\(\int \frac {390625 \log (x)-x \log ^2(x)+(390625+x \log ^2(x)) \log (x \log (3))+x \log ^2(x) \log ^2(x \log (3))}{(-390625 x \log (x)+x^2 \log ^2(x)) \log (x \log (3))+(x+x^2) \log ^2(x) \log ^2(x \log (3))} \, dx\) [7784]

Optimal result

Integrand size = 80, antiderivative size = 20 \[ \int \frac {390625 \log (x)-x \log ^2(x)+\left (390625+x \log ^2(x)\right ) \log (x \log (3))+x \log ^2(x) \log ^2(x \log (3))}{\left (-390625 x \log (x)+x^2 \log ^2(x)\right ) \log (x \log (3))+\left (x+x^2\right ) \log ^2(x) \log ^2(x \log (3))} \, dx=\log \left (1+x+\frac {x-\frac {390625}{\log (x)}}{\log (x \log (3))}\right ) \]

[Out]

ln(1+(x-390625/ln(x))/ln(x*ln(3))+x)

Rubi [F]

\[ \int \frac {390625 \log (x)-x \log ^2(x)+\left (390625+x \log ^2(x)\right ) \log (x \log (3))+x \log ^2(x) \log ^2(x \log (3))}{\left (-390625 x \log (x)+x^2 \log ^2(x)\right ) \log (x \log (3))+\left (x+x^2\right ) \log ^2(x) \log ^2(x \log (3))} \, dx=\int \frac {390625 \log (x)-x \log ^2(x)+\left (390625+x \log ^2(x)\right ) \log (x \log (3))+x \log ^2(x) \log ^2(x \log (3))}{\left (-390625 x \log (x)+x^2 \log ^2(x)\right ) \log (x \log (3))+\left (x+x^2\right ) \log ^2(x) \log ^2(x \log (3))} \, dx \]

[In]

Int[(390625*Log[x] - x*Log[x]^2 + (390625 + x*Log[x]^2)*Log[x*Log[3]] + x*Log[x]^2*Log[x*Log[3]]^2)/((-390625*
x*Log[x] + x^2*Log[x]^2)*Log[x*Log[3]] + (x + x^2)*Log[x]^2*Log[x*Log[3]]^2),x]

[Out]

