Integrand size = 108, antiderivative size = 21 \[ \int \frac {50+50 x+\left (51+103 x+2 x^2\right ) \log (x)+(1250+1250 x+(1300+2550 x) \log (x)) \log (1+x)+(625+625 x) \log (x) \log ^2(1+x)+(1250+1250 x+(1300+2550 x) \log (x)+(1250+1250 x) \log (x) \log (1+x)) \log (4 x \log (x))+(625+625 x) \log (x) \log ^2(4 x \log (x))}{(1+x) \log (x)} \, dx=x \left (x+(1+25 (\log (1+x)+\log (4 x \log (x))))^2\right ) \] Output:
(x+(25*ln(1+x)+25*ln(4*x*ln(x))+1)^2)*x
Leaf count is larger than twice the leaf count of optimal. \(45\) vs. \(2(21)=42\).
Time = 0.31 (sec) , antiderivative size = 45, normalized size of antiderivative = 2.14 \[ \int \frac {50+50 x+\left (51+103 x+2 x^2\right ) \log (x)+(1250+1250 x+(1300+2550 x) \log (x)) \log (1+x)+(625+625 x) \log (x) \log ^2(1+x)+(1250+1250 x+(1300+2550 x) \log (x)+(1250+1250 x) \log (x) \log (1+x)) \log (4 x \log (x))+(625+625 x) \log (x) \log ^2(4 x \log (x))}{(1+x) \log (x)} \, dx=x \left (1+x+50 \log (1+x)+625 \log ^2(1+x)+50 (1+25 \log (1+x)) \log (4 x \log (x))+625 \log ^2(4 x \log (x))\right ) \] Input:
Integrate[(50 + 50*x + (51 + 103*x + 2*x^2)*Log[x] + (1250 + 1250*x + (130 0 + 2550*x)*Log[x])*Log[1 + x] + (625 + 625*x)*Log[x]*Log[1 + x]^2 + (1250 + 1250*x + (1300 + 2550*x)*Log[x] + (1250 + 1250*x)*Log[x]*Log[1 + x])*Lo g[4*x*Log[x]] + (625 + 625*x)*Log[x]*Log[4*x*Log[x]]^2)/((1 + x)*Log[x]),x ]
Output:
x*(1 + x + 50*Log[1 + x] + 625*Log[1 + x]^2 + 50*(1 + 25*Log[1 + x])*Log[4 *x*Log[x]] + 625*Log[4*x*Log[x]]^2)
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.
\(\displaystyle \int \frac {\left (2 x^2+103 x+51\right ) \log (x)+50 x+(625 x+625) \log (x) \log ^2(x+1)+(625 x+625) \log (x) \log ^2(4 x \log (x))+(1250 x+(2550 x+1300) \log (x)+1250) \log (x+1)+(1250 x+(2550 x+1300) \log (x)+(1250 x+1250) \log (x) \log (x+1)+1250) \log (4 x \log (x))+50}{(x+1) \log (x)} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {2 x^2 \log (x)+50 x+625 x \log (x) \log ^2(x+1)+625 \log (x) \log ^2(x+1)+103 x \log (x)+2550 x \log (x) \log (x+1)+1250 x \log (x+1)+51 \log (x)+1300 \log (x) \log (x+1)+1250 \log (x+1)+50}{(x+1) \log (x)}+625 \log ^2(4 x \log (x))+\frac {50 (25 x+51 x \log (x)+25 x \log (x) \log (x+1)+26 \log (x)+25 \log (x) \log (x+1)+25) \log (4 x \log (x))}{(x+1) \log (x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -1250 \int \frac {\operatorname {LogIntegral}(x)}{x \log (x)}dx+625 \int \log ^2(4 x \log (x))dx+1250 \int \frac {\log (x+1)}{\log (x)}dx-1250 \int \frac {\log (4 x \log (x))}{x+1}dx+1250 \int \log (x+1) \log (4 x \log (x))dx-2500 \operatorname {LogIntegral}(x)-1250 \operatorname {LogIntegral}(x) \log (x)+1250 \operatorname {LogIntegral}(x) \log (4 x \log (x))+x^2-2499 x+625 (x+1) \log ^2(x+1)-625 \log ^2(x+1)+2550 x \log (4 x \log (x))+1300 (x+1) \log (x+1)-50 \log (x+1)\) |
Input:
Int[(50 + 50*x + (51 + 103*x + 2*x^2)*Log[x] + (1250 + 1250*x + (1300 + 25 50*x)*Log[x])*Log[1 + x] + (625 + 625*x)*Log[x]*Log[1 + x]^2 + (1250 + 125 0*x + (1300 + 2550*x)*Log[x] + (1250 + 1250*x)*Log[x]*Log[1 + x])*Log[4*x* Log[x]] + (625 + 625*x)*Log[x]*Log[4*x*Log[x]]^2)/((1 + x)*Log[x]),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(55\) vs. \(2(22)=44\).
