Integrand size = 187, antiderivative size = 26 \[ \int \frac {e^{2 \log ^2(x)} \left (-216+27 x+90 x^3+216 x^4-27 x^5-12 x^7-72 x^8+9 x^9-6 x^{11}+8 x^{12}-x^{13}+\left (-864-108 x-72 x^3+864 x^4+108 x^5+48 x^7-288 x^8-36 x^9-8 x^{11}+32 x^{12}+4 x^{13}\right ) \log (x)\right )}{-13824-5184 x-648 x^2-3483 x^3+12960 x^4+5130 x^5+360 x^6+2295 x^7-4032 x^8-1700 x^9-120 x^{10}-381 x^{11}+416 x^{12}+186 x^{13}+24 x^{14}+x^{15}} \, dx=\frac {e^{2 \log ^2(x)} x}{\left (8+x-\frac {2}{-\frac {3}{x^3}+x}\right )^2} \] Output:
x/(8+x-2/(x-3/x^3))^2*exp(ln(x)^2)^2
Time = 0.06 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.42 \[ \int \frac {e^{2 \log ^2(x)} \left (-216+27 x+90 x^3+216 x^4-27 x^5-12 x^7-72 x^8+9 x^9-6 x^{11}+8 x^{12}-x^{13}+\left (-864-108 x-72 x^3+864 x^4+108 x^5+48 x^7-288 x^8-36 x^9-8 x^{11}+32 x^{12}+4 x^{13}\right ) \log (x)\right )}{-13824-5184 x-648 x^2-3483 x^3+12960 x^4+5130 x^5+360 x^6+2295 x^7-4032 x^8-1700 x^9-120 x^{10}-381 x^{11}+416 x^{12}+186 x^{13}+24 x^{14}+x^{15}} \, dx=\frac {e^{2 \log ^2(x)} x \left (-3+x^4\right )^2}{\left (-24-3 x-2 x^3+8 x^4+x^5\right )^2} \] Input:
Integrate[(E^(2*Log[x]^2)*(-216 + 27*x + 90*x^3 + 216*x^4 - 27*x^5 - 12*x^ 7 - 72*x^8 + 9*x^9 - 6*x^11 + 8*x^12 - x^13 + (-864 - 108*x - 72*x^3 + 864 *x^4 + 108*x^5 + 48*x^7 - 288*x^8 - 36*x^9 - 8*x^11 + 32*x^12 + 4*x^13)*Lo g[x]))/(-13824 - 5184*x - 648*x^2 - 3483*x^3 + 12960*x^4 + 5130*x^5 + 360* x^6 + 2295*x^7 - 4032*x^8 - 1700*x^9 - 120*x^10 - 381*x^11 + 416*x^12 + 18 6*x^13 + 24*x^14 + x^15),x]
Output:
(E^(2*Log[x]^2)*x*(-3 + x^4)^2)/(-24 - 3*x - 2*x^3 + 8*x^4 + x^5)^2
Leaf count is larger than twice the leaf count of optimal. \(82\) vs. \(2(26)=52\).
Time = 0.71 (sec) , antiderivative size = 82, normalized size of antiderivative = 3.15, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.011, Rules used = {2463, 2726}
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.