Log[1 + x] - Log[Log[x*Log[3]]] + 390625*Defer[Int][1/((1 + x)*(-390625 + x*Log[x] + Log[x]*Log[x*Log[3]] + x*
Log[x]*Log[x*Log[3]])), x] + Defer[Int][Log[x]/((1 + x)*(-390625 + x*Log[x] + Log[x]*Log[x*Log[3]] + x*Log[x]*
Log[x*Log[3]])), x] + 390625*Defer[Int][1/(x*Log[x]*(-390625 + Log[x]*(x + (1 + x)*Log[x*Log[3]]))), x] + Defe
r[Int][Log[x]/(-390625 + Log[x]*(x + (1 + x)*Log[x*Log[3]])), x] + Defer[Int][Log[x]/(x*(-390625 + Log[x]*(x +
(1 + x)*Log[x*Log[3]]))), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {-390625 \log (x)+x \log ^2(x)-\left (390625+x \log ^2(x)\right ) \log (x \log (3))-x \log ^2(x) \log ^2(x \log (3))}{x \log (x) \log (x \log (3)) (390625-x \log (x)-\log (x) \log (x \log (3))-x \log (x) \log (x \log (3)))} \, dx \\ & = \int \left (\frac {1}{1+x}-\frac {1}{x \log (x \log (3))}+\frac {(1+x) \log (x)}{x (-390625+x \log (x)+\log (x) \log (x \log (3))+x \log (x) \log (x \log (3)))}+\frac {390625+390625 x+390625 x \log (x)+x \log ^2(x)}{x (1+x) \log (x) (-390625+x \log (x)+\log (x) \log (x \log (3))+x \log (x) \log (x \log (3)))}\right ) \, dx \\ & = \log (1+x)-\int \frac {1}{x \log (x \log (3))} \, dx+\int \frac {(1+x) \log (x)}{x (-390625+x \log (x)+\log (x) \log (x \log (3))+x \log (x) \log (x \log (3)))} \, dx+\int \frac {390625+390625 x+390625 x \log (x)+x \log ^2(x)}{x (1+x) \log (x) (-390625+x \log (x)+\log (x) \log (x \log (3))+x \log (x) \log (x \log (3)))} \, dx \\ & = \log (1+x)+\int \left (\frac {\log (x)}{-390625+x \log (x)+\log (x) \log (x \log (3))+x \log (x) \log (x \log (3))}+\frac {\log (x)}{x (-390625+x \log (x)+\log (x) \log (x \log (3))+x \log (x) \log (x \log (3)))}\right ) \, dx+\int \left (\frac {-390625-390625 x-390625 x \log (x)-x \log ^2(x)}{(1+x) \log (x) (-390625+x \log (x)+\log (x) \log (x \log (3))+x \log (x) \log (x \log (3)))}+\frac {390625+390625 x+390625 x \log (x)+x \log ^2(x)}{x \log (x) (-390625+x \log (x)+\log (x) \log (x \log (3))+x \log (x) \log (x \log (3)))}\right ) \, dx-\text {Subst}\left (\int \frac {1}{x} \, dx,x,\log (x \log (3))\right ) \\ & = \log (1+x)-\log (\log (x \log (3)))+\int \frac {\log (x)}{-390625+x \log (x)+\log (x) \log (x \log (3))+x \log (x) \log (x \log (3))} \, dx+\int \frac {\log (x)}{x (-390625+x \log (x)+\log (x) \log (x \log (3))+x \log (x) \log (x \log (3)))} \, dx+\int \frac {-390625-390625 x-390625 x \log (x)-x \log ^2(x)}{(1+x) \log (x) (-390625+x \log (x)+\log (x) \log (x \log (3))+x \log (x) \log (x \log (3)))} \, dx+\int \frac {390625+390625 x+390625 x \log (x)+x \log ^2(x)}{x \log (x) (-390625+x \log (x)+\log (x) \log (x \log (3))+x \log (x) \log (x \log (3)))} \, dx \\ & = \log (1+x)-\log (\log (x \log (3)))+\int \frac {\log (x)}{-390625+\log (x) (x+(1+x) \log (x \log (3)))} \, dx+\int \frac {\log (x)}{x (-390625+\log (x) (x+(1+x) \log (x \log (3))))} \, dx+\int \left (\frac {390625}{-390625+x \log (x)+\log (x) \log (x \log (3))+x \log (x) \log (x \log (3))}+\frac {390625}{\log (x) (-390625+x \log (x)+\log (x) \log (x \log (3))+x \log (x) \log (x \log (3)))}+\frac {390625}{x \log (x) (-390625+x \log (x)+\log (x) \log (x \log (3))+x \log (x) \log (x \log (3)))}+\frac {\log (x)}{-390625+x \log (x)+\log (x) \log (x \log (3))+x \log (x) \log (x \log (3))}\right ) \, dx+\int \left (-\frac {390625 x}{(1+x) (-390625+x \log (x)+\log (x) \log (x \log (3))+x \log (x) \log (x \log (3)))}-\frac {390625}{(1+x) \log (x) (-390625+x \log (x)+\log (x) \log (x \log (3))+x \log (x) \log (x \log (3)))}-\frac {390625 x}{(1+x) \log (x) (-390625+x \log (x)+\log (x) \log (x \log (3))+x \log (x) \log (x \log (3)))}-\frac {x \log (x)}{(1+x) (-390625+x \log (x)+\log (x) \log (x \log (3))+x \log (x) \log (x \log (3)))}\right ) \, dx \\ & = \log (1+x)-\log (\log (x \log (3)))+390625 \int \frac {1}{-390625+x \log (x)+\log (x) \log (x \log (3))+x \log (x) \log (x \log (3))} \, dx-390625 \int \frac {x}{(1+x) (-390625+x \log (x)+\log (x) \log (x \log (3))+x \log (x) \log (x \log (3)))} \, dx+390625 \int \frac {1}{\log (x) (-390625+x \log (x)+\log (x) \log (x \log (3))+x \log (x) \log (x \log (3)))} \, dx+390625 \int \frac {1}{x \log (x) (-390625+x \log (x)+\log (x) \log (x \log (3))+x \log (x) \log (x \log (3)))} \, dx-390625 \int \frac {1}{(1+x) \log (x) (-390625+x \log (x)+\log (x) \log (x \log (3))+x \log (x) \log (x \log (3)))} \, dx-390625 \int \frac {x}{(1+x) \log (x) (-390625+x \log (x)+\log (x) \log (x \log (3))+x \log (x) \log (x \log (3)))} \, dx+\int \frac {\log (x)}{-390625+x \log (x)+\log (x) \log (x \log (3))+x \log (x) \log (x \log (3))} \, dx-\int \frac {x \log (x)}{(1+x) (-390625+x \log (x)+\log (x) \log (x \log (3))+x \log (x) \log (x \log (3)))} \, dx+\int \frac {\log (x)}{-390625+\log (x) (x+(1+x) \log (x \log (3)))} \, dx+\int \frac {\log (x)}{x (-390625+\log (x) (x+(1+x) \log (x \log (3))))} \, dx \\ & = \log (1+x)-\log (\log (x \log (3)))-390625 \int \frac {1}{(1+x) \log (x) (-390625+x \log (x)+\log (x) \log (x \log (3))+x \log (x) \log (x \log (3)))} \, dx+390625 \int \frac {1}{-390625+\log (x) (x+(1+x) \log (x \log (3)))} \, dx+390625 \int \frac {1}{\log (x) (-390625+\log (x) (x+(1+x) \log (x \log (3))))} \, dx+390625 \int \frac {1}{x \log (x) (-390625+\log (x) (x+(1+x) \log (x \log (3))))} \, dx-390625 \int \left (\frac {1}{-390625+x \log (x)+\log (x) \log (x \log (3))+x \log (x) \log (x \log (3))}-\frac {1}{(1+x) (-390625+x \log (x)+\log (x) \log (x \log (3))+x \log (x) \log (x \log (3)))}\right ) \, dx-390625 \int \left (\frac {1}{\log (x) (-390625+x \log (x)+\log (x) \log (x \log (3))+x \log (x) \log (x \log (3)))}-\frac {1}{(1+x) \log (x) (-390625+x \log (x)+\log (x) \log (x \log (3))+x \log (x) \log (x \log (3)))}\right ) \, dx+2 \int \frac {\log (x)}{-390625+\log (x) (x+(1+x) \log (x \log (3)))} \, dx+\int \frac {\log (x)}{x (-390625+\log (x) (x+(1+x) \log (x \log (3))))} \, dx-\int \left (\frac {\log (x)}{-390625+x \log (x)+\log (x) \log (x \log (3))+x \log (x) \log (x \log (3))}-\frac {\log (x)}{(1+x) (-390625+x \log (x)+\log (x) \log (x \log (3))+x \log (x) \log (x \log (3)))}\right ) \, dx \\ & = \log (1+x)-\log (\log (x \log (3)))-390625 \int \frac {1}{-390625+x \log (x)+\log (x) \log (x \log (3))+x \log (x) \log (x \log (3))} \, dx+390625 \int \frac {1}{(1+x) (-390625+x \log (x)+\log (x) \log (x \log (3))+x \log (x) \log (x \log (3)))} \, dx-390625 \int \frac {1}{\log (x) (-390625+x \log (x)+\log (x) \log (x \log (3))+x \log (x) \log (x \log (3)))} \, dx+390625 \int \frac {1}{-390625+\log (x) (x+(1+x) \log (x \log (3)))} \, dx+390625 \int \frac {1}{\log (x) (-390625+\log (x) (x+(1+x) \log (x \log (3))))} \, dx+390625 \int \frac {1}{x \log (x) (-390625+\log (x) (x+(1+x) \log (x \log (3))))} \, dx-\int \frac {\log (x)}{-390625+x \log (x)+\log (x) \log (x \log (3))+x \log (x) \log (x \log (3))} \, dx+\int \frac {\log (x)}{(1+x) (-390625+x \log (x)+\log (x) \log (x \log (3))+x \log (x) \log (x \log (3)))} \, dx+2 \int \frac {\log (x)}{-390625+\log (x) (x+(1+x) \log (x \log (3)))} \, dx+\int \frac {\log (x)}{x (-390625+\log (x) (x+(1+x) \log (x \log (3))))} \, dx \\ & = \log (1+x)-\log (\log (x \log (3)))+390625 \int \frac {1}{(1+x) (-390625+x \log (x)+\log (x) \log (x \log (3))+x \log (x) \log (x \log (3)))} \, dx+390625 \int \frac {1}{x \log (x) (-390625+\log (x) (x+(1+x) \log (x \log (3))))} \, dx+\int \frac {\log (x)}{(1+x) (-390625+x \log (x)+\log (x) \log (x \log (3))+x \log (x) \log (x \log (3)))} \, dx+\int \frac {\log (x)}{-390625+\log (x) (x+(1+x) \log (x \log (3)))} \, dx+\int \frac {\log (x)}{x (-390625+\log (x) (x+(1+x) \log (x \log (3))))} \, dx \\ \end{align*}

