Integrand size = 65, antiderivative size = 24 \[ \int \frac {384 x^3+\left (-576 x^2+352 x^3\right ) \log (9)+\left (192 x-352 x^2\right ) \log ^2(9)}{e^{32} \left (7776+71280 x+261360 x^2+479160 x^3+439230 x^4+161051 x^5\right )} \, dx=\frac {x^2 (-x+\log (9))^2}{e^{32} \left (-3-\frac {11 x}{2}\right )^4} \] Output:
(2*ln(3)-x)^2/(-3-11/2*x)^4*x^2/exp(16)^2
Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.79 \[ \int \frac {384 x^3+\left (-576 x^2+352 x^3\right ) \log (9)+\left (192 x-352 x^2\right ) \log ^2(9)}{e^{32} \left (7776+71280 x+261360 x^2+479160 x^3+439230 x^4+161051 x^5\right )} \, dx=-\frac {16 \left (1296+9504 x+2662 x^3 (12+11 \log (9))-121 x^2 \left (-216+121 \log ^2(9)\right )\right )}{14641 e^{32} (6+11 x)^4} \] Input:
Integrate[(384*x^3 + (-576*x^2 + 352*x^3)*Log[9] + (192*x - 352*x^2)*Log[9 ]^2)/(E^32*(7776 + 71280*x + 261360*x^2 + 479160*x^3 + 439230*x^4 + 161051 *x^5)),x]
Output:
(-16*(1296 + 9504*x + 2662*x^3*(12 + 11*Log[9]) - 121*x^2*(-216 + 121*Log[ 9]^2)))/(14641*E^32*(6 + 11*x)^4)
Leaf count is larger than twice the leaf count of optimal. \(88\) vs. \(2(24)=48\).
Time = 0.35 (sec) , antiderivative size = 88, normalized size of antiderivative = 3.67, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 27, 2007, 2389, 2009}
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.
\(\displaystyle \int \frac {384 x^3+\left (192 x-352 x^2\right ) \log ^2(9)+\left (352 x^3-576 x^2\right ) \log (9)}{e^{32} \left (161051 x^5+439230 x^4+479160 x^3+261360 x^2+71280 x+7776\right )} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {32 \left (12 x^3+\left (6 x-11 x^2\right ) \log ^2(9)-\left (18 x^2-11 x^3\right ) \log (9)\right )}{161051 x^5+439230 x^4+479160 x^3+261360 x^2+71280 x+7776}dx}{e^{32}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {32 \int \frac {12 x^3+\left (6 x-11 x^2\right ) \log ^2(9)-\left (18 x^2-11 x^3\right ) \log (9)}{161051 x^5+439230 x^4+479160 x^3+261360 x^2+71280 x+7776}dx}{e^{32}}\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \frac {32 \int \frac {12 x^3+\left (6 x-11 x^2\right ) \log ^2(9)-\left (18 x^2-11 x^3\right ) \log (9)}{(11 x+6)^5}dx}{e^{32}}\) |
\(\Big \downarrow \) 2389 |
\(\displaystyle \frac {32 \int \left (\frac {12+11 \log (9)}{1331 (11 x+6)^2}+\frac {-216-396 \log (9)-121 \log ^2(9)}{1331 (11 x+6)^3}+\frac {18 \left (72+198 \log (9)+121 \log ^2(9)\right )}{1331 (11 x+6)^4}-\frac {72 (6+11 \log (9))^2}{1331 (11 x+6)^5}\right )dx}{e^{32}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {32 \left (\frac {216+121 \log ^2(9)+396 \log (9)}{29282 (11 x+6)^2}-\frac {6 \left (72+121 \log ^2(9)+198 \log (9)\right )}{14641 (11 x+6)^3}-\frac {12+11 \log (9)}{14641 (11 x+6)}+\frac {18 (6+11 \log (9))^2}{14641 (11 x+6)^4}\right )}{e^{32}}\) |
Input:
Int[(384*x^3 + (-576*x^2 + 352*x^3)*Log[9] + (192*x - 352*x^2)*Log[9]^2)/( E^32*(7776 + 71280*x + 261360*x^2 + 479160*x^3 + 439230*x^4 + 161051*x^5)) ,x]