Time = 1.84 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.67
method | result | size |
parallelrisch | \(625 \ln \left (1+x \right )^{2} x +1250 \ln \left (1+x \right ) \ln \left (4 x \ln \left (x \right )\right ) x +625 x \ln \left (4 x \ln \left (x \right )\right )^{2}-\frac {1}{2}+x^{2}+50 \ln \left (1+x \right ) x +50 \ln \left (4 x \ln \left (x \right )\right ) x +x\) | \(56\) |
risch | \(\text {Expression too large to display}\) | \(729\) |
Input:
int(((625*x+625)*ln(x)*ln(4*x*ln(x))^2+((1250*x+1250)*ln(x)*ln(1+x)+(2550* x+1300)*ln(x)+1250*x+1250)*ln(4*x*ln(x))+(625*x+625)*ln(x)*ln(1+x)^2+((255 0*x+1300)*ln(x)+1250*x+1250)*ln(1+x)+(2*x^2+103*x+51)*ln(x)+50*x+50)/ln(x) /(1+x),x,method=_RETURNVERBOSE)
Output:
625*ln(1+x)^2*x+1250*ln(1+x)*ln(4*x*ln(x))*x+625*x*ln(4*x*ln(x))^2-1/2+x^2 +50*ln(1+x)*x+50*ln(4*x*ln(x))*x+x
Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (22) = 44\).
Time = 0.10 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.33 \[ \int \frac {50+50 x+\left (51+103 x+2 x^2\right ) \log (x)+(1250+1250 x+(1300+2550 x) \log (x)) \log (1+x)+(625+625 x) \log (x) \log ^2(1+x)+(1250+1250 x+(1300+2550 x) \log (x)+(1250+1250 x) \log (x) \log (1+x)) \log (4 x \log (x))+(625+625 x) \log (x) \log ^2(4 x \log (x))}{(1+x) \log (x)} \, dx=625 \, x \log \left (4 \, x \log \left (x\right )\right )^{2} + 625 \, x \log \left (x + 1\right )^{2} + x^{2} + 50 \, {\left (25 \, x \log \left (x + 1\right ) + x\right )} \log \left (4 \, x \log \left (x\right )\right ) + 50 \, x \log \left (x + 1\right ) + x \] Input:
integrate(((625*x+625)*log(x)*log(4*x*log(x))^2+((1250*x+1250)*log(x)*log( 1+x)+(2550*x+1300)*log(x)+1250*x+1250)*log(4*x*log(x))+(625*x+625)*log(x)* log(1+x)^2+((2550*x+1300)*log(x)+1250*x+1250)*log(1+x)+(2*x^2+103*x+51)*lo g(x)+50*x+50)/log(x)/(1+x),x, algorithm="fricas")
Output:
625*x*log(4*x*log(x))^2 + 625*x*log(x + 1)^2 + x^2 + 50*(25*x*log(x + 1) + x)*log(4*x*log(x)) + 50*x*log(x + 1) + x
Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (22) = 44\).
Time = 4.76 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.67 \[ \int \frac {50+50 x+\left (51+103 x+2 x^2\right ) \log (x)+(1250+1250 x+(1300+2550 x) \log (x)) \log (1+x)+(625+625 x) \log (x) \log ^2(1+x)+(1250+1250 x+(1300+2550 x) \log (x)+(1250+1250 x) \log (x) \log (1+x)) \log (4 x \log (x))+(625+625 x) \log (x) \log ^2(4 x \log (x))}{(1+x) \log (x)} \, dx=x^{2} + 625 x \log {\left (4 x \log {\left (x \right )} \right )}^{2} + 625 x \log {\left (x + 1 \right )}^{2} + 50 x \log {\left (x + 1 \right )} + x + \left (1250 x \log {\left (x + 1 \right )} + 50 x\right ) \log {\left (4 x \log {\left (x \right )} \right )} \] Input:
integrate(((625*x+625)*ln(x)*ln(4*x*ln(x))**2+((1250*x+1250)*ln(x)*ln(1+x) +(2550*x+1300)*ln(x)+1250*x+1250)*ln(4*x*ln(x))+(625*x+625)*ln(x)*ln(1+x)* *2+((2550*x+1300)*ln(x)+1250*x+1250)*ln(1+x)+(2*x**2+103*x+51)*ln(x)+50*x+ 50)/ln(x)/(1+x),x)
Output:
x**2 + 625*x*log(4*x*log(x))**2 + 625*x*log(x + 1)**2 + 50*x*log(x + 1) + x + (1250*x*log(x + 1) + 50*x)*log(4*x*log(x))
Leaf count of result is larger than twice the leaf count of optimal. 98 vs. \(2 (22) = 44\).
