Integrand size = 71, antiderivative size = 29 \[ \int \frac {48 x-1168 x^2-1216 x^3+336 x^4+2000 x^5+\left (32-784 x-800 x^2-48 x^3+1600 x^4\right ) \log (x)+\left (16-384 x-800 x^2-400 x \log (x)\right ) \log \left (x^2\right )}{x} \, dx=16 \left (x-25 x^2\right ) (x+\log (x)) \left (-x^2+\frac {x+\log \left (x^2\right )}{x}\right ) \] Output:
16*(-25*x^2+x)*(x+ln(x))*((ln(x^2)+x)/x-x^2)
Leaf count is larger than twice the leaf count of optimal. \(73\) vs. \(2(29)=58\).
Time = 0.04 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.52 \[ \int \frac {48 x-1168 x^2-1216 x^3+336 x^4+2000 x^5+\left (32-784 x-800 x^2-48 x^3+1600 x^4\right ) \log (x)+\left (16-384 x-800 x^2-400 x \log (x)\right ) \log \left (x^2\right )}{x} \, dx=4 \left (4 x^2 \left (1-25 x-x^2+25 x^3\right )+4 \log ^2(x)+4 x \log (x) \left (1-25 x-x^2+25 x^3-25 \log \left (x^2\right )\right )+4 (1-25 x) x \log \left (x^2\right )+\log ^2\left (x^2\right )\right ) \] Input:
Integrate[(48*x - 1168*x^2 - 1216*x^3 + 336*x^4 + 2000*x^5 + (32 - 784*x - 800*x^2 - 48*x^3 + 1600*x^4)*Log[x] + (16 - 384*x - 800*x^2 - 400*x*Log[x ])*Log[x^2])/x,x]
Output:
4*(4*x^2*(1 - 25*x - x^2 + 25*x^3) + 4*Log[x]^2 + 4*x*Log[x]*(1 - 25*x - x ^2 + 25*x^3 - 25*Log[x^2]) + 4*(1 - 25*x)*x*Log[x^2] + Log[x^2]^2)
Leaf count is larger than twice the leaf count of optimal. \(86\) vs. \(2(29)=58\).
Time = 0.51 (sec) , antiderivative size = 86, normalized size of antiderivative = 2.97, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {2010, 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.
\(\displaystyle \int \frac {2000 x^5+336 x^4-1216 x^3-1168 x^2+\left (-800 x^2-384 x-400 x \log (x)+16\right ) \log \left (x^2\right )+\left (1600 x^4-48 x^3-800 x^2-784 x+32\right ) \log (x)+48 x}{x} \, dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \int \left (\frac {16 (x-1) \left (125 x^4+146 x^3+100 x^3 \log (x)+70 x^2+97 x^2 \log (x)-3 x+47 x \log (x)-2 \log (x)\right )}{x}-\frac {16 \left (50 x^2+24 x+25 x \log (x)-1\right ) \log \left (x^2\right )}{x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 400 x^5-16 x^4+400 x^4 \log (x)-400 x^3-16 x^3 \log (x)+16 x^2+4 \log ^2\left (x^2\right )-400 x^2 \log (x)-400 x^2 \log \left (x^2\right )-400 x \log (x) \log \left (x^2\right )+16 x \log \left (x^2\right )+16 \log ^2(x)+16 x \log (x)\) |
Input:
Int[(48*x - 1168*x^2 - 1216*x^3 + 336*x^4 + 2000*x^5 + (32 - 784*x - 800*x ^2 - 48*x^3 + 1600*x^4)*Log[x] + (16 - 384*x - 800*x^2 - 400*x*Log[x])*Log [x^2])/x,x]
Output:
16*x^2 - 400*x^3 - 16*x^4 + 400*x^5 + 16*x*Log[x] - 400*x^2*Log[x] - 16*x^ 3*Log[x] + 400*x^4*Log[x] + 16*Log[x]^2 + 16*x*Log[x^2] - 400*x^2*Log[x^2] - 400*x*Log[x]*Log[x^2] + 4*Log[x^2]^2
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] , x] /; FreeQ[{c, m}, x] && SumQ[u] && !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
Leaf count of result is larger than twice the leaf count of optimal. \(86\) vs. \(2(29)=58\).