\(\displaystyle \int \frac {e^{2 \log ^2(x)} \left (-x^{13}+8 x^{12}-6 x^{11}+9 x^9-72 x^8-12 x^7-27 x^5+216 x^4+90 x^3+\left (4 x^{13}+32 x^{12}-8 x^{11}-36 x^9-288 x^8+48 x^7+108 x^5+864 x^4-72 x^3-108 x-864\right ) \log (x)+27 x-216\right )}{x^{15}+24 x^{14}+186 x^{13}+416 x^{12}-381 x^{11}-120 x^{10}-1700 x^9-4032 x^8+2295 x^7+360 x^6+5130 x^5+12960 x^4-3483 x^3-648 x^2-5184 x-13824} \, dx\) |
\(\Big \downarrow \) 2463 |
\(\displaystyle \int \frac {e^{2 \log ^2(x)} \left (-x^{13}+8 x^{12}-6 x^{11}+9 x^9-72 x^8-12 x^7-27 x^5+216 x^4+90 x^3+\left (4 x^{13}+32 x^{12}-8 x^{11}-36 x^9-288 x^8+48 x^7+108 x^5+864 x^4-72 x^3-108 x-864\right ) \log (x)+27 x-216\right )}{\left (x^5+8 x^4-2 x^3-3 x-24\right )^3}dx\) |
\(\Big \downarrow \) 2726 |
\(\displaystyle \frac {x \left (-x^{13}-8 x^{12}+2 x^{11}+9 x^9+72 x^8-12 x^7-27 x^5-216 x^4+18 x^3+27 x+216\right ) e^{2 \log ^2(x)}}{\left (-x^5-8 x^4+2 x^3+3 x+24\right )^3}\) |
Input:
Int[(E^(2*Log[x]^2)*(-216 + 27*x + 90*x^3 + 216*x^4 - 27*x^5 - 12*x^7 - 72 *x^8 + 9*x^9 - 6*x^11 + 8*x^12 - x^13 + (-864 - 108*x - 72*x^3 + 864*x^4 + 108*x^5 + 48*x^7 - 288*x^8 - 36*x^9 - 8*x^11 + 32*x^12 + 4*x^13)*Log[x])) /(-13824 - 5184*x - 648*x^2 - 3483*x^3 + 12960*x^4 + 5130*x^5 + 360*x^6 + 2295*x^7 - 4032*x^8 - 1700*x^9 - 120*x^10 - 381*x^11 + 416*x^12 + 186*x^13 + 24*x^14 + x^15),x]
Output:
(E^(2*Log[x]^2)*x*(216 + 27*x + 18*x^3 - 216*x^4 - 27*x^5 - 12*x^7 + 72*x^ 8 + 9*x^9 + 2*x^11 - 8*x^12 - x^13))/(24 + 3*x + 2*x^3 - 8*x^4 - x^5)^3
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u, Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, x]))}, Simp[F^u*z, x] /; EqQ[D[z, x], w*y]] /; FreeQ[F, x]
Leaf count of result is larger than twice the leaf count of optimal. \(69\) vs. \(2(25)=50\).
Time = 67.04 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.69
method | result | size |
risch | \(\frac {x \left (x^{8}-6 x^{4}+9\right ) {\mathrm e}^{2 \ln \left (x \right )^{2}}}{x^{10}+16 x^{9}+60 x^{8}-32 x^{7}-2 x^{6}-96 x^{5}-372 x^{4}+96 x^{3}+9 x^{2}+144 x +576}\) | \(70\) |
parallelrisch | \(\frac {864 \,{\mathrm e}^{2 \ln \left (x \right )^{2}} x +96 \,{\mathrm e}^{2 \ln \left (x \right )^{2}} x^{9}-576 \,{\mathrm e}^{2 \ln \left (x \right )^{2}} x^{5}}{96 x^{10}+1536 x^{9}+5760 x^{8}-3072 x^{7}-192 x^{6}-9216 x^{5}-35712 x^{4}+9216 x^{3}+864 x^{2}+13824 x +55296}\) | \(88\) |
Input:
int(((4*x^13+32*x^12-8*x^11-36*x^9-288*x^8+48*x^7+108*x^5+864*x^4-72*x^3-1 08*x-864)*ln(x)-x^13+8*x^12-6*x^11+9*x^9-72*x^8-12*x^7-27*x^5+216*x^4+90*x ^3+27*x-216)*exp(ln(x)^2)^2/(x^15+24*x^14+186*x^13+416*x^12-381*x^11-120*x ^10-1700*x^9-4032*x^8+2295*x^7+360*x^6+5130*x^5+12960*x^4-3483*x^3-648*x^2 -5184*x-13824),x,method=_RETURNVERBOSE)
Output:
x*(x^8-6*x^4+9)/(x^10+16*x^9+60*x^8-32*x^7-2*x^6-96*x^5-372*x^4+96*x^3+9*x ^2+144*x+576)*exp(2*ln(x)^2)
Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (25) = 50\).
