Integrand size = 168, antiderivative size = 30 \[ \int \frac {100-105 x+31 x^2+3 e^{2 x} x^2+e^x \left (-30 x+24 x^2-x^3\right )+\left (30 x-19 x^2-6 e^x x^2\right ) \log (x)+3 x^2 \log ^2(x)}{75+10 x-83 x^2+30 x^3+e^{2 x} \left (3 x^2+3 x^3\right )+e^x \left (-30 x-17 x^2+19 x^3\right )+\left (30 x+17 x^2-19 x^3+e^x \left (-6 x^2-6 x^3\right )\right ) \log (x)+\left (3 x^2+3 x^3\right ) \log ^2(x)} \, dx=\log \left (x+\frac {1}{3} \left (3-\frac {-5+x}{-3-e^x+\frac {5}{x}+\log (x)}\right )\right ) \] Output:
ln(x-1/3*(-5+x)/(ln(x)-3-exp(x)+5/x)+1)
Leaf count is larger than twice the leaf count of optimal. \(70\) vs. \(2(30)=60\).
Time = 0.17 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.33 \[ \int \frac {100-105 x+31 x^2+3 e^{2 x} x^2+e^x \left (-30 x+24 x^2-x^3\right )+\left (30 x-19 x^2-6 e^x x^2\right ) \log (x)+3 x^2 \log ^2(x)}{75+10 x-83 x^2+30 x^3+e^{2 x} \left (3 x^2+3 x^3\right )+e^x \left (-30 x-17 x^2+19 x^3\right )+\left (30 x+17 x^2-19 x^3+e^x \left (-6 x^2-6 x^3\right )\right ) \log (x)+\left (3 x^2+3 x^3\right ) \log ^2(x)} \, dx=\log (x)+\log (1+x)-\log (x (1+x))-\log \left (5-3 x-e^x x+x \log (x)\right )+\log \left (15+11 x-3 e^x x-10 x^2-3 e^x x^2+3 x \log (x)+3 x^2 \log (x)\right ) \] Input:
Integrate[(100 - 105*x + 31*x^2 + 3*E^(2*x)*x^2 + E^x*(-30*x + 24*x^2 - x^ 3) + (30*x - 19*x^2 - 6*E^x*x^2)*Log[x] + 3*x^2*Log[x]^2)/(75 + 10*x - 83* x^2 + 30*x^3 + E^(2*x)*(3*x^2 + 3*x^3) + E^x*(-30*x - 17*x^2 + 19*x^3) + ( 30*x + 17*x^2 - 19*x^3 + E^x*(-6*x^2 - 6*x^3))*Log[x] + (3*x^2 + 3*x^3)*Lo g[x]^2),x]
Output:
Log[x] + Log[1 + x] - Log[x*(1 + x)] - Log[5 - 3*x - E^x*x + x*Log[x]] + L og[15 + 11*x - 3*E^x*x - 10*x^2 - 3*E^x*x^2 + 3*x*Log[x] + 3*x^2*Log[x]]
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.
\(\displaystyle \int \frac {3 e^{2 x} x^2+31 x^2+3 x^2 \log ^2(x)+\left (-6 e^x x^2-19 x^2+30 x\right ) \log (x)+e^x \left (-x^3+24 x^2-30 x\right )-105 x+100}{30 x^3-83 x^2+e^{2 x} \left (3 x^3+3 x^2\right )+e^x \left (19 x^3-17 x^2-30 x\right )+\left (3 x^3+3 x^2\right ) \log ^2(x)+\left (-19 x^3+17 x^2+e^x \left (-6 x^3-6 x^2\right )+30 x\right ) \log (x)+10 x+75} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {-3 x^2+x^2 \log (x)+4 x+5}{x \left (e^x x+3 x-x \log (x)-5\right )}+\frac {-10 x^4+3 x^4 \log (x)-2 x^3+6 x^3 \log (x)+41 x^2+3 x^2 \log (x)+42 x+15}{x (x+1) \left (3 e^x x^2+10 x^2-3 x^2 \log (x)+3 e^x x-11 x-3 x \log (x)-15\right )}+\frac {1}{x+1}\right )dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \left (-\frac {-3 x^2+x^2 \log (x)+4 x+5}{x \left (e^x x+3 x-x \log (x)-5\right )}+\frac {-10 x^4+3 x^4 \log (x)-2 x^3+6 x^3 \log (x)+41 x^2+3 x^2 \log (x)+42 x+15}{x (x+1) \left (3 e^x x^2+10 x^2-3 x^2 \log (x)+3 e^x x-11 x-3 x \log (x)-15\right )}+\frac {1}{x+1}\right )dx\) |
Input:
Int[(100 - 105*x + 31*x^2 + 3*E^(2*x)*x^2 + E^x*(-30*x + 24*x^2 - x^3) + ( 30*x - 19*x^2 - 6*E^x*x^2)*Log[x] + 3*x^2*Log[x]^2)/(75 + 10*x - 83*x^2 + 30*x^3 + E^(2*x)*(3*x^2 + 3*x^3) + E^x*(-30*x - 17*x^2 + 19*x^3) + (30*x + 17*x^2 - 19*x^3 + E^x*(-6*x^2 - 6*x^3))*Log[x] + (3*x^2 + 3*x^3)*Log[x]^2 ),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(51\) vs. \(2(24)=48\).
