Integrand size = 138, antiderivative size = 30 \[ \int \frac {e^{16} \left (-36-12 x^2+6 x^3\right )+e^{16} \left (54+18 x^2\right ) \log (x)+\left (e^{16} \left (-27+3 x^2-9 x^3\right )-18 e^{16} x^2 \log (x)\right ) \log \left (x^2\right )}{36 x+24 x^3-12 x^4+4 x^5-4 x^6+x^7+\left (-108 x-72 x^3+18 x^4-12 x^5+6 x^6\right ) \log (x)+\left (81 x+54 x^3+9 x^5\right ) \log ^2(x)} \, dx=\frac {e^{16} \log \left (x^2\right )}{-x+\left (3+x^2\right ) \left (\frac {1}{3} (-2+x)+\log (x)\right )} \] Output:
exp(16)*ln(x^2)/((ln(x)-2/3+1/3*x)*(x^2+3)-x)
Time = 0.44 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {e^{16} \left (-36-12 x^2+6 x^3\right )+e^{16} \left (54+18 x^2\right ) \log (x)+\left (e^{16} \left (-27+3 x^2-9 x^3\right )-18 e^{16} x^2 \log (x)\right ) \log \left (x^2\right )}{36 x+24 x^3-12 x^4+4 x^5-4 x^6+x^7+\left (-108 x-72 x^3+18 x^4-12 x^5+6 x^6\right ) \log (x)+\left (81 x+54 x^3+9 x^5\right ) \log ^2(x)} \, dx=\frac {3 e^{16} \log \left (x^2\right )}{-6-2 x^2+x^3+3 \left (3+x^2\right ) \log (x)} \] Input:
Integrate[(E^16*(-36 - 12*x^2 + 6*x^3) + E^16*(54 + 18*x^2)*Log[x] + (E^16 *(-27 + 3*x^2 - 9*x^3) - 18*E^16*x^2*Log[x])*Log[x^2])/(36*x + 24*x^3 - 12 *x^4 + 4*x^5 - 4*x^6 + x^7 + (-108*x - 72*x^3 + 18*x^4 - 12*x^5 + 6*x^6)*L og[x] + (81*x + 54*x^3 + 9*x^5)*Log[x]^2),x]
Output:
(3*E^16*Log[x^2])/(-6 - 2*x^2 + x^3 + 3*(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.
| \(\displaystyle \int \frac {e^{16} \left (18 x^2+54\right ) \log (x)+e^{16} \left (6 x^3-12 x^2-36\right )+\left (e^{16} \left (-9 x^3+3 x^2-27\right )-18 e^{16} x^2 \log (x)\right ) \log \left (x^2\right )}{x^7-4 x^6+4 x^5-12 x^4+24 x^3+\left (9 x^5+54 x^3+81 x\right ) \log ^2(x)+\left (6 x^6-12 x^5+18 x^4-72 x^3-108 x\right ) \log (x)+36 x} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {3 e^{16} \left (2 x^3-4 x^2-6 \log (x) \left (-x^2+x^2 \log \left (x^2\right )-3\right )-\left (3 x^3-x^2+9\right ) \log \left (x^2\right )-12\right )}{x \left (-x^3+2 x^2-3 \left (x^2+3\right ) \log (x)+6\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 3 e^{16} \int -\frac {-2 x^3+4 x^2+\left (3 x^3-x^2+9\right ) \log \left (x^2\right )-6 \log (x) \left (-\log \left (x^2\right ) x^2+x^2+3\right )+12}{x \left (-x^3+2 x^2-3 \left (x^2+3\right ) \log (x)+6\right )^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -3 e^{16} \int \frac {-2 x^3+4 x^2+\left (3 x^3-x^2+9\right ) \log \left (x^2\right )-6 \log (x) \left (-\log \left (x^2\right ) x^2+x^2+3\right )+12}{x \left (-x^3+2 x^2-3 \left (x^2+3\right ) \log (x)+6\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -3 e^{16} \int \left (-\frac {2 x^2}{\left (x^3+3 \log (x) x^2-2 x^2+9 \log (x)-6\right )^2}-\frac {6 \log (x) x}{\left (x^3+3 \log (x) x^2-2 x^2+9 \log (x)-6\right )^2}+\frac {4 x}{\left (x^3+3 \log (x) x^2-2 x^2+9 \log (x)-6\right )^2}+\frac {\left (3 x^3+6 \log (x) x^2-x^2+9\right ) \log \left (x^2\right )}{\left (x^3+3 \log (x) x^2-2 x^2+9 \log (x)-6\right )^2 x}-\frac {18 \log (x)}{\left (x^3+3 \log (x) x^2-2 x^2+9 \log (x)-6\right )^2 x}+\frac {12}{\left (x^3+3 \log (x) x^2-2 x^2+9 \log (x)-6\right )^2 x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -3 e^{16} \left (-2 \int \frac {1}{x \left (x^3+3 \log (x) x^2-2 x^2+9 \log (x)-6\right )}dx+9 \int \frac {\log \left (x^2\right )}{x \left (x^3+3 \log (x) x^2-2 x^2+9 \log (x)-6\right )^2}dx-\int \frac {x \log \left (x^2\right )}{\left (x^3+3 \log (x) x^2-2 x^2+9 \log (x)-6\right )^2}dx+3 \int \frac {x^2 \log \left (x^2\right )}{\left (x^3+3 \log (x) x^2-2 x^2+9 \log (x)-6\right )^2}dx+6 \int \frac {x \log (x) \log \left (x^2\right )}{\left (x^3+3 \log (x) x^2-2 x^2+9 \log (x)-6\right )^2}dx\right )\) |
Input:
Int[(E^16*(-36 - 12*x^2 + 6*x^3) + E^16*(54 + 18*x^2)*Log[x] + (E^16*(-27 + 3*x^2 - 9*x^3) - 18*E^16*x^2*Log[x])*Log[x^2])/(36*x + 24*x^3 - 12*x^4 + 4*x^5 - 4*x^6 + x^7 + (-108*x - 72*x^3 + 18*x^4 - 12*x^5 + 6*x^6)*Log[x] + (81*x + 54*x^3 + 9*x^5)*Log[x]^2),x]
Output:
$Aborted
Time = 4.20 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07
| method | result | size |
| parallelrisch | \(\frac {3 \ln \left (x^{2}\right ) {\mathrm e}^{16}}{3 x^{2} \ln \left (x \right )+x^{3}-2 x^{2}+9 \ln \left (x \right )-6}\) | \(32\) |
| risch | \(\frac {2 \,{\mathrm e}^{16}}{x^{2}+3}-\frac {\left (3 i \pi \,x^{2} \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-6 i \pi \,x^{2} \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+3 i \pi \,x^{2} \operatorname {csgn}\left (i x^{2}\right )^{3}+9 i \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-18 i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+9 i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+4 x^{3}-8 x^{2}-24\right ) {\mathrm e}^{16}}{2 \left (x^{2}+3\right ) \left (3 x^{2} \ln \left (x \right )+x^{3}-2 x^{2}+9 \ln \left (x \right )-6\right )}\) | \(166\) |
Input:
int(((-18*x^2*exp(16)*ln(x)+(-9*x^3+3*x^2-27)*exp(16))*ln(x^2)+(18*x^2+54) *exp(16)*ln(x)+(6*x^3-12*x^2-36)*exp(16))/((9*x^5+54*x^3+81*x)*ln(x)^2+(6* x^6-12*x^5+18*x^4-72*x^3-108*x)*ln(x)+x^7-4*x^6+4*x^5-12*x^4+24*x^3+36*x), x,method=_RETURNVERBOSE)
Output:
3*ln(x^2)*exp(16)/(3*x^2*ln(x)+x^3-2*x^2+9*ln(x)-6)
Time = 0.07 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.90 \[ \int \frac {e^{16} \left (-36-12 x^2+6 x^3\right )+e^{16} \left (54+18 x^2\right ) \log (x)+\left (e^{16} \left (-27+3 x^2-9 x^3\right )-18 e^{16} x^2 \log (x)\right ) \log \left (x^2\right )}{36 x+24 x^3-12 x^4+4 x^5-4 x^6+x^7+\left (-108 x-72 x^3+18 x^4-12 x^5+6 x^6\right ) \log (x)+\left (81 x+54 x^3+9 x^5\right ) \log ^2(x)} \, dx=\frac {6 \, e^{16} \log \left (x\right )}{x^{3} - 2 \, x^{2} + 3 \, {\left (x^{2} + 3\right )} \log \left (x\right ) - 6} \] Input:
integrate(((-18*x^2*exp(16)*log(x)+(-9*x^3+3*x^2-27)*exp(16))*log(x^2)+(18 *x^2+54)*exp(16)*log(x)+(6*x^3-12*x^2-36)*exp(16))/((9*x^5+54*x^3+81*x)*lo g(x)^2+(6*x^6-12*x^5+18*x^4-72*x^3-108*x)*log(x)+x^7-4*x^6+4*x^5-12*x^4+24 *x^3+36*x),x, algorithm="fricas")
Output:
6*e^16*log(x)/(x^3 - 2*x^2 + 3*(x^2 + 3)*log(x) - 6)
Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (24) = 48\).
