Integrand size = 146, antiderivative size = 30 \[ \int \frac {-9+288 x^2+\left (6-192 x^2\right ) \log (x)+\left (-1+32 x^2\right ) \log ^2(x)+e^{\frac {-15-3 \log (5)+5 \log (x)}{-3+\log (x)}} \left (-144 x^2-24 x^2 \log (5)+96 x^2 \log (x)-16 x^2 \log ^2(x)\right )+e^{\frac {2 (-15-3 \log (5)+5 \log (x))}{-3+\log (x)}} \left (18 x^2+6 x^2 \log (5)-12 x^2 \log (x)+2 x^2 \log ^2(x)\right )}{9 x-6 x \log (x)+x \log ^2(x)} \, dx=7+\left (-4 x+e^{5+\frac {3 \log (5)}{3-\log (x)}} x\right )^2-\log (x) \] Output:
7+(x*exp(5+3*ln(5)/(3-ln(x)))-4*x)^2-ln(x)
\[ \int \frac {-9+288 x^2+\left (6-192 x^2\right ) \log (x)+\left (-1+32 x^2\right ) \log ^2(x)+e^{\frac {-15-3 \log (5)+5 \log (x)}{-3+\log (x)}} \left (-144 x^2-24 x^2 \log (5)+96 x^2 \log (x)-16 x^2 \log ^2(x)\right )+e^{\frac {2 (-15-3 \log (5)+5 \log (x))}{-3+\log (x)}} \left (18 x^2+6 x^2 \log (5)-12 x^2 \log (x)+2 x^2 \log ^2(x)\right )}{9 x-6 x \log (x)+x \log ^2(x)} \, dx=\int \frac {-9+288 x^2+\left (6-192 x^2\right ) \log (x)+\left (-1+32 x^2\right ) \log ^2(x)+e^{\frac {-15-3 \log (5)+5 \log (x)}{-3+\log (x)}} \left (-144 x^2-24 x^2 \log (5)+96 x^2 \log (x)-16 x^2 \log ^2(x)\right )+e^{\frac {2 (-15-3 \log (5)+5 \log (x))}{-3+\log (x)}} \left (18 x^2+6 x^2 \log (5)-12 x^2 \log (x)+2 x^2 \log ^2(x)\right )}{9 x-6 x \log (x)+x \log ^2(x)} \, dx \] Input:
Integrate[(-9 + 288*x^2 + (6 - 192*x^2)*Log[x] + (-1 + 32*x^2)*Log[x]^2 + E^((-15 - 3*Log[5] + 5*Log[x])/(-3 + Log[x]))*(-144*x^2 - 24*x^2*Log[5] + 96*x^2*Log[x] - 16*x^2*Log[x]^2) + E^((2*(-15 - 3*Log[5] + 5*Log[x]))/(-3 + Log[x]))*(18*x^2 + 6*x^2*Log[5] - 12*x^2*Log[x] + 2*x^2*Log[x]^2))/(9*x - 6*x*Log[x] + x*Log[x]^2),x]
Output:
Integrate[(-9 + 288*x^2 + (6 - 192*x^2)*Log[x] + (-1 + 32*x^2)*Log[x]^2 + E^((-15 - 3*Log[5] + 5*Log[x])/(-3 + Log[x]))*(-144*x^2 - 24*x^2*Log[5] + 96*x^2*Log[x] - 16*x^2*Log[x]^2) + E^((2*(-15 - 3*Log[5] + 5*Log[x]))/(-3 + Log[x]))*(18*x^2 + 6*x^2*Log[5] - 12*x^2*Log[x] + 2*x^2*Log[x]^2))/(9*x - 6*x*Log[x] + x*Log[x]^2), 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.