Output:
(32*((18*(6 + 11*Log[9])^2)/(14641*(6 + 11*x)^4) - (12 + 11*Log[9])/(14641 *(6 + 11*x)) - (6*(72 + 198*Log[9] + 121*Log[9]^2))/(14641*(6 + 11*x)^3) + (216 + 396*Log[9] + 121*Log[9]^2)/(29282*(6 + 11*x)^2)))/E^32
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^(Ex pon[Px, x]*p), x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; IntegerQ[p] && Pol yQ[Px, x] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 0]
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand [Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p , 0] || EqQ[n, 1])
Time = 0.17 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.12
method | result | size |
risch | \(\frac {{\mathrm e}^{-32} \left (\left (-\frac {384}{161051}-\frac {64 \ln \left (3\right )}{14641}\right ) x^{3}+\left (\frac {64 \ln \left (3\right )^{2}}{14641}-\frac {3456}{1771561}\right ) x^{2}-\frac {13824 x}{19487171}-\frac {20736}{214358881}\right )}{x^{4}+\frac {24}{11} x^{3}+\frac {216}{121} x^{2}+\frac {864}{1331} x +\frac {1296}{14641}}\) | \(51\) |
norman | \(\frac {\left (-\frac {64 \left (11 \ln \left (3\right )+6\right ) {\mathrm e}^{-16} x^{3}}{11}+\frac {64 \left (121 \ln \left (3\right )^{2}-54\right ) {\mathrm e}^{-16} x^{2}}{121}-\frac {13824 \,{\mathrm e}^{-16} x}{1331}-\frac {20736 \,{\mathrm e}^{-16}}{14641}\right ) {\mathrm e}^{-16}}{\left (11 x +6\right )^{4}}\) | \(59\) |
gosper | \(\frac {64 \left (14641 x^{2} \ln \left (3\right )^{2}-14641 x^{3} \ln \left (3\right )-7986 x^{3}-6534 x^{2}-2376 x -324\right ) {\mathrm e}^{-32}}{14641 \left (14641 x^{4}+31944 x^{3}+26136 x^{2}+9504 x +1296\right )}\) | \(60\) |
parallelrisch | \(\frac {{\mathrm e}^{-32} \left (-20736+937024 x^{2} \ln \left (3\right )^{2}-937024 x^{3} \ln \left (3\right )-511104 x^{3}-418176 x^{2}-152064 x \right )}{214358881 x^{4}+467692104 x^{3}+382657176 x^{2}+139148064 x +18974736}\) | \(60\) |
default | \(2 \,{\mathrm e}^{-32} \left (-\frac {8 \left (-\frac {144 \ln \left (3\right )^{2}}{121}-\frac {864 \ln \left (3\right )}{1331}-\frac {1296}{14641}\right )}{\left (11 x +6\right )^{4}}-\frac {32 \left (\frac {\ln \left (3\right )}{1331}+\frac {6}{14641}\right )}{11 x +6}-\frac {16 \left (-\frac {2 \ln \left (3\right )^{2}}{121}-\frac {36 \ln \left (3\right )}{1331}-\frac {108}{14641}\right )}{\left (11 x +6\right )^{2}}-\frac {32 \left (\frac {36 \ln \left (3\right )^{2}}{121}+\frac {324 \ln \left (3\right )}{1331}+\frac {648}{14641}\right )}{3 \left (11 x +6\right )^{3}}\right )\) | \(86\) |
Input:
int((4*(-352*x^2+192*x)*ln(3)^2+2*(352*x^3-576*x^2)*ln(3)+384*x^3)/(161051 *x^5+439230*x^4+479160*x^3+261360*x^2+71280*x+7776)/exp(16)^2,x,method=_RE TURNVERBOSE)
Output:
exp(-32)*((-384/161051-64/14641*ln(3))*x^3+(64/14641*ln(3)^2-3456/1771561) *x^2-13824/19487171*x-20736/214358881)/(x^4+24/11*x^3+216/121*x^2+864/1331 *x+1296/14641)
Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (22) = 44\).
