\(\int \frac {-800-800 x+(1600-400 x-800 x^2) \log (x)+(-5 x+2 x^3) \log ^3(x)+((400+400 x) \log (x)+(-x-x^2) \log ^3(x)) \log (\frac {400-x \log ^2(x)}{\log ^2(x)})}{(-400-400 x) \log (x)+(x+x^2) \log ^3(x)} \, dx\) [3945]

Optimal result

Integrand size = 93, antiderivative size = 26 \[ \int \frac {-800-800 x+\left (1600-400 x-800 x^2\right ) \log (x)+\left (-5 x+2 x^3\right ) \log ^3(x)+\left ((400+400 x) \log (x)+\left (-x-x^2\right ) \log ^3(x)\right ) \log \left (\frac {400-x \log ^2(x)}{\log ^2(x)}\right )}{(-400-400 x) \log (x)+\left (x+x^2\right ) \log ^3(x)} \, dx=5-3 \log (1+x)+x \left (-1+x-\log \left (-x+\frac {400}{\log ^2(x)}\right )\right ) \]

[Out]

5+x*(x-ln(400/ln(x)^2-x)-1)-3*ln(1+x)

Rubi [F]

\[ \int \frac {-800-800 x+\left (1600-400 x-800 x^2\right ) \log (x)+\left (-5 x+2 x^3\right ) \log ^3(x)+\left ((400+400 x) \log (x)+\left (-x-x^2\right ) \log ^3(x)\right ) \log \left (\frac {400-x \log ^2(x)}{\log ^2(x)}\right )}{(-400-400 x) \log (x)+\left (x+x^2\right ) \log ^3(x)} \, dx=\int \frac {-800-800 x+\left (1600-400 x-800 x^2\right ) \log (x)+\left (-5 x+2 x^3\right ) \log ^3(x)+\left ((400+400 x) \log (x)+\left (-x-x^2\right ) \log ^3(x)\right ) \log \left (\frac {400-x \log ^2(x)}{\log ^2(x)}\right )}{(-400-400 x) \log (x)+\left (x+x^2\right ) \log ^3(x)} \, dx \]

[In]

Int[(-800 - 800*x + (1600 - 400*x - 800*x^2)*Log[x] + (-5*x + 2*x^3)*Log[x]^3 + ((400 + 400*x)*Log[x] + (-x -
x^2)*Log[x]^3)*Log[(400 - x*Log[x]^2)/Log[x]^2])/((-400 - 400*x)*Log[x] + (x + x^2)*Log[x]^3),x]

[Out]

