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

Optimal. Leaf size=30 \[ \frac {x-x^2 \log \left (1-\frac {e^{-4+x}}{x}\right ) \log \left (e^x x\right )}{x} \]

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Rubi [F]  time = 3.28, 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 {\left (1+x+e^{4-x} \left (-x-x^2\right )\right ) \log \left (\frac {e^{-4+x} \left (-1+e^{4-x} x\right )}{x}\right )+\log \left (e^x x\right ) \left (-1+x+\left (1-e^{4-x} x\right ) \log \left (\frac {e^{-4+x} \left (-1+e^{4-x} x\right )}{x}\right )\right )}{-1+e^{4-x} x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-\left ((-1+x) \log \left (e^x x\right )\right )-e^{-x} \left (e^x-e^4 x\right ) \log \left (1-\frac {e^{-4+x}}{x}\right ) \left (1+x+\log \left (e^x x\right )\right )}{1-e^{4-x} x} \, dx\\ &=\int \left (-\log \left (1-\frac {e^{-4+x}}{x}\right )-x \log \left (1-\frac {e^{-4+x}}{x}\right )+\log \left (e^x x\right )-x \log \left (e^x x\right )+\frac {e^4 (-1+x) x \log \left (e^x x\right )}{-e^x+e^4 x}-\log \left (1-\frac {e^{-4+x}}{x}\right ) \log \left (e^x x\right )\right ) \, dx\\ &=e^4 \int \frac {(-1+x) x \log \left (e^x x\right )}{-e^x+e^4 x} \, dx-\int \log \left (1-\frac {e^{-4+x}}{x}\right ) \, dx-\int x \log \left (1-\frac {e^{-4+x}}{x}\right ) \, dx+\int \log \left (e^x x\right ) \, dx-\int x \log \left (e^x x\right ) \, dx-\int \log \left (1-\frac {e^{-4+x}}{x}\right ) \log \left (e^x x\right ) \, dx\\ &=-x \log \left (1-\frac {e^{-4+x}}{x}\right )-\frac {1}{2} x^2 \log \left (1-\frac {e^{-4+x}}{x}\right )+x \log \left (e^x x\right )-\frac {1}{2} x^2 \log \left (e^x x\right )-x \log \left (1-\frac {e^{-4+x}}{x}\right ) \log \left (e^x x\right )+\frac {1}{2} \int x (1+x) \, dx+\frac {1}{2} \int \frac {e^x (-1+x) x}{e^x-e^4 x} \, dx-e^4 \int \frac {(1+x) \left (-\int \frac {x}{-e^x+e^4 x} \, dx+\int \frac {x^2}{-e^x+e^4 x} \, dx\right )}{x} \, dx-\left (e^4 \log \left (e^x x\right )\right ) \int \frac {x}{-e^x+e^4 x} \, dx+\left (e^4 \log \left (e^x x\right )\right ) \int \frac {x^2}{-e^x+e^4 x} \, dx-\int (1+x) \, dx+\int \frac {e^x (-1+x)}{e^x-e^4 x} \, dx+\int (1+x) \log \left (1-\frac {e^{-4+x}}{x}\right ) \, dx+\int \frac {e^x (-1+x) \log \left (e^x x\right )}{e^x-e^4 x} \, dx\\ &=-x-\frac {x^2}{2}-x \log \left (1-\frac {e^{-4+x}}{x}\right )-\frac {1}{2} x^2 \log \left (1-\frac {e^{-4+x}}{x}\right )+\frac {1}{2} (1+x)^2 \log \left (1-\frac {e^{-4+x}}{x}\right )+x \log \left (e^x x\right )-\frac {1}{2} x^2 \log \left (e^x x\right )-x \log \left (1-\frac {e^{-4+x}}{x}\right ) \log \left (e^x x\right )-\frac {1}{2} \int \frac {e^x (-1+x) (1+x)^2}{x \left (e^x-e^4 x\right )} \, dx+\frac {1}{2} \int \left (x+x^2\right ) \, dx+\frac {1}{2} \int \left (\frac {e^x x}{-e^x+e^4 x}-\frac {e^x x^2}{-e^x+e^4 x}\right ) \, dx-e^4 \int \left (-\frac {(1+x) \int \frac {x}{-e^x+e^4 x} \, dx}{x}+\frac {(1+x) \int \frac {x^2}{-e^x+e^4 x} \, dx}{x}\right ) \, dx-\log \left (e^x x\right ) \int \frac {e^x}{e^x-e^4 x} \, dx-\log \left (e^x x\right ) \int \frac {e^x x}{-e^x+e^4 x} \, dx-\left (e^4 \log \left (e^x x\right )\right ) \int \frac {x}{-e^x+e^4 x} \, dx+\left (e^4 \log \left (e^x x\right )\right ) \int \frac {x^2}{-e^x+e^4 x} \, dx+\int \left (-\frac {e^x}{e^x-e^4 x}-\frac {e^x x}{-e^x+e^4 x}\right ) \, dx-\int \frac {(1+x) \left (-\int \frac {e^x}{e^x-e^4 x} \, dx-\int \frac {e^x x}{-e^x+e^4 x} \, dx\right )}{x} \, dx\\ &=-x-\frac {x^2}{4}+\frac {x^3}{6}-x \log \left (1-\frac {e^{-4+x}}{x}\right )-\frac {1}{2} x^2 \log \left (1-\frac {e^{-4+x}}{x}\right )+\frac {1}{2} (1+x)^2 \log \left (1-\frac {e^{-4+x}}{x}\right )+x \log \left (e^x x\right )-\frac {1}{2} x^2 \log \left (e^x x\right )-x \log \left (1-\frac {e^{-4+x}}{x}\right ) \log \left (e^x x\right )+\frac {1}{2} \int \frac {e^x x}{-e^x+e^4 x} \, dx-\frac {1}{2} \int \frac {e^x x^2}{-e^x+e^4 x} \, dx-\frac {1}{2} \int \left (-\frac {e^x}{e^x-e^4 x}+\frac {e^x}{x \left (-e^x+e^4 x\right )}-\frac {e^x x}{-e^x+e^4 x}-\frac {e^x x^2}{-e^x+e^4 x}\right ) \, dx+e^4 \int \frac {(1+x) \int \frac {x}{-e^x+e^4 x} \, dx}{x} \, dx-e^4 \int \frac {(1+x) \int \frac {x^2}{-e^x+e^4 x} \, dx}{x} \, dx-\log \left (e^x x\right ) \int \frac {e^x}{e^x-e^4 x} \, dx-\log \left (e^x x\right ) \int \frac {e^x x}{-e^x+e^4 x} \, dx-\left (e^4 \log \left (e^x x\right )\right ) \int \frac {x}{-e^x+e^4 x} \, dx+\left (e^4 \log \left (e^x x\right )\right ) \int \frac {x^2}{-e^x+e^4 x} \, dx-\int \frac {e^x}{e^x-e^4 x} \, dx-\int \frac {e^x x}{-e^x+e^4 x} \, dx-\int \left (-\frac {(1+x) \int \frac {e^x}{e^x-e^4 x} \, dx}{x}-\frac {(1+x) \int \frac {e^x x}{-e^x+e^4 x} \, dx}{x}\right ) \, dx\\ &=-x-\frac {x^2}{4}+\frac {x^3}{6}-x \log \left (1-\frac {e^{-4+x}}{x}\right )-\frac {1}{2} x^2 \log \left (1-\frac {e^{-4+x}}{x}\right )+\frac {1}{2} (1+x)^2 \log \left (1-\frac {e^{-4+x}}{x}\right )+x \log \left (e^x x\right )-\frac {1}{2} x^2 \log \left (e^x x\right )-x \log \left (1-\frac {e^{-4+x}}{x}\right ) \log \left (e^x x\right )+\frac {1}{2} \int \frac {e^x}{e^x-e^4 x} \, dx-\frac {1}{2} \int \frac {e^x}{x \left (-e^x+e^4 x\right )} \, dx+2 \left (\frac {1}{2} \int \frac {e^x x}{-e^x+e^4 x} \, dx\right )+e^4 \int \left (\int \frac {x}{-e^x+e^4 x} \, dx+\frac {\int \frac {x}{-e^x+e^4 x} \, dx}{x}\right ) \, dx-e^4 \int \left (\int \frac {x^2}{-e^x+e^4 x} \, dx+\frac {\int \frac {x^2}{-e^x+e^4 x} \, dx}{x}\right ) \, dx-\log \left (e^x x\right ) \int \frac {e^x}{e^x-e^4 x} \, dx-\log \left (e^x x\right ) \int \frac {e^x x}{-e^x+e^4 x} \, dx-\left (e^4 \log \left (e^x x\right )\right ) \int \frac {x}{-e^x+e^4 x} \, dx+\left (e^4 \log \left (e^x x\right )\right ) \int \frac {x^2}{-e^x+e^4 x} \, dx-\int \frac {e^x}{e^x-e^4 x} \, dx-\int \frac {e^x x}{-e^x+e^4 x} \, dx+\int \frac {(1+x) \int \frac {e^x}{e^x-e^4 x} \, dx}{x} \, dx+\int \frac {(1+x) \int \frac {e^x x}{-e^x+e^4 x} \, dx}{x} \, dx\\ &=-x-\frac {x^2}{4}+\frac {x^3}{6}-x \log \left (1-\frac {e^{-4+x}}{x}\right )-\frac {1}{2} x^2 \log \left (1-\frac {e^{-4+x}}{x}\right )+\frac {1}{2} (1+x)^2 \log \left (1-\frac {e^{-4+x}}{x}\right )+x \log \left (e^x x\right )-\frac {1}{2} x^2 \log \left (e^x x\right )-x \log \left (1-\frac {e^{-4+x}}{x}\right ) \log \left (e^x x\right )+\frac {1}{2} \int \frac {e^x}{e^x-e^4 x} \, dx-\frac {1}{2} \int \frac {e^x}{x \left (-e^x+e^4 x\right )} \, dx+2 \left (\frac {1}{2} \int \frac {e^x x}{-e^x+e^4 x} \, dx\right )+e^4 \int \left (\int \frac {x}{-e^x+e^4 x} \, dx\right ) \, dx+e^4 \int \frac {\int \frac {x}{-e^x+e^4 x} \, dx}{x} \, dx-e^4 \int \left (\int \frac {x^2}{-e^x+e^4 x} \, dx\right ) \, dx-e^4 \int \frac {\int \frac {x^2}{-e^x+e^4 x} \, dx}{x} \, dx-\log \left (e^x x\right ) \int \frac {e^x}{e^x-e^4 x} \, dx-\log \left (e^x x\right ) \int \frac {e^x x}{-e^x+e^4 x} \, dx-\left (e^4 \log \left (e^x x\right )\right ) \int \frac {x}{-e^x+e^4 x} \, dx+\left (e^4 \log \left (e^x x\right )\right ) \int \frac {x^2}{-e^x+e^4 x} \, dx-\int \frac {e^x}{e^x-e^4 x} \, dx-\int \frac {e^x x}{-e^x+e^4 x} \, dx+\int \left (\int \frac {e^x}{e^x-e^4 x} \, dx+\frac {\int \frac {e^x}{e^x-e^4 x} \, dx}{x}\right ) \, dx+\int \left (\int \frac {e^x x}{-e^x+e^4 x} \, dx+\frac {\int \frac {e^x x}{-e^x+e^4 x} \, dx}{x}\right ) \, dx\\ &=-x-\frac {x^2}{4}+\frac {x^3}{6}-x \log \left (1-\frac {e^{-4+x}}{x}\right )-\frac {1}{2} x^2 \log \left (1-\frac {e^{-4+x}}{x}\right )+\frac {1}{2} (1+x)^2 \log \left (1-\frac {e^{-4+x}}{x}\right )+x \log \left (e^x x\right )-\frac {1}{2} x^2 \log \left (e^x x\right )-x \log \left (1-\frac {e^{-4+x}}{x}\right ) \log \left (e^x x\right )+\frac {1}{2} \int \frac {e^x}{e^x-e^4 x} \, dx-\frac {1}{2} \int \frac {e^x}{x \left (-e^x+e^4 x\right )} \, dx+2 \left (\frac {1}{2} \int \frac {e^x x}{-e^x+e^4 x} \, dx\right )+e^4 \int \left (\int \frac {x}{-e^x+e^4 x} \, dx\right ) \, dx+e^4 \int \frac {\int \frac {x}{-e^x+e^4 x} \, dx}{x} \, dx-e^4 \int \left (\int \frac {x^2}{-e^x+e^4 x} \, dx\right ) \, dx-e^4 \int \frac {\int \frac {x^2}{-e^x+e^4 x} \, dx}{x} \, dx-\log \left (e^x x\right ) \int \frac {e^x}{e^x-e^4 x} \, dx-\log \left (e^x x\right ) \int \frac {e^x x}{-e^x+e^4 x} \, dx-\left (e^4 \log \left (e^x x\right )\right ) \int \frac {x}{-e^x+e^4 x} \, dx+\left (e^4 \log \left (e^x x\right )\right ) \int \frac {x^2}{-e^x+e^4 x} \, dx-\int \frac {e^x}{e^x-e^4 x} \, dx-\int \frac {e^x x}{-e^x+e^4 x} \, dx+\int \left (\int \frac {e^x}{e^x-e^4 x} \, dx\right ) \, dx+\int \frac {\int \frac {e^x}{e^x-e^4 x} \, dx}{x} \, dx+\int \left (\int \frac {e^x x}{-e^x+e^4 x} \, dx\right ) \, dx+\int \frac {\int \frac {e^x x}{-e^x+e^4 x} \, dx}{x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.50, size = 22, normalized size = 0.73 \begin {gather*} -x \log \left (1-\frac {e^{-4+x}}{x}\right ) \log \left (e^x x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(x*Log[1 - E^(-4 + x)/x]*Log[E^x*x])

