\(\int \frac {165+1920 x+8320 x^2-61440 x^4+(-165-3840 x-30720 x^2-81920 x^3) \log (x)}{x^2+2 x \log (x)+\log ^2(x)} \, dx\) [7156]

Optimal result

Integrand size = 49, antiderivative size = 22 \[ \int \frac {165+1920 x+8320 x^2-61440 x^4+\left (-165-3840 x-30720 x^2-81920 x^3\right ) \log (x)}{x^2+2 x \log (x)+\log ^2(x)} \, dx=\frac {5 \left (1-x-(1+8 x)^4\right )}{x+\log (x)} \]

[Out]

5*(1-x-(8*x+1)^4)/(x+ln(x))

Rubi [F]

\[ \int \frac {165+1920 x+8320 x^2-61440 x^4+\left (-165-3840 x-30720 x^2-81920 x^3\right ) \log (x)}{x^2+2 x \log (x)+\log ^2(x)} \, dx=\int \frac {165+1920 x+8320 x^2-61440 x^4+\left (-165-3840 x-30720 x^2-81920 x^3\right ) \log (x)}{x^2+2 x \log (x)+\log ^2(x)} \, dx \]

[In]

Int[(165 + 1920*x + 8320*x^2 - 61440*x^4 + (-165 - 3840*x - 30720*x^2 - 81920*x^3)*Log[x])/(x^2 + 2*x*Log[x] +
Log[x]^2),x]

[Out]

165*Defer[Int][(x + Log[x])^(-2), x] + 2085*Defer[Int][x/(x + Log[x])^2, x] + 12160*Defer[Int][x^2/(x + Log[x]
)^2, x] + 30720*Defer[Int][x^3/(x + Log[x])^2, x] + 20480*Defer[Int][x^4/(x + Log[x])^2, x] - 165*Defer[Int][(
x + Log[x])^(-1), x] - 3840*Defer[Int][x/(x + Log[x]), x] - 30720*Defer[Int][x^2/(x + Log[x]), x] - 81920*Defe
r[Int][x^3/(x + Log[x]), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {165+1920 x+8320 x^2-61440 x^4+\left (-165-3840 x-30720 x^2-81920 x^3\right ) \log (x)}{(x+\log (x))^2} \, dx \\ & = \int \left (\frac {5 \left (33+417 x+2432 x^2+6144 x^3+4096 x^4\right )}{(x+\log (x))^2}-\frac {5 \left (33+768 x+6144 x^2+16384 x^3\right )}{x+\log (x)}\right ) \, dx \\ & = 5 \int \frac {33+417 x+2432 x^2+6144 x^3+4096 x^4}{(x+\log (x))^2} \, dx-5 \int \frac {33+768 x+6144 x^2+16384 x^3}{x+\log (x)} \, dx \\ & = 5 \int \left (\frac {33}{(x+\log (x))^2}+\frac {417 x}{(x+\log (x))^2}+\frac {2432 x^2}{(x+\log (x))^2}+\frac {6144 x^3}{(x+\log (x))^2}+\frac {4096 x^4}{(x+\log (x))^2}\right ) \, dx-5 \int \left (\frac {33}{x+\log (x)}+\frac {768 x}{x+\log (x)}+\frac {6144 x^2}{x+\log (x)}+\frac {16384 x^3}{x+\log (x)}\right ) \, dx \\ & = 165 \int \frac {1}{(x+\log (x))^2} \, dx-165 \int \frac {1}{x+\log (x)} \, dx+2085 \int \frac {x}{(x+\log (x))^2} \, dx-3840 \int \frac {x}{x+\log (x)} \, dx+12160 \int \frac {x^2}{(x+\log (x))^2} \, dx+20480 \int \frac {x^4}{(x+\log (x))^2} \, dx+30720 \int \frac {x^3}{(x+\log (x))^2} \, dx-30720 \int \frac {x^2}{x+\log (x)} \, dx-81920 \int \frac {x^3}{x+\log (x)} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {165+1920 x+8320 x^2-61440 x^4+\left (-165-3840 x-30720 x^2-81920 x^3\right ) \log (x)}{x^2+2 x \log (x)+\log ^2(x)} \, dx=-\frac {5 x \left (33+384 x+2048 x^2+4096 x^3\right )}{x+\log (x)} \]

[In]

Integrate[(165 + 1920*x + 8320*x^2 - 61440*x^4 + (-165 - 3840*x - 30720*x^2 - 81920*x^3)*Log[x])/(x^2 + 2*x*Lo
g[x] + Log[x]^2),x]

[Out]

(-5*x*(33 + 384*x + 2048*x^2 + 4096*x^3))/(x + Log[x])

Maple [A] (verified)

