\(\int \frac {15-8 \log (x^2)+\log ^2(x^2)+(-64+128 x-64 x^2+(16-32 x+16 x^2) \log (x^2)) \log ^3(-15+8 \log (x^2)-\log ^2(x^2))+(-30 x+30 x^2+(16 x-16 x^2) \log (x^2)+(-2 x+2 x^2) \log ^2(x^2)) \log ^4(-15+8 \log (x^2)-\log ^2(x^2))}{15 x-8 x \log (x^2)+x \log ^2(x^2)} \, dx\) [3045]

Optimal result

Integrand size = 140, antiderivative size = 24 \[ \int \frac {15-8 \log \left (x^2\right )+\log ^2\left (x^2\right )+\left (-64+128 x-64 x^2+\left (16-32 x+16 x^2\right ) \log \left (x^2\right )\right ) \log ^3\left (-15+8 \log \left (x^2\right )-\log ^2\left (x^2\right )\right )+\left (-30 x+30 x^2+\left (16 x-16 x^2\right ) \log \left (x^2\right )+\left (-2 x+2 x^2\right ) \log ^2\left (x^2\right )\right ) \log ^4\left (-15+8 \log \left (x^2\right )-\log ^2\left (x^2\right )\right )}{15 x-8 x \log \left (x^2\right )+x \log ^2\left (x^2\right )} \, dx=\log (x)+(-1+x)^2 \log ^4\left (1-\left (-4+\log \left (x^2\right )\right )^2\right ) \]

[Out]

(-1+x)^2*ln(1-(ln(x^2)-4)^2)^4+ln(x)

Rubi [F]

\[ \int \frac {15-8 \log \left (x^2\right )+\log ^2\left (x^2\right )+\left (-64+128 x-64 x^2+\left (16-32 x+16 x^2\right ) \log \left (x^2\right )\right ) \log ^3\left (-15+8 \log \left (x^2\right )-\log ^2\left (x^2\right )\right )+\left (-30 x+30 x^2+\left (16 x-16 x^2\right ) \log \left (x^2\right )+\left (-2 x+2 x^2\right ) \log ^2\left (x^2\right )\right ) \log ^4\left (-15+8 \log \left (x^2\right )-\log ^2\left (x^2\right )\right )}{15 x-8 x \log \left (x^2\right )+x \log ^2\left (x^2\right )} \, dx=\int \frac {15-8 \log \left (x^2\right )+\log ^2\left (x^2\right )+\left (-64+128 x-64 x^2+\left (16-32 x+16 x^2\right ) \log \left (x^2\right )\right ) \log ^3\left (-15+8 \log \left (x^2\right )-\log ^2\left (x^2\right )\right )+\left (-30 x+30 x^2+\left (16 x-16 x^2\right ) \log \left (x^2\right )+\left (-2 x+2 x^2\right ) \log ^2\left (x^2\right )\right ) \log ^4\left (-15+8 \log \left (x^2\right )-\log ^2\left (x^2\right )\right )}{15 x-8 x \log \left (x^2\right )+x \log ^2\left (x^2\right )} \, dx \]

[In]

Int[(15 - 8*Log[x^2] + Log[x^2]^2 + (-64 + 128*x - 64*x^2 + (16 - 32*x + 16*x^2)*Log[x^2])*Log[-15 + 8*Log[x^2
] - Log[x^2]^2]^3 + (-30*x + 30*x^2 + (16*x - 16*x^2)*Log[x^2] + (-2*x + 2*x^2)*Log[x^2]^2)*Log[-15 + 8*Log[x^
2] - Log[x^2]^2]^4)/(15*x - 8*x*Log[x^2] + x*Log[x^2]^2),x]

[Out]

