Integrand size = 177, antiderivative size = 20 \begin {dmath*} \int \frac {32 x+160 \log (3)-32 \log ^2(3)}{-x^4-9 x^3 \log (3)-27 x^2 \log ^2(3)-27 x \log ^3(3)+\left (-15 x^3 \log (3)+\left (-90 x^2+3 x^3\right ) \log ^2(3)+\left (-135 x+18 x^2\right ) \log ^3(3)+27 x \log ^4(3)\right ) \log (x)+\left (-75 x^2 \log ^2(3)+\left (-225 x+30 x^2\right ) \log ^3(3)+\left (90 x-3 x^2\right ) \log ^4(3)-9 x \log ^5(3)\right ) \log ^2(x)+\left (-125 x \log ^3(3)+75 x \log ^4(3)-15 x \log ^5(3)+x \log ^6(3)\right ) \log ^3(x)} \, dx=\frac {16}{(x+\log (3) (3+(5-\log (3)) \log (x)))^2} \end {dmath*}
Time = 0.04 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \begin {dmath*} \int \frac {32 x+160 \log (3)-32 \log ^2(3)}{-x^4-9 x^3 \log (3)-27 x^2 \log ^2(3)-27 x \log ^3(3)+\left (-15 x^3 \log (3)+\left (-90 x^2+3 x^3\right ) \log ^2(3)+\left (-135 x+18 x^2\right ) \log ^3(3)+27 x \log ^4(3)\right ) \log (x)+\left (-75 x^2 \log ^2(3)+\left (-225 x+30 x^2\right ) \log ^3(3)+\left (90 x-3 x^2\right ) \log ^4(3)-9 x \log ^5(3)\right ) \log ^2(x)+\left (-125 x \log ^3(3)+75 x \log ^4(3)-15 x \log ^5(3)+x \log ^6(3)\right ) \log ^3(x)} \, dx=\frac {16}{(x+\log (27)-(-5+\log (3)) \log (3) \log (x))^2} \end {dmath*}
Integrate[(32*x + 160*Log[3] - 32*Log[3]^2)/(-x^4 - 9*x^3*Log[3] - 27*x^2* Log[3]^2 - 27*x*Log[3]^3 + (-15*x^3*Log[3] + (-90*x^2 + 3*x^3)*Log[3]^2 + (-135*x + 18*x^2)*Log[3]^3 + 27*x*Log[3]^4)*Log[x] + (-75*x^2*Log[3]^2 + ( -225*x + 30*x^2)*Log[3]^3 + (90*x - 3*x^2)*Log[3]^4 - 9*x*Log[3]^5)*Log[x] ^2 + (-125*x*Log[3]^3 + 75*x*Log[3]^4 - 15*x*Log[3]^5 + x*Log[3]^6)*Log[x] ^3),x]
Time = 0.57 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {7239, 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.
| \(\displaystyle \int \frac {32 x-32 \log ^2(3)+160 \log (3)}{-x^4-9 x^3 \log (3)-27 x^2 \log ^2(3)+\left (\left (90 x-3 x^2\right ) \log ^4(3)+\left (30 x^2-225 x\right ) \log ^3(3)-75 x^2 \log ^2(3)-9 x \log ^5(3)\right ) \log ^2(x)+\left (-15 x^3 \log (3)+\left (18 x^2-135 x\right ) \log ^3(3)+\left (3 x^3-90 x^2\right ) \log ^2(3)+27 x \log ^4(3)\right ) \log (x)-27 x \log ^3(3)+\left (x \log ^6(3)-15 x \log ^5(3)+75 x \log ^4(3)-125 x \log ^3(3)\right ) \log ^3(x)} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {32 ((\log (3)-5) \log (3)-x)}{x (x-(\log (3)-5) \log (3) \log (x)+\log (27))^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 32 \int -\frac {x+(5-\log (3)) \log (3)}{x (x+(5-\log (3)) \log (3) \log (x)+\log (27))^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -32 \int \frac {x+(5-\log (3)) \log (3)}{x (x+(5-\log (3)) \log (3) \log (x)+\log (27))^3}dx\) |
| \(\Big \downarrow \) 7237 |
| \(\displaystyle \frac {16}{(x+(5-\log (3)) \log (3) \log (x)+\log (27))^2}\) |
Int[(32*x + 160*Log[3] - 32*Log[3]^2)/(-x^4 - 9*x^3*Log[3] - 27*x^2*Log[3] ^2 - 27*x*Log[3]^3 + (-15*x^3*Log[3] + (-90*x^2 + 3*x^3)*Log[3]^2 + (-135* x + 18*x^2)*Log[3]^3 + 27*x*Log[3]^4)*Log[x] + (-75*x^2*Log[3]^2 + (-225*x + 30*x^2)*Log[3]^3 + (90*x - 3*x^2)*Log[3]^4 - 9*x*Log[3]^5)*Log[x]^2 + ( -125*x*Log[3]^3 + 75*x*Log[3]^4 - 15*x*Log[3]^5 + x*Log[3]^6)*Log[x]^3),x]
