3.38.90 \(\int \frac {-8 e^{2 x}-8 x^2-32 x^3+x^4+e^x (16 x+44 x^2-13 x^3)+(3 x^4+e^x (-4 x^3+x^4)) \log (x)+(-8 x^3+e^x (12 x^2-4 x^3)) \log (x^2)}{2 e^{2 x}-4 e^x x+2 x^2} \, dx\)

Optimal. Leaf size=31 \[ 2 x \left (-2+\frac {x^2 \left (3-\frac {1}{4} x \log (x)+\log \left (x^2\right )\right )}{e^x-x}\right ) \]

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Rubi [F]  time = 2.83, 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 {-8 e^{2 x}-8 x^2-32 x^3+x^4+e^x \left (16 x+44 x^2-13 x^3\right )+\left (3 x^4+e^x \left (-4 x^3+x^4\right )\right ) \log (x)+\left (-8 x^3+e^x \left (12 x^2-4 x^3\right )\right ) \log \left (x^2\right )}{2 e^{2 x}-4 e^x x+2 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-8*E^(2*x) - 8*x^2 - 32*x^3 + x^4 + E^x*(16*x + 44*x^2 - 13*x^3) + (3*x^4 + E^x*(-4*x^3 + x^4))*Log[x] +
(-8*x^3 + E^x*(12*x^2 - 4*x^3))*Log[x^2])/(2*E^(2*x) - 4*E^x*x + 2*x^2),x]

[Out]