Log[x] + 4*Log[-5 + Log[x^2]]*Log[-15 + 8*Log[x^2] - Log[x^2]^2]^3 + 4*Log[-3 + Log[x^2]]*Log[-15 + 8*Log[x^2]
- Log[x^2]^2]^3 + 128*Defer[Int][Log[-15 + 8*Log[x^2] - Log[x^2]^2]^3/(15 - 8*Log[x^2] + Log[x^2]^2), x] - 32
*Defer[Int][(Log[x^2]*Log[-15 + 8*Log[x^2] - Log[x^2]^2]^3)/(15 - 8*Log[x^2] + Log[x^2]^2), x] - 2*Defer[Int][
Log[-15 + 8*Log[x^2] - Log[x^2]^2]^4, x] - 12*Defer[Subst][Defer[Int][(Log[-5 + x]*Log[-15 + 8*x - x^2]^2)/(-5
+ x), x], x, Log[x^2]] - 12*Defer[Subst][Defer[Int][(Log[-5 + x]*Log[-15 + 8*x - x^2]^2)/(-3 + x), x], x, Log
[x^2]] - 12*Defer[Subst][Defer[Int][(Log[-3 + x]*Log[-15 + 8*x - x^2]^2)/(-5 + x), x], x, Log[x^2]] - 12*Defer
[Subst][Defer[Int][(Log[-3 + x]*Log[-15 + 8*x - x^2]^2)/(-3 + x), x], x, Log[x^2]] - 32*Defer[Subst][Defer[Int
][Log[-15 + 8*Log[x] - Log[x]^2]^3/(15 - 8*Log[x] + Log[x]^2), x], x, x^2] + 8*Defer[Subst][Defer[Int][(Log[x]
*Log[-15 + 8*Log[x] - Log[x]^2]^3)/(15 - 8*Log[x] + Log[x]^2), x], x, x^2] + Defer[Subst][Defer[Int][Log[-15 +
8*Log[x] - Log[x]^2]^4, x], x, x^2]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{x}+\frac {16 (-1+x)^2 \left (-4+\log \left (x^2\right )\right ) \log ^3\left (-15+8 \log \left (x^2\right )-\log ^2\left (x^2\right )\right )}{x \left (-5+\log \left (x^2\right )\right ) \left (-3+\log \left (x^2\right )\right )}+2 (-1+x) \log ^4\left (-15+8 \log \left (x^2\right )-\log ^2\left (x^2\right )\right )\right ) \, dx \\ & = \log (x)+2 \int (-1+x) \log ^4\left (-15+8 \log \left (x^2\right )-\log ^2\left (x^2\right )\right ) \, dx+16 \int \frac {(-1+x)^2 \left (-4+\log \left (x^2\right )\right ) \log ^3\left (-15+8 \log \left (x^2\right )-\log ^2\left (x^2\right )\right )}{x \left (-5+\log \left (x^2\right )\right ) \left (-3+\log \left (x^2\right )\right )} \, dx \\ & = \log (x)+2 \int \left (-\log ^4\left (-15+8 \log \left (x^2\right )-\log ^2\left (x^2\right )\right )+x \log ^4\left (-15+8 \log \left (x^2\right )-\log ^2\left (x^2\right )\right )\right ) \, dx+16 \int \left (-\frac {2 \left (-4+\log \left (x^2\right )\right ) \log ^3\left (-15+8 \log \left (x^2\right )-\log ^2\left (x^2\right )\right )}{\left (-5+\log \left (x^2\right )\right ) \left (-3+\log \left (x^2\right )\right )}+\frac {\left (-4+\log \left (x^2\right )\right ) \log ^3\left (-15+8 \log \left (x^2\right )-\log ^2\left (x^2\right )\right )}{x \left (-5+\log \left (x^2\right )\right ) \left (-3+\log \left (x^2\right )\right )}+\frac {x \left (-4+\log \left (x^2\right )\right ) \log ^3\left (-15+8 \log \left (x^2\right )-\log ^2\left (x^2\right )\right )}{\left (-5+\log \left (x^2\right )\right ) \left (-3+\log \left (x^2\right )\right )}\right ) \, dx \\ & = \log (x)-2 \int \log ^4\left (-15+8 \log \left (x^2\right )-\log ^2\left (x^2\right )\right ) \, dx+2 \int x \log ^4\left (-15+8 \log \left (x^2\right )-\log ^2\left (x^2\right )\right ) \, dx+16 \int \frac {\left (-4+\log \left (x^2\right )\right ) \log ^3\left (-15+8 \log \left (x^2\right )-\log ^2\left (x^2\right )\right )}{x \left (-5+\log \left (x^2\right )\right ) \left (-3+\log \left (x^2\right )\right )} \, dx+16 \int \frac {x \left (-4+\log \left (x^2\right )\right ) \log ^3\left (-15+8 \log \left (x^2\right )-\log ^2\left (x^2\right )\right )}{\left (-5+\log \left (x^2\right )\right ) \left (-3+\log \left (x^2\right )\right )} \, dx-32 \int \frac {\left (-4+\log \left (x^2\right )\right ) \log ^3\left (-15+8 \log \left (x^2\right )-\log ^2\left (x^2\right )\right )}{\left (-5+\log \left (x^2\right )\right ) \left (-3+\log \left (x^2\right )\right )} \, dx \\ & = \log (x)-2 \int \log ^4\left (-15+8 \log \left (x^2\right )-\log ^2\left (x^2\right )\right ) \, dx+8 \text {Subst}\left (\int \frac {(-4+x) \log ^3\left (-15+8 x-x^2\right )}{(-5+x) (-3+x)} \, dx,x,\log \left (x^2\right )\right )+8 \text {Subst}\left (\int \frac {(-4+\log (x)) \log ^3\left (-15+8 \log (x)-\log ^2(x)\right )}{(-5+\log (x)) (-3+\log (x))} \, dx,x,x^2\right )-32 \int \left (-\frac {4 \log ^3\left (-15+8 \log \left (x^2\right )-\log ^2\left (x^2\right )\right )}{15-8 \log \left (x^2\right )+\log ^2\left (x^2\right )}+\frac {\log \left (x^2\right ) \log ^3\left (-15+8 \log \left (x^2\right )-\log ^2\left (x^2\right )\right )}{15-8 \log \left (x^2\right )+\log ^2\left (x^2\right )}\right ) \, dx+\text {Subst}\left (\int \log ^4\left (-15+8 \log (x)-\log ^2(x)\right ) \, dx,x,x^2\right ) \\ & = \log (x)-2 \int \log ^4\left (-15+8 \log \left (x^2\right )-\log ^2\left (x^2\right )\right ) \, dx+8 \text {Subst}\left (\int \left (\frac {\log ^3\left (-15+8 x-x^2\right )}{2 (-5+x)}+\frac {\log ^3\left (-15+8 x-x^2\right )}{2 (-3+x)}\right ) \, dx,x,\log \left (x^2\right )\right )+8 \text {Subst}\left (\int \left (-\frac {4 \log ^3\left (-15+8 \log (x)-\log ^2(x)\right )}{15-8 \log (x)+\log ^2(x)}+\frac {\log (x) \log ^3\left (-15+8 \log (x)-\log ^2(x)\right )}{15-8 \log (x)+\log ^2(x)}\right ) \, dx,x,x^2\right )-32 \int \frac {\log \left (x^2\right ) \log ^3\left (-15+8 \log \left (x^2\right )-\log ^2\left (x^2\right )\right )}{15-8 \log \left (x^2\right )+\log ^2\left (x^2\right )} \, dx+128 \int \frac {\log ^3\left (-15+8 \log \left (x^2\right )-\log ^2\left (x^2\right )\right )}{15-8 \log \left (x^2\right )+\log ^2\left (x^2\right )} \, dx+\text {Subst}\left (\int \log ^4\left (-15+8 \log (x)-\log ^2(x)\right ) \, dx,x,x^2\right ) \\ & = \log (x)-2 \int \log ^4\left (-15+8 \log \left (x^2\right )-\log ^2\left (x^2\right )\right ) \, dx+4 \text {Subst}\left (\int \frac {\log ^3\left (-15+8 x-x^2\right )}{-5+x} \, dx,x,\log \left (x^2\right )\right )+4 \text {Subst}\left (\int \frac {\log ^3\left (-15+8 x-x^2\right )}{-3+x} \, dx,x,\log \left (x^2\right )\right )+8 \text {Subst}\left (\int \frac {\log (x) \log ^3\left (-15+8 \log (x)-\log ^2(x)\right )}{15-8 \log (x)+\log ^2(x)} \, dx,x,x^2\right )-32 \int \frac {\log \left (x^2\right ) \log ^3\left (-15+8 \log \left (x^2\right )-\log ^2\left (x^2\right )\right )}{15-8 \log \left (x^2\right )+\log ^2\left (x^2\right )} \, dx-32 \text {Subst}\left (\int \frac {\log ^3\left (-15+8 \log (x)-\log ^2(x)\right )}{15-8 \log (x)+\log ^2(x)} \, dx,x,x^2\right )+128 \int \frac {\log ^3\left (-15+8 \log \left (x^2\right )-\log ^2\left (x^2\right )\right )}{15-8 \log \left (x^2\right )+\log ^2\left (x^2\right )} \, dx+\text {Subst}\left (\int \log ^4\left (-15+8 \log (x)-\log ^2(x)\right ) \, dx,x,x^2\right ) \\ & = \log (x)+4 \log \left (-5+\log \left (x^2\right )\right ) \log ^3\left (-15+8 \log \left (x^2\right )-\log ^2\left (x^2\right )\right )+4 \log \left (-3+\log \left (x^2\right )\right ) \log ^3\left (-15+8 \log \left (x^2\right )-\log ^2\left (x^2\right )\right )-2 \int \log ^4\left (-15+8 \log \left (x^2\right )-\log ^2\left (x^2\right )\right ) \, dx+8 \text {Subst}\left (\int \frac {\log (x) \log ^3\left (-15+8 \log (x)-\log ^2(x)\right )}{15-8 \log (x)+\log ^2(x)} \, dx,x,x^2\right )-12 \text {Subst}\left (\int \frac {(8-2 x) \log (-5+x) \log ^2\left (-15+8 x-x^2\right )}{-15+8 x-x^2} \, dx,x,\log \left (x^2\right )\right )-12 \text {Subst}\left (\int \frac {(8-2 x) \log (-3+x) \log ^2\left (-15+8 x-x^2\right )}{-15+8 x-x^2} \, dx,x,\log \left (x^2\right )\right )-32 \int \frac {\log \left (x^2\right ) \log ^3\left (-15+8 \log \left (x^2\right )-\log ^2\left (x^2\right )\right )}{15-8 \log \left (x^2\right )+\log ^2\left (x^2\right )} \, dx-32 \text {Subst}\left (\int \frac {\log ^3\left (-15+8 \log (x)-\log ^2(x)\right )}{15-8 \log (x)+\log ^2(x)} \, dx,x,x^2\right )+128 \int \frac {\log ^3\left (-15+8 \log \left (x^2\right )-\log ^2\left (x^2\right )\right )}{15-8 \log \left (x^2\right )+\log ^2\left (x^2\right )} \, dx+\text {Subst}\left (\int \log ^4\left (-15+8 \log (x)-\log ^2(x)\right ) \, dx,x,x^2\right ) \\ & = \log (x)+4 \log \left (-5+\log \left (x^2\right )\right ) \log ^3\left (-15+8 \log \left (x^2\right )-\log ^2\left (x^2\right )\right )+4 \log \left (-3+\log \left (x^2\right )\right ) \log ^3\left (-15+8 \log \left (x^2\right )-\log ^2\left (x^2\right )\right )-2 \int \log ^4\left (-15+8 \log \left (x^2\right )-\log ^2\left (x^2\right )\right ) \, dx+8 \text {Subst}\left (\int \frac {\log (x) \log ^3\left (-15+8 \log (x)-\log ^2(x)\right )}{15-8 \log (x)+\log ^2(x)} \, dx,x,x^2\right )-12 \text {Subst}\left (\int \left (\frac {\log (-5+x) \log ^2\left (-15+8 x-x^2\right )}{-5+x}+\frac {\log (-5+x) \log ^2\left (-15+8 x-x^2\right )}{-3+x}\right ) \, dx,x,\log \left (x^2\right )\right )-12 \text {Subst}\left (\int \left (\frac {\log (-3+x) \log ^2\left (-15+8 x-x^2\right )}{-5+x}+\frac {\log (-3+x) \log ^2\left (-15+8 x-x^2\right )}{-3+x}\right ) \, dx,x,\log \left (x^2\right )\right )-32 \int \frac {\log \left (x^2\right ) \log ^3\left (-15+8 \log \left (x^2\right )-\log ^2\left (x^2\right )\right )}{15-8 \log \left (x^2\right )+\log ^2\left (x^2\right )} \, dx-32 \text {Subst}\left (\int \frac {\log ^3\left (-15+8 \log (x)-\log ^2(x)\right )}{15-8 \log (x)+\log ^2(x)} \, dx,x,x^2\right )+128 \int \frac {\log ^3\left (-15+8 \log \left (x^2\right )-\log ^2\left (x^2\right )\right )}{15-8 \log \left (x^2\right )+\log ^2\left (x^2\right )} \, dx+\text {Subst}\left (\int \log ^4\left (-15+8 \log (x)-\log ^2(x)\right ) \, dx,x,x^2\right ) \\ & = \log (x)+4 \log \left (-5+\log \left (x^2\right )\right ) \log ^3\left (-15+8 \log \left (x^2\right )-\log ^2\left (x^2\right )\right )+4 \log \left (-3+\log \left (x^2\right )\right ) \log ^3\left (-15+8 \log \left (x^2\right )-\log ^2\left (x^2\right )\right )-2 \int \log ^4\left (-15+8 \log \left (x^2\right )-\log ^2\left (x^2\right )\right ) \, dx+8 \text {Subst}\left (\int \frac {\log (x) \log ^3\left (-15+8 \log (x)-\log ^2(x)\right )}{15-8 \log (x)+\log ^2(x)} \, dx,x,x^2\right )-12 \text {Subst}\left (\int \frac {\log (-5+x) \log ^2\left (-15+8 x-x^2\right )}{-5+x} \, dx,x,\log \left (x^2\right )\right )-12 \text {Subst}\left (\int \frac {\log (-5+x) \log ^2\left (-15+8 x-x^2\right )}{-3+x} \, dx,x,\log \left (x^2\right )\right )-12 \text {Subst}\left (\int \frac {\log (-3+x) \log ^2\left (-15+8 x-x^2\right )}{-5+x} \, dx,x,\log \left (x^2\right )\right )-12 \text {Subst}\left (\int \frac {\log (-3+x) \log ^2\left (-15+8 x-x^2\right )}{-3+x} \, dx,x,\log \left (x^2\right )\right )-32 \int \frac {\log \left (x^2\right ) \log ^3\left (-15+8 \log \left (x^2\right )-\log ^2\left (x^2\right )\right )}{15-8 \log \left (x^2\right )+\log ^2\left (x^2\right )} \, dx-32 \text {Subst}\left (\int \frac {\log ^3\left (-15+8 \log (x)-\log ^2(x)\right )}{15-8 \log (x)+\log ^2(x)} \, dx,x,x^2\right )+128 \int \frac {\log ^3\left (-15+8 \log \left (x^2\right )-\log ^2\left (x^2\right )\right )}{15-8 \log \left (x^2\right )+\log ^2\left (x^2\right )} \, dx+\text {Subst}\left (\int \log ^4\left (-15+8 \log (x)-\log ^2(x)\right ) \, dx,x,x^2\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.17 \[ \int \frac {15-8 \log \left (x^2\right )+\log ^2\left (x^2\right )+\left (-64+128 x-64 x^2+\left (16-32 x+16 x^2\right ) \log \left (x^2\right )\right ) \log ^3\left (-15+8 \log \left (x^2\right )-\log ^2\left (x^2\right )\right )+\left (-30 x+30 x^2+\left (16 x-16 x^2\right ) \log \left (x^2\right )+\left (-2 x+2 x^2\right ) \log ^2\left (x^2\right )\right ) \log ^4\left (-15+8 \log \left (x^2\right )-\log ^2\left (x^2\right )\right )}{15 x-8 x \log \left (x^2\right )+x \log ^2\left (x^2\right )} \, dx=\log (x)+(-1+x)^2 \log ^4\left (-15+8 \log \left (x^2\right )-\log ^2\left (x^2\right )\right ) \]