3.10.59.3.1 Defintions of rubi rules used
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]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 3.30 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.30
| method | result | size |
| default | \(\frac {16}{\left (\ln \left (3\right )^{2} \ln \left (x \right )-5 \ln \left (3\right ) \ln \left (x \right )-3 \ln \left (3\right )-x \right )^{2}}\) | \(26\) |
| norman | \(\frac {16}{\left (\ln \left (3\right )^{2} \ln \left (x \right )-5 \ln \left (3\right ) \ln \left (x \right )-3 \ln \left (3\right )-x \right )^{2}}\) | \(26\) |
| risch | \(\frac {16}{\left (\ln \left (3\right )^{2} \ln \left (x \right )-5 \ln \left (3\right ) \ln \left (x \right )-3 \ln \left (3\right )-x \right )^{2}}\) | \(26\) |
| parallelrisch | \(\frac {16}{\ln \left (x \right )^{2} \ln \left (3\right )^{4}-10 \ln \left (3\right )^{3} \ln \left (x \right )^{2}-6 \ln \left (x \right ) \ln \left (3\right )^{3}-2 x \ln \left (x \right ) \ln \left (3\right )^{2}+25 \ln \left (3\right )^{2} \ln \left (x \right )^{2}+30 \ln \left (3\right )^{2} \ln \left (x \right )+10 x \ln \left (3\right ) \ln \left (x \right )+9 \ln \left (3\right )^{2}+6 x \ln \left (3\right )+x^{2}}\) | \(81\) |
int((-32*ln(3)^2+160*ln(3)+32*x)/((x*ln(3)^6-15*x*ln(3)^5+75*x*ln(3)^4-125 *x*ln(3)^3)*ln(x)^3+(-9*x*ln(3)^5+(-3*x^2+90*x)*ln(3)^4+(30*x^2-225*x)*ln( 3)^3-75*x^2*ln(3)^2)*ln(x)^2+(27*x*ln(3)^4+(18*x^2-135*x)*ln(3)^3+(3*x^3-9 0*x^2)*ln(3)^2-15*x^3*ln(3))*ln(x)-27*x*ln(3)^3-27*x^2*ln(3)^2-9*x^3*ln(3) -x^4),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (20) = 40\).
Time = 0.27 (sec) , antiderivative size = 65, normalized size of antiderivative = 3.25 \begin {dmath*} \int \frac {32 x+160 \log (3)-32 \log ^2(3)}{-x^4-9 x^3 \log (3)-27 x^2 \log ^2(3)-27 x \log ^3(3)+\left (-15 x^3 \log (3)+\left (-90 x^2+3 x^3\right ) \log ^2(3)+\left (-135 x+18 x^2\right ) \log ^3(3)+27 x \log ^4(3)\right ) \log (x)+\left (-75 x^2 \log ^2(3)+\left (-225 x+30 x^2\right ) \log ^3(3)+\left (90 x-3 x^2\right ) \log ^4(3)-9 x \log ^5(3)\right ) \log ^2(x)+\left (-125 x \log ^3(3)+75 x \log ^4(3)-15 x \log ^5(3)+x \log ^6(3)\right ) \log ^3(x)} \, dx=\frac {16}{{\left (\log \left (3\right )^{4} - 10 \, \log \left (3\right )^{3} + 25 \, \log \left (3\right )^{2}\right )} \log \left (x\right )^{2} + x^{2} + 6 \, x \log \left (3\right ) + 9 \, \log \left (3\right )^{2} - 2 \, {\left ({\left (x - 15\right )} \log \left (3\right )^{2} + 3 \, \log \left (3\right )^{3} - 5 \, x \log \left (3\right )\right )} \log \left (x\right )} \end {dmath*}
integrate((-32*log(3)^2+160*log(3)+32*x)/((x*log(3)^6-15*x*log(3)^5+75*x*l og(3)^4-125*x*log(3)^3)*log(x)^3+(-9*x*log(3)^5+(-3*x^2+90*x)*log(3)^4+(30 *x^2-225*x)*log(3)^3-75*x^2*log(3)^2)*log(x)^2+(27*x*log(3)^4+(18*x^2-135* x)*log(3)^3+(3*x^3-90*x^2)*log(3)^2-15*x^3*log(3))*log(x)-27*x*log(3)^3-27 *x^2*log(3)^2-9*x^3*log(3)-x^4),x, algorithm=\
16/((log(3)^4 - 10*log(3)^3 + 25*log(3)^2)*log(x)^2 + x^2 + 6*x*log(3) + 9 *log(3)^2 - 2*((x - 15)*log(3)^2 + 3*log(3)^3 - 5*x*log(3))*log(x))
Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (17) = 34\).