Time = 0.14 (sec) , antiderivative size = 98, normalized size of antiderivative = 4.67 \[ \int \frac {50+50 x+\left (51+103 x+2 x^2\right ) \log (x)+(1250+1250 x+(1300+2550 x) \log (x)) \log (1+x)+(625+625 x) \log (x) \log ^2(1+x)+(1250+1250 x+(1300+2550 x) \log (x)+(1250+1250 x) \log (x) \log (1+x)) \log (4 x \log (x))+(625+625 x) \log (x) \log ^2(4 x \log (x))}{(1+x) \log (x)} \, dx=625 \, x \log \left (x + 1\right )^{2} + 50 \, x {\left (50 \, \log \left (2\right ) + 1\right )} \log \left (x\right ) + 625 \, x \log \left (x\right )^{2} + 625 \, x \log \left (\log \left (x\right )\right )^{2} + {\left (2500 \, \log \left (2\right )^{2} + 100 \, \log \left (2\right ) + 1\right )} x + x^{2} + 50 \, {\left (x {\left (50 \, \log \left (2\right ) + 1\right )} + 25 \, x \log \left (x\right ) + 25 \, x \log \left (\log \left (x\right )\right )\right )} \log \left (x + 1\right ) + 50 \, {\left (x {\left (50 \, \log \left (2\right ) + 1\right )} + 25 \, x \log \left (x\right )\right )} \log \left (\log \left (x\right )\right ) \] Input:
integrate(((625*x+625)*log(x)*log(4*x*log(x))^2+((1250*x+1250)*log(x)*log( 1+x)+(2550*x+1300)*log(x)+1250*x+1250)*log(4*x*log(x))+(625*x+625)*log(x)* log(1+x)^2+((2550*x+1300)*log(x)+1250*x+1250)*log(1+x)+(2*x^2+103*x+51)*lo g(x)+50*x+50)/log(x)/(1+x),x, algorithm="maxima")
Output:
625*x*log(x + 1)^2 + 50*x*(50*log(2) + 1)*log(x) + 625*x*log(x)^2 + 625*x* log(log(x))^2 + (2500*log(2)^2 + 100*log(2) + 1)*x + x^2 + 50*(x*(50*log(2 ) + 1) + 25*x*log(x) + 25*x*log(log(x)))*log(x + 1) + 50*(x*(50*log(2) + 1 ) + 25*x*log(x))*log(log(x))
Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (22) = 44\).
Time = 0.17 (sec) , antiderivative size = 70, normalized size of antiderivative = 3.33 \[ \int \frac {50+50 x+\left (51+103 x+2 x^2\right ) \log (x)+(1250+1250 x+(1300+2550 x) \log (x)) \log (1+x)+(625+625 x) \log (x) \log ^2(1+x)+(1250+1250 x+(1300+2550 x) \log (x)+(1250+1250 x) \log (x) \log (1+x)) \log (4 x \log (x))+(625+625 x) \log (x) \log ^2(4 x \log (x))}{(1+x) \log (x)} \, dx=625 \, x \log \left (x + 1\right )^{2} + 625 \, x \log \left (x\right )^{2} + 625 \, x \log \left (4 \, \log \left (x\right )\right )^{2} + x^{2} + 50 \, {\left (25 \, x \log \left (x\right ) + x\right )} \log \left (x + 1\right ) + 50 \, x \log \left (x\right ) + 50 \, {\left (25 \, x \log \left (x + 1\right ) + 25 \, x \log \left (x\right ) + x\right )} \log \left (4 \, \log \left (x\right )\right ) + x \] Input:
integrate(((625*x+625)*log(x)*log(4*x*log(x))^2+((1250*x+1250)*log(x)*log( 1+x)+(2550*x+1300)*log(x)+1250*x+1250)*log(4*x*log(x))+(625*x+625)*log(x)* log(1+x)^2+((2550*x+1300)*log(x)+1250*x+1250)*log(1+x)+(2*x^2+103*x+51)*lo g(x)+50*x+50)/log(x)/(1+x),x, algorithm="giac")
Output:
625*x*log(x + 1)^2 + 625*x*log(x)^2 + 625*x*log(4*log(x))^2 + x^2 + 50*(25 *x*log(x) + x)*log(x + 1) + 50*x*log(x) + 50*(25*x*log(x + 1) + 25*x*log(x ) + x)*log(4*log(x)) + x