Time = 1.58 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.73
method | result | size |
parallelrisch | \(\ln \left (x^{2} \ln \left (x \right )-{\mathrm e}^{x} x^{2}-\frac {10 x^{2}}{3}+x \ln \left (x \right )-{\mathrm e}^{x} x +\frac {11 x}{3}+5\right )-\ln \left (x \ln \left (x \right )-{\mathrm e}^{x} x -3 x +5\right )\) | \(52\) |
risch | \(\ln \left (1+x \right )+\ln \left (\ln \left (x \right )-\frac {3 \,{\mathrm e}^{x} x^{2}+10 x^{2}+3 \,{\mathrm e}^{x} x -11 x -15}{3 x \left (1+x \right )}\right )-\ln \left (\ln \left (x \right )-\frac {{\mathrm e}^{x} x +3 x -5}{x}\right )\) | \(62\) |
Input:
int((3*x^2*ln(x)^2+(-6*exp(x)*x^2-19*x^2+30*x)*ln(x)+3*exp(x)^2*x^2+(-x^3+ 24*x^2-30*x)*exp(x)+31*x^2-105*x+100)/((3*x^3+3*x^2)*ln(x)^2+((-6*x^3-6*x^ 2)*exp(x)-19*x^3+17*x^2+30*x)*ln(x)+(3*x^3+3*x^2)*exp(x)^2+(19*x^3-17*x^2- 30*x)*exp(x)+30*x^3-83*x^2+10*x+75),x,method=_RETURNVERBOSE)
Output:
ln(x^2*ln(x)-exp(x)*x^2-10/3*x^2+x*ln(x)-exp(x)*x+11/3*x+5)-ln(x*ln(x)-exp (x)*x-3*x+5)
Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (24) = 48\).
Time = 0.08 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.17 \[ \int \frac {100-105 x+31 x^2+3 e^{2 x} x^2+e^x \left (-30 x+24 x^2-x^3\right )+\left (30 x-19 x^2-6 e^x x^2\right ) \log (x)+3 x^2 \log ^2(x)}{75+10 x-83 x^2+30 x^3+e^{2 x} \left (3 x^2+3 x^3\right )+e^x \left (-30 x-17 x^2+19 x^3\right )+\left (30 x+17 x^2-19 x^3+e^x \left (-6 x^2-6 x^3\right )\right ) \log (x)+\left (3 x^2+3 x^3\right ) \log ^2(x)} \, dx=\log \left (x + 1\right ) + \log \left (-\frac {10 \, x^{2} + 3 \, {\left (x^{2} + x\right )} e^{x} - 3 \, {\left (x^{2} + x\right )} \log \left (x\right ) - 11 \, x - 15}{x^{2} + x}\right ) - \log \left (-\frac {x e^{x} - x \log \left (x\right ) + 3 \, x - 5}{x}\right ) \] Input:
integrate((3*x^2*log(x)^2+(-6*exp(x)*x^2-19*x^2+30*x)*log(x)+3*exp(x)^2*x^ 2+(-x^3+24*x^2-30*x)*exp(x)+31*x^2-105*x+100)/((3*x^3+3*x^2)*log(x)^2+((-6 *x^3-6*x^2)*exp(x)-19*x^3+17*x^2+30*x)*log(x)+(3*x^3+3*x^2)*exp(x)^2+(19*x ^3-17*x^2-30*x)*exp(x)+30*x^3-83*x^2+10*x+75),x, algorithm="fricas")
Output:
log(x + 1) + log(-(10*x^2 + 3*(x^2 + x)*e^x - 3*(x^2 + x)*log(x) - 11*x - 15)/(x^2 + x)) - log(-(x*e^x - x*log(x) + 3*x - 5)/x)
Exception generated. \[ \int \frac {100-105 x+31 x^2+3 e^{2 x} x^2+e^x \left (-30 x+24 x^2-x^3\right )+\left (30 x-19 x^2-6 e^x x^2\right ) \log (x)+3 x^2 \log ^2(x)}{75+10 x-83 x^2+30 x^3+e^{2 x} \left (3 x^2+3 x^3\right )+e^x \left (-30 x-17 x^2+19 x^3\right )+\left (30 x+17 x^2-19 x^3+e^x \left (-6 x^2-6 x^3\right )\right ) \log (x)+\left (3 x^2+3 x^3\right ) \log ^2(x)} \, dx=\text {Exception raised: PolynomialError} \] Input:
integrate((3*x**2*ln(x)**2+(-6*exp(x)*x**2-19*x**2+30*x)*ln(x)+3*exp(x)**2 *x**2+(-x**3+24*x**2-30*x)*exp(x)+31*x**2-105*x+100)/((3*x**3+3*x**2)*ln(x )**2+((-6*x**3-6*x**2)*exp(x)-19*x**3+17*x**2+30*x)*ln(x)+(3*x**3+3*x**2)* exp(x)**2+(19*x**3-17*x**2-30*x)*exp(x)+30*x**3-83*x**2+10*x+75),x)
Output:
Exception raised: PolynomialError >> 1/(3*x**4 + 6*x**3 + 3*x**2) contains an element of the set of generators.
Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (24) = 48\).
Time = 0.14 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.13 \[ \int \frac {100-105 x+31 x^2+3 e^{2 x} x^2+e^x \left (-30 x+24 x^2-x^3\right )+\left (30 x-19 x^2-6 e^x x^2\right ) \log (x)+3 x^2 \log ^2(x)}{75+10 x-83 x^2+30 x^3+e^{2 x} \left (3 x^2+3 x^3\right )+e^x \left (-30 x-17 x^2+19 x^3\right )+\left (30 x+17 x^2-19 x^3+e^x \left (-6 x^2-6 x^3\right )\right ) \log (x)+\left (3 x^2+3 x^3\right ) \log ^2(x)} \, dx=\log \left (x + 1\right ) + \log \left (\frac {10 \, x^{2} + 3 \, {\left (x^{2} + x\right )} e^{x} - 3 \, {\left (x^{2} + x\right )} \log \left (x\right ) - 11 \, x - 15}{3 \, {\left (x^{2} + x\right )}}\right ) - \log \left (\frac {x e^{x} - x \log \left (x\right ) + 3 \, x - 5}{x}\right ) \] Input:
integrate((3*x^2*log(x)^2+(-6*exp(x)*x^2-19*x^2+30*x)*log(x)+3*exp(x)^2*x^ 2+(-x^3+24*x^2-30*x)*exp(x)+31*x^2-105*x+100)/((3*x^3+3*x^2)*log(x)^2+((-6 *x^3-6*x^2)*exp(x)-19*x^3+17*x^2+30*x)*log(x)+(3*x^3+3*x^2)*exp(x)^2+(19*x ^3-17*x^2-30*x)*exp(x)+30*x^3-83*x^2+10*x+75),x, algorithm="maxima")
Output:
log(x + 1) + log(1/3*(10*x^2 + 3*(x^2 + x)*e^x - 3*(x^2 + x)*log(x) - 11*x - 15)/(x^2 + x)) - log((x*e^x - x*log(x) + 3*x - 5)/x)
Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (24) = 48\).
Time = 0.33 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.77 \[ \int \frac {100-105 x+31 x^2+3 e^{2 x} x^2+e^x \left (-30 x+24 x^2-x^3\right )+\left (30 x-19 x^2-6 e^x x^2\right ) \log (x)+3 x^2 \log ^2(x)}{75+10 x-83 x^2+30 x^3+e^{2 x} \left (3 x^2+3 x^3\right )+e^x \left (-30 x-17 x^2+19 x^3\right )+\left (30 x+17 x^2-19 x^3+e^x \left (-6 x^2-6 x^3\right )\right ) \log (x)+\left (3 x^2+3 x^3\right ) \log ^2(x)} \, dx=\log \left (-3 \, x^{2} e^{x} + 3 \, x^{2} \log \left (x\right ) - 10 \, x^{2} - 3 \, x e^{x} + 3 \, x \log \left (x\right ) + 11 \, x + 15\right ) - \log \left (-x e^{x} + x \log \left (x\right ) - 3 \, x + 5\right ) \] Input:
integrate((3*x^2*log(x)^2+(-6*exp(x)*x^2-19*x^2+30*x)*log(x)+3*exp(x)^2*x^ 2+(-x^3+24*x^2-30*x)*exp(x)+31*x^2-105*x+100)/((3*x^3+3*x^2)*log(x)^2+((-6 *x^3-6*x^2)*exp(x)-19*x^3+17*x^2+30*x)*log(x)+(3*x^3+3*x^2)*exp(x)^2+(19*x ^3-17*x^2-30*x)*exp(x)+30*x^3-83*x^2+10*x+75),x, algorithm="giac")
Output:
log(-3*x^2*e^x + 3*x^2*log(x) - 10*x^2 - 3*x*e^x + 3*x*log(x) + 11*x + 15) - log(-x*e^x + x*log(x) - 3*x + 5)