-2*x + x^2 - 3*Log[1 + x] + 2*Defer[Int][1/((1 + x)*Log[x]), x] + 2*Defer[Int][x/((1 + x)*Log[x]), x] - 400*De
fer[Int][(400 - x*Log[x]^2)^(-1), x] - 800*Defer[Int][(-400 + x*Log[x]^2)^(-1), x] - 2*Defer[Int][(x*Log[x])/(
-400 + x*Log[x]^2), x] - Defer[Int][Log[-x + 400/Log[x]^2], x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {800+800 x-\left (1600-400 x-800 x^2\right ) \log (x)-\left (-5 x+2 x^3\right ) \log ^3(x)-\left ((400+400 x) \log (x)+\left (-x-x^2\right ) \log ^3(x)\right ) \log \left (\frac {400-x \log ^2(x)}{\log ^2(x)}\right )}{(1+x) \log (x) \left (400-x \log ^2(x)\right )} \, dx \\ & = \int \left (-\frac {400 \left (-4+x+2 x^2\right )}{(1+x) \left (-400+x \log ^2(x)\right )}-\frac {800}{(1+x) \log (x) \left (-400+x \log ^2(x)\right )}-\frac {800 x}{(1+x) \log (x) \left (-400+x \log ^2(x)\right )}+\frac {x \left (-5+2 x^2\right ) \log ^2(x)}{(1+x) \left (-400+x \log ^2(x)\right )}-\log \left (-x+\frac {400}{\log ^2(x)}\right )\right ) \, dx \\ & = -\left (400 \int \frac {-4+x+2 x^2}{(1+x) \left (-400+x \log ^2(x)\right )} \, dx\right )-800 \int \frac {1}{(1+x) \log (x) \left (-400+x \log ^2(x)\right )} \, dx-800 \int \frac {x}{(1+x) \log (x) \left (-400+x \log ^2(x)\right )} \, dx+\int \frac {x \left (-5+2 x^2\right ) \log ^2(x)}{(1+x) \left (-400+x \log ^2(x)\right )} \, dx-\int \log \left (-x+\frac {400}{\log ^2(x)}\right ) \, dx \\ & = -\left (400 \int \left (\frac {1}{400-x \log ^2(x)}+\frac {2 x}{-400+x \log ^2(x)}-\frac {3}{(1+x) \left (-400+x \log ^2(x)\right )}\right ) \, dx\right )-800 \int \left (-\frac {1}{400 (1+x) \log (x)}+\frac {x \log (x)}{400 (1+x) \left (-400+x \log ^2(x)\right )}\right ) \, dx-800 \int \left (-\frac {x}{400 (1+x) \log (x)}+\frac {x^2 \log (x)}{400 (1+x) \left (-400+x \log ^2(x)\right )}\right ) \, dx+\int \left (\frac {-5+2 x^2}{1+x}+\frac {400 \left (-5+2 x^2\right )}{(1+x) \left (-400+x \log ^2(x)\right )}\right ) \, dx-\int \log \left (-x+\frac {400}{\log ^2(x)}\right ) \, dx \\ & = 2 \int \frac {1}{(1+x) \log (x)} \, dx+2 \int \frac {x}{(1+x) \log (x)} \, dx-2 \int \frac {x \log (x)}{(1+x) \left (-400+x \log ^2(x)\right )} \, dx-2 \int \frac {x^2 \log (x)}{(1+x) \left (-400+x \log ^2(x)\right )} \, dx-400 \int \frac {1}{400-x \log ^2(x)} \, dx+400 \int \frac {-5+2 x^2}{(1+x) \left (-400+x \log ^2(x)\right )} \, dx-800 \int \frac {x}{-400+x \log ^2(x)} \, dx+1200 \int \frac {1}{(1+x) \left (-400+x \log ^2(x)\right )} \, dx+\int \frac {-5+2 x^2}{1+x} \, dx-\int \log \left (-x+\frac {400}{\log ^2(x)}\right ) \, dx \\ & = 2 \int \frac {1}{(1+x) \log (x)} \, dx+2 \int \frac {x}{(1+x) \log (x)} \, dx-2 \int \left (\frac {\log (x)}{-400+x \log ^2(x)}-\frac {\log (x)}{(1+x) \left (-400+x \log ^2(x)\right )}\right ) \, dx-2 \int \left (-\frac {\log (x)}{-400+x \log ^2(x)}+\frac {x \log (x)}{-400+x \log ^2(x)}+\frac {\log (x)}{(1+x) \left (-400+x \log ^2(x)\right )}\right ) \, dx-400 \int \frac {1}{400-x \log ^2(x)} \, dx+400 \int \left (-\frac {2}{-400+x \log ^2(x)}+\frac {2 x}{-400+x \log ^2(x)}-\frac {3}{(1+x) \left (-400+x \log ^2(x)\right )}\right ) \, dx-800 \int \frac {x}{-400+x \log ^2(x)} \, dx+1200 \int \frac {1}{(1+x) \left (-400+x \log ^2(x)\right )} \, dx+\int \left (-2+2 x-\frac {3}{1+x}\right ) \, dx-\int \log \left (-x+\frac {400}{\log ^2(x)}\right ) \, dx \\ & = -2 x+x^2-3 \log (1+x)+2 \int \frac {1}{(1+x) \log (x)} \, dx+2 \int \frac {x}{(1+x) \log (x)} \, dx-2 \int \frac {x \log (x)}{-400+x \log ^2(x)} \, dx-400 \int \frac {1}{400-x \log ^2(x)} \, dx-800 \int \frac {1}{-400+x \log ^2(x)} \, dx-\int \log \left (-x+\frac {400}{\log ^2(x)}\right ) \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {-800-800 x+\left (1600-400 x-800 x^2\right ) \log (x)+\left (-5 x+2 x^3\right ) \log ^3(x)+\left ((400+400 x) \log (x)+\left (-x-x^2\right ) \log ^3(x)\right ) \log \left (\frac {400-x \log ^2(x)}{\log ^2(x)}\right )}{(-400-400 x) \log (x)+\left (x+x^2\right ) \log ^3(x)} \, dx=-x+x^2-3 \log (1+x)-x \log \left (-x+\frac {400}{\log ^2(x)}\right ) \]

[In]

Integrate[(-800 - 800*x + (1600 - 400*x - 800*x^2)*Log[x] + (-5*x + 2*x^3)*Log[x]^3 + ((400 + 400*x)*Log[x] +
(-x - x^2)*Log[x]^3)*Log[(400 - x*Log[x]^2)/Log[x]^2])/((-400 - 400*x)*Log[x] + (x + x^2)*Log[x]^3),x]