[In]

Integrate[(15 - 8*Log[x^2] + Log[x^2]^2 + (-64 + 128*x - 64*x^2 + (16 - 32*x + 16*x^2)*Log[x^2])*Log[-15 + 8*L
og[x^2] - Log[x^2]^2]^3 + (-30*x + 30*x^2 + (16*x - 16*x^2)*Log[x^2] + (-2*x + 2*x^2)*Log[x^2]^2)*Log[-15 + 8*
Log[x^2] - Log[x^2]^2]^4)/(15*x - 8*x*Log[x^2] + x*Log[x^2]^2),x]

[Out]

Log[x] + (-1 + x)^2*Log[-15 + 8*Log[x^2] - Log[x^2]^2]^4

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(67\) vs. \(2(24)=48\).

Time = 1.64 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.83

method	result	size
parallelrisch	\({\ln \left (-\ln \left (x^{2}\right )^{2}+8 \ln \left (x^{2}\right )-15\right )}^{4} x^{2}-2 x {\ln \left (-\ln \left (x^{2}\right )^{2}+8 \ln \left (x^{2}\right )-15\right )}^{4}+{\ln \left (-\ln \left (x^{2}\right )^{2}+8 \ln \left (x^{2}\right )-15\right )}^{4}+\ln \left (x \right )\)	\(68\)

[In]

int((((2*x^2-2*x)*ln(x^2)^2+(-16*x^2+16*x)*ln(x^2)+30*x^2-30*x)*ln(-ln(x^2)^2+8*ln(x^2)-15)^4+((16*x^2-32*x+16
)*ln(x^2)-64*x^2+128*x-64)*ln(-ln(x^2)^2+8*ln(x^2)-15)^3+ln(x^2)^2-8*ln(x^2)+15)/(x*ln(x^2)^2-8*x*ln(x^2)+15*x
),x,method=_RETURNVERBOSE)

[Out]

ln(-ln(x^2)^2+8*ln(x^2)-15)^4*x^2-2*x*ln(-ln(x^2)^2+8*ln(x^2)-15)^4+ln(-ln(x^2)^2+8*ln(x^2)-15)^4+ln(x)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.46 \[ \int \frac {15-8 \log \left (x^2\right )+\log ^2\left (x^2\right )+\left (-64+128 x-64 x^2+\left (16-32 x+16 x^2\right ) \log \left (x^2\right )\right ) \log ^3\left (-15+8 \log \left (x^2\right )-\log ^2\left (x^2\right )\right )+\left (-30 x+30 x^2+\left (16 x-16 x^2\right ) \log \left (x^2\right )+\left (-2 x+2 x^2\right ) \log ^2\left (x^2\right )\right ) \log ^4\left (-15+8 \log \left (x^2\right )-\log ^2\left (x^2\right )\right )}{15 x-8 x \log \left (x^2\right )+x \log ^2\left (x^2\right )} \, dx={\left (x^{2} - 2 \, x + 1\right )} \log \left (-\log \left (x^{2}\right )^{2} + 8 \, \log \left (x^{2}\right ) - 15\right )^{4} + \frac {1}{2} \, \log \left (x^{2}\right ) \]

[In]

integrate((((2*x^2-2*x)*log(x^2)^2+(-16*x^2+16*x)*log(x^2)+30*x^2-30*x)*log(-log(x^2)^2+8*log(x^2)-15)^4+((16*
x^2-32*x+16)*log(x^2)-64*x^2+128*x-64)*log(-log(x^2)^2+8*log(x^2)-15)^3+log(x^2)^2-8*log(x^2)+15)/(x*log(x^2)^
2-8*x*log(x^2)+15*x),x, algorithm="fricas")

[Out]

(x^2 - 2*x + 1)*log(-log(x^2)^2 + 8*log(x^2) - 15)^4 + 1/2*log(x^2)

Sympy [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.21 \[ \int \frac {15-8 \log \left (x^2\right )+\log ^2\left (x^2\right )+\left (-64+128 x-64 x^2+\left (16-32 x+16 x^2\right ) \log \left (x^2\right )\right ) \log ^3\left (-15+8 \log \left (x^2\right )-\log ^2\left (x^2\right )\right )+\left (-30 x+30 x^2+\left (16 x-16 x^2\right ) \log \left (x^2\right )+\left (-2 x+2 x^2\right ) \log ^2\left (x^2\right )\right ) \log ^4\left (-15+8 \log \left (x^2\right )-\log ^2\left (x^2\right )\right )}{15 x-8 x \log \left (x^2\right )+x \log ^2\left (x^2\right )} \, dx=\left (x^{2} - 2 x + 1\right ) \log {\left (- \log {\left (x^{2} \right )}^{2} + 8 \log {\left (x^{2} \right )} - 15 \right )}^{4} + \log {\left (x \right )} \]

[In]

integrate((((2*x**2-2*x)*ln(x**2)**2+(-16*x**2+16*x)*ln(x**2)+30*x**2-30*x)*ln(-ln(x**2)**2+8*ln(x**2)-15)**4+
((16*x**2-32*x+16)*ln(x**2)-64*x**2+128*x-64)*ln(-ln(x**2)**2+8*ln(x**2)-15)**3+ln(x**2)**2-8*ln(x**2)+15)/(x*
ln(x**2)**2-8*x*ln(x**2)+15*x),x)

[Out]

(x**2 - 2*x + 1)*log(-log(x**2)**2 + 8*log(x**2) - 15)**4 + log(x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 284 vs. \(2 (24) = 48\).