Time = 0.08 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.38 \[ \int \frac {384 x^3+\left (-576 x^2+352 x^3\right ) \log (9)+\left (192 x-352 x^2\right ) \log ^2(9)}{e^{32} \left (7776+71280 x+261360 x^2+479160 x^3+439230 x^4+161051 x^5\right )} \, dx=-\frac {64 \, {\left (14641 \, x^{3} \log \left (3\right ) - 14641 \, x^{2} \log \left (3\right )^{2} + 7986 \, x^{3} + 6534 \, x^{2} + 2376 \, x + 324\right )} e^{\left (-32\right )}}{14641 \, {\left (14641 \, x^{4} + 31944 \, x^{3} + 26136 \, x^{2} + 9504 \, x + 1296\right )}} \] Input:
integrate((4*(-352*x^2+192*x)*log(3)^2+2*(352*x^3-576*x^2)*log(3)+384*x^3) /(161051*x^5+439230*x^4+479160*x^3+261360*x^2+71280*x+7776)/exp(16)^2,x, a lgorithm="fricas")
Output:
-64/14641*(14641*x^3*log(3) - 14641*x^2*log(3)^2 + 7986*x^3 + 6534*x^2 + 2 376*x + 324)*e^(-32)/(14641*x^4 + 31944*x^3 + 26136*x^2 + 9504*x + 1296)
Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (24) = 48\).
Time = 0.75 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.71 \[ \int \frac {384 x^3+\left (-576 x^2+352 x^3\right ) \log (9)+\left (192 x-352 x^2\right ) \log ^2(9)}{e^{32} \left (7776+71280 x+261360 x^2+479160 x^3+439230 x^4+161051 x^5\right )} \, dx=\frac {x^{3} \left (- 937024 \log {\left (3 \right )} - 511104\right ) + x^{2} \left (-418176 + 937024 \log {\left (3 \right )}^{2}\right ) - 152064 x - 20736}{214358881 x^{4} e^{32} + 467692104 x^{3} e^{32} + 382657176 x^{2} e^{32} + 139148064 x e^{32} + 18974736 e^{32}} \] Input:
integrate((4*(-352*x**2+192*x)*ln(3)**2+2*(352*x**3-576*x**2)*ln(3)+384*x* *3)/(161051*x**5+439230*x**4+479160*x**3+261360*x**2+71280*x+7776)/exp(16) **2,x)
Output:
(x**3*(-937024*log(3) - 511104) + x**2*(-418176 + 937024*log(3)**2) - 1520 64*x - 20736)/(214358881*x**4*exp(32) + 467692104*x**3*exp(32) + 382657176 *x**2*exp(32) + 139148064*x*exp(32) + 18974736*exp(32))
Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (22) = 44\).
Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.29 \[ \int \frac {384 x^3+\left (-576 x^2+352 x^3\right ) \log (9)+\left (192 x-352 x^2\right ) \log ^2(9)}{e^{32} \left (7776+71280 x+261360 x^2+479160 x^3+439230 x^4+161051 x^5\right )} \, dx=-\frac {64 \, {\left (1331 \, x^{3} {\left (11 \, \log \left (3\right ) + 6\right )} - 121 \, {\left (121 \, \log \left (3\right )^{2} - 54\right )} x^{2} + 2376 \, x + 324\right )} e^{\left (-32\right )}}{14641 \, {\left (14641 \, x^{4} + 31944 \, x^{3} + 26136 \, x^{2} + 9504 \, x + 1296\right )}} \] Input:
integrate((4*(-352*x^2+192*x)*log(3)^2+2*(352*x^3-576*x^2)*log(3)+384*x^3) /(161051*x^5+439230*x^4+479160*x^3+261360*x^2+71280*x+7776)/exp(16)^2,x, a lgorithm="maxima")
Output:
-64/14641*(1331*x^3*(11*log(3) + 6) - 121*(121*log(3)^2 - 54)*x^2 + 2376*x + 324)*e^(-32)/(14641*x^4 + 31944*x^3 + 26136*x^2 + 9504*x + 1296)
