Integrand size = 89, antiderivative size = 24 \[ \int \frac {2400+1176 x+102 x^2-6 x^3+\left (800+392 x+34 x^2-2 x^3\right ) \log \left (\frac {x+x^2+(-100-25 x) \log (x)}{100+25 x}\right )}{-4 x^2-5 x^3-x^4+\left (400 x+200 x^2+25 x^3\right ) \log (x)} \, dx=\left (3+\log \left (\frac {x+x^2}{25 (4+x)}-\log (x)\right )\right )^2 \] Output:
(3+ln(1/25*(x^2+x)/(4+x)-ln(x)))^2
Time = 0.03 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96 \[ \int \frac {2400+1176 x+102 x^2-6 x^3+\left (800+392 x+34 x^2-2 x^3\right ) \log \left (\frac {x+x^2+(-100-25 x) \log (x)}{100+25 x}\right )}{-4 x^2-5 x^3-x^4+\left (400 x+200 x^2+25 x^3\right ) \log (x)} \, dx=\left (3+\log \left (\frac {x (1+x)}{25 (4+x)}-\log (x)\right )\right )^2 \] Input:
Integrate[(2400 + 1176*x + 102*x^2 - 6*x^3 + (800 + 392*x + 34*x^2 - 2*x^3 )*Log[(x + x^2 + (-100 - 25*x)*Log[x])/(100 + 25*x)])/(-4*x^2 - 5*x^3 - x^ 4 + (400*x + 200*x^2 + 25*x^3)*Log[x]),x]
Output:
(3 + Log[(x*(1 + x))/(25*(4 + x)) - Log[x]])^2
Time = 0.58 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {7292, 27, 25, 7237}
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.
\(\displaystyle \int \frac {-6 x^3+102 x^2+\left (-2 x^3+34 x^2+392 x+800\right ) \log \left (\frac {x^2+x+(-25 x-100) \log (x)}{25 x+100}\right )+1176 x+2400}{-x^4-5 x^3-4 x^2+\left (25 x^3+200 x^2+400 x\right ) \log (x)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {2 \left (-x^3+17 x^2+196 x+400\right ) \left (-\log \left (\frac {x (x+1)}{25 (x+4)}-\log (x)\right )-3\right )}{x (x+4) \left (x^2+x-25 x \log (x)-100 \log (x)\right )}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 \int -\frac {\left (-x^3+17 x^2+196 x+400\right ) \left (\log \left (\frac {x (x+1)}{25 (x+4)}-\log (x)\right )+3\right )}{x (x+4) \left (x^2-25 \log (x) x+x-100 \log (x)\right )}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -2 \int \frac {\left (-x^3+17 x^2+196 x+400\right ) \left (\log \left (\frac {x (x+1)}{25 (x+4)}-\log (x)\right )+3\right )}{x (x+4) \left (x^2-25 \log (x) x+x-100 \log (x)\right )}dx\) |
\(\Big \downarrow \) 7237 |
\(\displaystyle \left (\log \left (\frac {x (x+1)}{25 (x+4)}-\log (x)\right )+3\right )^2\) |
Input:
Int[(2400 + 1176*x + 102*x^2 - 6*x^3 + (800 + 392*x + 34*x^2 - 2*x^3)*Log[ (x + x^2 + (-100 - 25*x)*Log[x])/(100 + 25*x)])/(-4*x^2 - 5*x^3 - x^4 + (4 00*x + 200*x^2 + 25*x^3)*Log[x]),x]
Output:
(3 + Log[(x*(1 + x))/(25*(4 + x)) - Log[x]])^2
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Si mp[q*(y^(m + 1)/(m + 1)), x] /; !FalseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(81\) vs. \(2(22)=44\).
Time = 1.96 (sec) , antiderivative size = 82, normalized size of antiderivative = 3.42
method | result | size |
default | \(6 \ln \left (25 x \ln \left (x \right )-x^{2}+100 \ln \left (x \right )-x \right )-6 \ln \left (4+x \right )+4 \ln \left (5\right ) \left (-\ln \left (25 x \ln \left (x \right )-x^{2}+100 \ln \left (x \right )-x \right )+\ln \left (4+x \right )\right )+\ln \left (\frac {-25 x \ln \left (x \right )+x^{2}-100 \ln \left (x \right )+x}{4+x}\right )^{2}\) | \(82\) |
Input:
int(((-2*x^3+34*x^2+392*x+800)*ln(((-25*x-100)*ln(x)+x^2+x)/(25*x+100))-6* x^3+102*x^2+1176*x+2400)/((25*x^3+200*x^2+400*x)*ln(x)-x^4-5*x^3-4*x^2),x, method=_RETURNVERBOSE)
Output:
6*ln(25*x*ln(x)-x^2+100*ln(x)-x)-6*ln(4+x)+4*ln(5)*(-ln(25*x*ln(x)-x^2+100 *ln(x)-x)+ln(4+x))+ln((-25*x*ln(x)+x^2-100*ln(x)+x)/(4+x))^2
Leaf count of result is larger than twice the leaf count of optimal. 45 vs. \(2 (22) = 44\).