Time = 0.39 (sec) , antiderivative size = 284, normalized size of antiderivative = 11.83 \[ \int \frac {15-8 \log \left (x^2\right )+\log ^2\left (x^2\right )+\left (-64+128 x-64 x^2+\left (16-32 x+16 x^2\right ) \log \left (x^2\right )\right ) \log ^3\left (-15+8 \log \left (x^2\right )-\log ^2\left (x^2\right )\right )+\left (-30 x+30 x^2+\left (16 x-16 x^2\right ) \log \left (x^2\right )+\left (-2 x+2 x^2\right ) \log ^2\left (x^2\right )\right ) \log ^4\left (-15+8 \log \left (x^2\right )-\log ^2\left (x^2\right )\right )}{15 x-8 x \log \left (x^2\right )+x \log ^2\left (x^2\right )} \, dx={\left (x^{2} - 2 \, x + 1\right )} \log \left (2 \, \log \left (x\right ) - 3\right )^{4} + 4 \, {\left (x^{2} - 2 \, x + 1\right )} \log \left (2 \, \log \left (x\right ) - 3\right )^{3} \log \left (-2 \, \log \left (x\right ) + 5\right ) + 6 \, {\left (x^{2} - 2 \, x + 1\right )} \log \left (2 \, \log \left (x\right ) - 3\right )^{2} \log \left (-2 \, \log \left (x\right ) + 5\right )^{2} + 4 \, {\left (x^{2} - 2 \, x + 1\right )} \log \left (2 \, \log \left (x\right ) - 3\right ) \log \left (-2 \, \log \left (x\right ) + 5\right )^{3} + {\left (x^{2} - 2 \, x + 1\right )} \log \left (-2 \, \log \left (x\right ) + 5\right )^{4} - \frac {1}{4} \, {\left (\log \left (\log \left (x\right ) - \frac {3}{2}\right ) - \log \left (\log \left (x\right ) - \frac {5}{2}\right )\right )} \log \left (x^{2}\right )^{2} + \frac {1}{2} \, {\left ({\left (2 \, \log \left (x\right ) - 3\right )} \log \left (\log \left (x\right ) - \frac {3}{2}\right ) - {\left (2 \, \log \left (x\right ) - 5\right )} \log \left (\log \left (x\right ) - \frac {5}{2}\right ) - 2\right )} \log \left (x^{2}\right ) + 2 \, {\left (\log \left (\log \left (x\right ) - \frac {3}{2}\right ) - \log \left (\log \left (x\right ) - \frac {5}{2}\right )\right )} \log \left (x^{2}\right ) - {\left (\log \left (x\right )^{2} - 3 \, \log \left (x\right )\right )} \log \left (\log \left (x\right ) - \frac {3}{2}\right ) - 2 \, {\left (2 \, \log \left (x\right ) - 3\right )} \log \left (\log \left (x\right ) - \frac {3}{2}\right ) + {\left (\log \left (x\right )^{2} - 5 \, \log \left (x\right )\right )} \log \left (\log \left (x\right ) - \frac {5}{2}\right ) + 2 \, {\left (2 \, \log \left (x\right ) - 5\right )} \log \left (\log \left (x\right ) - \frac {5}{2}\right ) + 3 \, \log \left (x\right ) - \frac {9}{4} \, \log \left (2 \, \log \left (x\right ) - 3\right ) + \frac {25}{4} \, \log \left (2 \, \log \left (x\right ) - 5\right ) - \frac {15}{4} \, \log \left (\log \left (x\right ) - \frac {3}{2}\right ) + \frac {15}{4} \, \log \left (\log \left (x\right ) - \frac {5}{2}\right ) + 4 \]

[In]

integrate((((2*x^2-2*x)*log(x^2)^2+(-16*x^2+16*x)*log(x^2)+30*x^2-30*x)*log(-log(x^2)^2+8*log(x^2)-15)^4+((16*
x^2-32*x+16)*log(x^2)-64*x^2+128*x-64)*log(-log(x^2)^2+8*log(x^2)-15)^3+log(x^2)^2-8*log(x^2)+15)/(x*log(x^2)^
2-8*x*log(x^2)+15*x),x, algorithm="maxima")

[Out]