Time = 0.10 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.69 \[ \int \frac {e^{2 \log ^2(x)} \left (-216+27 x+90 x^3+216 x^4-27 x^5-12 x^7-72 x^8+9 x^9-6 x^{11}+8 x^{12}-x^{13}+\left (-864-108 x-72 x^3+864 x^4+108 x^5+48 x^7-288 x^8-36 x^9-8 x^{11}+32 x^{12}+4 x^{13}\right ) \log (x)\right )}{-13824-5184 x-648 x^2-3483 x^3+12960 x^4+5130 x^5+360 x^6+2295 x^7-4032 x^8-1700 x^9-120 x^{10}-381 x^{11}+416 x^{12}+186 x^{13}+24 x^{14}+x^{15}} \, dx=\frac {{\left (x^{9} - 6 \, x^{5} + 9 \, x\right )} e^{\left (2 \, \log \left (x\right )^{2}\right )}}{x^{10} + 16 \, x^{9} + 60 \, x^{8} - 32 \, x^{7} - 2 \, x^{6} - 96 \, x^{5} - 372 \, x^{4} + 96 \, x^{3} + 9 \, x^{2} + 144 \, x + 576} \] Input:
integrate(((4*x^13+32*x^12-8*x^11-36*x^9-288*x^8+48*x^7+108*x^5+864*x^4-72 *x^3-108*x-864)*log(x)-x^13+8*x^12-6*x^11+9*x^9-72*x^8-12*x^7-27*x^5+216*x ^4+90*x^3+27*x-216)*exp(log(x)^2)^2/(x^15+24*x^14+186*x^13+416*x^12-381*x^ 11-120*x^10-1700*x^9-4032*x^8+2295*x^7+360*x^6+5130*x^5+12960*x^4-3483*x^3 -648*x^2-5184*x-13824),x, algorithm="fricas")
Output:
(x^9 - 6*x^5 + 9*x)*e^(2*log(x)^2)/(x^10 + 16*x^9 + 60*x^8 - 32*x^7 - 2*x^ 6 - 96*x^5 - 372*x^4 + 96*x^3 + 9*x^2 + 144*x + 576)
Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (22) = 44\).
Time = 0.27 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.62 \[ \int \frac {e^{2 \log ^2(x)} \left (-216+27 x+90 x^3+216 x^4-27 x^5-12 x^7-72 x^8+9 x^9-6 x^{11}+8 x^{12}-x^{13}+\left (-864-108 x-72 x^3+864 x^4+108 x^5+48 x^7-288 x^8-36 x^9-8 x^{11}+32 x^{12}+4 x^{13}\right ) \log (x)\right )}{-13824-5184 x-648 x^2-3483 x^3+12960 x^4+5130 x^5+360 x^6+2295 x^7-4032 x^8-1700 x^9-120 x^{10}-381 x^{11}+416 x^{12}+186 x^{13}+24 x^{14}+x^{15}} \, dx=\frac {\left (x^{9} - 6 x^{5} + 9 x\right ) e^{2 \log {\left (x \right )}^{2}}}{x^{10} + 16 x^{9} + 60 x^{8} - 32 x^{7} - 2 x^{6} - 96 x^{5} - 372 x^{4} + 96 x^{3} + 9 x^{2} + 144 x + 576} \] Input:
integrate(((4*x**13+32*x**12-8*x**11-36*x**9-288*x**8+48*x**7+108*x**5+864 *x**4-72*x**3-108*x-864)*ln(x)-x**13+8*x**12-6*x**11+9*x**9-72*x**8-12*x** 7-27*x**5+216*x**4+90*x**3+27*x-216)*exp(ln(x)**2)**2/(x**15+24*x**14+186* x**13+416*x**12-381*x**11-120*x**10-1700*x**9-4032*x**8+2295*x**7+360*x**6 +5130*x**5+12960*x**4-3483*x**3-648*x**2-5184*x-13824),x)
Output:
(x**9 - 6*x**5 + 9*x)*exp(2*log(x)**2)/(x**10 + 16*x**9 + 60*x**8 - 32*x** 7 - 2*x**6 - 96*x**5 - 372*x**4 + 96*x**3 + 9*x**2 + 144*x + 576)
Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (25) = 50\).
