Integrand size = 162, antiderivative size = 32 \[ \int \frac {2 \log (x)+(3+x-x \log (x)) \log \left (x^2\right )+\left (-14 x-4 x^2\right ) \log ^2\left (x^2\right )-3 x \log ^3\left (x^2\right )+\left (-\log \left (x^2\right )+4 x \log ^2\left (x^2\right )+x \log ^3\left (x^2\right )\right ) \log \left (\log \left (x^2\right )\right )}{\left (\left (3 x+x^2\right ) \log (x)+5 x \log ^2(x)\right ) \log \left (x^2\right )+\left (-3 x^2-x^3-10 x^2 \log (x)\right ) \log ^3\left (x^2\right )+5 x^3 \log ^5\left (x^2\right )+\left (-x \log (x) \log \left (x^2\right )+x^2 \log ^3\left (x^2\right )\right ) \log \left (\log \left (x^2\right )\right )} \, dx=\log \left (\frac {1}{5-\frac {3+x-\log \left (\log \left (x^2\right )\right )}{-\log (x)+x \log ^2\left (x^2\right )}}\right ) \]
\[ \int \frac {2 \log (x)+(3+x-x \log (x)) \log \left (x^2\right )+\left (-14 x-4 x^2\right ) \log ^2\left (x^2\right )-3 x \log ^3\left (x^2\right )+\left (-\log \left (x^2\right )+4 x \log ^2\left (x^2\right )+x \log ^3\left (x^2\right )\right ) \log \left (\log \left (x^2\right )\right )}{\left (\left (3 x+x^2\right ) \log (x)+5 x \log ^2(x)\right ) \log \left (x^2\right )+\left (-3 x^2-x^3-10 x^2 \log (x)\right ) \log ^3\left (x^2\right )+5 x^3 \log ^5\left (x^2\right )+\left (-x \log (x) \log \left (x^2\right )+x^2 \log ^3\left (x^2\right )\right ) \log \left (\log \left (x^2\right )\right )} \, dx=\int \frac {2 \log (x)+(3+x-x \log (x)) \log \left (x^2\right )+\left (-14 x-4 x^2\right ) \log ^2\left (x^2\right )-3 x \log ^3\left (x^2\right )+\left (-\log \left (x^2\right )+4 x \log ^2\left (x^2\right )+x \log ^3\left (x^2\right )\right ) \log \left (\log \left (x^2\right )\right )}{\left (\left (3 x+x^2\right ) \log (x)+5 x \log ^2(x)\right ) \log \left (x^2\right )+\left (-3 x^2-x^3-10 x^2 \log (x)\right ) \log ^3\left (x^2\right )+5 x^3 \log ^5\left (x^2\right )+\left (-x \log (x) \log \left (x^2\right )+x^2 \log ^3\left (x^2\right )\right ) \log \left (\log \left (x^2\right )\right )} \, dx \]
Integrate[(2*Log[x] + (3 + x - x*Log[x])*Log[x^2] + (-14*x - 4*x^2)*Log[x^ 2]^2 - 3*x*Log[x^2]^3 + (-Log[x^2] + 4*x*Log[x^2]^2 + x*Log[x^2]^3)*Log[Lo g[x^2]])/(((3*x + x^2)*Log[x] + 5*x*Log[x]^2)*Log[x^2] + (-3*x^2 - x^3 - 1 0*x^2*Log[x])*Log[x^2]^3 + 5*x^3*Log[x^2]^5 + (-(x*Log[x]*Log[x^2]) + x^2* Log[x^2]^3)*Log[Log[x^2]]),x]
Integrate[(2*Log[x] + (3 + x - x*Log[x])*Log[x^2] + (-14*x - 4*x^2)*Log[x^ 2]^2 - 3*x*Log[x^2]^3 + (-Log[x^2] + 4*x*Log[x^2]^2 + x*Log[x^2]^3)*Log[Lo g[x^2]])/(((3*x + x^2)*Log[x] + 5*x*Log[x]^2)*Log[x^2] + (-3*x^2 - x^3 - 1 0*x^2*Log[x])*Log[x^2]^3 + 5*x^3*Log[x^2]^5 + (-(x*Log[x]*Log[x^2]) + x^2* Log[x^2]^3)*Log[Log[x^2]]), x]
