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

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

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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 = {} $$\begin{array}{c}\int \frac{-8e^{2x}-8x^2-32x^3+x^4+e^x(16x+44x^2-13x^3)+(3x^4+e^x(-4x^3+x^4))\log (x)+(-8x^3+e^x(12x^2-4x^3))\log \left(x^2\right)}{2e^{2x}-4e^{x}x+2x^2}dx\end{array}$$

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{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{-8e^{2x}-8x^2-32x^3+x^4+e^x(16x+44x^2-13x^3)+(3x^4+e^x(-4x^3+x^4))\log (x)+(-8x^3+e^x(12x^2-4x^3))\log \left(x^2\right)}{2{(e^x-x)}^2}dx\\ & =\frac{1}{2}\int \frac{-8e^{2x}-8x^2-32x^3+x^4+e^x(16x+44x^2-13x^3)+(3x^4+e^x(-4x^3+x^4))\log (x)+(-8x^3+e^x(12x^2-4x^3))\log \left(x^2\right)}{{(e^x-x)}^2}dx\\ & =\frac{1}{2}\int (-8+\frac{(-1+x)x^3(-12+x\log (x)-4\log \left(x^2\right))}{{(e^x-x)}^2}+\frac{x^2(44-13x-4x\log (x)+x^2\log (x)+12\log \left(x^2\right)-4x\log \left(x^2\right))}{e^x-x})dx\\ & =-4x+\frac{1}{2}\int \frac{(-1+x)x^3(-12+x\log (x)-4\log \left(x^2\right))}{{(e^x-x)}^2}dx+\frac{1}{2}\int \frac{x^2(44-13x-4x\log (x)+x^2\log (x)+12\log \left(x^2\right)-4x\log \left(x^2\right))}{e^x-x}dx\\ & =-4x+\frac{1}{2}\int (-\frac{x^3(-12+x\log (x)-4\log \left(x^2\right))}{{(e^x-x)}^2}+\frac{x^4(-12+x\log (x)-4\log \left(x^2\right))}{{(e^x-x)}^2})dx+\frac{1}{2}\int \frac{x^2(44-13x+(-4+x)x\log (x)-4(-3+x)\log \left(x^2\right))}{e^x-x}dx\\ & =-4x-\frac{1}{2}\int \frac{x^3(-12+x\log (x)-4\log \left(x^2\right))}{{(e^x-x)}^2}dx+\frac{1}{2}\int \frac{x^4(-12+x\log (x)-4\log \left(x^2\right))}{{(e^x-x)}^2}dx+\frac{1}{2}\int (\frac{44x^2}{e^x-x}-\frac{13x^3}{e^x-x}-\frac{4x^3\log (x)}{e^x-x}+\frac{x^4\log (x)}{e^x-x}+\frac{12x^2\log \left(x^2\right)}{e^x-x}-\frac{4x^3\log \left(x^2\right)}{e^x-x})dx\\ & =-4x+\frac{1}{2}\int \frac{x^4\log (x)}{e^x-x}dx-\frac{1}{2}\int (-\frac{12x^3}{{(e^x-x)}^2}+\frac{x^4\log (x)}{{(e^x-x)}^2}-\frac{4x^3\log \left(x^2\right)}{{(e^x-x)}^2})dx+\frac{1}{2}\int (-\frac{12x^4}{{(e^x-x)}^2}+\frac{x^5\log (x)}{{(e^x-x)}^2}-\frac{4x^4\log \left(x^2\right)}{{(e^x-x)}^2})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\\ & =-4x-\frac{1}{2}\int \frac{x^4\log (x)}{{(e^x-x)}^2}dx+\frac{1}{2}\int \frac{x^5\log (x)}{{(e^x-x)}^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)}{{(e^x-x)}^2}dx-2\int \frac{x^4\log \left(x^2\right)}{{(e^x-x)}^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}{{(e^x-x)}^2}dx-6\int \frac{x^4}{{(e^x-x)}^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-(2\log \left(x^2\right))\int \frac{x^3}{e^x-x}dx+(6\log \left(x^2\right))\int \frac{x^2}{e^x-x}dx\\ & =-4x+\frac{1}{2}\int \frac{\int \frac{x^4}{{(e^x-x)}^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}{{(e^x-x)}^2}dx}{x}dx-2\int \frac{2\int \frac{x^3}{{(e^x-x)}^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}{{(e^x-x)}^2}dx}{x}dx+4\int \frac{\int \frac{x^3}{e^x-x}dx}{x}dx+6\int \frac{x^3}{{(e^x-x)}^2}dx-6\int \frac{x^4}{{(e^x-x)}^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}{{(e^x-x)}^2}dx+\frac{1}{2}\log (x)\int \frac{x^4}{e^x-x}dx+\frac{1}{2}\log (x)\int \frac{x^5}{{(e^x-x)}^2}dx-(2\log (x))\int \frac{x^3}{e^x-x}dx+(2\log \left(x^2\right))\int \frac{x^3}{{(e^x-x)}^2}dx-(2\log \left(x^2\right))\int \frac{x^3}{e^x-x}dx-(2\log \left(x^2\right))\int \frac{x^4}{{(e^x-x)}^2}dx+(6\log \left(x^2\right))\int \frac{x^2}{e^x-x}dx\\ & =-4x+\frac{1}{2}\int \frac{\int \frac{x^4}{{(e^x-x)}^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}{{(e^x-x)}^2}dx}{x}dx+2\int \frac{\int \frac{x^3}{e^x-x}dx}{x}dx-4\int \frac{\int \frac{x^3}{{(e^x-x)}^2}dx}{x}dx+4\int \frac{\int \frac{x^3}{e^x-x}dx}{x}dx+4\int \frac{\int \frac{x^4}{{(e^x-x)}^2}dx}{x}dx+6\int \frac{x^3}{{(e^x-x)}^2}dx-6\int \frac{x^4}{{(e^x-x)}^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}{{(e^x-x)}^2}dx+\frac{1}{2}\log (x)\int \frac{x^4}{e^x-x}dx+\frac{1}{2}\log (x)\int \frac{x^5}{{(e^x-x)}^2}dx-(2\log (x))\int \frac{x^3}{e^x-x}dx+(2\log \left(x^2\right))\int \frac{x^3}{{(e^x-x)}^2}dx-(2\log \left(x^2\right))\int \frac{x^3}{e^x-x}dx-(2\log \left(x^2\right))\int \frac{x^4}{{(e^x-x)}^2}dx+(6\log \left(x^2\right))\int \frac{x^2}{e^x-x}dx\end{array}\end{array}$$