-4*x + 22*Defer[Int][x^2/(E^x - x), x] + 6*Log[x^2]*Defer[Int][x^2/(E^x - x), x] + 6*Defer[Int][x^3/(E^x - x)^
2, x] + 2*Log[x^2]*Defer[Int][x^3/(E^x - x)^2, x] - (13*Defer[Int][x^3/(E^x - x), x])/2 - 2*Log[x]*Defer[Int][
x^3/(E^x - x), x] - 2*Log[x^2]*Defer[Int][x^3/(E^x - x), x] - 6*Defer[Int][x^4/(E^x - x)^2, x] - (Log[x]*Defer
[Int][x^4/(E^x - x)^2, x])/2 - 2*Log[x^2]*Defer[Int][x^4/(E^x - x)^2, x] + (Log[x]*Defer[Int][x^4/(E^x - x), x
])/2 + (Log[x]*Defer[Int][x^5/(E^x - x)^2, x])/2 - 12*Defer[Int][Defer[Int][x^2/(E^x - x), x]/x, x] - 4*Defer[
Int][Defer[Int][x^3/(E^x - x)^2, x]/x, x] + 6*Defer[Int][Defer[Int][x^3/(E^x - x), x]/x, x] + (9*Defer[Int][De
fer[Int][x^4/(E^x - x)^2, x]/x, x])/2 - Defer[Int][Defer[Int][x^4/(E^x - x), x]/x, x]/2 - Defer[Int][Defer[Int
][x^5/(E^x - x)^2, x]/x, x]/2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-8 e^{2 x}-8 x^2-32 x^3+x^4+e^x \left (16 x+44 x^2-13 x^3\right )+\left (3 x^4+e^x \left (-4 x^3+x^4\right )\right ) \log (x)+\left (-8 x^3+e^x \left (12 x^2-4 x^3\right )\right ) \log \left (x^2\right )}{2 \left (e^x-x\right )^2} \, dx\\ &=\frac {1}{2} \int \frac {-8 e^{2 x}-8 x^2-32 x^3+x^4+e^x \left (16 x+44 x^2-13 x^3\right )+\left (3 x^4+e^x \left (-4 x^3+x^4\right )\right ) \log (x)+\left (-8 x^3+e^x \left (12 x^2-4 x^3\right )\right ) \log \left (x^2\right )}{\left (e^x-x\right )^2} \, dx\\ &=\frac {1}{2} \int \left (-8+\frac {(-1+x) x^3 \left (-12+x \log (x)-4 \log \left (x^2\right )\right )}{\left (e^x-x\right )^2}+\frac {x^2 \left (44-13 x-4 x \log (x)+x^2 \log (x)+12 \log \left (x^2\right )-4 x \log \left (x^2\right )\right )}{e^x-x}\right ) \, dx\\ &=-4 x+\frac {1}{2} \int \frac {(-1+x) x^3 \left (-12+x \log (x)-4 \log \left (x^2\right )\right )}{\left (e^x-x\right )^2} \, dx+\frac {1}{2} \int \frac {x^2 \left (44-13 x-4 x \log (x)+x^2 \log (x)+12 \log \left (x^2\right )-4 x \log \left (x^2\right )\right )}{e^x-x} \, dx\\ &=-4 x+\frac {1}{2} \int \left (-\frac {x^3 \left (-12+x \log (x)-4 \log \left (x^2\right )\right )}{\left (e^x-x\right )^2}+\frac {x^4 \left (-12+x \log (x)-4 \log \left (x^2\right )\right )}{\left (e^x-x\right )^2}\right ) \, dx+\frac {1}{2} \int \frac {x^2 \left (44-13 x+(-4+x) x \log (x)-4 (-3+x) \log \left (x^2\right )\right )}{e^x-x} \, dx\\ &=-4 x-\frac {1}{2} \int \frac {x^3 \left (-12+x \log (x)-4 \log \left (x^2\right )\right )}{\left (e^x-x\right )^2} \, dx+\frac {1}{2} \int \frac {x^4 \left (-12+x \log (x)-4 \log \left (x^2\right )\right )}{\left (e^x-x\right )^2} \, dx+\frac {1}{2} \int \left (\frac {44 x^2}{e^x-x}-\frac {13 x^3}{e^x-x}-\frac {4 x^3 \log (x)}{e^x-x}+\frac {x^4 \log (x)}{e^x-x}+\frac {12 x^2 \log \left (x^2\right )}{e^x-x}-\frac {4 x^3 \log \left (x^2\right )}{e^x-x}\right ) \, dx\\ &=-4 x+\frac {1}{2} \int \frac {x^4 \log (x)}{e^x-x} \, dx-\frac {1}{2} \int \left (-\frac {12 x^3}{\left (e^x-x\right )^2}+\frac {x^4 \log (x)}{\left (e^x-x\right )^2}-\frac {4 x^3 \log \left (x^2\right )}{\left (e^x-x\right )^2}\right ) \, dx+\frac {1}{2} \int \left (-\frac {12 x^4}{\left (e^x-x\right )^2}+\frac {x^5 \log (x)}{\left (e^x-x\right )^2}-\frac {4 x^4 \log \left (x^2\right )}{\left (e^x-x\right )^2}\right ) \, dx-2 \int \frac {x^3 \log (x)}{e^x-x} \, dx-2 \int \frac {x^3 \log \left (x^2\right )}{e^x-x} \, dx+6 \int \frac {x^2 \log \left (x^2\right )}{e^x-x} \, dx-\frac {13}{2} \int \frac {x^3}{e^x-x} \, dx+22 \int \frac {x^2}{e^x-x} \, dx\\ &=-4 x-\frac {1}{2} \int \frac {x^4 \log (x)}{\left (e^x-x\right )^2} \, dx+\frac {1}{2} \int \frac {x^5 \log (x)}{\left (e^x-x\right )^2} \, dx-\frac {1}{2} \int \frac {\int \frac {x^4}{e^x-x} \, dx}{x} \, dx+2 \int \frac {x^3 \log \left (x^2\right )}{\left (e^x-x\right )^2} \, dx-2 \int \frac {x^4 \log \left (x^2\right )}{\left (e^x-x\right )^2} \, dx+2 \int \frac {\int \frac {x^3}{e^x-x} \, dx}{x} \, dx+2 \int \frac {2 \int \frac {x^3}{e^x-x} \, dx}{x} \, dx+6 \int \frac {x^3}{\left (e^x-x\right )^2} \, dx-6 \int \frac {x^4}{\left (e^x-x\right )^2} \, dx-6 \int \frac {2 \int \frac {x^2}{e^x-x} \, dx}{x} \, dx-\frac {13}{2} \int \frac {x^3}{e^x-x} \, dx+22 \int \frac {x^2}{e^x-x} \, dx+\frac {1}{2} \log (x) \int \frac {x^4}{e^x-x} \, dx-(2 \log (x)) \int \frac {x^3}{e^x-x} \, dx-\left (2 \log \left (x^2\right )\right ) \int \frac {x^3}{e^x-x} \, dx+\left (6 \log \left (x^2\right )\right ) \int \frac {x^2}{e^x-x} \, dx\\ &=-4 x+\frac {1}{2} \int \frac {\int \frac {x^4}{\left (e^x-x\right )^2} \, dx}{x} \, dx-\frac {1}{2} \int \frac {\int \frac {x^4}{e^x-x} \, dx}{x} \, dx-\frac {1}{2} \int \frac {\int \frac {x^5}{\left (e^x-x\right )^2} \, dx}{x} \, dx-2 \int \frac {2 \int \frac {x^3}{\left (e^x-x\right )^2} \, dx}{x} \, dx+2 \int \frac {\int \frac {x^3}{e^x-x} \, dx}{x} \, dx+2 \int \frac {2 \int \frac {x^4}{\left (e^x-x\right )^2} \, dx}{x} \, dx+4 \int \frac {\int \frac {x^3}{e^x-x} \, dx}{x} \, dx+6 \int \frac {x^3}{\left (e^x-x\right )^2} \, dx-6 \int \frac {x^4}{\left (e^x-x\right )^2} \, dx-\frac {13}{2} \int \frac {x^3}{e^x-x} \, dx-12 \int \frac {\int \frac {x^2}{e^x-x} \, dx}{x} \, dx+22 \int \frac {x^2}{e^x-x} \, dx-\frac {1}{2} \log (x) \int \frac {x^4}{\left (e^x-x\right )^2} \, dx+\frac {1}{2} \log (x) \int \frac {x^4}{e^x-x} \, dx+\frac {1}{2} \log (x) \int \frac {x^5}{\left (e^x-x\right )^2} \, dx-(2 \log (x)) \int \frac {x^3}{e^x-x} \, dx+\left (2 \log \left (x^2\right )\right ) \int \frac {x^3}{\left (e^x-x\right )^2} \, dx-\left (2 \log \left (x^2\right )\right ) \int \frac {x^3}{e^x-x} \, dx-\left (2 \log \left (x^2\right )\right ) \int \frac {x^4}{\left (e^x-x\right )^2} \, dx+\left (6 \log \left (x^2\right )\right ) \int \frac {x^2}{e^x-x} \, dx\\ &=-4 x+\frac {1}{2} \int \frac {\int \frac {x^4}{\left (e^x-x\right )^2} \, dx}{x} \, dx-\frac {1}{2} \int \frac {\int \frac {x^4}{e^x-x} \, dx}{x} \, dx-\frac {1}{2} \int \frac {\int \frac {x^5}{\left (e^x-x\right )^2} \, dx}{x} \, dx+2 \int \frac {\int \frac {x^3}{e^x-x} \, dx}{x} \, dx-4 \int \frac {\int \frac {x^3}{\left (e^x-x\right )^2} \, dx}{x} \, dx+4 \int \frac {\int \frac {x^3}{e^x-x} \, dx}{x} \, dx+4 \int \frac {\int \frac {x^4}{\left (e^x-x\right )^2} \, dx}{x} \, dx+6 \int \frac {x^3}{\left (e^x-x\right )^2} \, dx-6 \int \frac {x^4}{\left (e^x-x\right )^2} \, dx-\frac {13}{2} \int \frac {x^3}{e^x-x} \, dx-12 \int \frac {\int \frac {x^2}{e^x-x} \, dx}{x} \, dx+22 \int \frac {x^2}{e^x-x} \, dx-\frac {1}{2} \log (x) \int \frac {x^4}{\left (e^x-x\right )^2} \, dx+\frac {1}{2} \log (x) \int \frac {x^4}{e^x-x} \, dx+\frac {1}{2} \log (x) \int \frac {x^5}{\left (e^x-x\right )^2} \, dx-(2 \log (x)) \int \frac {x^3}{e^x-x} \, dx+\left (2 \log \left (x^2\right )\right ) \int \frac {x^3}{\left (e^x-x\right )^2} \, dx-\left (2 \log \left (x^2\right )\right ) \int \frac {x^3}{e^x-x} \, dx-\left (2 \log \left (x^2\right )\right ) \int \frac {x^4}{\left (e^x-x\right )^2} \, dx+\left (6 \log \left (x^2\right )\right ) \int \frac {x^2}{e^x-x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.10, size = 44, normalized size = 1.42 \begin {gather*} \frac {x \left (-8 e^x+8 x+12 x^2-x^3 \log (x)+4 x^2 \log \left (x^2\right )\right )}{2 \left (e^x-x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-8*E^(2*x) - 8*x^2 - 32*x^3 + x^4 + E^x*(16*x + 44*x^2 - 13*x^3) + (3*x^4 + E^x*(-4*x^3 + x^4))*Log
[x] + (-8*x^3 + E^x*(12*x^2 - 4*x^3))*Log[x^2])/(2*E^(2*x) - 4*E^x*x + 2*x^2),x]

