2 \ln \relax (2) {\mathrm e}^{-x}+x}\right ) \mathrm {csgn}\left (\frac {i x \left (2 \ln \relax (2) {\mathrm e}^{-x}+x +1\right )}{2 \ln \relax (2) {\mathrm e}^{-x}+x}\right )-i \pi \ln \relax (x ) \mathrm {csgn}\left (\frac {i}{2 \ln \relax (2) {\mathrm e}^{-x}+x}\right ) \mathrm {csgn}\left (i \left (2 \ln \relax (2) {\mathrm e}^{-x}+x +1\right )\right ) \mathrm {csgn}\left (\frac {i \left (2 \ln \relax (2) {\mathrm e}^{-x}+x +1\right )}{2 \ln \relax (2) {\mathrm e}^{-x}+x}\right )\)	\(1475\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*ln(x)*exp(ln(2*ln(2))-x)^2+(6*x+2)*ln(x)*exp(ln(2*ln(2))-x)+2*x^2*ln(x))*ln((x*exp(ln(2*ln(2))-x)+x^2+
x)/(exp(ln(2*ln(2))-x)+x))+(x*ln(x)-1)*exp(ln(2*ln(2))-x)^2+((2*x^2+x)*ln(x)-2*x-1)*exp(ln(2*ln(2))-x)+(x^3+x^
2)*ln(x)-x^2-x)/(x*ln(x)*exp(ln(2*ln(2))-x)^2+(2*x^2+x)*ln(x)*exp(ln(2*ln(2))-x)+(x^3+x^2)*ln(x)),x,method=_RE
TURNVERBOSE)

[Out]

x-2*ln(ln(2))^2-2*ln(2)^2-ln(ln(x))+ln(x)^2+I*Pi*ln(x)*csgn(I/(2*ln(2)*exp(-x)+x))*csgn(I/(2*ln(2)*exp(-x)+x)*
(2*ln(2)*exp(-x)+x+1))^2+I*Pi*ln(x)*csgn(I*(2*ln(2)*exp(-x)+x+1))*csgn(I/(2*ln(2)*exp(-x)+x)*(2*ln(2)*exp(-x)+
x+1))^2+I*Pi*ln(x)*csgn(I/(2*ln(2)*exp(-x)+x)*(2*ln(2)*exp(-x)+x+1))*csgn(I*x/(2*ln(2)*exp(-x)+x)*(2*ln(2)*exp
(-x)+x+1))^2-I*Pi*ln(2*ln(2)*exp(-x)+x+1)*csgn(I/(2*ln(2)*exp(-x)+x)*(2*ln(2)*exp(-x)+x+1))^3-I*Pi*ln(2*ln(2)*
exp(-x)+x+1)*csgn(I*x/(2*ln(2)*exp(-x)+x)*(2*ln(2)*exp(-x)+x+1))^3-I*Pi*ln(x)*csgn(I/(2*ln(2)*exp(-x)+x)*(2*ln
(2)*exp(-x)+x+1))^3-I*Pi*ln(x)*csgn(I*x/(2*ln(2)*exp(-x)+x)*(2*ln(2)*exp(-x)+x+1))^3+I*Pi*ln(2*ln(2)*exp(-x)+x
)*csgn(I*x/(2*ln(2)*exp(-x)+x)*(2*ln(2)*exp(-x)+x+1))^3+I*Pi*ln(2*ln(2)*exp(-x)+x)*csgn(I/(2*ln(2)*exp(-x)+x)*
(2*ln(2)*exp(-x)+x+1))^3+ln(2*ln(2)*exp(-x)+x)^2-4*ln(2)*ln(ln(2))+I*Pi*ln(2*ln(2)*exp(-x)+x)*csgn(I*x)*csgn(I
/(2*ln(2)*exp(-x)+x)*(2*ln(2)*exp(-x)+x+1))*csgn(I*x/(2*ln(2)*exp(-x)+x)*(2*ln(2)*exp(-x)+x+1))+ln(2*ln(2)*exp
(-x)+x+1)^2-2*ln(x)*ln(2*ln(2)*exp(-x)+x)+(2*ln(x)-2*ln(2*ln(2)*exp(-x)+x))*ln(2*ln(2)*exp(-x)+x+1)-I*Pi*ln(2*
ln(2)*exp(-x)+x+1)*csgn(I*x)*csgn(I/(2*ln(2)*exp(-x)+x)*(2*ln(2)*exp(-x)+x+1))*csgn(I*x/(2*ln(2)*exp(-x)+x)*(2
*ln(2)*exp(-x)+x+1))-I*Pi*ln(2*ln(2)*exp(-x)+x+1)*csgn(I/(2*ln(2)*exp(-x)+x))*csgn(I*(2*ln(2)*exp(-x)+x+1))*cs
gn(I/(2*ln(2)*exp(-x)+x)*(2*ln(2)*exp(-x)+x+1))+I*Pi*ln(2*ln(2)*exp(-x)+x)*csgn(I/(2*ln(2)*exp(-x)+x))*csgn(I*
(2*ln(2)*exp(-x)+x+1))*csgn(I/(2*ln(2)*exp(-x)+x)*(2*ln(2)*exp(-x)+x+1))-I*Pi*ln(x)*csgn(I*x)*csgn(I/(2*ln(2)*
exp(-x)+x)*(2*ln(2)*exp(-x)+x+1))*csgn(I*x/(2*ln(2)*exp(-x)+x)*(2*ln(2)*exp(-x)+x+1))-I*Pi*ln(x)*csgn(I/(2*ln(
2)*exp(-x)+x))*csgn(I*(2*ln(2)*exp(-x)+x+1))*csgn(I/(2*ln(2)*exp(-x)+x)*(2*ln(2)*exp(-x)+x+1))-I*Pi*ln(2*ln(2)
*exp(-x)+x)*csgn(I*x)*csgn(I*x/(2*ln(2)*exp(-x)+x)*(2*ln(2)*exp(-x)+x+1))^2-I*Pi*ln(2*ln(2)*exp(-x)+x)*csgn(I/
(2*ln(2)*exp(-x)+x)*(2*ln(2)*exp(-x)+x+1))*csgn(I*x/(2*ln(2)*exp(-x)+x)*(2*ln(2)*exp(-x)+x+1))^2+I*Pi*ln(x)*cs
gn(I*x)*csgn(I*x/(2*ln(2)*exp(-x)+x)*(2*ln(2)*exp(-x)+x+1))^2-I*Pi*ln(2*ln(2)*exp(-x)+x)*csgn(I/(2*ln(2)*exp(-
x)+x))*csgn(I/(2*ln(2)*exp(-x)+x)*(2*ln(2)*exp(-x)+x+1))^2-I*Pi*ln(2*ln(2)*exp(-x)+x)*csgn(I*(2*ln(2)*exp(-x)+
x+1))*csgn(I/(2*ln(2)*exp(-x)+x)*(2*ln(2)*exp(-x)+x+1))^2+I*Pi*ln(2*ln(2)*exp(-x)+x+1)*csgn(I*x)*csgn(I*x/(2*l
n(2)*exp(-x)+x)*(2*ln(2)*exp(-x)+x+1))^2+I*Pi*ln(2*ln(2)*exp(-x)+x+1)*csgn(I/(2*ln(2)*exp(-x)+x))*csgn(I/(2*ln
(2)*exp(-x)+x)*(2*ln(2)*exp(-x)+x+1))^2+I*Pi*ln(2*ln(2)*exp(-x)+x+1)*csgn(I*(2*ln(2)*exp(-x)+x+1))*csgn(I/(2*l
n(2)*exp(-x)+x)*(2*ln(2)*exp(-x)+x+1))^2+I*Pi*ln(2*ln(2)*exp(-x)+x+1)*csgn(I/(2*ln(2)*exp(-x)+x)*(2*ln(2)*exp(
-x)+x+1))*csgn(I*x/(2*ln(2)*exp(-x)+x)*(2*ln(2)*exp(-x)+x+1))^2