Time = 2.54 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.25, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {7292, 7293, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {-3 x \log ^3\left (x^2\right )+\left (-4 x^2-14 x\right ) \log ^2\left (x^2\right )+\left (x \log ^3\left (x^2\right )+4 x \log ^2\left (x^2\right )-\log \left (x^2\right )\right ) \log \left (\log \left (x^2\right )\right )+(x+x (-\log (x))+3) \log \left (x^2\right )+2 \log (x)}{\left (x^2 \log ^3\left (x^2\right )-x \log (x) \log \left (x^2\right )\right ) \log \left (\log \left (x^2\right )\right )+\left (\left (x^2+3 x\right ) \log (x)+5 x \log ^2(x)\right ) \log \left (x^2\right )+5 x^3 \log ^5\left (x^2\right )+\left (-x^3-3 x^2-10 x^2 \log (x)\right ) \log ^3\left (x^2\right )} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-3 x \log ^3\left (x^2\right )+\left (-4 x^2-14 x\right ) \log ^2\left (x^2\right )+\left (x \log ^3\left (x^2\right )+4 x \log ^2\left (x^2\right )-\log \left (x^2\right )\right ) \log \left (\log \left (x^2\right )\right )+(x+x (-\log (x))+3) \log \left (x^2\right )+2 \log (x)}{x \log \left (x^2\right ) \left (\log (x)-x \log ^2\left (x^2\right )\right ) \left (-5 x \log ^2\left (x^2\right )-\log \left (\log \left (x^2\right )\right )+x+5 \log (x)+3\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {x \log ^2\left (x^2\right )+4 x \log \left (x^2\right )-1}{x \left (x \log ^2\left (x^2\right )-\log (x)\right )}+\frac {-5 x \log ^3\left (x^2\right )-20 x \log ^2\left (x^2\right )+x \log \left (x^2\right )+5 \log \left (x^2\right )-2}{x \log \left (x^2\right ) \left (5 x \log ^2\left (x^2\right )+\log \left (\log \left (x^2\right )\right )-x-5 \log (x)-3\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \log \left (\log (x)-x \log ^2\left (x^2\right )\right )-\log \left (-5 x \log ^2\left (x^2\right )-\log \left (\log \left (x^2\right )\right )+x+5 \log (x)+3\right )\) |
Int[(2*Log[x] + (3 + x - x*Log[x])*Log[x^2] + (-14*x - 4*x^2)*Log[x^2]^2 - 3*x*Log[x^2]^3 + (-Log[x^2] + 4*x*Log[x^2]^2 + x*Log[x^2]^3)*Log[Log[x^2] ])/(((3*x + x^2)*Log[x] + 5*x*Log[x]^2)*Log[x^2] + (-3*x^2 - x^3 - 10*x^2* Log[x])*Log[x^2]^3 + 5*x^3*Log[x^2]^5 + (-(x*Log[x]*Log[x^2]) + x^2*Log[x^ 2]^3)*Log[Log[x^2]]),x]
3.22.30.3.1 Defintions of rubi rules used
Time = 34.00 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.34
method | result | size |
parallelrisch | \(-\ln \left (x \ln \left (x^{2}\right )^{2}-\frac {x}{5}-\ln \left (x \right )+\frac {\ln \left (\ln \left (x^{2}\right )\right )}{5}-\frac {3}{5}\right )+\ln \left (x \ln \left (x^{2}\right )^{2}-\ln \left (x \right )\right )\) | \(43\) |
default | \(\ln \left (x \right )+\ln \left (-\frac {x \,\pi ^{2} \operatorname {csgn}\left (i x \right )^{4} \operatorname {csgn}\left (i x^{2}\right )^{2}-4 x \,\pi ^{2} \operatorname {csgn}\left (i x \right )^{3} \operatorname {csgn}\left (i x^{2}\right )^{3}+6 x \,\pi ^{2} \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )^{4}-4 x \,\pi ^{2} \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{5}+x \,\pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{6}+4 \ln \left (x \right )}{16 x}-\frac {i \pi \,\operatorname {csgn}\left (i x^{2}\right ) \left (\operatorname {csgn}\left (i x \right )^{2}-2 \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )+\operatorname {csgn}\left (i x^{2}\right )^{2}\right ) \ln \left (x \right )}{2}+\ln \left (x \right )^{2}\right )-\ln \left (\ln \left (2 \ln \left (x \right )-\frac {i \pi \,\operatorname {csgn}\left (i x^{2}\right ) {\left (-\operatorname {csgn}\left (i x^{2}\right )+\operatorname {csgn}\left (i x \right )\right )}^{2}}{2}\right )-\frac {5 x \,\pi ^{2} \operatorname {csgn}\left (i x \right )^{4} \operatorname {csgn}\left (i x^{2}\right )^{2}}{4}+5 x \,\pi ^{2} \operatorname {csgn}\left (i x \right )^{3} \operatorname {csgn}\left (i x^{2}\right )^{3}-\frac {15 x \,\pi ^{2} \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )^{4}}{2}+5 x \,\pi ^{2} \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{5}-\frac {5 x \,\pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{6}}{4}-10 i x \ln \left (x \right ) \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )+20 i x \ln \left (x \right ) \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}-10 i x \ln \left (x \right ) \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+20 x \ln \left (x \right )^{2}-5 \ln \left (x \right )-x -3\right )\) | \(375\) |
risch | \(\text {Expression too large to display}\) | \(740\) |
int(((x*ln(x^2)^3+4*x*ln(x^2)^2-ln(x^2))*ln(ln(x^2))-3*x*ln(x^2)^3+(-4*x^2 -14*x)*ln(x^2)^2+(-x*ln(x)+3+x)*ln(x^2)+2*ln(x))/((x^2*ln(x^2)^3-x*ln(x)*l n(x^2))*ln(ln(x^2))+5*x^3*ln(x^2)^5+(-10*x^2*ln(x)-x^3-3*x^2)*ln(x^2)^3+(5 *x*ln(x)^2+(x^2+3*x)*ln(x))*ln(x^2)),x,method=_RETURNVERBOSE)
Time = 0.27 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.31 \[ \int \frac {2 \log (x)+(3+x-x \log (x)) \log \left (x^2\right )+\left (-14 x-4 x^2\right ) \log ^2\left (x^2\right )-3 x \log ^3\left (x^2\right )+\left (-\log \left (x^2\right )+4 x \log ^2\left (x^2\right )+x \log ^3\left (x^2\right )\right ) \log \left (\log \left (x^2\right )\right )}{\left (\left (3 x+x^2\right ) \log (x)+5 x \log ^2(x)\right ) \log \left (x^2\right )+\left (-3 x^2-x^3-10 x^2 \log (x)\right ) \log ^3\left (x^2\right )+5 x^3 \log ^5\left (x^2\right )+\left (-x \log (x) \log \left (x^2\right )+x^2 \log ^3\left (x^2\right )\right ) \log \left (\log \left (x^2\right )\right )} \, dx=-\log \left (20 \, x \log \left (x\right )^{2} - x - 5 \, \log \left (x\right ) + \log \left (2 \, \log \left (x\right )\right ) - 3\right ) + \log \left (x\right ) + \log \left (\frac {4 \, x \log \left (x\right ) - 1}{x}\right ) + \log \left (\log \left (x\right )\right ) \]
integrate(((x*log(x^2)^3+4*x*log(x^2)^2-log(x^2))*log(log(x^2))-3*x*log(x^ 2)^3+(-4*x^2-14*x)*log(x^2)^2+(-x*log(x)+3+x)*log(x^2)+2*log(x))/((x^2*log (x^2)^3-x*log(x)*log(x^2))*log(log(x^2))+5*x^3*log(x^2)^5+(-10*x^2*log(x)- x^3-3*x^2)*log(x^2)^3+(5*x*log(x)^2+(x^2+3*x)*log(x))*log(x^2)),x, algorit hm=\
-log(20*x*log(x)^2 - x - 5*log(x) + log(2*log(x)) - 3) + log(x) + log((4*x *log(x) - 1)/x) + log(log(x))