[Out]

(x*(-8*E^x + 8*x + 12*x^2 - x^3*Log[x] + 4*x^2*Log[x^2]))/(2*(E^x - x))

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fricas [A]  time = 0.57, size = 39, normalized size = 1.26 \begin {gather*} -\frac {12 \, x^{3} + 8 \, x^{2} - 8 \, x e^{x} - {\left (x^{4} - 8 \, x^{3}\right )} \log \relax (x)}{2 \, {\left (x - e^{x}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*x^3+12*x^2)*exp(x)-8*x^3)*log(x^2)+((x^4-4*x^3)*exp(x)+3*x^4)*log(x)-8*exp(x)^2+(-13*x^3+44*x^
2+16*x)*exp(x)+x^4-32*x^3-8*x^2)/(2*exp(x)^2-4*exp(x)*x+2*x^2),x, algorithm="fricas")

[Out]

-1/2*(12*x^3 + 8*x^2 - 8*x*e^x - (x^4 - 8*x^3)*log(x))/(x - e^x)

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giac [A]  time = 0.21, size = 39, normalized size = 1.26 \begin {gather*} \frac {x^{4} \log \relax (x) - 8 \, x^{3} \log \relax (x) - 12 \, x^{3} - 8 \, x^{2} + 8 \, x e^{x}}{2 \, {\left (x - e^{x}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*x^3+12*x^2)*exp(x)-8*x^3)*log(x^2)+((x^4-4*x^3)*exp(x)+3*x^4)*log(x)-8*exp(x)^2+(-13*x^3+44*x^
2+16*x)*exp(x)+x^4-32*x^3-8*x^2)/(2*exp(x)^2-4*exp(x)*x+2*x^2),x, algorithm="giac")

[Out]

1/2*(x^4*log(x) - 8*x^3*log(x) - 12*x^3 - 8*x^2 + 8*x*e^x)/(x - e^x)

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maple [C]  time = 0.10, size = 102, normalized size = 3.29

method	result	size
risch	\(\frac {\left (-8+x \right ) x^{3} \ln \relax (x )}{-2 \,{\mathrm e}^{x}+2 x}-\frac {x \left (i \pi \,x^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 i \pi \,x^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+i \pi \,x^{2} \mathrm {csgn}\left (i x^{2}\right )^{3}-6 x^{2}-4 x +4 \,{\mathrm e}^{x}\right )}{{\mathrm e}^{x}-x}\)	\(102\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-4*x^3+12*x^2)*exp(x)-8*x^3)*ln(x^2)+((x^4-4*x^3)*exp(x)+3*x^4)*ln(x)-8*exp(x)^2+(-13*x^3+44*x^2+16*x)*
exp(x)+x^4-32*x^3-8*x^2)/(2*exp(x)^2-4*exp(x)*x+2*x^2),x,method=_RETURNVERBOSE)

[Out]

1/2*(-8+x)*x^3/(x-exp(x))*ln(x)-x*(I*Pi*x^2*csgn(I*x)^2*csgn(I*x^2)-2*I*Pi*x^2*csgn(I*x)*csgn(I*x^2)^2+I*Pi*x^
2*csgn(I*x^2)^3-6*x^2-4*x+4*exp(x))/(exp(x)-x)

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maxima [A]  time = 0.51, size = 39, normalized size = 1.26 \begin {gather*} -\frac {12 \, x^{3} + 8 \, x^{2} - 8 \, x e^{x} - {\left (x^{4} - 8 \, x^{3}\right )} \log \relax (x)}{2 \, {\left (x - e^{x}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*x^3+12*x^2)*exp(x)-8*x^3)*log(x^2)+((x^4-4*x^3)*exp(x)+3*x^4)*log(x)-8*exp(x)^2+(-13*x^3+44*x^
2+16*x)*exp(x)+x^4-32*x^3-8*x^2)/(2*exp(x)^2-4*exp(x)*x+2*x^2),x, algorithm="maxima")

[Out]

-1/2*(12*x^3 + 8*x^2 - 8*x*e^x - (x^4 - 8*x^3)*log(x))/(x - e^x)

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mupad [B]  time = 2.42, size = 42, normalized size = 1.35 \begin {gather*} -\frac {x\,\left (8\,x-8\,{\mathrm {e}}^x-x^3\,\ln \relax (x)+4\,x^2\,\ln \left (x^2\right )+12\,x^2\right )}{2\,\left (x-{\mathrm {e}}^x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(8*exp(2*x) - log(x^2)*(exp(x)*(12*x^2 - 4*x^3) - 8*x^3) + 8*x^2 + 32*x^3 - x^4 - exp(x)*(16*x + 44*x^2 -
13*x^3) + log(x)*(exp(x)*(4*x^3 - x^4) - 3*x^4))/(2*exp(2*x) - 4*x*exp(x) + 2*x^2),x)

[Out]

-(x*(8*x - 8*exp(x) - x^3*log(x) + 4*x^2*log(x^2) + 12*x^2))/(2*(x - exp(x)))

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sympy [A]  time = 0.28, size = 31, normalized size = 1.00 \begin {gather*} - 4 x + \frac {- x^{4} \log {\relax (x )} + 8 x^{3} \log {\relax (x )} + 12 x^{3}}{- 2 x + 2 e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*x**3+12*x**2)*exp(x)-8*x**3)*ln(x**2)+((x**4-4*x**3)*exp(x)+3*x**4)*ln(x)-8*exp(x)**2+(-13*x**
3+44*x**2+16*x)*exp(x)+x**4-32*x**3-8*x**2)/(2*exp(x)**2-4*exp(x)*x+2*x**2),x)

[Out]

-4*x + (-x**4*log(x) + 8*x**3*log(x) + 12*x**3)/(-2*x + 2*exp(x))

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