\(\int \frac {-x+e^2 x+x^2-16 x^6+(2-2 e^2-4 x+192 x^5) \log (x-e^2 x-x^2+16 x^6)}{-x+e^2 x+x^2-16 x^6} \, dx\) [7915]

Optimal result

Integrand size = 72, antiderivative size = 24 \[ \int \frac {-x+e^2 x+x^2-16 x^6+\left (2-2 e^2-4 x+192 x^5\right ) \log \left (x-e^2 x-x^2+16 x^6\right )}{-x+e^2 x+x^2-16 x^6} \, dx=x-\log ^2\left (x \left (1-e^2-x+16 x^5\right )\right ) \]

[Out]

x-ln((16*x^5-x-exp(1)^2+1)*x)^2

Rubi [F]

\[ \int \frac {-x+e^2 x+x^2-16 x^6+\left (2-2 e^2-4 x+192 x^5\right ) \log \left (x-e^2 x-x^2+16 x^6\right )}{-x+e^2 x+x^2-16 x^6} \, dx=\int \frac {-x+e^2 x+x^2-16 x^6+\left (2-2 e^2-4 x+192 x^5\right ) \log \left (x-e^2 x-x^2+16 x^6\right )}{-x+e^2 x+x^2-16 x^6} \, dx \]

[In]

Int[(-x + E^2*x + x^2 - 16*x^6 + (2 - 2*E^2 - 4*x + 192*x^5)*Log[x - E^2*x - x^2 + 16*x^6])/(-x + E^2*x + x^2
- 16*x^6),x]

[Out]