Time = 0.22 (sec) , antiderivative size = 73, normalized size of antiderivative = 3.65 \begin {dmath*} \int \frac {32 x+160 \log (3)-32 \log ^2(3)}{-x^4-9 x^3 \log (3)-27 x^2 \log ^2(3)-27 x \log ^3(3)+\left (-15 x^3 \log (3)+\left (-90 x^2+3 x^3\right ) \log ^2(3)+\left (-135 x+18 x^2\right ) \log ^3(3)+27 x \log ^4(3)\right ) \log (x)+\left (-75 x^2 \log ^2(3)+\left (-225 x+30 x^2\right ) \log ^3(3)+\left (90 x-3 x^2\right ) \log ^4(3)-9 x \log ^5(3)\right ) \log ^2(x)+\left (-125 x \log ^3(3)+75 x \log ^4(3)-15 x \log ^5(3)+x \log ^6(3)\right ) \log ^3(x)} \, dx=\frac {16}{x^{2} + 6 x \log {\left (3 \right )} + \left (- 2 x \log {\left (3 \right )}^{2} + 10 x \log {\left (3 \right )} - 6 \log {\left (3 \right )}^{3} + 30 \log {\left (3 \right )}^{2}\right ) \log {\left (x \right )} + \left (- 10 \log {\left (3 \right )}^{3} + \log {\left (3 \right )}^{4} + 25 \log {\left (3 \right )}^{2}\right ) \log {\left (x \right )}^{2} + 9 \log {\left (3 \right )}^{2}} \end {dmath*}
integrate((-32*ln(3)**2+160*ln(3)+32*x)/((x*ln(3)**6-15*x*ln(3)**5+75*x*ln (3)**4-125*x*ln(3)**3)*ln(x)**3+(-9*x*ln(3)**5+(-3*x**2+90*x)*ln(3)**4+(30 *x**2-225*x)*ln(3)**3-75*x**2*ln(3)**2)*ln(x)**2+(27*x*ln(3)**4+(18*x**2-1 35*x)*ln(3)**3+(3*x**3-90*x**2)*ln(3)**2-15*x**3*ln(3))*ln(x)-27*x*ln(3)** 3-27*x**2*ln(3)**2-9*x**3*ln(3)-x**4),x)
16/(x**2 + 6*x*log(3) + (-2*x*log(3)**2 + 10*x*log(3) - 6*log(3)**3 + 30*l og(3)**2)*log(x) + (-10*log(3)**3 + log(3)**4 + 25*log(3)**2)*log(x)**2 + 9*log(3)**2)
Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (20) = 40\).
