3.13.70 \(\int \frac {(-32 x-32 e^{-x} x-192 x^6) \log (1-e^{-x}+x+x^6)+(16-16 e^{-x}+16 x+16 x^6) \log ^2(1-e^{-x}+x+x^6)}{-x^2+e^{-x} x^2-x^3-x^8} \, dx\)

Optimal. Leaf size=21 \[ \frac {16 \log ^2\left (1-e^{-x}+x+x^6\right )}{x} \]

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Rubi [F]  time = 21.72, 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 (-32 x-32 e^{-x} x-192 x^6\right ) \log \left (1-e^{-x}+x+x^6\right )+\left (16-16 e^{-x}+16 x+16 x^6\right ) \log ^2\left (1-e^{-x}+x+x^6\right )}{-x^2+e^{-x} x^2-x^3-x^8} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-32*x - (32*x)/E^x - 192*x^6)*Log[1 - E^(-x) + x + x^6] + (16 - 16/E^x + 16*x + 16*x^6)*Log[1 - E^(-x) +
x + x^6]^2)/(-x^2 + x^2/E^x - x^3 - x^8),x]

[Out]

-32*Log[1 - E^(-x) + x + x^6]*Defer[Int][(1 + x + x^6)^(-1), x] + 192*Log[1 - E^(-x) + x + x^6]*Defer[Int][x^4
/(1 + x + x^6), x] - 32*Log[1 - E^(-x) + x + x^6]*Defer[Int][1/((1 + x + x^6)*(-1 + E^x + E^x*x + E^x*x^6)), x
] + 192*Log[1 - E^(-x) + x + x^6]*Defer[Int][x^4/((1 + x + x^6)*(-1 + E^x + E^x*x + E^x*x^6)), x] - 32*Log[1 -
E^(-x) + x + x^6]*Defer[Int][x^5/((1 + x + x^6)*(-1 + E^x + E^x*x + E^x*x^6)), x] + 64*Log[1 - E^(-x) + x + x
^6]*Defer[Int][1/(x*(-1 + E^x*(1 + x + x^6))), x] + 32*Defer[Int][Log[1 - E^(-x) + x + x^6]/x, x] - 32*Defer[I
nt][(x^5*Log[1 - E^(-x) + x + x^6])/(1 + x + x^6), x] - 16*Defer[Int][Log[1 - E^(-x) + x + x^6]^2/x^2, x] + 32
*Defer[Int][Defer[Int][(1 + x + x^6)^(-1), x]/(1 + x + x^6), x] + 192*Defer[Int][(x^5*Defer[Int][(1 + x + x^6)
^(-1), x])/(1 + x + x^6), x] + 32*Defer[Int][Defer[Int][(1 + x + x^6)^(-1), x]/((1 + x + x^6)*(-1 + E^x + E^x*
x + E^x*x^6)), x] + 192*Defer[Int][(x^5*Defer[Int][(1 + x + x^6)^(-1), x])/((1 + x + x^6)*(-1 + E^x + E^x*x +
E^x*x^6)), x] + 32*Defer[Int][Defer[Int][(1 + x + x^6)^(-1), x]/(-1 + E^x*(1 + x + x^6)), x] - 192*Defer[Int][
Defer[Int][x^4/(1 + x + x^6), x]/(1 + x + x^6), x] - 1152*Defer[Int][(x^5*Defer[Int][x^4/(1 + x + x^6), x])/(1
+ x + x^6), x] - 192*Defer[Int][Defer[Int][x^4/(1 + x + x^6), x]/((1 + x + x^6)*(-1 + E^x + E^x*x + E^x*x^6))
, x] - 1152*Defer[Int][(x^5*Defer[Int][x^4/(1 + x + x^6), x])/((1 + x + x^6)*(-1 + E^x + E^x*x + E^x*x^6)), x]
- 192*Defer[Int][Defer[Int][x^4/(1 + x + x^6), x]/(-1 + E^x*(1 + x + x^6)), x] - 64*Defer[Int][Defer[Int][1/(
x*(-1 + E^x*(1 + x + x^6))), x]/(1 + x + x^6), x] - 384*Defer[Int][(x^5*Defer[Int][1/(x*(-1 + E^x*(1 + x + x^6
))), x])/(1 + x + x^6), x] - 64*Defer[Int][Defer[Int][1/(x*(-1 + E^x*(1 + x + x^6))), x]/((1 + x + x^6)*(-1 +
E^x + E^x*x + E^x*x^6)), x] - 384*Defer[Int][(x^5*Defer[Int][1/(x*(-1 + E^x*(1 + x + x^6))), x])/((1 + x + x^6
)*(-1 + E^x + E^x*x + E^x*x^6)), x] - 64*Defer[Int][Defer[Int][1/(x*(-1 + E^x*(1 + x + x^6))), x]/(-1 + E^x*(1
+ x + x^6)), x] + 32*Defer[Int][Defer[Int][1/((1 + x + x^6)*(-1 + E^x*(1 + x + x^6))), x]/(1 + x + x^6), x] +