(x^2 - 2*x + 1)*log(2*log(x) - 3)^4 + 4*(x^2 - 2*x + 1)*log(2*log(x) - 3)^3*log(-2*log(x) + 5) + 6*(x^2 - 2*x
+ 1)*log(2*log(x) - 3)^2*log(-2*log(x) + 5)^2 + 4*(x^2 - 2*x + 1)*log(2*log(x) - 3)*log(-2*log(x) + 5)^3 + (x^
2 - 2*x + 1)*log(-2*log(x) + 5)^4 - 1/4*(log(log(x) - 3/2) - log(log(x) - 5/2))*log(x^2)^2 + 1/2*((2*log(x) -
3)*log(log(x) - 3/2) - (2*log(x) - 5)*log(log(x) - 5/2) - 2)*log(x^2) + 2*(log(log(x) - 3/2) - log(log(x) - 5/
2))*log(x^2) - (log(x)^2 - 3*log(x))*log(log(x) - 3/2) - 2*(2*log(x) - 3)*log(log(x) - 3/2) + (log(x)^2 - 5*lo
g(x))*log(log(x) - 5/2) + 2*(2*log(x) - 5)*log(log(x) - 5/2) + 3*log(x) - 9/4*log(2*log(x) - 3) + 25/4*log(2*l
og(x) - 5) - 15/4*log(log(x) - 3/2) + 15/4*log(log(x) - 5/2) + 4

Giac [A] (verification not implemented)

none

Time = 1.73 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.29 \[ \int \frac {15-8 \log \left (x^2\right )+\log ^2\left (x^2\right )+\left (-64+128 x-64 x^2+\left (16-32 x+16 x^2\right ) \log \left (x^2\right )\right ) \log ^3\left (-15+8 \log \left (x^2\right )-\log ^2\left (x^2\right )\right )+\left (-30 x+30 x^2+\left (16 x-16 x^2\right ) \log \left (x^2\right )+\left (-2 x+2 x^2\right ) \log ^2\left (x^2\right )\right ) \log ^4\left (-15+8 \log \left (x^2\right )-\log ^2\left (x^2\right )\right )}{15 x-8 x \log \left (x^2\right )+x \log ^2\left (x^2\right )} \, dx={\left (x^{2} - 2 \, x + 1\right )} \log \left (-\log \left (x^{2}\right )^{2} + 8 \, \log \left (x^{2}\right ) - 15\right )^{4} + \log \left (x\right ) \]

[In]

integrate((((2*x^2-2*x)*log(x^2)^2+(-16*x^2+16*x)*log(x^2)+30*x^2-30*x)*log(-log(x^2)^2+8*log(x^2)-15)^4+((16*
x^2-32*x+16)*log(x^2)-64*x^2+128*x-64)*log(-log(x^2)^2+8*log(x^2)-15)^3+log(x^2)^2-8*log(x^2)+15)/(x*log(x^2)^
2-8*x*log(x^2)+15*x),x, algorithm="giac")

[Out]

(x^2 - 2*x + 1)*log(-log(x^2)^2 + 8*log(x^2) - 15)^4 + log(x)

Mupad [B] (verification not implemented)

Time = 9.15 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.21 \[ \int \frac {15-8 \log \left (x^2\right )+\log ^2\left (x^2\right )+\left (-64+128 x-64 x^2+\left (16-32 x+16 x^2\right ) \log \left (x^2\right )\right ) \log ^3\left (-15+8 \log \left (x^2\right )-\log ^2\left (x^2\right )\right )+\left (-30 x+30 x^2+\left (16 x-16 x^2\right ) \log \left (x^2\right )+\left (-2 x+2 x^2\right ) \log ^2\left (x^2\right )\right ) \log ^4\left (-15+8 \log \left (x^2\right )-\log ^2\left (x^2\right )\right )}{15 x-8 x \log \left (x^2\right )+x \log ^2\left (x^2\right )} \, dx=\left (x^2-2\,x+1\right )\,{\ln \left (\ln \left (x^{16}\right )-{\ln \left (x^2\right )}^2-15\right )}^4+\ln \left (x\right ) \]

[In]

int((log(x^2)^2 - log(8*log(x^2) - log(x^2)^2 - 15)^4*(30*x - log(x^2)*(16*x - 16*x^2) + log(x^2)^2*(2*x - 2*x
^2) - 30*x^2) - 8*log(x^2) + log(8*log(x^2) - log(x^2)^2 - 15)^3*(128*x + log(x^2)*(16*x^2 - 32*x + 16) - 64*x
^2 - 64) + 15)/(15*x - 8*x*log(x^2) + x*log(x^2)^2),x)

[Out]

log(x) + log(log(x^16) - log(x^2)^2 - 15)^4*(x^2 - 2*x + 1)