Time = 0.20 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.27 \[ \int \frac {e^{16} \left (-36-12 x^2+6 x^3\right )+e^{16} \left (54+18 x^2\right ) \log (x)+\left (e^{16} \left (-27+3 x^2-9 x^3\right )-18 e^{16} x^2 \log (x)\right ) \log \left (x^2\right )}{36 x+24 x^3-12 x^4+4 x^5-4 x^6+x^7+\left (-108 x-72 x^3+18 x^4-12 x^5+6 x^6\right ) \log (x)+\left (81 x+54 x^3+9 x^5\right ) \log ^2(x)} \, dx=\frac {- 2 x^{3} e^{16} + 4 x^{2} e^{16} + 12 e^{16}}{x^{5} - 2 x^{4} + 3 x^{3} - 12 x^{2} + \left (3 x^{4} + 18 x^{2} + 27\right ) \log {\left (x \right )} - 18} + \frac {4 e^{16}}{2 x^{2} + 6} \] Input:
integrate(((-18*x**2*exp(16)*ln(x)+(-9*x**3+3*x**2-27)*exp(16))*ln(x**2)+( 18*x**2+54)*exp(16)*ln(x)+(6*x**3-12*x**2-36)*exp(16))/((9*x**5+54*x**3+81 *x)*ln(x)**2+(6*x**6-12*x**5+18*x**4-72*x**3-108*x)*ln(x)+x**7-4*x**6+4*x* *5-12*x**4+24*x**3+36*x),x)
Output:
(-2*x**3*exp(16) + 4*x**2*exp(16) + 12*exp(16))/(x**5 - 2*x**4 + 3*x**3 - 12*x**2 + (3*x**4 + 18*x**2 + 27)*log(x) - 18) + 4*exp(16)/(2*x**2 + 6)
Time = 0.08 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.90 \[ \int \frac {e^{16} \left (-36-12 x^2+6 x^3\right )+e^{16} \left (54+18 x^2\right ) \log (x)+\left (e^{16} \left (-27+3 x^2-9 x^3\right )-18 e^{16} x^2 \log (x)\right ) \log \left (x^2\right )}{36 x+24 x^3-12 x^4+4 x^5-4 x^6+x^7+\left (-108 x-72 x^3+18 x^4-12 x^5+6 x^6\right ) \log (x)+\left (81 x+54 x^3+9 x^5\right ) \log ^2(x)} \, dx=\frac {6 \, e^{16} \log \left (x\right )}{x^{3} - 2 \, x^{2} + 3 \, {\left (x^{2} + 3\right )} \log \left (x\right ) - 6} \] Input:
integrate(((-18*x^2*exp(16)*log(x)+(-9*x^3+3*x^2-27)*exp(16))*log(x^2)+(18 *x^2+54)*exp(16)*log(x)+(6*x^3-12*x^2-36)*exp(16))/((9*x^5+54*x^3+81*x)*lo g(x)^2+(6*x^6-12*x^5+18*x^4-72*x^3-108*x)*log(x)+x^7-4*x^6+4*x^5-12*x^4+24 *x^3+36*x),x, algorithm="maxima")
Output:
6*e^16*log(x)/(x^3 - 2*x^2 + 3*(x^2 + 3)*log(x) - 6)
Time = 0.14 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97 \[ \int \frac {e^{16} \left (-36-12 x^2+6 x^3\right )+e^{16} \left (54+18 x^2\right ) \log (x)+\left (e^{16} \left (-27+3 x^2-9 x^3\right )-18 e^{16} x^2 \log (x)\right ) \log \left (x^2\right )}{36 x+24 x^3-12 x^4+4 x^5-4 x^6+x^7+\left (-108 x-72 x^3+18 x^4-12 x^5+6 x^6\right ) \log (x)+\left (81 x+54 x^3+9 x^5\right ) \log ^2(x)} \, dx=\frac {6 \, e^{16} \log \left (x\right )}{x^{3} + 3 \, x^{2} \log \left (x\right ) - 2 \, x^{2} + 9 \, \log \left (x\right ) - 6} \] Input:
integrate(((-18*x^2*exp(16)*log(x)+(-9*x^3+3*x^2-27)*exp(16))*log(x^2)+(18 *x^2+54)*exp(16)*log(x)+(6*x^3-12*x^2-36)*exp(16))/((9*x^5+54*x^3+81*x)*lo g(x)^2+(6*x^6-12*x^5+18*x^4-72*x^3-108*x)*log(x)+x^7-4*x^6+4*x^5-12*x^4+24 *x^3+36*x),x, algorithm="giac")
Output:
6*e^16*log(x)/(x^3 + 3*x^2*log(x) - 2*x^2 + 9*log(x) - 6)
Timed out. \[ \int \frac {e^{16} \left (-36-12 x^2+6 x^3\right )+e^{16} \left (54+18 x^2\right ) \log (x)+\left (e^{16} \left (-27+3 x^2-9 x^3\right )-18 e^{16} x^2 \log (x)\right ) \log \left (x^2\right )}{36 x+24 x^3-12 x^4+4 x^5-4 x^6+x^7+\left (-108 x-72 x^3+18 x^4-12 x^5+6 x^6\right ) \log (x)+\left (81 x+54 x^3+9 x^5\right ) \log ^2(x)} \, dx=\int -\frac {\ln \left (x^2\right )\,\left ({\mathrm {e}}^{16}\,\left (9\,x^3-3\,x^2+27\right )+18\,x^2\,{\mathrm {e}}^{16}\,\ln \left (x\right )\right )+{\mathrm {e}}^{16}\,\left (-6\,x^3+12\,x^2+36\right )-{\mathrm {e}}^{16}\,\ln \left (x\right )\,\left (18\,x^2+54\right )}{36\,x+{\ln \left (x\right )}^2\,\left (9\,x^5+54\,x^3+81\,x\right )-\ln \left (x\right )\,\left (-6\,x^6+12\,x^5-18\,x^4+72\,x^3+108\,x\right )+24\,x^3-12\,x^4+4\,x^5-4\,x^6+x^7} \,d x \] Input:
int(-(log(x^2)*(exp(16)*(9*x^3 - 3*x^2 + 27) + 18*x^2*exp(16)*log(x)) + ex p(16)*(12*x^2 - 6*x^3 + 36) - exp(16)*log(x)*(18*x^2 + 54))/(36*x + log(x) ^2*(81*x + 54*x^3 + 9*x^5) - log(x)*(108*x + 72*x^3 - 18*x^4 + 12*x^5 - 6* x^6) + 24*x^3 - 12*x^4 + 4*x^5 - 4*x^6 + x^7),x)
Output:
int(-(log(x^2)*(exp(16)*(9*x^3 - 3*x^2 + 27) + 18*x^2*exp(16)*log(x)) + ex p(16)*(12*x^2 - 6*x^3 + 36) - exp(16)*log(x)*(18*x^2 + 54))/(36*x + log(x) ^2*(81*x + 54*x^3 + 9*x^5) - log(x)*(108*x + 72*x^3 - 18*x^4 + 12*x^5 - 6* x^6) + 24*x^3 - 12*x^4 + 4*x^5 - 4*x^6 + x^7), x)
Time = 0.22 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {e^{16} \left (-36-12 x^2+6 x^3\right )+e^{16} \left (54+18 x^2\right ) \log (x)+\left (e^{16} \left (-27+3 x^2-9 x^3\right )-18 e^{16} x^2 \log (x)\right ) \log \left (x^2\right )}{36 x+24 x^3-12 x^4+4 x^5-4 x^6+x^7+\left (-108 x-72 x^3+18 x^4-12 x^5+6 x^6\right ) \log (x)+\left (81 x+54 x^3+9 x^5\right ) \log ^2(x)} \, dx=\frac {3 \,\mathrm {log}\left (x^{2}\right ) e^{16}}{3 \,\mathrm {log}\left (x \right ) x^{2}+9 \,\mathrm {log}\left (x \right )+x^{3}-2 x^{2}-6} \] Input:
int(((-18*x^2*exp(16)*log(x)+(-9*x^3+3*x^2-27)*exp(16))*log(x^2)+(18*x^2+5 4)*exp(16)*log(x)+(6*x^3-12*x^2-36)*exp(16))/((9*x^5+54*x^3+81*x)*log(x)^2 +(6*x^6-12*x^5+18*x^4-72*x^3-108*x)*log(x)+x^7-4*x^6+4*x^5-12*x^4+24*x^3+3 6*x),x)
Output:
(3*log(x**2)*e**16)/(3*log(x)*x**2 + 9*log(x) + x**3 - 2*x**2 - 6)