Time = 0.07 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.88 \[ \int \frac {2400+1176 x+102 x^2-6 x^3+\left (800+392 x+34 x^2-2 x^3\right ) \log \left (\frac {x+x^2+(-100-25 x) \log (x)}{100+25 x}\right )}{-4 x^2-5 x^3-x^4+\left (400 x+200 x^2+25 x^3\right ) \log (x)} \, dx=\log \left (\frac {x^{2} - 25 \, {\left (x + 4\right )} \log \left (x\right ) + x}{25 \, {\left (x + 4\right )}}\right )^{2} + 6 \, \log \left (\frac {x^{2} - 25 \, {\left (x + 4\right )} \log \left (x\right ) + x}{25 \, {\left (x + 4\right )}}\right ) \] Input:
integrate(((-2*x^3+34*x^2+392*x+800)*log(((-25*x-100)*log(x)+x^2+x)/(25*x+ 100))-6*x^3+102*x^2+1176*x+2400)/((25*x^3+200*x^2+400*x)*log(x)-x^4-5*x^3- 4*x^2),x, algorithm="fricas")
Output:
log(1/25*(x^2 - 25*(x + 4)*log(x) + x)/(x + 4))^2 + 6*log(1/25*(x^2 - 25*( x + 4)*log(x) + x)/(x + 4))
Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (19) = 38\).
Time = 0.40 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.71 \[ \int \frac {2400+1176 x+102 x^2-6 x^3+\left (800+392 x+34 x^2-2 x^3\right ) \log \left (\frac {x+x^2+(-100-25 x) \log (x)}{100+25 x}\right )}{-4 x^2-5 x^3-x^4+\left (400 x+200 x^2+25 x^3\right ) \log (x)} \, dx=\log {\left (\frac {x^{2} + x + \left (- 25 x - 100\right ) \log {\left (x \right )}}{25 x + 100} \right )}^{2} + 6 \log {\left (\log {\left (x \right )} + \frac {- x^{2} - x}{25 x + 100} \right )} \] Input:
integrate(((-2*x**3+34*x**2+392*x+800)*ln(((-25*x-100)*ln(x)+x**2+x)/(25*x +100))-6*x**3+102*x**2+1176*x+2400)/((25*x**3+200*x**2+400*x)*ln(x)-x**4-5 *x**3-4*x**2),x)
Output:
log((x**2 + x + (-25*x - 100)*log(x))/(25*x + 100))**2 + 6*log(log(x) + (- x**2 - x)/(25*x + 100))
\[ \int \frac {2400+1176 x+102 x^2-6 x^3+\left (800+392 x+34 x^2-2 x^3\right ) \log \left (\frac {x+x^2+(-100-25 x) \log (x)}{100+25 x}\right )}{-4 x^2-5 x^3-x^4+\left (400 x+200 x^2+25 x^3\right ) \log (x)} \, dx=\int { \frac {2 \, {\left (3 \, x^{3} - 51 \, x^{2} + {\left (x^{3} - 17 \, x^{2} - 196 \, x - 400\right )} \log \left (\frac {x^{2} - 25 \, {\left (x + 4\right )} \log \left (x\right ) + x}{25 \, {\left (x + 4\right )}}\right ) - 588 \, x - 1200\right )}}{x^{4} + 5 \, x^{3} + 4 \, x^{2} - 25 \, {\left (x^{3} + 8 \, x^{2} + 16 \, x\right )} \log \left (x\right )} \,d x } \] Input:
integrate(((-2*x^3+34*x^2+392*x+800)*log(((-25*x-100)*log(x)+x^2+x)/(25*x+ 100))-6*x^3+102*x^2+1176*x+2400)/((25*x^3+200*x^2+400*x)*log(x)-x^4-5*x^3- 4*x^2),x, algorithm="maxima")
Output:
2*integrate((3*x^3 - 51*x^2 + (x^3 - 17*x^2 - 196*x - 400)*log(1/25*(x^2 - 25*(x + 4)*log(x) + x)/(x + 4)) - 588*x - 1200)/(x^4 + 5*x^3 + 4*x^2 - 25 *(x^3 + 8*x^2 + 16*x)*log(x)), x)