| \(\displaystyle \int \frac {288 x^2+\left (32 x^2-1\right ) \log ^2(x)+e^{\frac {5 \log (x)-15-3 \log (5)}{\log (x)-3}} \left (-144 x^2-16 x^2 \log ^2(x)+96 x^2 \log (x)-24 x^2 \log (5)\right )+e^{\frac {2 (5 \log (x)-15-3 \log (5))}{\log (x)-3}} \left (18 x^2+2 x^2 \log ^2(x)-12 x^2 \log (x)+6 x^2 \log (5)\right )+\left (6-192 x^2\right ) \log (x)-9}{9 x+x \log ^2(x)-6 x \log (x)} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {288 x^2+\left (32 x^2-1\right ) \log ^2(x)+e^{\frac {5 \log (x)-15-3 \log (5)}{\log (x)-3}} \left (-144 x^2-16 x^2 \log ^2(x)+96 x^2 \log (x)-24 x^2 \log (5)\right )+e^{\frac {2 (5 \log (x)-15-3 \log (5))}{\log (x)-3}} \left (18 x^2+2 x^2 \log ^2(x)-12 x^2 \log (x)+6 x^2 \log (5)\right )+\left (6-192 x^2\right ) \log (x)-9}{x (3-\log (x))^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {8\ 5^{-\frac {3}{\log (x)-3}} e^{-\frac {15}{\log (x)-3}} \left (-2 \log ^2(x)+12 \log (x)-18 \left (1+\frac {\log (5)}{6}\right )\right ) x^{\frac {\log (x)+2}{\log (x)-3}}}{(3-\log (x))^2}+\frac {2\ 5^{-\frac {6}{\log (x)-3}} e^{-\frac {30}{\log (x)-3}} \left (\log ^2(x)-6 \log (x)+9 \left (1+\frac {\log (5)}{3}\right )\right ) x^{\frac {\log (x)+7}{\log (x)-3}}}{(3-\log (x))^2}+\frac {\left (32 x^2-1\right ) \log ^2(x)}{x (\log (x)-3)^2}-\frac {6 \left (32 x^2-1\right ) \log (x)}{x (\log (x)-3)^2}+\frac {288 x}{(\log (x)-3)^2}-\frac {9}{x (\log (x)-3)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -16 \int 5^{-\frac {3}{\log (x)-3}} e^{-\frac {15}{\log (x)-3}} x^{\frac {\log (x)+2}{\log (x)-3}}dx+2 \int 5^{-\frac {6}{\log (x)-3}} e^{-\frac {30}{\log (x)-3}} x^{\frac {\log (x)+7}{\log (x)-3}}dx-24 \log (5) \int \frac {5^{-\frac {3}{\log (x)-3}} e^{-\frac {15}{\log (x)-3}} x^{\frac {\log (x)+2}{\log (x)-3}}}{(\log (x)-3)^2}dx+6 \log (5) \int \frac {5^{-\frac {6}{\log (x)-3}} e^{-\frac {30}{\log (x)-3}} x^{\frac {\log (x)+7}{\log (x)-3}}}{(\log (x)-3)^2}dx+16 x^2-\log (x)\) |
Input:
Int[(-9 + 288*x^2 + (6 - 192*x^2)*Log[x] + (-1 + 32*x^2)*Log[x]^2 + E^((-1 5 - 3*Log[5] + 5*Log[x])/(-3 + Log[x]))*(-144*x^2 - 24*x^2*Log[5] + 96*x^2 *Log[x] - 16*x^2*Log[x]^2) + E^((2*(-15 - 3*Log[5] + 5*Log[x]))/(-3 + Log[ x]))*(18*x^2 + 6*x^2*Log[5] - 12*x^2*Log[x] + 2*x^2*Log[x]^2))/(9*x - 6*x* Log[x] + x*Log[x]^2),x]
Output:
$Aborted
Time = 2.17 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.40
| method | result | size |
| risch | \(16 x^{2}-\ln \left (x \right )+{\mathrm e}^{10} 125^{-\frac {2}{\ln \left (x \right )-3}} x^{2}-8 \,{\mathrm e}^{5} \left (\frac {1}{125}\right )^{\frac {1}{\ln \left (x \right )-3}} x^{2}\) | \(42\) |