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fricas [A]  time = 0.59, size = 24, normalized size = 0.80 \begin {gather*} -x \log \left (x e^{x}\right ) \log \left (\frac {{\left (x e^{4} - e^{x}\right )} e^{\left (-4\right )}}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x*exp(-x+4)+1)*log((x*exp(-x+4)-1)/x/exp(-x+4))+x-1)*log(exp(x)*x)+((-x^2-x)*exp(-x+4)+x+1)*log(
(x*exp(-x+4)-1)/x/exp(-x+4)))/(x*exp(-x+4)-1),x, algorithm="fricas")

[Out]

-x*log(x*e^x)*log((x*e^4 - e^x)*e^(-4)/x)

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giac [A]  time = 0.14, size = 53, normalized size = 1.77 \begin {gather*} -x^{2} \log \left (x e^{4} - e^{x}\right ) + x^{2} \log \relax (x) - x \log \left (x e^{4} - e^{x}\right ) \log \relax (x) + x \log \relax (x)^{2} + 4 \, x^{2} + 4 \, x \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x*exp(-x+4)+1)*log((x*exp(-x+4)-1)/x/exp(-x+4))+x-1)*log(exp(x)*x)+((-x^2-x)*exp(-x+4)+x+1)*log(
(x*exp(-x+4)-1)/x/exp(-x+4)))/(x*exp(-x+4)-1),x, algorithm="giac")

[Out]

-x^2*log(x*e^4 - e^x) + x^2*log(x) - x*log(x*e^4 - e^x)*log(x) + x*log(x)^2 + 4*x^2 + 4*x*log(x)

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maple [C]  time = 0.34, size = 1357, normalized size = 45.23