192*Defer[Int][(x^5*Defer[Int][1/((1 + x + x^6)*(-1 + E^x*(1 + x + x^6))), x])/(1 + x + x^6), x] + 32*Defer[I
nt][Defer[Int][1/((1 + x + x^6)*(-1 + E^x*(1 + x + x^6))), x]/((1 + x + x^6)*(-1 + E^x + E^x*x + E^x*x^6)), x]
+ 192*Defer[Int][(x^5*Defer[Int][1/((1 + x + x^6)*(-1 + E^x*(1 + x + x^6))), x])/((1 + x + x^6)*(-1 + E^x + E
^x*x + E^x*x^6)), x] + 32*Defer[Int][Defer[Int][1/((1 + x + x^6)*(-1 + E^x*(1 + x + x^6))), x]/(-1 + E^x*(1 +
x + x^6)), x] - 192*Defer[Int][Defer[Int][x^4/((1 + x + x^6)*(-1 + E^x*(1 + x + x^6))), x]/(1 + x + x^6), x] -
1152*Defer[Int][(x^5*Defer[Int][x^4/((1 + x + x^6)*(-1 + E^x*(1 + x + x^6))), x])/(1 + x + x^6), x] - 192*Def
er[Int][Defer[Int][x^4/((1 + x + x^6)*(-1 + E^x*(1 + x + x^6))), x]/((1 + x + x^6)*(-1 + E^x + E^x*x + E^x*x^6
)), x] - 1152*Defer[Int][(x^5*Defer[Int][x^4/((1 + x + x^6)*(-1 + E^x*(1 + x + x^6))), x])/((1 + x + x^6)*(-1
+ E^x + E^x*x + E^x*x^6)), x] - 192*Defer[Int][Defer[Int][x^4/((1 + x + x^6)*(-1 + E^x*(1 + x + x^6))), x]/(-1
+ E^x*(1 + x + x^6)), x] + 32*Defer[Int][Defer[Int][x^5/((1 + x + x^6)*(-1 + E^x*(1 + x + x^6))), x]/(1 + x +
x^6), x] + 192*Defer[Int][(x^5*Defer[Int][x^5/((1 + x + x^6)*(-1 + E^x*(1 + x + x^6))), x])/(1 + x + x^6), x]
+ 32*Defer[Int][Defer[Int][x^5/((1 + x + x^6)*(-1 + E^x*(1 + x + x^6))), x]/((1 + x + x^6)*(-1 + E^x + E^x*x
+ E^x*x^6)), x] + 192*Defer[Int][(x^5*Defer[Int][x^5/((1 + x + x^6)*(-1 + E^x*(1 + x + x^6))), x])/((1 + x + x
^6)*(-1 + E^x + E^x*x + E^x*x^6)), x] + 32*Defer[Int][Defer[Int][x^5/((1 + x + x^6)*(-1 + E^x*(1 + x + x^6))),
x]/(-1 + E^x*(1 + x + x^6)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {16 \left (\frac {2 x \left (1+e^x \left (1+6 x^5\right )\right )}{-1+e^x \left (1+x+x^6\right )}-\log \left (1-e^{-x}+x+x^6\right )\right ) \log \left (1-e^{-x}+x+x^6\right )}{x^2} \, dx\\ &=16 \int \frac {\left (\frac {2 x \left (1+e^x \left (1+6 x^5\right )\right )}{-1+e^x \left (1+x+x^6\right )}-\log \left (1-e^{-x}+x+x^6\right )\right ) \log \left (1-e^{-x}+x+x^6\right )}{x^2} \, dx\\ &=16 \int \left (\frac {2 \left (2+x+6 x^5+x^6\right ) \log \left (1-e^{-x}+x+x^6\right )}{x \left (1+x+x^6\right ) \left (-1+e^x+e^x x+e^x x^6\right )}-\frac {\log \left (1-e^{-x}+x+x^6\right ) \left (-2 x-12 x^6+\log \left (1-e^{-x}+x+x^6\right )+x \log \left (1-e^{-x}+x+x^6\right )+x^6 \log \left (1-e^{-x}+x+x^6\right )\right )}{x^2 \left (1+x+x^6\right )}\right ) \, dx\\ &=-\left (16 \int \frac {\log \left (1-e^{-x}+x+x^6\right ) \left (-2 x-12 x^6+\log \left (1-e^{-x}+x+x^6\right )+x \log \left (1-e^{-x}+x+x^6\right )+x^6 \log \left (1-e^{-x}+x+x^6\right )\right )}{x^2 \left (1+x+x^6\right )} \, dx\right )+32 \int \frac {\left (2+x+6 x^5+x^6\right ) \log \left (1-e^{-x}+x+x^6\right )}{x \left (1+x+x^6\right ) \left (-1+e^x+e^x x+e^x x^6\right )} \, dx\\ &=-\left (16 \int \frac {\log \left (1-e^{-x}+x+x^6\right ) \left (-\frac {2 \left (x+6 x^6\right )}{1+x+x^6}+\log \left (1-e^{-x}+x+x^6\right )\right )}{x^2} \, dx\right )-32 \int \frac {\left (1+e^x \left (1+6 x^5\right )\right ) \left (-2 \int \frac {1}{x \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx+\int \frac {1}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx-6 \int \frac {x^4}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx+\int \frac {x^5}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx\right )}{1-e^x \left (1+x+x^6\right )} \, dx-\left (32 \log \left (1-e^{-x}+x+x^6\right )\right ) \int \frac {1}{\left (1+x+x^6\right ) \left (-1+e^x+e^x x+e^x x^6\right )} \, dx-\left (32 \log \left (1-e^{-x}+x+x^6\right )\right ) \int \frac {x^5}{\left (1+x+x^6\right ) \left (-1+e^x+e^x x+e^x x^6\right )} \, dx+\left (64 \log \left (1-e^{-x}+x+x^6\right )\right ) \int \frac {1}{x \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx+\left (192 \log \left (1-e^{-x}+x+x^6\right )\right ) \int \frac {x^4}{\left (1+x+x^6\right ) \left (-1+e^x+e^x x+e^x x^6\right )} \, dx\\ &=-\left (16 \int \left (-\frac {2 \left (1+6 x^5\right ) \log \left (1-e^{-x}+x+x^6\right )}{x \left (1+x+x^6\right )}+\frac {\log ^2\left (1-e^{-x}+x+x^6\right )}{x^2}\right ) \, dx\right )-32 \int \left (\frac {\left (1+6 x^5\right ) \left (2 \int \frac {1}{x \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx-\int \frac {1}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx+6 \int \frac {x^4}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx-\int \frac {x^5}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx\right )}{1+x+x^6}+\frac {\left (2+x+6 x^5+x^6\right ) \left (2 \int \frac {1}{x \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx-\int \frac {1}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx+6 \int \frac {x^4}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx-\int \frac {x^5}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx\right )}{\left (1+x+x^6\right ) \left (-1+e^x+e^x x+e^x x^6\right )}\right ) \, dx-\left (32 \log \left (1-e^{-x}+x+x^6\right )\right ) \int \frac {1}{\left (1+x+x^6\right ) \left (-1+e^x+e^x x+e^x x^6\right )} \, dx-\left (32 \log \left (1-e^{-x}+x+x^6\right )\right ) \int \frac {x^5}{\left (1+x+x^6\right ) \left (-1+e^x+e^x x+e^x x^6\right )} \, dx+\left (64 \log \left (1-e^{-x}+x+x^6\right )\right ) \int \frac {1}{x \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx+\left (192 \log \left (1-e^{-x}+x+x^6\right )\right ) \int \frac {x^4}{\left (1+x+x^6\right ) \left (-1+e^x+e^x x+e^x x^6\right )} \, dx\\ &=-\left (16 \int \frac {\log ^2\left (1-e^{-x}+x+x^6\right )}{x^2} \, dx\right )+32 \int \frac {\left (1+6 x^5\right ) \log \left (1-e^{-x}+x+x^6\right )}{x \left (1+x+x^6\right )} \, dx-32 \int \frac {\left (1+6 x^5\right ) \left (2 \int \frac {1}{x \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx-\int \frac {1}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx+6 \int \frac {x^4}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx-\int \frac {x^5}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx\right )}{1+x+x^6} \, dx-32 \int \frac {\left (2+x+6 x^5+x^6\right ) \left (2 \int \frac {1}{x \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx-\int \frac {1}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx+6 \int \frac {x^4}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx-\int \frac {x^5}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx\right )}{\left (1+x+x^6\right ) \left (-1+e^x+e^x x+e^x x^6\right )} \, dx-\left (32 \log \left (1-e^{-x}+x+x^6\right )\right ) \int \frac {1}{\left (1+x+x^6\right ) \left (-1+e^x+e^x x+e^x x^6\right )} \, dx-\left (32 \log \left (1-e^{-x}+x+x^6\right )\right ) \int \frac {x^5}{\left (1+x+x^6\right ) \left (-1+e^x+e^x x+e^x x^6\right )} \, dx+\left (64 \log \left (1-e^{-x}+x+x^6\right )\right ) \int \frac {1}{x \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx+\left (192 \log \left (1-e^{-x}+x+x^6\right )\right ) \int \frac {x^4}{\left (1+x+x^6\right ) \left (-1+e^x+e^x x+e^x x^6\right )} \, dx\\ &=-\left (16 \int \frac {\log ^2\left (1-e^{-x}+x+x^6\right )}{x^2} \, dx\right )+32 \int \left (\frac {\log \left (1-e^{-x}+x+x^6\right )}{x}-\frac {\left (1-6 x^4+x^5\right ) \log \left (1-e^{-x}+x+x^6\right )}{1+x+x^6}\right ) \, dx-32 \int \left (\frac {2 \int \frac {1}{x \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx-\int \frac {1}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx+6 \int \frac {x^4}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx-\int \frac {x^5}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx}{-1+e^x+e^x x+e^x x^6}+\frac {\left (1+6 x^5\right ) \left (2 \int \frac {1}{x \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx-\int \frac {1}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx+6 \int \frac {x^4}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx-\int \frac {x^5}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx\right )}{\left (1+x+x^6\right ) \left (-1+e^x+e^x x+e^x x^6\right )}\right ) \, dx-32 \int \left (\frac {\left (1+6 x^5\right ) \left (2 \int \frac {1}{x \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx-\int \frac {1}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx+6 \int \frac {x^4}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx\right )}{1+x+x^6}-\frac {\left (1+6 x^5\right ) \int \frac {x^5}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx}{1+x+x^6}\right ) \, dx-\left (32 \log \left (1-e^{-x}+x+x^6\right )\right ) \int \frac {1}{\left (1+x+x^6\right ) \left (-1+e^x+e^x x+e^x x^6\right )} \, dx-\left (32 \log \left (1-e^{-x}+x+x^6\right )\right ) \int \frac {x^5}{\left (1+x+x^6\right ) \left (-1+e^x+e^x x+e^x x^6\right )} \, dx+\left (64 \log \left (1-e^{-x}+x+x^6\right )\right ) \int \frac {1}{x \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx+\left (192 \log \left (1-e^{-x}+x+x^6\right )\right ) \int \frac {x^4}{\left (1+x+x^6\right ) \left (-1+e^x+e^x x+e^x x^6\right )} \, dx\\ &=-\left (16 \int \frac {\log ^2\left (1-e^{-x}+x+x^6\right )}{x^2} \, dx\right )+32 \int \frac {\log \left (1-e^{-x}+x+x^6\right )}{x} \, dx-32 \int \frac {\left (1-6 x^4+x^5\right ) \log \left (1-e^{-x}+x+x^6\right )}{1+x+x^6} \, dx-32 \int \frac {\left (1+6 x^5\right ) \left (2 \int \frac {1}{x \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx-\int \frac {1}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx+6 \int \frac {x^4}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx\right )}{1+x+x^6} \, dx-32 \int \frac {2 \int \frac {1}{x \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx-\int \frac {1}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx+6 \int \frac {x^4}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx-\int \frac {x^5}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx}{-1+e^x+e^x x+e^x x^6} \, dx-32 \int \frac {\left (1+6 x^5\right ) \left (2 \int \frac {1}{x \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx-\int \frac {1}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx+6 \int \frac {x^4}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx-\int \frac {x^5}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx\right )}{\left (1+x+x^6\right ) \left (-1+e^x+e^x x+e^x x^6\right )} \, dx+32 \int \frac {\left (1+6 x^5\right ) \int \frac {x^5}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx}{1+x+x^6} \, dx-\left (32 \log \left (1-e^{-x}+x+x^6\right )\right ) \int \frac {1}{\left (1+x+x^6\right ) \left (-1+e^x+e^x x+e^x x^6\right )} \, dx-\left (32 \log \left (1-e^{-x}+x+x^6\right )\right ) \int \frac {x^5}{\left (1+x+x^6\right ) \left (-1+e^x+e^x x+e^x x^6\right )} \, dx+\left (64 \log \left (1-e^{-x}+x+x^6\right )\right ) \int \frac {1}{x \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx+\left (192 \log \left (1-e^{-x}+x+x^6\right )\right ) \int \frac {x^4}{\left (1+x+x^6\right ) \left (-1+e^x+e^x x+e^x x^6\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.79, size = 58, normalized size = 2.76 \begin {gather*} 16 \left (2 x+2 \log \left (1-e^{-x}+x+x^6\right )+\frac {\log ^2\left (1-e^{-x}+x+x^6\right )}{x}-2 \log \left (1-e^x \left (1+x+x^6\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-32*x - (32*x)/E^x - 192*x^6)*Log[1 - E^(-x) + x + x^6] + (16 - 16/E^x + 16*x + 16*x^6)*Log[1 - E^
(-x) + x + x^6]^2)/(-x^2 + x^2/E^x - x^3 - x^8),x]