Mathematica [F]

\[ \int \frac {390625 \log (x)-x \log ^2(x)+\left (390625+x \log ^2(x)\right ) \log (x \log (3))+x \log ^2(x) \log ^2(x \log (3))}{\left (-390625 x \log (x)+x^2 \log ^2(x)\right ) \log (x \log (3))+\left (x+x^2\right ) \log ^2(x) \log ^2(x \log (3))} \, dx=\int \frac {390625 \log (x)-x \log ^2(x)+\left (390625+x \log ^2(x)\right ) \log (x \log (3))+x \log ^2(x) \log ^2(x \log (3))}{\left (-390625 x \log (x)+x^2 \log ^2(x)\right ) \log (x \log (3))+\left (x+x^2\right ) \log ^2(x) \log ^2(x \log (3))} \, dx \]

[In]

Integrate[(390625*Log[x] - x*Log[x]^2 + (390625 + x*Log[x]^2)*Log[x*Log[3]] + x*Log[x]^2*Log[x*Log[3]]^2)/((-3
90625*x*Log[x] + x^2*Log[x]^2)*Log[x*Log[3]] + (x + x^2)*Log[x]^2*Log[x*Log[3]]^2),x]

[Out]

Integrate[(390625*Log[x] - x*Log[x]^2 + (390625 + x*Log[x]^2)*Log[x*Log[3]] + x*Log[x]^2*Log[x*Log[3]]^2)/((-3
90625*x*Log[x] + x^2*Log[x]^2)*Log[x*Log[3]] + (x + x^2)*Log[x]^2*Log[x*Log[3]]^2), x]

Maple [A] (verified)

Time = 1.71 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.95

method	result	size
parallelrisch	\(-\ln \left (\ln \left (x \right )\right )-\ln \left (\ln \left (x \ln \left (3\right )\right )\right )+\ln \left (\ln \left (x \right ) \ln \left (x \ln \left (3\right )\right ) x +\ln \left (x \right ) \ln \left (x \ln \left (3\right )\right )+x \ln \left (x \right )-390625\right )\)	\(39\)
default	\(-\ln \left (\ln \left (x \right )\right )-\ln \left (\ln \left (\ln \left (3\right )\right )+\ln \left (x \right )\right )+\ln \left (x \ln \left (x \right ) \ln \left (\ln \left (3\right )\right )+x \ln \left (x \right )^{2}+\ln \left (\ln \left (3\right )\right ) \ln \left (x \right )+\ln \left (x \right )^{2}+x \ln \left (x \right )-390625\right )\)	\(46\)
risch	\(\ln \left (1+x \right )-\ln \left (\ln \left (\ln \left (3\right )\right ) \ln \left (x \right )+\ln \left (x \right )^{2}\right )+\ln \left (\ln \left (x \right )^{2}+\frac {\left (2 \ln \left (\ln \left (3\right )\right ) x +2 \ln \left (\ln \left (3\right )\right )+2 x \right ) \ln \left (x \right )}{2+2 x}-\frac {390625}{1+x}\right )\)	\(57\)