[Out]

-x + x^2 - 3*Log[1 + x] - x*Log[-x + 400/Log[x]^2]

Maple [A] (verified)

Time = 9.55 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23

method	result	size
parallelrisch	\(x^{2}-x \ln \left (-\frac {x \ln \left (x \right )^{2}-400}{\ln \left (x \right )^{2}}\right )-3 \ln \left (1+x \right )-x\)	\(32\)
risch	\(-x \ln \left (x \ln \left (x \right )^{2}-400\right )+2 x \ln \left (\ln \left (x \right )\right )+i \pi x \operatorname {csgn}\left (i \ln \left (x \right )^{2}\right )^{2} \operatorname {csgn}\left (i \ln \left (x \right )\right )+\frac {i \pi x \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )^{2}}\right ) \operatorname {csgn}\left (i \left (x \ln \left (x \right )^{2}-400\right )\right ) \operatorname {csgn}\left (\frac {i \left (x \ln \left (x \right )^{2}-400\right )}{\ln \left (x \right )^{2}}\right )}{2}-\frac {i \pi x \operatorname {csgn}\left (i \ln \left (x \right )^{2}\right )^{3}}{2}-\frac {i \pi x \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )^{2}}\right ) {\operatorname {csgn}\left (\frac {i \left (x \ln \left (x \right )^{2}-400\right )}{\ln \left (x \right )^{2}}\right )}^{2}}{2}-\frac {i \pi x \,\operatorname {csgn}\left (i \left (x \ln \left (x \right )^{2}-400\right )\right ) {\operatorname {csgn}\left (\frac {i \left (x \ln \left (x \right )^{2}-400\right )}{\ln \left (x \right )^{2}}\right )}^{2}}{2}-\frac {i \pi x {\operatorname {csgn}\left (\frac {i \left (x \ln \left (x \right )^{2}-400\right )}{\ln \left (x \right )^{2}}\right )}^{3}}{2}+i \pi x {\operatorname {csgn}\left (\frac {i \left (x \ln \left (x \right )^{2}-400\right )}{\ln \left (x \right )^{2}}\right )}^{2}-i \pi x -\frac {i \pi x \,\operatorname {csgn}\left (i \ln \left (x \right )^{2}\right ) \operatorname {csgn}\left (i \ln \left (x \right )\right )^{2}}{2}+x^{2}-x -3 \ln \left (1+x \right )\)	\(247\)

[In]

int((((-x^2-x)*ln(x)^3+(400*x+400)*ln(x))*ln((-x*ln(x)^2+400)/ln(x)^2)+(2*x^3-5*x)*ln(x)^3+(-800*x^2-400*x+160
0)*ln(x)-800*x-800)/((x^2+x)*ln(x)^3+(-400*x-400)*ln(x)),x,method=_RETURNVERBOSE)

[Out]

x^2-x*ln(-(x*ln(x)^2-400)/ln(x)^2)-3*ln(1+x)-x

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19 \[ \int \frac {-800-800 x+\left (1600-400 x-800 x^2\right ) \log (x)+\left (-5 x+2 x^3\right ) \log ^3(x)+\left ((400+400 x) \log (x)+\left (-x-x^2\right ) \log ^3(x)\right ) \log \left (\frac {400-x \log ^2(x)}{\log ^2(x)}\right )}{(-400-400 x) \log (x)+\left (x+x^2\right ) \log ^3(x)} \, dx=x^{2} - x \log \left (-\frac {x \log \left (x\right )^{2} - 400}{\log \left (x\right )^{2}}\right ) - x - 3 \, \log \left (x + 1\right ) \]

[In]

integrate((((-x^2-x)*log(x)^3+(400*x+400)*log(x))*log((-x*log(x)^2+400)/log(x)^2)+(2*x^3-5*x)*log(x)^3+(-800*x
^2-400*x+1600)*log(x)-800*x-800)/((x^2+x)*log(x)^3+(-400*x-400)*log(x)),x, algorithm="fricas")

[Out]

x^2 - x*log(-(x*log(x)^2 - 400)/log(x)^2) - x - 3*log(x + 1)