x + Log[x]^2 - 2*Log[x]*Log[x*(1 - E^2 - x + 16*x^5)] - 2*Log[x*(1 - E^2 - x + 16*x^5)]*Defer[Int][(-1 + E^2 +
x - 16*x^5)^(-1), x] + 2*Defer[Int][Log[x]/(-1 + E^2 + x - 16*x^5), x] - 160*Defer[Int][(x^4*Log[x])/(-1 + E^
2 + x - 16*x^5), x] + 160*Defer[Int][(x^4*Log[x*(1 - E^2 - x + 16*x^5)])/(-1 + E^2 + x - 16*x^5), x] + 2*Defer
[Int][Defer[Int][(-1 + E^2 + x - 16*x^5)^(-1), x]/x, x] + 2*Defer[Int][Defer[Int][(-1 + E^2 + x - 16*x^5)^(-1)
, x]/(-1 + E^2 + x - 16*x^5), x] - 160*Defer[Int][(x^4*Defer[Int][(-1 + E^2 + x - 16*x^5)^(-1), x])/(-1 + E^2
+ x - 16*x^5), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {-x+e^2 x+x^2-16 x^6+\left (2-2 e^2-4 x+192 x^5\right ) \log \left (x-e^2 x-x^2+16 x^6\right )}{\left (-1+e^2\right ) x+x^2-16 x^6} \, dx \\ & = \int \frac {\left (-1+e^2\right ) x+x^2-16 x^6+\left (2-2 e^2-4 x+192 x^5\right ) \log \left (x-e^2 x-x^2+16 x^6\right )}{\left (-1+e^2\right ) x+x^2-16 x^6} \, dx \\ & = \int \frac {\left (-1+e^2\right ) x+x^2-16 x^6+\left (2-2 e^2-4 x+192 x^5\right ) \log \left (x-e^2 x-x^2+16 x^6\right )}{x \left (-1+e^2+x-16 x^5\right )} \, dx \\ & = \int \left (1-\frac {2 \left (1-e^2-2 x+96 x^5\right ) \log \left (x \left (1-e^2-x+16 x^5\right )\right )}{x \left (1-e^2-x+16 x^5\right )}\right ) \, dx \\ & = x-2 \int \frac {\left (1-e^2-2 x+96 x^5\right ) \log \left (x \left (1-e^2-x+16 x^5\right )\right )}{x \left (1-e^2-x+16 x^5\right )} \, dx \\ & = x-2 \int \left (\frac {\log \left (x \left (1-e^2-x+16 x^5\right )\right )}{x}+\frac {\left (1-80 x^4\right ) \log \left (x \left (1-e^2-x+16 x^5\right )\right )}{-1+e^2+x-16 x^5}\right ) \, dx \\ & = x-2 \int \frac {\log \left (x \left (1-e^2-x+16 x^5\right )\right )}{x} \, dx-2 \int \frac {\left (1-80 x^4\right ) \log \left (x \left (1-e^2-x+16 x^5\right )\right )}{-1+e^2+x-16 x^5} \, dx \\ & = x-2 \log (x) \log \left (x \left (1-e^2-x+16 x^5\right )\right )+2 \int \frac {\left (1-e^2-x+16 x^5+x \left (-1+80 x^4\right )\right ) \log (x)}{x \left (1-e^2-x+16 x^5\right )} \, dx-2 \int \left (\frac {\log \left (x \left (1-e^2-x+16 x^5\right )\right )}{-1+e^2+x-16 x^5}-\frac {80 x^4 \log \left (x \left (1-e^2-x+16 x^5\right )\right )}{-1+e^2+x-16 x^5}\right ) \, dx \\ & = x-2 \log (x) \log \left (x \left (1-e^2-x+16 x^5\right )\right )+2 \int \left (\frac {\log (x)}{x}+\frac {\left (1-80 x^4\right ) \log (x)}{-1+e^2+x-16 x^5}\right ) \, dx-2 \int \frac {\log \left (x \left (1-e^2-x+16 x^5\right )\right )}{-1+e^2+x-16 x^5} \, dx+160 \int \frac {x^4 \log \left (x \left (1-e^2-x+16 x^5\right )\right )}{-1+e^2+x-16 x^5} \, dx \\ & = x-2 \log (x) \log \left (x \left (1-e^2-x+16 x^5\right )\right )+2 \int \frac {\log (x)}{x} \, dx+2 \int \frac {\left (1-80 x^4\right ) \log (x)}{-1+e^2+x-16 x^5} \, dx+2 \int \frac {\left (1-e^2-2 x+96 x^5\right ) \int \frac {1}{-1+e^2+x-16 x^5} \, dx}{x \left (1-e^2-x+16 x^5\right )} \, dx+160 \int \frac {x^4 \log \left (x \left (1-e^2-x+16 x^5\right )\right )}{-1+e^2+x-16 x^5} \, dx-\left (2 \log \left (x \left (1-e^2-x+16 x^5\right )\right )\right ) \int \frac {1}{-1+e^2+x-16 x^5} \, dx \\ & = x+\log ^2(x)-2 \log (x) \log \left (x \left (1-e^2-x+16 x^5\right )\right )+2 \int \left (\frac {\log (x)}{-1+e^2+x-16 x^5}-\frac {80 x^4 \log (x)}{-1+e^2+x-16 x^5}\right ) \, dx+2 \int \left (\frac {\int \frac {1}{-1+e^2+x-16 x^5} \, dx}{x}+\frac {\left (-1+80 x^4\right ) \int \frac {1}{-1+e^2+x-16 x^5} \, dx}{1-e^2-x+16 x^5}\right ) \, dx+160 \int \frac {x^4 \log \left (x \left (1-e^2-x+16 x^5\right )\right )}{-1+e^2+x-16 x^5} \, dx-\left (2 \log \left (x \left (1-e^2-x+16 x^5\right )\right )\right ) \int \frac {1}{-1+e^2+x-16 x^5} \, dx \\ & = x+\log ^2(x)-2 \log (x) \log \left (x \left (1-e^2-x+16 x^5\right )\right )+2 \int \frac {\log (x)}{-1+e^2+x-16 x^5} \, dx+2 \int \frac {\int \frac {1}{-1+e^2+x-16 x^5} \, dx}{x} \, dx+2 \int \frac {\left (-1+80 x^4\right ) \int \frac {1}{-1+e^2+x-16 x^5} \, dx}{1-e^2-x+16 x^5} \, dx-160 \int \frac {x^4 \log (x)}{-1+e^2+x-16 x^5} \, dx+160 \int \frac {x^4 \log \left (x \left (1-e^2-x+16 x^5\right )\right )}{-1+e^2+x-16 x^5} \, dx-\left (2 \log \left (x \left (1-e^2-x+16 x^5\right )\right )\right ) \int \frac {1}{-1+e^2+x-16 x^5} \, dx \\ & = x+\log ^2(x)-2 \log (x) \log \left (x \left (1-e^2-x+16 x^5\right )\right )+2 \int \frac {\log (x)}{-1+e^2+x-16 x^5} \, dx+2 \int \frac {\int \frac {1}{-1+e^2+x-16 x^5} \, dx}{x} \, dx+2 \int \left (\frac {\int \frac {1}{-1+e^2+x-16 x^5} \, dx}{-1+e^2+x-16 x^5}-\frac {80 x^4 \int \frac {1}{-1+e^2+x-16 x^5} \, dx}{-1+e^2+x-16 x^5}\right ) \, dx-160 \int \frac {x^4 \log (x)}{-1+e^2+x-16 x^5} \, dx+160 \int \frac {x^4 \log \left (x \left (1-e^2-x+16 x^5\right )\right )}{-1+e^2+x-16 x^5} \, dx-\left (2 \log \left (x \left (1-e^2-x+16 x^5\right )\right )\right ) \int \frac {1}{-1+e^2+x-16 x^5} \, dx \\ & = x+\log ^2(x)-2 \log (x) \log \left (x \left (1-e^2-x+16 x^5\right )\right )+2 \int \frac {\log (x)}{-1+e^2+x-16 x^5} \, dx+2 \int \frac {\int \frac {1}{-1+e^2+x-16 x^5} \, dx}{x} \, dx+2 \int \frac {\int \frac {1}{-1+e^2+x-16 x^5} \, dx}{-1+e^2+x-16 x^5} \, dx-160 \int \frac {x^4 \log (x)}{-1+e^2+x-16 x^5} \, dx+160 \int \frac {x^4 \log \left (x \left (1-e^2-x+16 x^5\right )\right )}{-1+e^2+x-16 x^5} \, dx-160 \int \frac {x^4 \int \frac {1}{-1+e^2+x-16 x^5} \, dx}{-1+e^2+x-16 x^5} \, dx-\left (2 \log \left (x \left (1-e^2-x+16 x^5\right )\right )\right ) \int \frac {1}{-1+e^2+x-16 x^5} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {-x+e^2 x+x^2-16 x^6+\left (2-2 e^2-4 x+192 x^5\right ) \log \left (x-e^2 x-x^2+16 x^6\right )}{-x+e^2 x+x^2-16 x^6} \, dx=x-\log ^2\left (x \left (1-e^2-x+16 x^5\right )\right ) \]