[In]

int((x*ln(x)^2*ln(x*ln(3))^2+(x*ln(x)^2+390625)*ln(x*ln(3))-x*ln(x)^2+390625*ln(x))/((x^2+x)*ln(x)^2*ln(x*ln(3
))^2+(x^2*ln(x)^2-390625*x*ln(x))*ln(x*ln(3))),x,method=_RETURNVERBOSE)

[Out]

-ln(ln(x))-ln(ln(x*ln(3)))+ln(ln(x)*ln(x*ln(3))*x+ln(x)*ln(x*ln(3))+x*ln(x)-390625)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (20) = 40\).

Time = 0.28 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.45 \[ \int \frac {390625 \log (x)-x \log ^2(x)+\left (390625+x \log ^2(x)\right ) \log (x \log (3))+x \log ^2(x) \log ^2(x \log (3))}{\left (-390625 x \log (x)+x^2 \log ^2(x)\right ) \log (x \log (3))+\left (x+x^2\right ) \log ^2(x) \log ^2(x \log (3))} \, dx=\log \left (x + 1\right ) + \log \left (\frac {{\left (x + 1\right )} \log \left (x\right )^{2} + {\left (x + 1\right )} \log \left (x\right ) \log \left (\log \left (3\right )\right ) + x \log \left (x\right ) - 390625}{x + 1}\right ) - \log \left (\log \left (x\right ) + \log \left (\log \left (3\right )\right )\right ) - \log \left (\log \left (x\right )\right ) \]

[In]

integrate((x*log(x)^2*log(x*log(3))^2+(x*log(x)^2+390625)*log(x*log(3))-x*log(x)^2+390625*log(x))/((x^2+x)*log
(x)^2*log(x*log(3))^2+(x^2*log(x)^2-390625*x*log(x))*log(x*log(3))),x, algorithm="fricas")

[Out]

log(x + 1) + log(((x + 1)*log(x)^2 + (x + 1)*log(x)*log(log(3)) + x*log(x) - 390625)/(x + 1)) - log(log(x) + l
og(log(3))) - log(log(x))

Sympy [F(-2)]

Exception generated. \[ \int \frac {390625 \log (x)-x \log ^2(x)+\left (390625+x \log ^2(x)\right ) \log (x \log (3))+x \log ^2(x) \log ^2(x \log (3))}{\left (-390625 x \log (x)+x^2 \log ^2(x)\right ) \log (x \log (3))+\left (x+x^2\right ) \log ^2(x) \log ^2(x \log (3))} \, dx=\text {Exception raised: PolynomialError} \]

[In]

integrate((x*ln(x)**2*ln(x*ln(3))**2+(x*ln(x)**2+390625)*ln(x*ln(3))-x*ln(x)**2+390625*ln(x))/((x**2+x)*ln(x)*
*2*ln(x*ln(3))**2+(x**2*ln(x)**2-390625*x*ln(x))*ln(x*ln(3))),x)

[Out]

Exception raised: PolynomialError >> 1/(x**6 + 4*x**5 + 6*x**4 + 4*x**3 + x**2) contains an element of the set
of generators.

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (20) = 40\).

Time = 0.34 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.50 \[ \int \frac {390625 \log (x)-x \log ^2(x)+\left (390625+x \log ^2(x)\right ) \log (x \log (3))+x \log ^2(x) \log ^2(x \log (3))}{\left (-390625 x \log (x)+x^2 \log ^2(x)\right ) \log (x \log (3))+\left (x+x^2\right ) \log ^2(x) \log ^2(x \log (3))} \, dx=\log \left (x + 1\right ) + \log \left (\frac {{\left (x + 1\right )} \log \left (x\right )^{2} + {\left (x {\left (\log \left (\log \left (3\right )\right ) + 1\right )} + \log \left (\log \left (3\right )\right )\right )} \log \left (x\right ) - 390625}{x + 1}\right ) - \log \left (\log \left (x\right ) + \log \left (\log \left (3\right )\right )\right ) - \log \left (\log \left (x\right )\right ) \]