Time = 6.45 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.00
method | result | size |
parallelrisch | \(400 x^{4} \ln \left (x \right )-400 x^{2} \ln \left (x^{2}\right )-16 x^{3} \ln \left (x \right )+16 x \ln \left (x^{2}\right )-400 x \ln \left (x \right ) \ln \left (x^{2}\right )+4 \ln \left (x^{2}\right )^{2}-400 x^{2} \ln \left (x \right )+16 x \ln \left (x \right )+400 x^{5}+16 \ln \left (x \right )^{2}+16 x^{2}-400 x^{3}-16 x^{4}\) | \(87\) |
default | \(816 x \ln \left (x \right )-800 x \ln \left (x \right )^{2}-400 \left (\ln \left (x^{2}\right )-2 \ln \left (x \right )\right ) \left (x \ln \left (x \right )-x \right )-400 x^{2} \ln \left (x \right )+16 x^{2}-16 x^{3} \ln \left (x \right )-400 x^{3}+400 x^{4} \ln \left (x \right )-16 x^{4}+400 x^{5}-400 x^{2} \ln \left (x^{2}\right )-384 x \ln \left (x^{2}\right )+16 \ln \left (x \right ) \ln \left (x^{2}\right )\) | \(98\) |
parts | \(816 x \ln \left (x \right )-800 x \ln \left (x \right )^{2}-400 \left (\ln \left (x^{2}\right )-2 \ln \left (x \right )\right ) \left (x \ln \left (x \right )-x \right )-400 x^{2} \ln \left (x \right )+16 x^{2}-16 x^{3} \ln \left (x \right )-400 x^{3}+400 x^{4} \ln \left (x \right )-16 x^{4}+400 x^{5}-400 x^{2} \ln \left (x^{2}\right )-384 x \ln \left (x^{2}\right )+16 \ln \left (x \right ) \ln \left (x^{2}\right )\) | \(98\) |
risch | \(\left (-800 x +32\right ) \ln \left (x \right )^{2}+8 x \left (25 i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}-50 i \pi \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right )+25 i \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )^{2}+50 x^{3}-2 x^{2}-150 x +6\right ) \ln \left (x \right )+200 i \pi \,x^{2} \operatorname {csgn}\left (i x^{2}\right )^{3}-400 i \pi \,x^{2} \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right )+200 i \pi \,x^{2} \operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )^{2}+400 x^{5}-8 i \pi x \operatorname {csgn}\left (i x^{2}\right )^{3}+16 i \pi x \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right )-8 i \pi x \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )^{2}-16 x^{4}-400 x^{3}+16 x^{2}-8 i \ln \left (x \right ) \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+16 i \ln \left (x \right ) \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}-8 i \ln \left (x \right ) \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )\) | \(266\) |
orering | \(\frac {\left (x -\frac {1}{25}\right ) \left (\left (-400 x \ln \left (x \right )-800 x^{2}-384 x +16\right ) \ln \left (x^{2}\right )+\left (1600 x^{4}-48 x^{3}-800 x^{2}-784 x +32\right ) \ln \left (x \right )+2000 x^{5}+336 x^{4}-1216 x^{3}-1168 x^{2}+48 x \right )}{x}-\frac {x \left (39375000 x^{10}+21262500 x^{9}-72708750 x^{8}-124420440 x^{7}-30578310 x^{6}+15668343 x^{4}-3596800 x^{2}-423164 x +1440\right ) \left (\frac {\left (-400 \ln \left (x \right )-784-1600 x \right ) \ln \left (x^{2}\right )+\frac {-800 x \ln \left (x \right )-1600 x^{2}-768 x +32}{x}+\left (6400 x^{3}-144 x^{2}-1600 x -784\right ) \ln \left (x \right )+\frac {1600 x^{4}-48 x^{3}-800 x^{2}-784 x +32}{x}+10000 x^{4}+1344 x^{3}-3648 x^{2}-2336 x +48}{x}-\frac {\left (-400 x \ln \left (x \right )-800 x^{2}-384 x +16\right ) \ln \left (x^{2}\right )+\left (1600 x^{4}-48 x^{3}-800 x^{2}-784 x +32\right ) \ln \left (x \right )+2000 x^{5}+336 x^{4}-1216 x^{3}-1168 x^{2}+48 x}{x^{2}}\right )}{1500 \left (75000 x^{9}+42000 x^{8}-91539 x^{7}-102532 x^{6}-3944 x^{5}+21864 x^{4}+19865 x^{3}+7092 x^{2}+794 x +8\right )}+\frac {\left (5625000 x^{10}+3543750 x^{9}-14541750 x^{8}-31105110 x^{7}-10192770 x^{6}+15668343 x^{4}+12156975 x^{3}+3596800 x^{2}+211582 x -480\right ) x^{2} \left (\frac {\left (-\frac {400}{x}-1600\right ) \ln \left (x^{2}\right )+\frac {-1600 \ln \left (x \right )-3136-6400 x}{x}-\frac {2 \left (-400 x \ln \left (x \right )-800 x^{2}-384 x +16\right )}{x^{2}}+\left (19200 x^{2}-288 x -1600\right ) \ln \left (x \right )+\frac {12800 x^{3}-288 x^{2}-3200 x -1568}{x}-\frac {1600 x^{4}-48 x^{3}-800 x^{2}-784 x +32}{x^{2}}+40000 x^{3}+4032 x^{2}-7296 x -2336}{x}-\frac {2 \left (\left (-400 \ln \left (x \right )-784-1600 x \right ) \ln \left (x^{2}\right )+\frac {-800 x \ln \left (x \right )-1600 x^{2}-768 x +32}{x}+\left (6400 x^{3}-144 x^{2}-1600 x -784\right ) \ln \left (x \right )+\frac {1600 x^{4}-48 x^{3}-800 x^{2}-784 x +32}{x}+10000 x^{4}+1344 x^{3}-3648 x^{2}-2336 x +48\right )}{x^{2}}+\frac {2 \left (-400 x \ln \left (x \right )-800 x^{2}-384 x +16\right ) \ln \left (x^{2}\right )+2 \left (1600 x^{4}-48 x^{3}-800 x^{2}-784 x +32\right ) \ln \left (x \right )+4000 x^{5}+672 x^{4}-2432 x^{3}-2336 x^{2}+96 x}{x^{3}}\right )}{112500000 x^{9}+63000000 x^{8}-137308500 x^{7}-153798000 x^{6}-5916000 x^{5}+32796000 x^{4}+29797500 x^{3}+10638000 x^{2}+1191000 x +12000}\) | \(733\) |
Input:
int(((-400*x*ln(x)-800*x^2-384*x+16)*ln(x^2)+(1600*x^4-48*x^3-800*x^2-784* x+32)*ln(x)+2000*x^5+336*x^4-1216*x^3-1168*x^2+48*x)/x,x,method=_RETURNVER BOSE)
Output:
400*x^4*ln(x)-400*x^2*ln(x^2)-16*x^3*ln(x)+16*x*ln(x^2)-400*x*ln(x)*ln(x^2 )+4*ln(x^2)^2-400*x^2*ln(x)+16*x*ln(x)+400*x^5+16*ln(x)^2+16*x^2-400*x^3-1 6*x^4
Time = 0.09 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.90 \[ \int \frac {48 x-1168 x^2-1216 x^3+336 x^4+2000 x^5+\left (32-784 x-800 x^2-48 x^3+1600 x^4\right ) \log (x)+\left (16-384 x-800 x^2-400 x \log (x)\right ) \log \left (x^2\right )}{x} \, dx=400 \, x^{5} - 16 \, x^{4} - 400 \, x^{3} - 32 \, {\left (25 \, x - 1\right )} \log \left (x\right )^{2} + 16 \, x^{2} + 16 \, {\left (25 \, x^{4} - x^{3} - 75 \, x^{2} + 3 \, x\right )} \log \left (x\right ) \] Input:
integrate(((-400*x*log(x)-800*x^2-384*x+16)*log(x^2)+(1600*x^4-48*x^3-800* x^2-784*x+32)*log(x)+2000*x^5+336*x^4-1216*x^3-1168*x^2+48*x)/x,x, algorit hm="fricas")
Output:
400*x^5 - 16*x^4 - 400*x^3 - 32*(25*x - 1)*log(x)^2 + 16*x^2 + 16*(25*x^4 - x^3 - 75*x^2 + 3*x)*log(x)
Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (24) = 48\).