Time = 0.34 (sec) , antiderivative size = 69, normalized size of antiderivative = 3.45 \begin {dmath*} \int \frac {32 x+160 \log (3)-32 \log ^2(3)}{-x^4-9 x^3 \log (3)-27 x^2 \log ^2(3)-27 x \log ^3(3)+\left (-15 x^3 \log (3)+\left (-90 x^2+3 x^3\right ) \log ^2(3)+\left (-135 x+18 x^2\right ) \log ^3(3)+27 x \log ^4(3)\right ) \log (x)+\left (-75 x^2 \log ^2(3)+\left (-225 x+30 x^2\right ) \log ^3(3)+\left (90 x-3 x^2\right ) \log ^4(3)-9 x \log ^5(3)\right ) \log ^2(x)+\left (-125 x \log ^3(3)+75 x \log ^4(3)-15 x \log ^5(3)+x \log ^6(3)\right ) \log ^3(x)} \, dx=\frac {16}{{\left (\log \left (3\right )^{4} - 10 \, \log \left (3\right )^{3} + 25 \, \log \left (3\right )^{2}\right )} \log \left (x\right )^{2} + x^{2} + 6 \, x \log \left (3\right ) + 9 \, \log \left (3\right )^{2} - 2 \, {\left (3 \, \log \left (3\right )^{3} + {\left (\log \left (3\right )^{2} - 5 \, \log \left (3\right )\right )} x - 15 \, \log \left (3\right )^{2}\right )} \log \left (x\right )} \end {dmath*}
integrate((-32*log(3)^2+160*log(3)+32*x)/((x*log(3)^6-15*x*log(3)^5+75*x*l og(3)^4-125*x*log(3)^3)*log(x)^3+(-9*x*log(3)^5+(-3*x^2+90*x)*log(3)^4+(30 *x^2-225*x)*log(3)^3-75*x^2*log(3)^2)*log(x)^2+(27*x*log(3)^4+(18*x^2-135* x)*log(3)^3+(3*x^3-90*x^2)*log(3)^2-15*x^3*log(3))*log(x)-27*x*log(3)^3-27 *x^2*log(3)^2-9*x^3*log(3)-x^4),x, algorithm=\
16/((log(3)^4 - 10*log(3)^3 + 25*log(3)^2)*log(x)^2 + x^2 + 6*x*log(3) + 9 *log(3)^2 - 2*(3*log(3)^3 + (log(3)^2 - 5*log(3))*x - 15*log(3)^2)*log(x))
Leaf count of result is larger than twice the leaf count of optimal. 206 vs. \(2 (20) = 40\).
Time = 0.29 (sec) , antiderivative size = 206, normalized size of antiderivative = 10.30 \begin {dmath*} \int \frac {32 x+160 \log (3)-32 \log ^2(3)}{-x^4-9 x^3 \log (3)-27 x^2 \log ^2(3)-27 x \log ^3(3)+\left (-15 x^3 \log (3)+\left (-90 x^2+3 x^3\right ) \log ^2(3)+\left (-135 x+18 x^2\right ) \log ^3(3)+27 x \log ^4(3)\right ) \log (x)+\left (-75 x^2 \log ^2(3)+\left (-225 x+30 x^2\right ) \log ^3(3)+\left (90 x-3 x^2\right ) \log ^4(3)-9 x \log ^5(3)\right ) \log ^2(x)+\left (-125 x \log ^3(3)+75 x \log ^4(3)-15 x \log ^5(3)+x \log ^6(3)\right ) \log ^3(x)} \, dx=\frac {16 \, {\left (\log \left (3\right )^{2} - x - 5 \, \log \left (3\right )\right )}}{\log \left (3\right )^{6} \log \left (x\right )^{2} - x \log \left (3\right )^{4} \log \left (x\right )^{2} - 15 \, \log \left (3\right )^{5} \log \left (x\right )^{2} - 2 \, x \log \left (3\right )^{4} \log \left (x\right ) - 6 \, \log \left (3\right )^{5} \log \left (x\right ) + 10 \, x \log \left (3\right )^{3} \log \left (x\right )^{2} + 75 \, \log \left (3\right )^{4} \log \left (x\right )^{2} + 2 \, x^{2} \log \left (3\right )^{2} \log \left (x\right ) + 26 \, x \log \left (3\right )^{3} \log \left (x\right ) + 60 \, \log \left (3\right )^{4} \log \left (x\right ) - 25 \, x \log \left (3\right )^{2} \log \left (x\right )^{2} - 125 \, \log \left (3\right )^{3} \log \left (x\right )^{2} + x^{2} \log \left (3\right )^{2} + 6 \, x \log \left (3\right )^{3} + 9 \, \log \left (3\right )^{4} - 10 \, x^{2} \log \left (3\right ) \log \left (x\right ) - 80 \, x \log \left (3\right )^{2} \log \left (x\right ) - 150 \, \log \left (3\right )^{3} \log \left (x\right ) - x^{3} - 11 \, x^{2} \log \left (3\right ) - 39 \, x \log \left (3\right )^{2} - 45 \, \log \left (3\right )^{3}} \end {dmath*}