[Out]

16*(2*x + 2*Log[1 - E^(-x) + x + x^6] + Log[1 - E^(-x) + x + x^6]^2/x - 2*Log[1 - E^x*(1 + x + x^6)])

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fricas [A]  time = 0.68, size = 20, normalized size = 0.95 \begin {gather*} \frac {16 \, \log \left (x^{6} + x - e^{\left (-x\right )} + 1\right )^{2}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*exp(-x)+16*x^6+16*x+16)*log(-exp(-x)+x^6+x+1)^2+(-32*x*exp(-x)-192*x^6-32*x)*log(-exp(-x)+x^6+
x+1))/(x^2*exp(-x)-x^8-x^3-x^2),x, algorithm="fricas")

[Out]

16*log(x^6 + x - e^(-x) + 1)^2/x

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giac [A]  time = 0.99, size = 20, normalized size = 0.95 \begin {gather*} \frac {16 \, \log \left (x^{6} + x - e^{\left (-x\right )} + 1\right )^{2}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*exp(-x)+16*x^6+16*x+16)*log(-exp(-x)+x^6+x+1)^2+(-32*x*exp(-x)-192*x^6-32*x)*log(-exp(-x)+x^6+
x+1))/(x^2*exp(-x)-x^8-x^3-x^2),x, algorithm="giac")

[Out]

16*log(x^6 + x - e^(-x) + 1)^2/x

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maple [A]  time = 0.02, size = 21, normalized size = 1.00

method	result	size
risch	\(\frac {16 \ln \left (-{\mathrm e}^{-x}+x^{6}+x +1\right )^{2}}{x}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-16*exp(-x)+16*x^6+16*x+16)*ln(-exp(-x)+x^6+x+1)^2+(-32*x*exp(-x)-192*x^6-32*x)*ln(-exp(-x)+x^6+x+1))/(x
^2*exp(-x)-x^8-x^3-x^2),x,method=_RETURNVERBOSE)

[Out]

16*ln(-exp(-x)+x^6+x+1)^2/x

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maxima [B]  time = 0.45, size = 56, normalized size = 2.67 \begin {gather*} \frac {16 \, {\left (x^{2} + \log \left ({\left (x^{6} + x + 1\right )} e^{x} - 1\right )^{2}\right )}}{x} - 32 \, \log \left (x^{6} + x + 1\right ) - 32 \, \log \left (\frac {{\left (x^{6} + x + 1\right )} e^{x} - 1}{x^{6} + x + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*exp(-x)+16*x^6+16*x+16)*log(-exp(-x)+x^6+x+1)^2+(-32*x*exp(-x)-192*x^6-32*x)*log(-exp(-x)+x^6+
x+1))/(x^2*exp(-x)-x^8-x^3-x^2),x, algorithm="maxima")

[Out]

16*(x^2 + log((x^6 + x + 1)*e^x - 1)^2)/x - 32*log(x^6 + x + 1) - 32*log(((x^6 + x + 1)*e^x - 1)/(x^6 + x + 1)
)

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mupad [B]  time = 1.08, size = 20, normalized size = 0.95 \begin {gather*} \frac {16\,{\ln \left (x-{\mathrm {e}}^{-x}+x^6+1\right )}^2}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x - exp(-x) + x^6 + 1)^2*(16*x - 16*exp(-x) + 16*x^6 + 16) - log(x - exp(-x) + x^6 + 1)*(32*x + 32*x
*exp(-x) + 192*x^6))/(x^2 - x^2*exp(-x) + x^3 + x^8),x)

[Out]

(16*log(x - exp(-x) + x^6 + 1)^2)/x

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sympy [A]  time = 0.39, size = 17, normalized size = 0.81 \begin {gather*} \frac {16 \log {\left (x^{6} + x + 1 - e^{- x} \right )}^{2}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*exp(-x)+16*x**6+16*x+16)*ln(-exp(-x)+x**6+x+1)**2+(-32*x*exp(-x)-192*x**6-32*x)*ln(-exp(-x)+x*
*6+x+1))/(x**2*exp(-x)-x**8-x**3-x**2),x)

[Out]

16*log(x**6 + x + 1 - exp(-x))**2/x

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