Time = 0.09 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.76 \[ \int \frac {48 x-1168 x^2-1216 x^3+336 x^4+2000 x^5+\left (32-784 x-800 x^2-48 x^3+1600 x^4\right ) \log (x)+\left (16-384 x-800 x^2-400 x \log (x)\right ) \log \left (x^2\right )}{x} \, dx=400 x^{5} - 16 x^{4} - 400 x^{3} + 16 x^{2} + \left (32 - 800 x\right ) \log {\left (x \right )}^{2} + \left (400 x^{4} - 16 x^{3} - 1200 x^{2} + 48 x\right ) \log {\left (x \right )} \] Input:
integrate(((-400*x*ln(x)-800*x**2-384*x+16)*ln(x**2)+(1600*x**4-48*x**3-80 0*x**2-784*x+32)*ln(x)+2000*x**5+336*x**4-1216*x**3-1168*x**2+48*x)/x,x)
Output:
400*x**5 - 16*x**4 - 400*x**3 + 16*x**2 + (32 - 800*x)*log(x)**2 + (400*x* *4 - 16*x**3 - 1200*x**2 + 48*x)*log(x)
Leaf count of result is larger than twice the leaf count of optimal. 91 vs. \(2 (30) = 60\).
Time = 0.03 (sec) , antiderivative size = 91, normalized size of antiderivative = 3.14 \[ \int \frac {48 x-1168 x^2-1216 x^3+336 x^4+2000 x^5+\left (32-784 x-800 x^2-48 x^3+1600 x^4\right ) \log (x)+\left (16-384 x-800 x^2-400 x \log (x)\right ) \log \left (x^2\right )}{x} \, dx=400 \, x^{5} + 400 \, x^{4} \log \left (x\right ) - 16 \, x^{4} - 16 \, x^{3} \log \left (x\right ) - 400 \, x^{3} - 400 \, x^{2} \log \left (x^{2}\right ) - 400 \, x^{2} \log \left (x\right ) + 16 \, x^{2} - 384 \, x \log \left (x^{2}\right ) + 4 \, \log \left (x^{2}\right )^{2} - 400 \, {\left (x \log \left (x^{2}\right ) - 2 \, x\right )} \log \left (x\right ) + 16 \, x \log \left (x\right ) + 16 \, \log \left (x\right )^{2} \] Input:
integrate(((-400*x*log(x)-800*x^2-384*x+16)*log(x^2)+(1600*x^4-48*x^3-800* x^2-784*x+32)*log(x)+2000*x^5+336*x^4-1216*x^3-1168*x^2+48*x)/x,x, algorit hm="maxima")
Output:
400*x^5 + 400*x^4*log(x) - 16*x^4 - 16*x^3*log(x) - 400*x^3 - 400*x^2*log( x^2) - 400*x^2*log(x) + 16*x^2 - 384*x*log(x^2) + 4*log(x^2)^2 - 400*(x*lo g(x^2) - 2*x)*log(x) + 16*x*log(x) + 16*log(x)^2
Time = 0.12 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.90 \[ \int \frac {48 x-1168 x^2-1216 x^3+336 x^4+2000 x^5+\left (32-784 x-800 x^2-48 x^3+1600 x^4\right ) \log (x)+\left (16-384 x-800 x^2-400 x \log (x)\right ) \log \left (x^2\right )}{x} \, dx=400 \, x^{5} - 16 \, x^{4} - 400 \, x^{3} - 32 \, {\left (25 \, x - 1\right )} \log \left (x\right )^{2} + 16 \, x^{2} + 16 \, {\left (25 \, x^{4} - x^{3} - 75 \, x^{2} + 3 \, x\right )} \log \left (x\right ) \] Input:
integrate(((-400*x*log(x)-800*x^2-384*x+16)*log(x^2)+(1600*x^4-48*x^3-800* x^2-784*x+32)*log(x)+2000*x^5+336*x^4-1216*x^3-1168*x^2+48*x)/x,x, algorit hm="giac")
Output:
400*x^5 - 16*x^4 - 400*x^3 - 32*(25*x - 1)*log(x)^2 + 16*x^2 + 16*(25*x^4 - x^3 - 75*x^2 + 3*x)*log(x)
Time = 3.14 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.76 \[ \int \frac {48 x-1168 x^2-1216 x^3+336 x^4+2000 x^5+\left (32-784 x-800 x^2-48 x^3+1600 x^4\right ) \log (x)+\left (16-384 x-800 x^2-400 x \log (x)\right ) \log \left (x^2\right )}{x} \, dx=-16\,\left (25\,x-1\right )\,\left (x+\ln \left (x\right )\right )\,\left (x+\ln \left (x^2\right )-x^3\right ) \] Input:
int(-(log(x)*(784*x + 800*x^2 + 48*x^3 - 1600*x^4 - 32) - 48*x + log(x^2)* (384*x + 400*x*log(x) + 800*x^2 - 16) + 1168*x^2 + 1216*x^3 - 336*x^4 - 20 00*x^5)/x,x)
Output:
-16*(25*x - 1)*(x + log(x))*(x + log(x^2) - x^3)
Time = 0.19 (sec) , antiderivative size = 86, normalized size of antiderivative = 2.97 \[ \int \frac {48 x-1168 x^2-1216 x^3+336 x^4+2000 x^5+\left (32-784 x-800 x^2-48 x^3+1600 x^4\right ) \log (x)+\left (16-384 x-800 x^2-400 x \log (x)\right ) \log \left (x^2\right )}{x} \, dx=4 \mathrm {log}\left (x^{2}\right )^{2}-400 \,\mathrm {log}\left (x^{2}\right ) \mathrm {log}\left (x \right ) x -400 \,\mathrm {log}\left (x^{2}\right ) x^{2}+16 \,\mathrm {log}\left (x^{2}\right ) x +16 \mathrm {log}\left (x \right )^{2}+400 \,\mathrm {log}\left (x \right ) x^{4}-16 \,\mathrm {log}\left (x \right ) x^{3}-400 \,\mathrm {log}\left (x \right ) x^{2}+16 \,\mathrm {log}\left (x \right ) x +400 x^{5}-16 x^{4}-400 x^{3}+16 x^{2} \] Input:
int(((-400*x*log(x)-800*x^2-384*x+16)*log(x^2)+(1600*x^4-48*x^3-800*x^2-78 4*x+32)*log(x)+2000*x^5+336*x^4-1216*x^3-1168*x^2+48*x)/x,x)
Output:
4*(log(x**2)**2 - 100*log(x**2)*log(x)*x - 100*log(x**2)*x**2 + 4*log(x**2 )*x + 4*log(x)**2 + 100*log(x)*x**4 - 4*log(x)*x**3 - 100*log(x)*x**2 + 4* log(x)*x + 100*x**5 - 4*x**4 - 100*x**3 + 4*x**2)