Time = 0.80 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14

method	result	size
risch	\(-\frac {5 \left (4096 x^{3}+2048 x^{2}+384 x +33\right ) x}{x +\ln \left (x \right )}\)	\(25\)
parallelrisch	\(\frac {-20480 x^{4}-10240 x^{3}-1920 x^{2}-165 x}{x +\ln \left (x \right )}\)	\(27\)
default	\(-\frac {5 \left (4096 x^{4}+2048 x^{3}+384 x^{2}+33 x \right )}{x +\ln \left (x \right )}\)	\(28\)
norman	\(\frac {165 \ln \left (x \right )-1920 x^{2}-10240 x^{3}-20480 x^{4}}{x +\ln \left (x \right )}\)	\(28\)

[In]

int(((-81920*x^3-30720*x^2-3840*x-165)*ln(x)-61440*x^4+8320*x^2+1920*x+165)/(ln(x)^2+2*x*ln(x)+x^2),x,method=_
RETURNVERBOSE)

[Out]

-5*(4096*x^3+2048*x^2+384*x+33)*x/(x+ln(x))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.23 \[ \int \frac {165+1920 x+8320 x^2-61440 x^4+\left (-165-3840 x-30720 x^2-81920 x^3\right ) \log (x)}{x^2+2 x \log (x)+\log ^2(x)} \, dx=-\frac {5 \, {\left (4096 \, x^{4} + 2048 \, x^{3} + 384 \, x^{2} + 33 \, x\right )}}{x + \log \left (x\right )} \]

[In]

integrate(((-81920*x^3-30720*x^2-3840*x-165)*log(x)-61440*x^4+8320*x^2+1920*x+165)/(log(x)^2+2*x*log(x)+x^2),x
, algorithm="fricas")

[Out]

-5*(4096*x^4 + 2048*x^3 + 384*x^2 + 33*x)/(x + log(x))

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {165+1920 x+8320 x^2-61440 x^4+\left (-165-3840 x-30720 x^2-81920 x^3\right ) \log (x)}{x^2+2 x \log (x)+\log ^2(x)} \, dx=\frac {- 20480 x^{4} - 10240 x^{3} - 1920 x^{2} - 165 x}{x + \log {\left (x \right )}} \]

[In]

integrate(((-81920*x**3-30720*x**2-3840*x-165)*ln(x)-61440*x**4+8320*x**2+1920*x+165)/(ln(x)**2+2*x*ln(x)+x**2
),x)

[Out]

(-20480*x**4 - 10240*x**3 - 1920*x**2 - 165*x)/(x + log(x))

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.23 \[ \int \frac {165+1920 x+8320 x^2-61440 x^4+\left (-165-3840 x-30720 x^2-81920 x^3\right ) \log (x)}{x^2+2 x \log (x)+\log ^2(x)} \, dx=-\frac {5 \, {\left (4096 \, x^{4} + 2048 \, x^{3} + 384 \, x^{2} + 33 \, x\right )}}{x + \log \left (x\right )} \]

[In]

integrate(((-81920*x^3-30720*x^2-3840*x-165)*log(x)-61440*x^4+8320*x^2+1920*x+165)/(log(x)^2+2*x*log(x)+x^2),x
, algorithm="maxima")

[Out]

-5*(4096*x^4 + 2048*x^3 + 384*x^2 + 33*x)/(x + log(x))

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.23 \[ \int \frac {165+1920 x+8320 x^2-61440 x^4+\left (-165-3840 x-30720 x^2-81920 x^3\right ) \log (x)}{x^2+2 x \log (x)+\log ^2(x)} \, dx=-\frac {5 \, {\left (4096 \, x^{4} + 2048 \, x^{3} + 384 \, x^{2} + 33 \, x\right )}}{x + \log \left (x\right )} \]

[In]

integrate(((-81920*x^3-30720*x^2-3840*x-165)*log(x)-61440*x^4+8320*x^2+1920*x+165)/(log(x)^2+2*x*log(x)+x^2),x
, algorithm="giac")

[Out]

-5*(4096*x^4 + 2048*x^3 + 384*x^2 + 33*x)/(x + log(x))

Mupad [B] (verification not implemented)

Time = 12.39 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {165+1920 x+8320 x^2-61440 x^4+\left (-165-3840 x-30720 x^2-81920 x^3\right ) \log (x)}{x^2+2 x \log (x)+\log ^2(x)} \, dx=-\frac {5\,x\,\left (4096\,x^3+2048\,x^2+384\,x+33\right )}{x+\ln \left (x\right )} \]

[In]

int((1920*x + 8320*x^2 - 61440*x^4 - log(x)*(3840*x + 30720*x^2 + 81920*x^3 + 165) + 165)/(log(x)^2 + 2*x*log(
x) + x^2),x)

[Out]

-(5*x*(384*x + 2048*x^2 + 4096*x^3 + 33))/(x + log(x))