method	result	size
risch	\(x \ln \relax (x )^{2}+4 x \ln \relax (x )+\left (-x \ln \left (x \,{\mathrm e}^{4}-{\mathrm e}^{x}\right )+x \ln \relax (x )+\frac {i \pi x \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \left (x \,{\mathrm e}^{4}-{\mathrm e}^{x}\right )\right ) \mathrm {csgn}\left (\frac {i \left (x \,{\mathrm e}^{4}-{\mathrm e}^{x}\right )}{x}\right )}{2}-\frac {i \pi x \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (x \,{\mathrm e}^{4}-{\mathrm e}^{x}\right )}{x}\right )^{2}}{2}-\frac {i \pi x \,\mathrm {csgn}\left (i \left (x \,{\mathrm e}^{4}-{\mathrm e}^{x}\right )\right ) \mathrm {csgn}\left (\frac {i \left (x \,{\mathrm e}^{4}-{\mathrm e}^{x}\right )}{x}\right )^{2}}{2}+\frac {i \pi x \mathrm {csgn}\left (\frac {i \left (x \,{\mathrm e}^{4}-{\mathrm e}^{x}\right )}{x}\right )^{3}}{2}+4 x \right ) \ln \left ({\mathrm e}^{x}\right )+\frac {\pi ^{2} x \mathrm {csgn}\left (\frac {i \left (x \,{\mathrm e}^{4}-{\mathrm e}^{x}\right )}{x}\right )^{3} \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )^{3}}{4}-2 i x \pi \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )^{3}+\frac {\pi ^{2} x \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \left (x \,{\mathrm e}^{4}-{\mathrm e}^{x}\right )\right ) \mathrm {csgn}\left (\frac {i \left (x \,{\mathrm e}^{4}-{\mathrm e}^{x}\right )}{x}\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )}{4}+\frac {i x \ln \left (x \,{\mathrm e}^{4}-{\mathrm e}^{x}\right ) \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )}{2}+\frac {i \pi x \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \left (x \,{\mathrm e}^{4}-{\mathrm e}^{x}\right )\right ) \mathrm {csgn}\left (\frac {i \left (x \,{\mathrm e}^{4}-{\mathrm e}^{x}\right )}{x}\right ) \ln \relax (x )}{2}-\frac {i x \ln \relax (x ) \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )}{2}-\frac {\pi ^{2} x \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \left (x \,{\mathrm e}^{4}-{\mathrm e}^{x}\right )\right ) \mathrm {csgn}\left (\frac {i \left (x \,{\mathrm e}^{4}-{\mathrm e}^{x}\right )}{x}\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )^{2}}{4}-\frac {\pi ^{2} x \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \left (x \,{\mathrm e}^{4}-{\mathrm e}^{x}\right )\right ) \mathrm {csgn}\left (\frac {i \left (x \,{\mathrm e}^{4}-{\mathrm e}^{x}\right )}{x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )^{2}}{4}-\frac {\pi ^{2} x \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (x \,{\mathrm e}^{4}-{\mathrm e}^{x}\right )}{x}\right )^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )}{4}-\frac {\pi ^{2} x \,\mathrm {csgn}\left (i \left (x \,{\mathrm e}^{4}-{\mathrm e}^{x}\right )\right ) \mathrm {csgn}\left (\frac {i \left (x \,{\mathrm e}^{4}-{\mathrm e}^{x}\right )}{x}\right )^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )}{4}-x \ln \left (x \,{\mathrm e}^{4}-{\mathrm e}^{x}\right ) \ln \relax (x )+\frac {\pi ^{2} x \mathrm {csgn}\left (\frac {i \left (x \,{\mathrm e}^{4}-{\mathrm e}^{x}\right )}{x}\right )^{3} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )}{4}-\frac {i x \ln \left (x \,{\mathrm e}^{4}-{\mathrm e}^{x}\right ) \pi \,\mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )^{2}}{2}+\frac {i x \ln \relax (x ) \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )^{2}}{2}+\frac {i x \ln \relax (x ) \pi \,\mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )^{2}}{2}-\frac {i x \ln \left (x \,{\mathrm e}^{4}-{\mathrm e}^{x}\right ) \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )^{2}}{2}+\frac {\pi ^{2} x \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (x \,{\mathrm e}^{4}-{\mathrm e}^{x}\right )}{x}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )^{2}}{4}+\frac {\pi ^{2} x \,\mathrm {csgn}\left (i \left (x \,{\mathrm e}^{4}-{\mathrm e}^{x}\right )\right ) \mathrm {csgn}\left (\frac {i \left (x \,{\mathrm e}^{4}-{\mathrm e}^{x}\right )}{x}\right )^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )^{2}}{4}+\frac {\pi ^{2} x \,\mathrm {csgn}\left (i \left (x \,{\mathrm e}^{4}-{\mathrm e}^{x}\right )\right ) \mathrm {csgn}\left (\frac {i \left (x \,{\mathrm e}^{4}-{\mathrm e}^{x}\right )}{x}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )^{2}}{4}+\frac {\pi ^{2} x \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \left (x \,{\mathrm e}^{4}-{\mathrm e}^{x}\right )\right ) \mathrm {csgn}\left (\frac {i \left (x \,{\mathrm e}^{4}-{\mathrm e}^{x}\right )}{x}\right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )^{3}}{4}+\frac {\pi ^{2} x \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (x \,{\mathrm e}^{4}-{\mathrm e}^{x}\right )}{x}\right )^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )^{2}}{4}-2 i x \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )-\frac {i \pi x \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (x \,{\mathrm e}^{4}-{\mathrm e}^{x}\right )}{x}\right )^{2} \ln \relax (x )}{2}-\frac {i \pi x \,\mathrm {csgn}\left (i \left (x \,{\mathrm e}^{4}-{\mathrm e}^{x}\right )\right ) \mathrm {csgn}\left (\frac {i \left (x \,{\mathrm e}^{4}-{\mathrm e}^{x}\right )}{x}\right )^{2} \ln \relax (x )}{2}+2 i x \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )^{2}+2 i x \pi \,\mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )^{2}+\frac {i x \ln \left (x \,{\mathrm e}^{4}-{\mathrm e}^{x}\right ) \pi \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )^{3}}{2}+\frac {i \pi x \mathrm {csgn}\left (\frac {i \left (x \,{\mathrm e}^{4}-{\mathrm e}^{x}\right )}{x}\right )^{3} \ln \relax (x )}{2}-\frac {i x \ln \relax (x ) \pi \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )^{3}}{2}-\frac {\pi ^{2} x \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (x \,{\mathrm e}^{4}-{\mathrm e}^{x}\right )}{x}\right )^{2} \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )^{3}}{4}-\frac {\pi ^{2} x \,\mathrm {csgn}\left (i \left (x \,{\mathrm e}^{4}-{\mathrm e}^{x}\right )\right ) \mathrm {csgn}\left (\frac {i \left (x \,{\mathrm e}^{4}-{\mathrm e}^{x}\right )}{x}\right )^{2} \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )^{3}}{4}-\frac {\pi ^{2} x \mathrm {csgn}\left (\frac {i \left (x \,{\mathrm e}^{4}-{\mathrm e}^{x}\right )}{x}\right )^{3} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )^{2}}{4}-\frac {\pi ^{2} x \mathrm {csgn}\left (\frac {i \left (x \,{\mathrm e}^{4}-{\mathrm e}^{x}\right )}{x}\right )^{3} \mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )^{2}}{4}\)	\(1357\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-x*exp(-x+4)+1)*ln((x*exp(-x+4)-1)/x/exp(-x+4))+x-1)*ln(exp(x)*x)+((-x^2-x)*exp(-x+4)+x+1)*ln((x*exp(-x
+4)-1)/x/exp(-x+4)))/(x*exp(-x+4)-1),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.51, size = 42, normalized size = 1.40 \begin {gather*} x \log \relax (x)^{2} + 4 \, x^{2} - {\left (x^{2} + x \log \relax (x)\right )} \log \left (x e^{4} - e^{x}\right ) + {\left (x^{2} + 4 \, x\right )} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x*exp(-x+4)+1)*log((x*exp(-x+4)-1)/x/exp(-x+4))+x-1)*log(exp(x)*x)+((-x^2-x)*exp(-x+4)+x+1)*log(
(x*exp(-x+4)-1)/x/exp(-x+4)))/(x*exp(-x+4)-1),x, algorithm="maxima")

[Out]

x*log(x)^2 + 4*x^2 - (x^2 + x*log(x))*log(x*e^4 - e^x) + (x^2 + 4*x)*log(x)

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mupad [B]  time = 8.07, size = 20, normalized size = 0.67 \begin {gather*} -x\,\ln \left (\frac {x-{\mathrm {e}}^{-4}\,{\mathrm {e}}^x}{x}\right )\,\left (x+\ln \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log((exp(x - 4)*(x*exp(4 - x) - 1))/x)*(x - exp(4 - x)*(x + x^2) + 1) - log(x*exp(x))*(log((exp(x - 4)*(x
*exp(4 - x) - 1))/x)*(x*exp(4 - x) - 1) - x + 1))/(x*exp(4 - x) - 1),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x*exp(-x+4)+1)*ln((x*exp(-x+4)-1)/x/exp(-x+4))+x-1)*ln(exp(x)*x)+((-x**2-x)*exp(-x+4)+x+1)*ln((x
*exp(-x+4)-1)/x/exp(-x+4)))/(x*exp(-x+4)-1),x)

[Out]

Exception raised: CoercionFailed

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