Integrand size = 85, antiderivative size = 22 \[ \int \frac {\left (-32+16 x^5 \log ^8(5)-2 x^{10} \log ^{16}(5)\right ) \log (x)+\left (16-8 x^5 \log ^8(5)+x^{10} \log ^{16}(5)+\left (-128+24 x^5 \log ^8(5)+2 x^{10} \log ^{16}(5)\right ) \log (x)\right ) \log \left (x^2\right )}{x^9 \log ^{16}(5) \log ^2\left (x^2\right )} \, dx=\frac {\left (x-\frac {4}{x^4 \log ^8(5)}\right )^2 \log (x)}{\log \left (x^2\right )} \] Output:
ln(x)/ln(x^2)*(x-4/x^4/ln(5)^8)^2
Time = 0.18 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.27 \[ \int \frac {\left (-32+16 x^5 \log ^8(5)-2 x^{10} \log ^{16}(5)\right ) \log (x)+\left (16-8 x^5 \log ^8(5)+x^{10} \log ^{16}(5)+\left (-128+24 x^5 \log ^8(5)+2 x^{10} \log ^{16}(5)\right ) \log (x)\right ) \log \left (x^2\right )}{x^9 \log ^{16}(5) \log ^2\left (x^2\right )} \, dx=\frac {\left (-4+x^5 \log ^8(5)\right )^2 \log (x)}{x^8 \log ^{16}(5) \log \left (x^2\right )} \] Input:
Integrate[((-32 + 16*x^5*Log[5]^8 - 2*x^10*Log[5]^16)*Log[x] + (16 - 8*x^5 *Log[5]^8 + x^10*Log[5]^16 + (-128 + 24*x^5*Log[5]^8 + 2*x^10*Log[5]^16)*L og[x])*Log[x^2])/(x^9*Log[5]^16*Log[x^2]^2),x]
Output:
((-4 + x^5*Log[5]^8)^2*Log[x])/(x^8*Log[5]^16*Log[x^2])
Leaf count is larger than twice the leaf count of optimal. \(54\) vs. \(2(22)=44\).
Time = 2.11 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.45, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {27, 25, 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.
\(\displaystyle \int \frac {\left (-2 x^{10} \log ^{16}(5)+16 x^5 \log ^8(5)-32\right ) \log (x)+\left (x^{10} \log ^{16}(5)-8 x^5 \log ^8(5)+\left (2 x^{10} \log ^{16}(5)+24 x^5 \log ^8(5)-128\right ) \log (x)+16\right ) \log \left (x^2\right )}{x^9 \log ^{16}(5) \log ^2\left (x^2\right )} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int -\frac {2 \left (\log ^{16}(5) x^{10}-8 \log ^8(5) x^5+16\right ) \log (x)-\left (\log ^{16}(5) x^{10}-8 \log ^8(5) x^5-2 \left (-\log ^{16}(5) x^{10}-12 \log ^8(5) x^5+64\right ) \log (x)+16\right ) \log \left (x^2\right )}{x^9 \log ^2\left (x^2\right )}dx}{\log ^{16}(5)}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int \frac {2 \left (\log ^{16}(5) x^{10}-8 \log ^8(5) x^5+16\right ) \log (x)-\left (\log ^{16}(5) x^{10}-8 \log ^8(5) x^5-2 \left (-\log ^{16}(5) x^{10}-12 \log ^8(5) x^5+64\right ) \log (x)+16\right ) \log \left (x^2\right )}{x^9 \log ^2\left (x^2\right )}dx}{\log ^{16}(5)}\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle -\frac {\int \frac {\left (4-x^5 \log ^8(5)\right ) \left (-2 \log ^8(5) \log (x) x^5+2 \log ^8(5) \log (x) \log \left (x^2\right ) x^5+\log ^8(5) \log \left (x^2\right ) x^5+8 \log (x)+32 \log (x) \log \left (x^2\right )-4 \log \left (x^2\right )\right )}{x^9 \log ^2\left (x^2\right )}dx}{\log ^{16}(5)}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {\int \left (\frac {2 \left (x^5 \log ^8(5)-4\right )^2 \log (x)}{x^9 \log ^2\left (x^2\right )}-\frac {\left (x^5 \log ^8(5)-4\right ) \left (2 \log ^8(5) \log (x) x^5+\log ^8(5) x^5+32 \log (x)-4\right )}{x^9 \log \left (x^2\right )}\right )dx}{\log ^{16}(5)}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {-\frac {x^2 \log ^{16}(5) \log (x)}{\log \left (x^2\right )}-\frac {16 \log (x)}{x^8 \log \left (x^2\right )}+\frac {8 \log ^8(5) \log (x)}{x^3 \log \left (x^2\right )}}{\log ^{16}(5)}\) |
Input:
Int[((-32 + 16*x^5*Log[5]^8 - 2*x^10*Log[5]^16)*Log[x] + (16 - 8*x^5*Log[5 ]^8 + x^10*Log[5]^16 + (-128 + 24*x^5*Log[5]^8 + 2*x^10*Log[5]^16)*Log[x]) *Log[x^2])/(x^9*Log[5]^16*Log[x^2]^2),x]
Output:
-(((-16*Log[x])/(x^8*Log[x^2]) + (8*Log[5]^8*Log[x])/(x^3*Log[x^2]) - (x^2 *Log[5]^16*Log[x])/Log[x^2])/Log[5]^16)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Time = 40.94 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.86
method | result | size |
parallelrisch | \(\frac {\ln \left (x \right ) \ln \left (5\right )^{16} x^{10}-8 x^{5} \ln \left (5\right )^{8} \ln \left (x \right )+16 \ln \left (x \right )}{\ln \left (5\right )^{16} x^{8} \ln \left (x^{2}\right )}\) | \(41\) |
risch | \(\frac {x^{10} \ln \left (5\right )^{16}-8 x^{5} \ln \left (5\right )^{8}+16}{2 \ln \left (5\right )^{16} x^{8}}-\frac {\pi \,\operatorname {csgn}\left (i x^{2}\right ) \left (\ln \left (5\right )^{16} x^{10} \operatorname {csgn}\left (i x \right )^{2}-2 \ln \left (5\right )^{16} x^{10} \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )+\ln \left (5\right )^{16} x^{10} \operatorname {csgn}\left (i x^{2}\right )^{2}-8 \ln \left (5\right )^{8} x^{5} \operatorname {csgn}\left (i x \right )^{2}+16 \ln \left (5\right )^{8} x^{5} \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )-8 \ln \left (5\right )^{8} x^{5} \operatorname {csgn}\left (i x^{2}\right )^{2}+16 \operatorname {csgn}\left (i x \right )^{2}-32 \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )+16 \operatorname {csgn}\left (i x^{2}\right )^{2}\right )}{2 \ln \left (5\right )^{16} x^{8} \left (\pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-2 \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+\pi \operatorname {csgn}\left (i x^{2}\right )^{3}+4 i \ln \left (x \right )\right )}\) | \(242\) |
Input:
int((((2*x^10*ln(5)^16+24*x^5*ln(5)^8-128)*ln(x)+x^10*ln(5)^16-8*x^5*ln(5) ^8+16)*ln(x^2)+(-2*x^10*ln(5)^16+16*x^5*ln(5)^8-32)*ln(x))/x^9/ln(5)^16/ln (x^2)^2,x,method=_RETURNVERBOSE)
Output:
1/ln(5)^16/x^8*(ln(x)*ln(5)^16*x^10-8*x^5*ln(5)^8*ln(x)+16*ln(x))/ln(x^2)
Time = 0.08 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.27 \[ \int \frac {\left (-32+16 x^5 \log ^8(5)-2 x^{10} \log ^{16}(5)\right ) \log (x)+\left (16-8 x^5 \log ^8(5)+x^{10} \log ^{16}(5)+\left (-128+24 x^5 \log ^8(5)+2 x^{10} \log ^{16}(5)\right ) \log (x)\right ) \log \left (x^2\right )}{x^9 \log ^{16}(5) \log ^2\left (x^2\right )} \, dx=\frac {x^{10} \log \left (5\right )^{16} - 8 \, x^{5} \log \left (5\right )^{8} + 16}{2 \, x^{8} \log \left (5\right )^{16}} \] Input:
integrate((((2*x^10*log(5)^16+24*x^5*log(5)^8-128)*log(x)+x^10*log(5)^16-8 *x^5*log(5)^8+16)*log(x^2)+(-2*x^10*log(5)^16+16*x^5*log(5)^8-32)*log(x))/ x^9/log(5)^16/log(x^2)^2,x, algorithm="fricas")
Output:
1/2*(x^10*log(5)^16 - 8*x^5*log(5)^8 + 16)/(x^8*log(5)^16)
Time = 0.12 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.32 \[ \int \frac {\left (-32+16 x^5 \log ^8(5)-2 x^{10} \log ^{16}(5)\right ) \log (x)+\left (16-8 x^5 \log ^8(5)+x^{10} \log ^{16}(5)+\left (-128+24 x^5 \log ^8(5)+2 x^{10} \log ^{16}(5)\right ) \log (x)\right ) \log \left (x^2\right )}{x^9 \log ^{16}(5) \log ^2\left (x^2\right )} \, dx=\frac {\frac {x^{2} \log {\left (5 \right )}^{16}}{2} + \frac {- 4 x^{5} \log {\left (5 \right )}^{8} + 8}{x^{8}}}{\log {\left (5 \right )}^{16}} \] Input:
integrate((((2*x**10*ln(5)**16+24*x**5*ln(5)**8-128)*ln(x)+x**10*ln(5)**16 -8*x**5*ln(5)**8+16)*ln(x**2)+(-2*x**10*ln(5)**16+16*x**5*ln(5)**8-32)*ln( x))/x**9/ln(5)**16/ln(x**2)**2,x)
Output:
(x**2*log(5)**16/2 + (-4*x**5*log(5)**8 + 8)/x**8)/log(5)**16
Time = 0.15 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.27 \[ \int \frac {\left (-32+16 x^5 \log ^8(5)-2 x^{10} \log ^{16}(5)\right ) \log (x)+\left (16-8 x^5 \log ^8(5)+x^{10} \log ^{16}(5)+\left (-128+24 x^5 \log ^8(5)+2 x^{10} \log ^{16}(5)\right ) \log (x)\right ) \log \left (x^2\right )}{x^9 \log ^{16}(5) \log ^2\left (x^2\right )} \, dx=\frac {x^{10} \log \left (5\right )^{16} - 8 \, x^{5} \log \left (5\right )^{8} + 16}{2 \, x^{8} \log \left (5\right )^{16}} \] Input:
integrate((((2*x^10*log(5)^16+24*x^5*log(5)^8-128)*log(x)+x^10*log(5)^16-8 *x^5*log(5)^8+16)*log(x^2)+(-2*x^10*log(5)^16+16*x^5*log(5)^8-32)*log(x))/ x^9/log(5)^16/log(x^2)^2,x, algorithm="maxima")
Output:
1/2*(x^10*log(5)^16 - 8*x^5*log(5)^8 + 16)/(x^8*log(5)^16)
Time = 0.14 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.36 \[ \int \frac {\left (-32+16 x^5 \log ^8(5)-2 x^{10} \log ^{16}(5)\right ) \log (x)+\left (16-8 x^5 \log ^8(5)+x^{10} \log ^{16}(5)+\left (-128+24 x^5 \log ^8(5)+2 x^{10} \log ^{16}(5)\right ) \log (x)\right ) \log \left (x^2\right )}{x^9 \log ^{16}(5) \log ^2\left (x^2\right )} \, dx=\frac {x^{2} \log \left (5\right )^{16} - \frac {8 \, {\left (x^{5} \log \left (5\right )^{8} - 2\right )}}{x^{8}}}{2 \, \log \left (5\right )^{16}} \] Input:
integrate((((2*x^10*log(5)^16+24*x^5*log(5)^8-128)*log(x)+x^10*log(5)^16-8 *x^5*log(5)^8+16)*log(x^2)+(-2*x^10*log(5)^16+16*x^5*log(5)^8-32)*log(x))/ x^9/log(5)^16/log(x^2)^2,x, algorithm="giac")
Output:
1/2*(x^2*log(5)^16 - 8*(x^5*log(5)^8 - 2)/x^8)/log(5)^16
Time = 3.16 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.27 \[ \int \frac {\left (-32+16 x^5 \log ^8(5)-2 x^{10} \log ^{16}(5)\right ) \log (x)+\left (16-8 x^5 \log ^8(5)+x^{10} \log ^{16}(5)+\left (-128+24 x^5 \log ^8(5)+2 x^{10} \log ^{16}(5)\right ) \log (x)\right ) \log \left (x^2\right )}{x^9 \log ^{16}(5) \log ^2\left (x^2\right )} \, dx=\frac {\ln \left (x\right )\,{\left (x^5\,{\ln \left (5\right )}^8-4\right )}^2}{x^8\,\ln \left (x^2\right )\,{\ln \left (5\right )}^{16}} \] Input:
int(-(log(x)*(2*x^10*log(5)^16 - 16*x^5*log(5)^8 + 32) - log(x^2)*(x^10*lo g(5)^16 - 8*x^5*log(5)^8 + log(x)*(24*x^5*log(5)^8 + 2*x^10*log(5)^16 - 12 8) + 16))/(x^9*log(x^2)^2*log(5)^16),x)
Output:
(log(x)*(x^5*log(5)^8 - 4)^2)/(x^8*log(x^2)*log(5)^16)
Time = 0.15 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.59 \[ \int \frac {\left (-32+16 x^5 \log ^8(5)-2 x^{10} \log ^{16}(5)\right ) \log (x)+\left (16-8 x^5 \log ^8(5)+x^{10} \log ^{16}(5)+\left (-128+24 x^5 \log ^8(5)+2 x^{10} \log ^{16}(5)\right ) \log (x)\right ) \log \left (x^2\right )}{x^9 \log ^{16}(5) \log ^2\left (x^2\right )} \, dx=\frac {\mathrm {log}\left (x \right ) \left (\mathrm {log}\left (5\right )^{16} x^{10}-8 \mathrm {log}\left (5\right )^{8} x^{5}+16\right )}{\mathrm {log}\left (x^{2}\right ) \mathrm {log}\left (5\right )^{16} x^{8}} \] Input:
int((((2*x^10*log(5)^16+24*x^5*log(5)^8-128)*log(x)+x^10*log(5)^16-8*x^5*l og(5)^8+16)*log(x^2)+(-2*x^10*log(5)^16+16*x^5*log(5)^8-32)*log(x))/x^9/lo g(5)^16/log(x^2)^2,x)
Output:
(log(x)*(log(5)**16*x**10 - 8*log(5)**8*x**5 + 16))/(log(x**2)*log(5)**16* x**8)