\[ \int \frac {2400+1176 x+102 x^2-6 x^3+\left (800+392 x+34 x^2-2 x^3\right ) \log \left (\frac {x+x^2+(-100-25 x) \log (x)}{100+25 x}\right )}{-4 x^2-5 x^3-x^4+\left (400 x+200 x^2+25 x^3\right ) \log (x)} \, dx=\int { \frac {2 \, {\left (3 \, x^{3} - 51 \, x^{2} + {\left (x^{3} - 17 \, x^{2} - 196 \, x - 400\right )} \log \left (\frac {x^{2} - 25 \, {\left (x + 4\right )} \log \left (x\right ) + x}{25 \, {\left (x + 4\right )}}\right ) - 588 \, x - 1200\right )}}{x^{4} + 5 \, x^{3} + 4 \, x^{2} - 25 \, {\left (x^{3} + 8 \, x^{2} + 16 \, x\right )} \log \left (x\right )} \,d x } \] Input:
integrate(((-2*x^3+34*x^2+392*x+800)*log(((-25*x-100)*log(x)+x^2+x)/(25*x+ 100))-6*x^3+102*x^2+1176*x+2400)/((25*x^3+200*x^2+400*x)*log(x)-x^4-5*x^3- 4*x^2),x, algorithm="giac")
Output:
integrate(2*(3*x^3 - 51*x^2 + (x^3 - 17*x^2 - 196*x - 400)*log(1/25*(x^2 - 25*(x + 4)*log(x) + x)/(x + 4)) - 588*x - 1200)/(x^4 + 5*x^3 + 4*x^2 - 25 *(x^3 + 8*x^2 + 16*x)*log(x)), x)
Timed out. \[ \int \frac {2400+1176 x+102 x^2-6 x^3+\left (800+392 x+34 x^2-2 x^3\right ) \log \left (\frac {x+x^2+(-100-25 x) \log (x)}{100+25 x}\right )}{-4 x^2-5 x^3-x^4+\left (400 x+200 x^2+25 x^3\right ) \log (x)} \, dx=-\int \frac {1176\,x+\ln \left (\frac {x-\ln \left (x\right )\,\left (25\,x+100\right )+x^2}{25\,x+100}\right )\,\left (-2\,x^3+34\,x^2+392\,x+800\right )+102\,x^2-6\,x^3+2400}{4\,x^2+5\,x^3+x^4-\ln \left (x\right )\,\left (25\,x^3+200\,x^2+400\,x\right )} \,d x \] Input:
int(-(1176*x + log((x - log(x)*(25*x + 100) + x^2)/(25*x + 100))*(392*x + 34*x^2 - 2*x^3 + 800) + 102*x^2 - 6*x^3 + 2400)/(4*x^2 + 5*x^3 + x^4 - log (x)*(400*x + 200*x^2 + 25*x^3)),x)
Output:
-int((1176*x + log((x - log(x)*(25*x + 100) + x^2)/(25*x + 100))*(392*x + 34*x^2 - 2*x^3 + 800) + 102*x^2 - 6*x^3 + 2400)/(4*x^2 + 5*x^3 + x^4 - log (x)*(400*x + 200*x^2 + 25*x^3)), x)
Time = 0.22 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.21 \[ \int \frac {2400+1176 x+102 x^2-6 x^3+\left (800+392 x+34 x^2-2 x^3\right ) \log \left (\frac {x+x^2+(-100-25 x) \log (x)}{100+25 x}\right )}{-4 x^2-5 x^3-x^4+\left (400 x+200 x^2+25 x^3\right ) \log (x)} \, dx=6 \,\mathrm {log}\left (25 \,\mathrm {log}\left (x \right ) x +100 \,\mathrm {log}\left (x \right )-x^{2}-x \right )-6 \,\mathrm {log}\left (x +4\right )+\mathrm {log}\left (\frac {-25 \,\mathrm {log}\left (x \right ) x -100 \,\mathrm {log}\left (x \right )+x^{2}+x}{25 x +100}\right )^{2} \] Input:
int(((-2*x^3+34*x^2+392*x+800)*log(((-25*x-100)*log(x)+x^2+x)/(25*x+100))- 6*x^3+102*x^2+1176*x+2400)/((25*x^3+200*x^2+400*x)*log(x)-x^4-5*x^3-4*x^2) ,x)
Output:
6*log(25*log(x)*x + 100*log(x) - x**2 - x) - 6*log(x + 4) + log(( - 25*log (x)*x - 100*log(x) + x**2 + x)/(25*x + 100))**2