Time = 0.21 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.28 \[ \int \frac {2 \log (x)+(3+x-x \log (x)) \log \left (x^2\right )+\left (-14 x-4 x^2\right ) \log ^2\left (x^2\right )-3 x \log ^3\left (x^2\right )+\left (-\log \left (x^2\right )+4 x \log ^2\left (x^2\right )+x \log ^3\left (x^2\right )\right ) \log \left (\log \left (x^2\right )\right )}{\left (\left (3 x+x^2\right ) \log (x)+5 x \log ^2(x)\right ) \log \left (x^2\right )+\left (-3 x^2-x^3-10 x^2 \log (x)\right ) \log ^3\left (x^2\right )+5 x^3 \log ^5\left (x^2\right )+\left (-x \log (x) \log \left (x^2\right )+x^2 \log ^3\left (x^2\right )\right ) \log \left (\log \left (x^2\right )\right )} \, dx=\log {\left (x \right )} + \log {\left (\log {\left (x \right )}^{2} - \frac {\log {\left (x \right )}}{4 x} \right )} - \log {\left (20 x \log {\left (x \right )}^{2} - x - 5 \log {\left (x \right )} + \log {\left (2 \log {\left (x \right )} \right )} - 3 \right )} \]
integrate(((x*ln(x**2)**3+4*x*ln(x**2)**2-ln(x**2))*ln(ln(x**2))-3*x*ln(x* *2)**3+(-4*x**2-14*x)*ln(x**2)**2+(-x*ln(x)+3+x)*ln(x**2)+2*ln(x))/((x**2* ln(x**2)**3-x*ln(x)*ln(x**2))*ln(ln(x**2))+5*x**3*ln(x**2)**5+(-10*x**2*ln (x)-x**3-3*x**2)*ln(x**2)**3+(5*x*ln(x)**2+(x**2+3*x)*ln(x))*ln(x**2)),x)
log(x) + log(log(x)**2 - log(x)/(4*x)) - log(20*x*log(x)**2 - x - 5*log(x) + log(2*log(x)) - 3)
Time = 0.36 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.34 \[ \int \frac {2 \log (x)+(3+x-x \log (x)) \log \left (x^2\right )+\left (-14 x-4 x^2\right ) \log ^2\left (x^2\right )-3 x \log ^3\left (x^2\right )+\left (-\log \left (x^2\right )+4 x \log ^2\left (x^2\right )+x \log ^3\left (x^2\right )\right ) \log \left (\log \left (x^2\right )\right )}{\left (\left (3 x+x^2\right ) \log (x)+5 x \log ^2(x)\right ) \log \left (x^2\right )+\left (-3 x^2-x^3-10 x^2 \log (x)\right ) \log ^3\left (x^2\right )+5 x^3 \log ^5\left (x^2\right )+\left (-x \log (x) \log \left (x^2\right )+x^2 \log ^3\left (x^2\right )\right ) \log \left (\log \left (x^2\right )\right )} \, dx=-\log \left (20 \, x \log \left (x\right )^{2} - x + \log \left (2\right ) - 5 \, \log \left (x\right ) + \log \left (\log \left (x\right )\right ) - 3\right ) + \log \left (x\right ) + \log \left (\frac {4 \, x \log \left (x\right ) - 1}{4 \, x}\right ) + \log \left (\log \left (x\right )\right ) \]
integrate(((x*log(x^2)^3+4*x*log(x^2)^2-log(x^2))*log(log(x^2))-3*x*log(x^ 2)^3+(-4*x^2-14*x)*log(x^2)^2+(-x*log(x)+3+x)*log(x^2)+2*log(x))/((x^2*log (x^2)^3-x*log(x)*log(x^2))*log(log(x^2))+5*x^3*log(x^2)^5+(-10*x^2*log(x)- x^3-3*x^2)*log(x^2)^3+(5*x*log(x)^2+(x^2+3*x)*log(x))*log(x^2)),x, algorit hm=\
-log(20*x*log(x)^2 - x + log(2) - 5*log(x) + log(log(x)) - 3) + log(x) + l og(1/4*(4*x*log(x) - 1)/x) + log(log(x))