Time = 0.18 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.69 \[ \int \frac {e^{2 \log ^2(x)} \left (-216+27 x+90 x^3+216 x^4-27 x^5-12 x^7-72 x^8+9 x^9-6 x^{11}+8 x^{12}-x^{13}+\left (-864-108 x-72 x^3+864 x^4+108 x^5+48 x^7-288 x^8-36 x^9-8 x^{11}+32 x^{12}+4 x^{13}\right ) \log (x)\right )}{-13824-5184 x-648 x^2-3483 x^3+12960 x^4+5130 x^5+360 x^6+2295 x^7-4032 x^8-1700 x^9-120 x^{10}-381 x^{11}+416 x^{12}+186 x^{13}+24 x^{14}+x^{15}} \, dx=\frac {{\left (x^{9} - 6 \, x^{5} + 9 \, x\right )} e^{\left (2 \, \log \left (x\right )^{2}\right )}}{x^{10} + 16 \, x^{9} + 60 \, x^{8} - 32 \, x^{7} - 2 \, x^{6} - 96 \, x^{5} - 372 \, x^{4} + 96 \, x^{3} + 9 \, x^{2} + 144 \, x + 576} \] Input:
integrate(((4*x^13+32*x^12-8*x^11-36*x^9-288*x^8+48*x^7+108*x^5+864*x^4-72 *x^3-108*x-864)*log(x)-x^13+8*x^12-6*x^11+9*x^9-72*x^8-12*x^7-27*x^5+216*x ^4+90*x^3+27*x-216)*exp(log(x)^2)^2/(x^15+24*x^14+186*x^13+416*x^12-381*x^ 11-120*x^10-1700*x^9-4032*x^8+2295*x^7+360*x^6+5130*x^5+12960*x^4-3483*x^3 -648*x^2-5184*x-13824),x, algorithm="maxima")
Output:
(x^9 - 6*x^5 + 9*x)*e^(2*log(x)^2)/(x^10 + 16*x^9 + 60*x^8 - 32*x^7 - 2*x^ 6 - 96*x^5 - 372*x^4 + 96*x^3 + 9*x^2 + 144*x + 576)
\[ \int \frac {e^{2 \log ^2(x)} \left (-216+27 x+90 x^3+216 x^4-27 x^5-12 x^7-72 x^8+9 x^9-6 x^{11}+8 x^{12}-x^{13}+\left (-864-108 x-72 x^3+864 x^4+108 x^5+48 x^7-288 x^8-36 x^9-8 x^{11}+32 x^{12}+4 x^{13}\right ) \log (x)\right )}{-13824-5184 x-648 x^2-3483 x^3+12960 x^4+5130 x^5+360 x^6+2295 x^7-4032 x^8-1700 x^9-120 x^{10}-381 x^{11}+416 x^{12}+186 x^{13}+24 x^{14}+x^{15}} \, dx=\int { -\frac {{\left (x^{13} - 8 \, x^{12} + 6 \, x^{11} - 9 \, x^{9} + 72 \, x^{8} + 12 \, x^{7} + 27 \, x^{5} - 216 \, x^{4} - 90 \, x^{3} - 4 \, {\left (x^{13} + 8 \, x^{12} - 2 \, x^{11} - 9 \, x^{9} - 72 \, x^{8} + 12 \, x^{7} + 27 \, x^{5} + 216 \, x^{4} - 18 \, x^{3} - 27 \, x - 216\right )} \log \left (x\right ) - 27 \, x + 