Sympy [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {-800-800 x+\left (1600-400 x-800 x^2\right ) \log (x)+\left (-5 x+2 x^3\right ) \log ^3(x)+\left ((400+400 x) \log (x)+\left (-x-x^2\right ) \log ^3(x)\right ) \log \left (\frac {400-x \log ^2(x)}{\log ^2(x)}\right )}{(-400-400 x) \log (x)+\left (x+x^2\right ) \log ^3(x)} \, dx=x^{2} - x \log {\left (\frac {- x \log {\left (x \right )}^{2} + 400}{\log {\left (x \right )}^{2}} \right )} - x - 3 \log {\left (x + 1 \right )} \]

[In]

integrate((((-x**2-x)*ln(x)**3+(400*x+400)*ln(x))*ln((-x*ln(x)**2+400)/ln(x)**2)+(2*x**3-5*x)*ln(x)**3+(-800*x
**2-400*x+1600)*ln(x)-800*x-800)/((x**2+x)*ln(x)**3+(-400*x-400)*ln(x)),x)

[Out]

x**2 - x*log((-x*log(x)**2 + 400)/log(x)**2) - x - 3*log(x + 1)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23 \[ \int \frac {-800-800 x+\left (1600-400 x-800 x^2\right ) \log (x)+\left (-5 x+2 x^3\right ) \log ^3(x)+\left ((400+400 x) \log (x)+\left (-x-x^2\right ) \log ^3(x)\right ) \log \left (\frac {400-x \log ^2(x)}{\log ^2(x)}\right )}{(-400-400 x) \log (x)+\left (x+x^2\right ) \log ^3(x)} \, dx=x^{2} - x \log \left (-x \log \left (x\right )^{2} + 400\right ) + 2 \, x \log \left (\log \left (x\right )\right ) - x - 3 \, \log \left (x + 1\right ) \]

[In]

integrate((((-x^2-x)*log(x)^3+(400*x+400)*log(x))*log((-x*log(x)^2+400)/log(x)^2)+(2*x^3-5*x)*log(x)^3+(-800*x
^2-400*x+1600)*log(x)-800*x-800)/((x^2+x)*log(x)^3+(-400*x-400)*log(x)),x, algorithm="maxima")

[Out]

x^2 - x*log(-x*log(x)^2 + 400) + 2*x*log(log(x)) - x - 3*log(x + 1)

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.27 \[ \int \frac {-800-800 x+\left (1600-400 x-800 x^2\right ) \log (x)+\left (-5 x+2 x^3\right ) \log ^3(x)+\left ((400+400 x) \log (x)+\left (-x-x^2\right ) \log ^3(x)\right ) \log \left (\frac {400-x \log ^2(x)}{\log ^2(x)}\right )}{(-400-400 x) \log (x)+\left (x+x^2\right ) \log ^3(x)} \, dx=x^{2} - x \log \left (-x \log \left (x\right )^{2} + 400\right ) + x \log \left (\log \left (x\right )^{2}\right ) - x - 3 \, \log \left (x + 1\right ) \]

[In]

integrate((((-x^2-x)*log(x)^3+(400*x+400)*log(x))*log((-x*log(x)^2+400)/log(x)^2)+(2*x^3-5*x)*log(x)^3+(-800*x
^2-400*x+1600)*log(x)-800*x-800)/((x^2+x)*log(x)^3+(-400*x-400)*log(x)),x, algorithm="giac")

[Out]

x^2 - x*log(-x*log(x)^2 + 400) + x*log(log(x)^2) - x - 3*log(x + 1)

Mupad [B] (verification not implemented)

Time = 9.91 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19 \[ \int \frac {-800-800 x+\left (1600-400 x-800 x^2\right ) \log (x)+\left (-5 x+2 x^3\right ) \log ^3(x)+\left ((400+400 x) \log (x)+\left (-x-x^2\right ) \log ^3(x)\right ) \log \left (\frac {400-x \log ^2(x)}{\log ^2(x)}\right )}{(-400-400 x) \log (x)+\left (x+x^2\right ) \log ^3(x)} \, dx=x^2-3\,\ln \left (x+1\right )-x\,\ln \left (-\frac {x\,{\ln \left (x\right )}^2-400}{{\ln \left (x\right )}^2}\right )-x \]

[In]

int(-(800*x + log(x)^3*(5*x - 2*x^3) + log(-(x*log(x)^2 - 400)/log(x)^2)*(log(x)^3*(x + x^2) - log(x)*(400*x +
400)) + log(x)*(400*x + 800*x^2 - 1600) + 800)/(log(x)^3*(x + x^2) - log(x)*(400*x + 400)),x)

[Out]

x^2 - 3*log(x + 1) - x*log(-(x*log(x)^2 - 400)/log(x)^2) - x