\[ \int \frac {2 \log (x)+(3+x-x \log (x)) \log \left (x^2\right )+\left (-14 x-4 x^2\right ) \log ^2\left (x^2\right )-3 x \log ^3\left (x^2\right )+\left (-\log \left (x^2\right )+4 x \log ^2\left (x^2\right )+x \log ^3\left (x^2\right )\right ) \log \left (\log \left (x^2\right )\right )}{\left (\left (3 x+x^2\right ) \log (x)+5 x \log ^2(x)\right ) \log \left (x^2\right )+\left (-3 x^2-x^3-10 x^2 \log (x)\right ) \log ^3\left (x^2\right )+5 x^3 \log ^5\left (x^2\right )+\left (-x \log (x) \log \left (x^2\right )+x^2 \log ^3\left (x^2\right )\right ) \log \left (\log \left (x^2\right )\right )} \, dx=\int { -\frac {3 \, x \log \left (x^{2}\right )^{3} + 2 \, {\left (2 \, x^{2} + 7 \, x\right )} \log \left (x^{2}\right )^{2} + {\left (x \log \left (x\right ) - x - 3\right )} \log \left (x^{2}\right ) - {\left (x \log \left (x^{2}\right )^{3} + 4 \, x \log \left (x^{2}\right )^{2} - \log \left (x^{2}\right )\right )} \log \left (\log \left (x^{2}\right )\right ) - 2 \, \log \left (x\right )}{5 \, x^{3} \log \left (x^{2}\right )^{5} - {\left (x^{3} + 10 \, x^{2} \log \left (x\right ) + 3 \, x^{2}\right )} \log \left (x^{2}\right )^{3} + {\left (5 \, x \log \left (x\right )^{2} + {\left (x^{2} + 3 \, x\right )} \log \left (x\right )\right )} \log \left (x^{2}\right ) + {\left (x^{2} \log \left (x^{2}\right )^{3} - x \log \left (x^{2}\right ) \log \left (x\right )\right )} \log \left (\log \left (x^{2}\right )\right )} \,d x } \]
integrate(((x*log(x^2)^3+4*x*log(x^2)^2-log(x^2))*log(log(x^2))-3*x*log(x^ 2)^3+(-4*x^2-14*x)*log(x^2)^2+(-x*log(x)+3+x)*log(x^2)+2*log(x))/((x^2*log (x^2)^3-x*log(x)*log(x^2))*log(log(x^2))+5*x^3*log(x^2)^5+(-10*x^2*log(x)- x^3-3*x^2)*log(x^2)^3+(5*x*log(x)^2+(x^2+3*x)*log(x))*log(x^2)),x, algorit hm=\
integrate(-(3*x*log(x^2)^3 + 2*(2*x^2 + 7*x)*log(x^2)^2 + (x*log(x) - x - 3)*log(x^2) - (x*log(x^2)^3 + 4*x*log(x^2)^2 - log(x^2))*log(log(x^2)) - 2 *log(x))/(5*x^3*log(x^2)^5 - (x^3 + 10*x^2*log(x) + 3*x^2)*log(x^2)^3 + (5 *x*log(x)^2 + (x^2 + 3*x)*log(x))*log(x^2) + (x^2*log(x^2)^3 - x*log(x^2)* log(x))*log(log(x^2))), x)
Time = 13.87 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.44 \[ \int \frac {2 \log (x)+(3+x-x \log (x)) \log \left (x^2\right )+\left (-14 x-4 x^2\right ) \log ^2\left (x^2\right )-3 x \log ^3\left (x^2\right )+\left (-\log \left (x^2\right )+4 x \log ^2\left (x^2\right )+x \log ^3\left (x^2\right )\right ) \log \left (\log \left (x^2\right )\right )}{\left (\left (3 x+x^2\right ) \log (x)+5 x \log ^2(x)\right ) \log \left (x^2\right )+\left (-3 x^2-x^3-10 x^2 \log (x)\right ) \log ^3\left (x^2\right )+5 x^3 \log ^5\left (x^2\right )+\left (-x \log (x) \log \left (x^2\right )+x^2 \log ^3\left (x^2\right )\right ) \log \left (\log \left (x^2\right )\right )} \, dx=\ln \left (\frac {\ln \left (x\right )-x\,{\ln \left (x^2\right )}^2}{x}\right )-\ln \left (\ln \left (\ln \left (x^2\right )\right )-x-5\,\ln \left (x\right )+5\,x\,{\ln \left (x^2\right )}^2-3\right )+\ln \left (x\right ) \]
int((2*log(x) + log(log(x^2))*(4*x*log(x^2)^2 - log(x^2) + x*log(x^2)^3) + log(x^2)*(x - x*log(x) + 3) - log(x^2)^2*(14*x + 4*x^2) - 3*x*log(x^2)^3) /(log(log(x^2))*(x^2*log(x^2)^3 - x*log(x^2)*log(x)) + log(x^2)*(5*x*log(x )^2 + log(x)*(3*x + x^2)) - log(x^2)^3*(10*x^2*log(x) + 3*x^2 + x^3) + 5*x ^3*log(x^2)^5),x)