| parallelrisch | \(x^{2} {\mathrm e}^{\frac {10 \ln \left (x \right )+2 \ln \left (\frac {1}{125}\right )-30}{\ln \left (x \right )-3}}-8 x^{2} {\mathrm e}^{\frac {5 \ln \left (x \right )-3 \ln \left (5\right )-15}{\ln \left (x \right )-3}}+16 x^{2}-\ln \left (x \right )-12\) | \(59\) |
| default | \(16 x^{2}-\ln \left (x \right )+\frac {24 x^{2} {\mathrm e}^{\frac {5 \ln \left (x \right )-3 \ln \left (5\right )-15}{\ln \left (x \right )-3}}-8 \ln \left (x \right ) {\mathrm e}^{\frac {5 \ln \left (x \right )-3 \ln \left (5\right )-15}{\ln \left (x \right )-3}} x^{2}}{\ln \left (x \right )-3}+\frac {\ln \left (x \right ) {\mathrm e}^{\frac {10 \ln \left (x \right )-6 \ln \left (5\right )-30}{\ln \left (x \right )-3}} x^{2}-3 \,{\mathrm e}^{\frac {10 \ln \left (x \right )-6 \ln \left (5\right )-30}{\ln \left (x \right )-3}} x^{2}}{\ln \left (x \right )-3}\) | \(124\) |
Input:
int(((2*x^2*ln(x)^2-12*x^2*ln(x)+6*x^2*ln(5)+18*x^2)*exp((5*ln(x)-3*ln(5)- 15)/(ln(x)-3))^2+(-16*x^2*ln(x)^2+96*x^2*ln(x)-24*x^2*ln(5)-144*x^2)*exp(( 5*ln(x)-3*ln(5)-15)/(ln(x)-3))+(32*x^2-1)*ln(x)^2+(-192*x^2+6)*ln(x)+288*x ^2-9)/(x*ln(x)^2-6*x*ln(x)+9*x),x,method=_RETURNVERBOSE)
Output:
16*x^2-ln(x)+exp(10)*((1/125)^(1/(ln(x)-3)))^2*x^2-8*exp(5)*(1/125)^(1/(ln (x)-3))*x^2
Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (27) = 54\).
Time = 0.10 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.90 \[ \int \frac {-9+288 x^2+\left (6-192 x^2\right ) \log (x)+\left (-1+32 x^2\right ) \log ^2(x)+e^{\frac {-15-3 \log (5)+5 \log (x)}{-3+\log (x)}} \left (-144 x^2-24 x^2 \log (5)+96 x^2 \log (x)-16 x^2 \log ^2(x)\right )+e^{\frac {2 (-15-3 \log (5)+5 \log (x))}{-3+\log (x)}} \left (18 x^2+6 x^2 \log (5)-12 x^2 \log (x)+2 x^2 \log ^2(x)\right )}{9 x-6 x \log (x)+x \log ^2(x)} \, dx=-8 \, x^{2} e^{\left (-\frac {3 \, \log \left (5\right ) - 5 \, \log \left (x\right ) + 15}{\log \left (x\right ) - 3}\right )} + x^{2} e^{\left (-\frac {2 \, {\left (3 \, \log \left (5\right ) - 5 \, \log \left (x\right ) + 15\right )}}{\log \left (x\right ) - 3}\right )} + 16 \, x^{2} - \log \left (x\right ) \] Input:
integrate(((2*x^2*log(x)^2-12*x^2*log(x)+6*x^2*log(5)+18*x^2)*exp((5*log(x )-3*log(5)-15)/(log(x)-3))^2+(-16*x^2*log(x)^2+96*x^2*log(x)-24*x^2*log(5) -144*x^2)*exp((5*log(x)-3*log(5)-15)/(log(x)-3))+(32*x^2-1)*log(x)^2+(-192 *x^2+6)*log(x)+288*x^2-9)/(x*log(x)^2-6*x*log(x)+9*x),x, algorithm="fricas ")
Output:
-8*x^2*e^(-(3*log(5) - 5*log(x) + 15)/(log(x) - 3)) + x^2*e^(-2*(3*log(5) - 5*log(x) + 15)/(log(x) - 3)) + 16*x^2 - log(x)
Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (24) = 48\).