[In]

integrate((x*log(x)^2*log(x*log(3))^2+(x*log(x)^2+390625)*log(x*log(3))-x*log(x)^2+390625*log(x))/((x^2+x)*log
(x)^2*log(x*log(3))^2+(x^2*log(x)^2-390625*x*log(x))*log(x*log(3))),x, algorithm="maxima")

[Out]

log(x + 1) + log(((x + 1)*log(x)^2 + (x*(log(log(3)) + 1) + log(log(3)))*log(x) - 390625)/(x + 1)) - log(log(x
) + log(log(3))) - log(log(x))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 45 vs. \(2 (20) = 40\).

Time = 0.28 (sec) , antiderivative size = 45, normalized size of antiderivative = 2.25 \[ \int \frac {390625 \log (x)-x \log ^2(x)+\left (390625+x \log ^2(x)\right ) \log (x \log (3))+x \log ^2(x) \log ^2(x \log (3))}{\left (-390625 x \log (x)+x^2 \log ^2(x)\right ) \log (x \log (3))+\left (x+x^2\right ) \log ^2(x) \log ^2(x \log (3))} \, dx=\log \left (x \log \left (x\right )^{2} + x \log \left (x\right ) \log \left (\log \left (3\right )\right ) + x \log \left (x\right ) + \log \left (x\right )^{2} + \log \left (x\right ) \log \left (\log \left (3\right )\right ) - 390625\right ) - \log \left (\log \left (x\right ) + \log \left (\log \left (3\right )\right )\right ) - \log \left (\log \left (x\right )\right ) \]

[In]

integrate((x*log(x)^2*log(x*log(3))^2+(x*log(x)^2+390625)*log(x*log(3))-x*log(x)^2+390625*log(x))/((x^2+x)*log
(x)^2*log(x*log(3))^2+(x^2*log(x)^2-390625*x*log(x))*log(x*log(3))),x, algorithm="giac")

[Out]

log(x*log(x)^2 + x*log(x)*log(log(3)) + x*log(x) + log(x)^2 + log(x)*log(log(3)) - 390625) - log(log(x) + log(
log(3))) - log(log(x))

Mupad [F(-1)]

Timed out. \[ \int \frac {390625 \log (x)-x \log ^2(x)+\left (390625+x \log ^2(x)\right ) \log (x \log (3))+x \log ^2(x) \log ^2(x \log (3))}{\left (-390625 x \log (x)+x^2 \log ^2(x)\right ) \log (x \log (3))+\left (x+x^2\right ) \log ^2(x) \log ^2(x \log (3))} \, dx=\int \frac {390625\,\ln \left (x\right )-x\,{\ln \left (x\right )}^2+\ln \left (x\,\ln \left (3\right )\right )\,\left (x\,{\ln \left (x\right )}^2+390625\right )+x\,{\ln \left (x\,\ln \left (3\right )\right )}^2\,{\ln \left (x\right )}^2}{\ln \left (x\,\ln \left (3\right )\right )\,\left (x^2\,{\ln \left (x\right )}^2-390625\,x\,\ln \left (x\right )\right )+{\ln \left (x\,\ln \left (3\right )\right )}^2\,{\ln \left (x\right )}^2\,\left (x^2+x\right )} \,d x \]

[In]

int((390625*log(x) - x*log(x)^2 + log(x*log(3))*(x*log(x)^2 + 390625) + x*log(x*log(3))^2*log(x)^2)/(log(x*log
(3))*(x^2*log(x)^2 - 390625*x*log(x)) + log(x*log(3))^2*log(x)^2*(x + x^2)),x)

[Out]

int((390625*log(x) - x*log(x)^2 + log(x*log(3))*(x*log(x)^2 + 390625) + x*log(x*log(3))^2*log(x)^2)/(log(x*log
(3))*(x^2*log(x)^2 - 390625*x*log(x)) + log(x*log(3))^2*log(x)^2*(x + x^2)), x)