Time = 0.11 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.75 \[ \int \frac {384 x^3+\left (-576 x^2+352 x^3\right ) \log (9)+\left (192 x-352 x^2\right ) \log ^2(9)}{e^{32} \left (7776+71280 x+261360 x^2+479160 x^3+439230 x^4+161051 x^5\right )} \, dx=-\frac {64 \, {\left (14641 \, x^{3} \log \left (3\right ) - 14641 \, x^{2} \log \left (3\right )^{2} + 7986 \, x^{3} + 6534 \, x^{2} + 2376 \, x + 324\right )} e^{\left (-32\right )}}{14641 \, {\left (11 \, x + 6\right )}^{4}} \] Input:
integrate((4*(-352*x^2+192*x)*log(3)^2+2*(352*x^3-576*x^2)*log(3)+384*x^3) /(161051*x^5+439230*x^4+479160*x^3+261360*x^2+71280*x+7776)/exp(16)^2,x, a lgorithm="giac")
Output:
-64/14641*(14641*x^3*log(3) - 14641*x^2*log(3)^2 + 7986*x^3 + 6534*x^2 + 2 376*x + 324)*e^(-32)/(11*x + 6)^4
Time = 3.22 (sec) , antiderivative size = 83, normalized size of antiderivative = 3.46 \[ \int \frac {384 x^3+\left (-576 x^2+352 x^3\right ) \log (9)+\left (192 x-352 x^2\right ) \log ^2(9)}{e^{32} \left (7776+71280 x+261360 x^2+479160 x^3+439230 x^4+161051 x^5\right )} \, dx=\frac {64\,{\mathrm {e}}^{-32}\,\left (198\,\ln \left (3\right )+121\,{\ln \left (3\right )}^2+54\right )}{14641\,{\left (11\,x+6\right )}^2}-\frac {768\,{\mathrm {e}}^{-32}\,\left (99\,\ln \left (3\right )+121\,{\ln \left (3\right )}^2+18\right )}{14641\,{\left (11\,x+6\right )}^3}-\frac {64\,{\mathrm {e}}^{-32}\,\left (11\,\ln \left (3\right )+6\right )}{14641\,\left (11\,x+6\right )}+\frac {2304\,{\mathrm {e}}^{-32}\,{\left (11\,\ln \left (3\right )+3\right )}^2}{14641\,{\left (11\,x+6\right )}^4} \] Input:
int((exp(-32)*(4*log(3)^2*(192*x - 352*x^2) - 2*log(3)*(576*x^2 - 352*x^3) + 384*x^3))/(71280*x + 261360*x^2 + 479160*x^3 + 439230*x^4 + 161051*x^5 + 7776),x)
Output:
(64*exp(-32)*(198*log(3) + 121*log(3)^2 + 54))/(14641*(11*x + 6)^2) - (768 *exp(-32)*(99*log(3) + 121*log(3)^2 + 18))/(14641*(11*x + 6)^3) - (64*exp( -32)*(11*log(3) + 6))/(14641*(11*x + 6)) + (2304*exp(-32)*(11*log(3) + 3)^ 2)/(14641*(11*x + 6)^4)
Time = 0.15 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.71 \[ \int \frac {384 x^3+\left (-576 x^2+352 x^3\right ) \log (9)+\left (192 x-352 x^2\right ) \log ^2(9)}{e^{32} \left (7776+71280 x+261360 x^2+479160 x^3+439230 x^4+161051 x^5\right )} \, dx=\frac {64 \mathrm {log}\left (3\right )^{2} x^{2}+\frac {88 \,\mathrm {log}\left (3\right ) x^{4}}{3}+\frac {576 \,\mathrm {log}\left (3\right ) x^{2}}{11}+\frac {2304 \,\mathrm {log}\left (3\right ) x}{121}+\frac {3456 \,\mathrm {log}\left (3\right )}{1331}+16 x^{4}}{e^{32} \left (14641 x^{4}+31944 x^{3}+26136 x^{2}+9504 x +1296\right )} \] Input:
int((4*(-352*x^2+192*x)*log(3)^2+2*(352*x^3-576*x^2)*log(3)+384*x^3)/(1610 51*x^5+439230*x^4+479160*x^3+261360*x^2+71280*x+7776)/exp(16)^2,x)
Output:
(8*(31944*log(3)**2*x**2 + 14641*log(3)*x**4 + 26136*log(3)*x**2 + 9504*lo g(3)*x + 1296*log(3) + 7986*x**4))/(3993*e**32*(14641*x**4 + 31944*x**3 + 26136*x**2 + 9504*x + 1296))