Time = 3.72 (sec) , antiderivative size = 127, normalized size of antiderivative = 6.05 \[ \int \frac {50+50 x+\left (51+103 x+2 x^2\right ) \log (x)+(1250+1250 x+(1300+2550 x) \log (x)) \log (1+x)+(625+625 x) \log (x) \log ^2(1+x)+(1250+1250 x+(1300+2550 x) \log (x)+(1250+1250 x) \log (x) \log (1+x)) \log (4 x \log (x))+(625+625 x) \log (x) \log ^2(4 x \log (x))}{(1+x) \log (x)} \, dx=x+\ln \left (4\,x\,\ln \left (x\right )\right )\,\left (\frac {\ln \left (x+1\right )\,\left (1250\,x^2+1250\,x\right )}{x+1}-\frac {1250\,x^4+2500\,x^3+1250\,x^2}{x\,{\left (x+1\right )}^2}+\frac {1300\,x^4+2600\,x^3+1300\,x^2}{x\,{\left (x+1\right )}^2}\right )+50\,x\,\ln \left (x+1\right )+625\,x\,{\ln \left (x+1\right )}^2+x^2+\frac {{\ln \left (4\,x\,\ln \left (x\right )\right )}^2\,\left (625\,x^3+625\,x^2\right )}{x\,\left (x+1\right )} \] Input:
int((50*x + log(x + 1)*(1250*x + log(x)*(2550*x + 1300) + 1250) + log(x)*( 103*x + 2*x^2 + 51) + log(4*x*log(x))*(1250*x + log(x)*(2550*x + 1300) + l og(x + 1)*log(x)*(1250*x + 1250) + 1250) + log(4*x*log(x))^2*log(x)*(625*x + 625) + log(x + 1)^2*log(x)*(625*x + 625) + 50)/(log(x)*(x + 1)),x)
Output:
x + log(4*x*log(x))*((log(x + 1)*(1250*x + 1250*x^2))/(x + 1) - (1250*x^2 + 2500*x^3 + 1250*x^4)/(x*(x + 1)^2) + (1300*x^2 + 2600*x^3 + 1300*x^4)/(x *(x + 1)^2)) + 50*x*log(x + 1) + 625*x*log(x + 1)^2 + x^2 + (log(4*x*log(x ))^2*(625*x^2 + 625*x^3))/(x*(x + 1))
Time = 0.17 (sec) , antiderivative size = 71, normalized size of antiderivative = 3.38 \[ \int \frac {50+50 x+\left (51+103 x+2 x^2\right ) \log (x)+(1250+1250 x+(1300+2550 x) \log (x)) \log (1+x)+(625+625 x) \log (x) \log ^2(1+x)+(1250+1250 x+(1300+2550 x) \log (x)+(1250+1250 x) \log (x) \log (1+x)) \log (4 x \log (x))+(625+625 x) \log (x) \log ^2(4 x \log (x))}{(1+x) \log (x)} \, dx=1300 \,\mathrm {log}\left (\mathrm {log}\left (x \right )\right )+625 \mathrm {log}\left (x +1\right )^{2} x +1250 \,\mathrm {log}\left (x +1\right ) \mathrm {log}\left (4 \,\mathrm {log}\left (x \right ) x \right ) x +50 \,\mathrm {log}\left (x +1\right ) x +625 \mathrm {log}\left (4 \,\mathrm {log}\left (x \right ) x \right )^{2} x +50 \,\mathrm {log}\left (4 \,\mathrm {log}\left (x \right ) x \right ) x -1300 \,\mathrm {log}\left (4 \,\mathrm {log}\left (x \right ) x \right )+1300 \,\mathrm {log}\left (x \right )+x^{2}+x \] Input:
int(((625*x+625)*log(x)*log(4*x*log(x))^2+((1250*x+1250)*log(x)*log(1+x)+( 2550*x+1300)*log(x)+1250*x+1250)*log(4*x*log(x))+(625*x+625)*log(x)*log(1+ x)^2+((2550*x+1300)*log(x)+1250*x+1250)*log(1+x)+(2*x^2+103*x+51)*log(x)+5 0*x+50)/log(x)/(1+x),x)
Output:
1300*log(log(x)) + 625*log(x + 1)**2*x + 1250*log(x + 1)*log(4*log(x)*x)*x + 50*log(x + 1)*x + 625*log(4*log(x)*x)**2*x + 50*log(4*log(x)*x)*x - 130 0*log(4*log(x)*x) + 1300*log(x) + x**2 + x