Timed out. \[ \int \frac {100-105 x+31 x^2+3 e^{2 x} x^2+e^x \left (-30 x+24 x^2-x^3\right )+\left (30 x-19 x^2-6 e^x x^2\right ) \log (x)+3 x^2 \log ^2(x)}{75+10 x-83 x^2+30 x^3+e^{2 x} \left (3 x^2+3 x^3\right )+e^x \left (-30 x-17 x^2+19 x^3\right )+\left (30 x+17 x^2-19 x^3+e^x \left (-6 x^2-6 x^3\right )\right ) \log (x)+\left (3 x^2+3 x^3\right ) \log ^2(x)} \, dx=\int \frac {3\,x^2\,{\mathrm {e}}^{2\,x}-105\,x+3\,x^2\,{\ln \left (x\right )}^2-{\mathrm {e}}^x\,\left (x^3-24\,x^2+30\,x\right )-\ln \left (x\right )\,\left (6\,x^2\,{\mathrm {e}}^x-30\,x+19\,x^2\right )+31\,x^2+100}{10\,x+{\mathrm {e}}^{2\,x}\,\left (3\,x^3+3\,x^2\right )+{\ln \left (x\right )}^2\,\left (3\,x^3+3\,x^2\right )-83\,x^2+30\,x^3-{\mathrm {e}}^x\,\left (-19\,x^3+17\,x^2+30\,x\right )+\ln \left (x\right )\,\left (30\,x-{\mathrm {e}}^x\,\left (6\,x^3+6\,x^2\right )+17\,x^2-19\,x^3\right )+75} \,d x \] Input:
int((3*x^2*exp(2*x) - 105*x + 3*x^2*log(x)^2 - exp(x)*(30*x - 24*x^2 + x^3 ) - log(x)*(6*x^2*exp(x) - 30*x + 19*x^2) + 31*x^2 + 100)/(10*x + exp(2*x) *(3*x^2 + 3*x^3) + log(x)^2*(3*x^2 + 3*x^3) - 83*x^2 + 30*x^3 - exp(x)*(30 *x + 17*x^2 - 19*x^3) + log(x)*(30*x - exp(x)*(6*x^2 + 6*x^3) + 17*x^2 - 1 9*x^3) + 75),x)
Output:
int((3*x^2*exp(2*x) - 105*x + 3*x^2*log(x)^2 - exp(x)*(30*x - 24*x^2 + x^3 ) - log(x)*(6*x^2*exp(x) - 30*x + 19*x^2) + 31*x^2 + 100)/(10*x + exp(2*x) *(3*x^2 + 3*x^3) + log(x)^2*(3*x^2 + 3*x^3) - 83*x^2 + 30*x^3 - exp(x)*(30 *x + 17*x^2 - 19*x^3) + log(x)*(30*x - exp(x)*(6*x^2 + 6*x^3) + 17*x^2 - 1 9*x^3) + 75), x)
Time = 0.26 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.87 \[ \int \frac {100-105 x+31 x^2+3 e^{2 x} x^2+e^x \left (-30 x+24 x^2-x^3\right )+\left (30 x-19 x^2-6 e^x x^2\right ) \log (x)+3 x^2 \log ^2(x)}{75+10 x-83 x^2+30 x^3+e^{2 x} \left (3 x^2+3 x^3\right )+e^x \left (-30 x-17 x^2+19 x^3\right )+\left (30 x+17 x^2-19 x^3+e^x \left (-6 x^2-6 x^3\right )\right ) \log (x)+\left (3 x^2+3 x^3\right ) \log ^2(x)} \, dx=-\mathrm {log}\left (e^{x} x -\mathrm {log}\left (x \right ) x +3 x -5\right )+\mathrm {log}\left (3 e^{x} x^{2}+3 e^{x} x -3 \,\mathrm {log}\left (x \right ) x^{2}-3 \,\mathrm {log}\left (x \right ) x +10 x^{2}-11 x -15\right ) \] Input:
int((3*x^2*log(x)^2+(-6*exp(x)*x^2-19*x^2+30*x)*log(x)+3*exp(x)^2*x^2+(-x^ 3+24*x^2-30*x)*exp(x)+31*x^2-105*x+100)/((3*x^3+3*x^2)*log(x)^2+((-6*x^3-6 *x^2)*exp(x)-19*x^3+17*x^2+30*x)*log(x)+(3*x^3+3*x^2)*exp(x)^2+(19*x^3-17* x^2-30*x)*exp(x)+30*x^3-83*x^2+10*x+75),x)
Output:
- log(e**x*x - log(x)*x + 3*x - 5) + log(3*e**x*x**2 + 3*e**x*x - 3*log(x )*x**2 - 3*log(x)*x + 10*x**2 - 11*x - 15)