Time = 1.62 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.77 \[ \int \frac {-9+288 x^2+\left (6-192 x^2\right ) \log (x)+\left (-1+32 x^2\right ) \log ^2(x)+e^{\frac {-15-3 \log (5)+5 \log (x)}{-3+\log (x)}} \left (-144 x^2-24 x^2 \log (5)+96 x^2 \log (x)-16 x^2 \log ^2(x)\right )+e^{\frac {2 (-15-3 \log (5)+5 \log (x))}{-3+\log (x)}} \left (18 x^2+6 x^2 \log (5)-12 x^2 \log (x)+2 x^2 \log ^2(x)\right )}{9 x-6 x \log (x)+x \log ^2(x)} \, dx=x^{2} e^{\frac {2 \cdot \left (5 \log {\left (x \right )} - 15 - 3 \log {\left (5 \right )}\right )}{\log {\left (x \right )} - 3}} - 8 x^{2} e^{\frac {5 \log {\left (x \right )} - 15 - 3 \log {\left (5 \right )}}{\log {\left (x \right )} - 3}} + 16 x^{2} - \log {\left (x \right )} \] Input:
integrate(((2*x**2*ln(x)**2-12*x**2*ln(x)+6*x**2*ln(5)+18*x**2)*exp((5*ln( x)-3*ln(5)-15)/(ln(x)-3))**2+(-16*x**2*ln(x)**2+96*x**2*ln(x)-24*x**2*ln(5 )-144*x**2)*exp((5*ln(x)-3*ln(5)-15)/(ln(x)-3))+(32*x**2-1)*ln(x)**2+(-192 *x**2+6)*ln(x)+288*x**2-9)/(x*ln(x)**2-6*x*ln(x)+9*x),x)
Output:
x**2*exp(2*(5*log(x) - 15 - 3*log(5))/(log(x) - 3)) - 8*x**2*exp((5*log(x) - 15 - 3*log(5))/(log(x) - 3)) + 16*x**2 - log(x)
\[ \int \frac {-9+288 x^2+\left (6-192 x^2\right ) \log (x)+\left (-1+32 x^2\right ) \log ^2(x)+e^{\frac {-15-3 \log (5)+5 \log (x)}{-3+\log (x)}} \left (-144 x^2-24 x^2 \log (5)+96 x^2 \log (x)-16 x^2 \log ^2(x)\right )+e^{\frac {2 (-15-3 \log (5)+5 \log (x))}{-3+\log (x)}} \left (18 x^2+6 x^2 \log (5)-12 x^2 \log (x)+2 x^2 \log ^2(x)\right )}{9 x-6 x \log (x)+x \log ^2(x)} \, dx=\int { \frac {{\left (32 \, x^{2} - 1\right )} \log \left (x\right )^{2} + 288 \, x^{2} - 8 \, {\left (2 \, x^{2} \log \left (x\right )^{2} + 3 \, x^{2} \log \left (5\right ) - 12 \, x^{2} \log \left (x\right ) + 18 \, x^{2}\right )} e^{\left (-\frac {3 \, \log \left (5\right ) - 5 \, \log \left (x\right ) + 15}{\log \left (x\right ) - 3}\right )} + 2 \, {\left (x^{2} \log \left (x\right )^{2} + 3 \, x^{2} \log \left (5\right ) - 6 \, x^{2} \log \left (x\right ) + 9 \, x^{2}\right )} e^{\left (-\frac {2 \, {\left (3 \, \log \left (5\right ) - 5 \, \log \left (x\right ) + 15\right )}}{\log \left (x\right ) - 3}\right )} - 6 \, {\left (32 \, x^{2} - 1\right )} \log \left (x\right ) - 9}{x \log \left (x\right )^{2} - 6 \, x \log \left (x\right ) + 9 \, x} \,d x } \] Input:
integrate(((2*x^2*log(x)^2-12*x^2*log(x)+6*x^2*log(5)+18*x^2)*exp((5*log(x )-3*log(5)-15)/(log(x)-3))^2+(-16*x^2*log(x)^2+96*x^2*log(x)-24*x^2*log(5) -144*x^2)*exp((5*log(x)-3*log(5)-15)/(log(x)-3))+(32*x^2-1)*log(x)^2+(-192 *x^2+6)*log(x)+288*x^2-9)/(x*log(x)^2-6*x*log(x)+9*x),x, algorithm="maxima ")
Output:
(16*x^2*log(x) - 48*x^2 - 9)/(log(x) - 3) + 9/(log(x) - 3) - integrate(8*( 2*x*log(x)^2 + 3*x*(log(5) + 6) - 12*x*log(x))*e^(-3*log(5)/(log(x) - 3) + 5*log(x)/(log(x) - 3) - 15/(log(x) - 3))/(log(x)^2 - 6*log(x) + 9), x) + integrate(2*(x*log(x)^2 + 3*x*(log(5) + 3) - 6*x*log(x))*e^(-6*log(5)/(log (x) - 3) + 10*log(x)/(log(x) - 3) - 30/(log(x) - 3))/(log(x)^2 - 6*log(x) + 9), x) - log(x)
Time = 0.81 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.67 \[ \int \frac {-9+288 x^2+\left (6-192 x^2\right ) \log (x)+\left (-1+32 x^2\right ) \log ^2(x)+e^{\frac {-15-3 \log (5)+5 \log (x)}{-3+\log (x)}} \left (-144 x^2-24 x^2 \log (5)+96 x^2 \log (x)-16 x^2 \log ^2(x)\right )+e^{\frac {2 (-15-3 \log (5)+5 \log (x))}{-3+\log (x)}} \left (18 x^2+6 x^2 \log (5)-12 x^2 \log (x)+2 x^2 \log ^2(x)\right )}{9 x-6 x \log (x)+x \log ^2(x)} \, dx=-40 \, x^{2} e^{\left (-\frac {\log \left (5\right ) \log \left (x\right )}{\log \left (x\right ) - 3} + 5\right )} + 25 \, x^{2} e^{\left (-\frac {2 \, \log \left (5\right ) \log \left (x\right )}{\log \left (x\right ) - 3} + 10\right )} + 16 \, x^{2} - \log \left (x\right ) \] Input:
integrate(((2*x^2*log(x)^2-12*x^2*log(x)+6*x^2*log(5)+18*x^2)*exp((5*log(x )-3*log(5)-15)/(log(x)-3))^2+(-16*x^2*log(x)^2+96*x^2*log(x)-24*x^2*log(5) -144*x^2)*exp((5*log(x)-3*log(5)-15)/(log(x)-3))+(32*x^2-1)*log(x)^2+(-192 *x^2+6)*log(x)+288*x^2-9)/(x*log(x)^2-6*x*log(x)+9*x),x, algorithm="giac")
Output:
-40*x^2*e^(-log(5)*log(x)/(log(x) - 3) + 5) + 25*x^2*e^(-2*log(5)*log(x)/( log(x) - 3) + 10) + 16*x^2 - log(x)
Timed out. \[ \int \frac {-9+288 x^2+\left (6-192 x^2\right ) \log (x)+\left (-1+32 x^2\right ) \log ^2(x)+e^{\frac {-15-3 \log (5)+5 \log (x)}{-3+\log (x)}} \left (-144 x^2-24 x^2 \log (5)+96 x^2 \log (x)-16 x^2 \log ^2(x)\right )+e^{\frac {2 (-15-3 \log (5)+5 \log (x))}{-3+\log (x)}} \left (18 x^2+6 x^2 \log (5)-12 x^2 \log (x)+2 x^2 \log ^2(x)\right )}{9 x-6 x \log (x)+x \log ^2(x)} \, dx=\int \frac {{\mathrm {e}}^{-\frac {2\,\left (3\,\ln \left (5\right )-5\,\ln \left (x\right )+15\right )}{\ln \left (x\right )-3}}\,\left (2\,x^2\,{\ln \left (x\right )}^2-12\,x^2\,\ln \left (x\right )+6\,x^2\,\ln \left (5\right )+18\,x^2\right )-{\mathrm {e}}^{-\frac {3\,\ln \left (5\right )-5\,\ln \left (x\right )+15}{\ln \left (x\right )-3}}\,\left (16\,x^2\,{\ln \left (x\right )}^2-96\,x^2\,\ln \left (x\right )+24\,x^2\,\ln \left (5\right )+144\,x^2\right )+{\ln \left (x\right )}^2\,\left (32\,x^2-1\right )+288\,x^2-\ln \left (x\right )\,\left (192\,x^2-6\right )-9}{x\,{\ln \left (x\right )}^2-6\,x\,\ln \left (x\right )+9\,x} \,d x \] Input:
int((exp(-(2*(3*log(5) - 5*log(x) + 15))/(log(x) - 3))*(2*x^2*log(x)^2 - 1 2*x^2*log(x) + 6*x^2*log(5) + 18*x^2) - exp(-(3*log(5) - 5*log(x) + 15)/(l og(x) - 3))*(16*x^2*log(x)^2 - 96*x^2*log(x) + 24*x^2*log(5) + 144*x^2) + log(x)^2*(32*x^2 - 1) + 288*x^2 - log(x)*(192*x^2 - 6) - 9)/(9*x + x*log(x )^2 - 6*x*log(x)),x)
Output:
int((exp(-(2*(3*log(5) - 5*log(x) + 15))/(log(x) - 3))*(2*x^2*log(x)^2 - 1 2*x^2*log(x) + 6*x^2*log(5) + 18*x^2) - exp(-(3*log(5) - 5*log(x) + 15)/(l og(x) - 3))*(16*x^2*log(x)^2 - 96*x^2*log(x) + 24*x^2*log(5) + 144*x^2) + log(x)^2*(32*x^2 - 1) + 288*x^2 - log(x)*(192*x^2 - 6) - 9)/(9*x + x*log(x )^2 - 6*x*log(x)), x)
Time = 0.16 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.53 \[ \int \frac {-9+288 x^2+\left (6-192 x^2\right ) \log (x)+\left (-1+32 x^2\right ) \log ^2(x)+e^{\frac {-15-3 \log (5)+5 \log (x)}{-3+\log (x)}} \left (-144 x^2-24 x^2 \log (5)+96 x^2 \log (x)-16 x^2 \log ^2(x)\right )+e^{\frac {2 (-15-3 \log (5)+5 \log (x))}{-3+\log (x)}} \left (18 x^2+6 x^2 \log (5)-12 x^2 \log (x)+2 x^2 \log ^2(x)\right )}{9 x-6 x \log (x)+x \log ^2(x)} \, dx=\frac {-e^{\frac {6 \,\mathrm {log}\left (5\right )}{\mathrm {log}\left (x \right )-3}} \mathrm {log}\left (x \right )+16 e^{\frac {6 \,\mathrm {log}\left (5\right )}{\mathrm {log}\left (x \right )-3}} x^{2}-8 e^{\frac {3 \,\mathrm {log}\left (5\right )}{\mathrm {log}\left (x \right )-3}} e^{5} x^{2}+e^{10} x^{2}}{e^{\frac {6 \,\mathrm {log}\left (5\right )}{\mathrm {log}\left (x \right )-3}}} \] Input:
int(((2*x^2*log(x)^2-12*x^2*log(x)+6*x^2*log(5)+18*x^2)*exp((5*log(x)-3*lo g(5)-15)/(log(x)-3))^2+(-16*x^2*log(x)^2+96*x^2*log(x)-24*x^2*log(5)-144*x ^2)*exp((5*log(x)-3*log(5)-15)/(log(x)-3))+(32*x^2-1)*log(x)^2+(-192*x^2+6 )*log(x)+288*x^2-9)/(x*log(x)^2-6*x*log(x)+9*x),x)
Output:
( - e**((6*log(5))/(log(x) - 3))*log(x) + 16*e**((6*log(5))/(log(x) - 3))* x**2 - 8*e**((3*log(5))/(log(x) - 3))*e**5*x**2 + e**10*x**2)/e**((6*log(5 ))/(log(x) - 3))