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int -\frac {4 \, {\left (x \log \relax (x) - 1\right )} e^{\left (-2 \, x\right )} \log \relax (2)^{2} + 2 \, {\left ({\left (2 \, x^{2} + x\right )} \log \relax (x) - 2 \, x - 1\right )} e^{\left (-x\right )} \log \relax (2) - x^{2} + {\left (x^{3} + x^{2}\right )} \log \relax (x) + 2 \, {\left (2 \, {\left (3 \, x + 1\right )} e^{\left (-x\right )} \log \relax (2) \log \relax (x) + 4 \, e^{\left (-2 \, x\right )} \log \relax (2)^{2} \log \relax (x) + x^{2} \log \relax (x)\right )} \log \left (\frac {2 \, x e^{\left (-x\right )} \log \relax (2) + x^{2} + x}{2 \, e^{\left (-x\right )} \log \relax (2) + x}\right ) - x}{4 \, x e^{\left (-2 \, x\right )} \log \relax (2)^{2} \log \relax (x) + 2 \, {\left (2 \, x^{2} + x\right )} e^{\left (-x\right )} \log \relax (2) \log \relax (x) + {\left (x^{3} + x^{2}\right )} \log \relax (x)}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 4.54, size = 38, normalized size = 1.46 \begin {gather*} {\ln \left (\frac {x^2\,{\mathrm {e}}^x+2\,x\,\ln \relax (2)+x\,{\mathrm {e}}^x}{2\,\ln \relax (2)+x\,{\mathrm {e}}^x}\right )}^2+x-\ln \left (\ln \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.91, size = 34, normalized size = 1.31 \begin {gather*} x + \log {\left (\frac {x^{2} + x + 2 x e^{- x} \log {\relax (2 )}}{x + 2 e^{- x} \log {\relax (2 )}} \right )}^{2} - \log {\left (\log {\relax (x )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*ln(x)*exp(ln(2*ln(2))-x)**2+(6*x+2)*ln(x)*exp(ln(2*ln(2))-x)+2*x**2*ln(x))*ln((x*exp(ln(2*ln(2))
-x)+x**2+x)/(exp(ln(2*ln(2))-x)+x))+(x*ln(x)-1)*exp(ln(2*ln(2))-x)**2+((2*x**2+x)*ln(x)-2*x-1)*exp(ln(2*ln(2))
-x)+(x**3+x**2)*ln(x)-x**2-x)/(x*ln(x)*exp(ln(2*ln(2))-x)**2+(2*x**2+x)*ln(x)*exp(ln(2*ln(2))-x)+(x**3+x**2)*l
n(x)),x)

[Out]

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

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