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

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 $$\begin{array}{c}-\frac{12x^3+8x^2-8x{e}^x-(x^4-8x^3)\log (x)}{2(x-e^x)}\end{array}$$

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 $$\begin{array}{c}\frac{x^4\log (x)-8x^3\log (x)-12x^3-8x^2+8x{e}^x}{2(x-e^x)}\end{array}$$

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	$\frac{(-8+x)x^3\ln (x)}{-2{\mathrm{e}}^x+2x}-\frac{x(i\pi x^2\mathrm{csgn}{\left(ix\right)}^2\mathrm{csgn}\left(i{x}^2\right)-2i\pi x^2\mathrm{csgn}\left(ix\right)\mathrm{csgn}{\left(i{x}^2\right)}^2+i\pi x^2\mathrm{csgn}{\left(i{x}^2\right)}^3-6x^2-4x+4{\mathrm{e}}^x)}{{\mathrm{e}}^x-x}$	$102$

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 $$\begin{array}{c}-\frac{12x^3+8x^2-8x{e}^x-(x^4-8x^3)\log (x)}{2(x-e^x)}\end{array}$$

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 $$\begin{array}{c}-\frac{x(8x-8{\mathrm{e}}^x-x^3\ln (x)+4x^2\ln \left(x^2\right)+12x^2)}{2(x-{\mathrm{e}}^x)}\end{array}$$

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 $$\begin{array}{c}-4x+\frac{-x^4\log (x)+8x^3\log (x)+12x^3}{-2x+2e^x}\end{array}$$

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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