[In]

Integrate[(-x + E^2*x + x^2 - 16*x^6 + (2 - 2*E^2 - 4*x + 192*x^5)*Log[x - E^2*x - x^2 + 16*x^6])/(-x + E^2*x
+ x^2 - 16*x^6),x]

[Out]

x - Log[x*(1 - E^2 - x + 16*x^5)]^2

Maple [A] (verified)

Time = 0.22 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04

method	result	size
risch	\(x -\ln \left (-{\mathrm e}^{2} x +16 x^{6}-x^{2}+x \right )^{2}\)	\(25\)
default	\(x -\ln \left (-{\mathrm e}^{2} x +16 x^{6}-x^{2}+x \right )^{2}\)	\(27\)
norman	\(x -\ln \left (-{\mathrm e}^{2} x +16 x^{6}-x^{2}+x \right )^{2}\)	\(27\)
parts	\(x -\ln \left (-{\mathrm e}^{2} x +16 x^{6}-x^{2}+x \right )^{2}\)	\(27\)

[In]

int(((-2*exp(1)^2+192*x^5-4*x+2)*ln(-x*exp(1)^2+16*x^6-x^2+x)+x*exp(1)^2-16*x^6+x^2-x)/(x*exp(1)^2-16*x^6+x^2-
x),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {-x+e^2 x+x^2-16 x^6+\left (2-2 e^2-4 x+192 x^5\right ) \log \left (x-e^2 x-x^2+16 x^6\right )}{-x+e^2 x+x^2-16 x^6} \, dx=-\log \left (16 \, x^{6} - x^{2} - x e^{2} + x\right )^{2} + x \]