integrate((-32*log(3)^2+160*log(3)+32*x)/((x*log(3)^6-15*x*log(3)^5+75*x*l og(3)^4-125*x*log(3)^3)*log(x)^3+(-9*x*log(3)^5+(-3*x^2+90*x)*log(3)^4+(30 *x^2-225*x)*log(3)^3-75*x^2*log(3)^2)*log(x)^2+(27*x*log(3)^4+(18*x^2-135* x)*log(3)^3+(3*x^3-90*x^2)*log(3)^2-15*x^3*log(3))*log(x)-27*x*log(3)^3-27 *x^2*log(3)^2-9*x^3*log(3)-x^4),x, algorithm=\
16*(log(3)^2 - x - 5*log(3))/(log(3)^6*log(x)^2 - x*log(3)^4*log(x)^2 - 15 *log(3)^5*log(x)^2 - 2*x*log(3)^4*log(x) - 6*log(3)^5*log(x) + 10*x*log(3) ^3*log(x)^2 + 75*log(3)^4*log(x)^2 + 2*x^2*log(3)^2*log(x) + 26*x*log(3)^3 *log(x) + 60*log(3)^4*log(x) - 25*x*log(3)^2*log(x)^2 - 125*log(3)^3*log(x )^2 + x^2*log(3)^2 + 6*x*log(3)^3 + 9*log(3)^4 - 10*x^2*log(3)*log(x) - 80 *x*log(3)^2*log(x) - 150*log(3)^3*log(x) - x^3 - 11*x^2*log(3) - 39*x*log( 3)^2 - 45*log(3)^3)
Timed out. \begin {dmath*} \int \frac {32 x+160 \log (3)-32 \log ^2(3)}{-x^4-9 x^3 \log (3)-27 x^2 \log ^2(3)-27 x \log ^3(3)+\left (-15 x^3 \log (3)+\left (-90 x^2+3 x^3\right ) \log ^2(3)+\left (-135 x+18 x^2\right ) \log ^3(3)+27 x \log ^4(3)\right ) \log (x)+\left (-75 x^2 \log ^2(3)+\left (-225 x+30 x^2\right ) \log ^3(3)+\left (90 x-3 x^2\right ) \log ^4(3)-9 x \log ^5(3)\right ) \log ^2(x)+\left (-125 x \log ^3(3)+75 x \log ^4(3)-15 x \log ^5(3)+x \log ^6(3)\right ) \log ^3(x)} \, dx=\int -\frac {32\,x+160\,\ln \left (3\right )-32\,{\ln \left (3\right )}^2}{27\,x^2\,{\ln \left (3\right )}^2+\ln \left (x\right )\,\left ({\ln \left (3\right )}^3\,\left (135\,x-18\,x^2\right )+15\,x^3\,\ln \left (3\right )-27\,x\,{\ln \left (3\right )}^4+{\ln \left (3\right )}^2\,\left (90\,x^2-3\,x^3\right )\right )+{\ln \left (x\right )}^3\,\left (125\,x\,{\ln \left (3\right )}^3-75\,x\,{\ln \left (3\right )}^4+15\,x\,{\ln \left (3\right )}^5-x\,{\ln \left (3\right )}^6\right )+27\,x\,{\ln \left (3\right )}^3+9\,x^3\,\ln \left (3\right )+x^4+{\ln \left (x\right )}^2\,\left (75\,x^2\,{\ln \left (3\right )}^2-{\ln \left (3\right )}^4\,\left (90\,x-3\,x^2\right )+{\ln \left (3\right )}^3\,\left (225\,x-30\,x^2\right )+9\,x\,{\ln \left (3\right )}^5\right )} \,d x \end {dmath*}
int(-(32*x + 160*log(3) - 32*log(3)^2)/(27*x^2*log(3)^2 + log(x)*(log(3)^3 *(135*x - 18*x^2) + 15*x^3*log(3) - 27*x*log(3)^4 + log(3)^2*(90*x^2 - 3*x ^3)) + log(x)^3*(125*x*log(3)^3 - 75*x*log(3)^4 + 15*x*log(3)^5 - x*log(3) ^6) + 27*x*log(3)^3 + 9*x^3*log(3) + x^4 + log(x)^2*(75*x^2*log(3)^2 - log (3)^4*(90*x - 3*x^2) + log(3)^3*(225*x - 30*x^2) + 9*x*log(3)^5)),x)
int(-(32*x + 160*log(3) - 32*log(3)^2)/(27*x^2*log(3)^2 + log(x)*(log(3)^3 *(135*x - 18*x^2) + 15*x^3*log(3) - 27*x*log(3)^4 + log(3)^2*(90*x^2 - 3*x ^3)) + log(x)^3*(125*x*log(3)^3 - 75*x*log(3)^4 + 15*x*log(3)^5 - x*log(3) ^6) + 27*x*log(3)^3 + 9*x^3*log(3) + x^4 + log(x)^2*(75*x^2*log(3)^2 - log (3)^4*(90*x - 3*x^2) + log(3)^3*(225*x - 30*x^2) + 9*x*log(3)^5)), x)