[In]

integrate(((-2*exp(1)^2+192*x^5-4*x+2)*log(-x*exp(1)^2+16*x^6-x^2+x)+x*exp(1)^2-16*x^6+x^2-x)/(x*exp(1)^2-16*x
^6+x^2-x),x, algorithm="fricas")

[Out]

-log(16*x^6 - x^2 - x*e^2 + x)^2 + x

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79 \[ \int \frac {-x+e^2 x+x^2-16 x^6+\left (2-2 e^2-4 x+192 x^5\right ) \log \left (x-e^2 x-x^2+16 x^6\right )}{-x+e^2 x+x^2-16 x^6} \, dx=x - \log {\left (16 x^{6} - x^{2} - x e^{2} + x \right )}^{2} \]

[In]

integrate(((-2*exp(1)**2+192*x**5-4*x+2)*ln(-x*exp(1)**2+16*x**6-x**2+x)+x*exp(1)**2-16*x**6+x**2-x)/(x*exp(1)
**2-16*x**6+x**2-x),x)

[Out]

x - log(16*x**6 - x**2 - x*exp(2) + x)**2

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.92 \[ \int \frac {-x+e^2 x+x^2-16 x^6+\left (2-2 e^2-4 x+192 x^5\right ) \log \left (x-e^2 x-x^2+16 x^6\right )}{-x+e^2 x+x^2-16 x^6} \, dx=-\log \left (16 \, x^{5} - x - e^{2} + 1\right )^{2} - 2 \, \log \left (16 \, x^{5} - x - e^{2} + 1\right ) \log \left (x\right ) - \log \left (x\right )^{2} + x \]

[In]

integrate(((-2*exp(1)^2+192*x^5-4*x+2)*log(-x*exp(1)^2+16*x^6-x^2+x)+x*exp(1)^2-16*x^6+x^2-x)/(x*exp(1)^2-16*x
^6+x^2-x),x, algorithm="maxima")

[Out]

-log(16*x^5 - x - e^2 + 1)^2 - 2*log(16*x^5 - x - e^2 + 1)*log(x) - log(x)^2 + x

Giac [F]

\[ \int \frac {-x+e^2 x+x^2-16 x^6+\left (2-2 e^2-4 x+192 x^5\right ) \log \left (x-e^2 x-x^2+16 x^6\right )}{-x+e^2 x+x^2-16 x^6} \, dx=\int { \frac {16 \, x^{6} - x^{2} - x e^{2} - 2 \, {\left (96 \, x^{5} - 2 \, x - e^{2} + 1\right )} \log \left (16 \, x^{6} - x^{2} - x e^{2} + x\right ) + x}{16 \, x^{6} - x^{2} - x e^{2} + x} \,d x } \]

[In]

integrate(((-2*exp(1)^2+192*x^5-4*x+2)*log(-x*exp(1)^2+16*x^6-x^2+x)+x*exp(1)^2-16*x^6+x^2-x)/(x*exp(1)^2-16*x
^6+x^2-x),x, algorithm="giac")

[Out]

integrate((16*x^6 - x^2 - x*e^2 - 2*(96*x^5 - 2*x - e^2 + 1)*log(16*x^6 - x^2 - x*e^2 + x) + x)/(16*x^6 - x^2
- x*e^2 + x), x)

Mupad [B] (verification not implemented)

Time = 13.55 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {-x+e^2 x+x^2-16 x^6+\left (2-2 e^2-4 x+192 x^5\right ) \log \left (x-e^2 x-x^2+16 x^6\right )}{-x+e^2 x+x^2-16 x^6} \, dx=x-{\ln \left (x-x\,{\mathrm {e}}^2-x^2+16\,x^6\right )}^2 \]

[In]

int((x - x*exp(2) + log(x - x*exp(2) - x^2 + 16*x^6)*(4*x + 2*exp(2) - 192*x^5 - 2) - x^2 + 16*x^6)/(x - x*exp
(2) - x^2 + 16*x^6),x)

[Out]

x - log(x - x*exp(2) - x^2 + 16*x^6)^2