216\right )} e^{\left (2 \, \log \left (x\right )^{2}\right )}}{x^{15} + 24 \, x^{14} + 186 \, x^{13} + 416 \, x^{12} - 381 \, x^{11} - 120 \, x^{10} - 1700 \, x^{9} - 4032 \, x^{8} + 2295 \, x^{7} + 360 \, x^{6} + 5130 \, x^{5} + 12960 \, x^{4} - 3483 \, x^{3} - 648 \, x^{2} - 5184 \, x - 13824} \,d x } \] Input:
integrate(((4*x^13+32*x^12-8*x^11-36*x^9-288*x^8+48*x^7+108*x^5+864*x^4-72 *x^3-108*x-864)*log(x)-x^13+8*x^12-6*x^11+9*x^9-72*x^8-12*x^7-27*x^5+216*x ^4+90*x^3+27*x-216)*exp(log(x)^2)^2/(x^15+24*x^14+186*x^13+416*x^12-381*x^ 11-120*x^10-1700*x^9-4032*x^8+2295*x^7+360*x^6+5130*x^5+12960*x^4-3483*x^3 -648*x^2-5184*x-13824),x, algorithm="giac")
Output:
integrate(-(x^13 - 8*x^12 + 6*x^11 - 9*x^9 + 72*x^8 + 12*x^7 + 27*x^5 - 21 6*x^4 - 90*x^3 - 4*(x^13 + 8*x^12 - 2*x^11 - 9*x^9 - 72*x^8 + 12*x^7 + 27* x^5 + 216*x^4 - 18*x^3 - 27*x - 216)*log(x) - 27*x + 216)*e^(2*log(x)^2)/( x^15 + 24*x^14 + 186*x^13 + 416*x^12 - 381*x^11 - 120*x^10 - 1700*x^9 - 40 32*x^8 + 2295*x^7 + 360*x^6 + 5130*x^5 + 12960*x^4 - 3483*x^3 - 648*x^2 - 5184*x - 13824), x)
Time = 2.70 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.46 \[ \int \frac {e^{2 \log ^2(x)} \left (-216+27 x+90 x^3+216 x^4-27 x^5-12 x^7-72 x^8+9 x^9-6 x^{11}+8 x^{12}-x^{13}+\left (-864-108 x-72 x^3+864 x^4+108 x^5+48 x^7-288 x^8-36 x^9-8 x^{11}+32 x^{12}+4 x^{13}\right ) \log (x)\right )}{-13824-5184 x-648 x^2-3483 x^3+12960 x^4+5130 x^5+360 x^6+2295 x^7-4032 x^8-1700 x^9-120 x^{10}-381 x^{11}+416 x^{12}+186 x^{13}+24 x^{14}+x^{15}} \, dx=\frac {x\,{\mathrm {e}}^{2\,{\ln \left (x\right )}^2}\,{\left (x^4-3\right )}^2}{{\left (-x^5-8\,x^4+2\,x^3+3\,x+24\right )}^2} \] Input:
int((exp(2*log(x)^2)*(log(x)*(108*x + 72*x^3 - 864*x^4 - 108*x^5 - 48*x^7 + 288*x^8 + 36*x^9 + 8*x^11 - 32*x^12 - 4*x^13 + 864) - 27*x - 90*x^3 - 21 6*x^4 + 27*x^5 + 12*x^7 + 72*x^8 - 9*x^9 + 6*x^11 - 8*x^12 + x^13 + 216))/ (5184*x + 648*x^2 + 3483*x^3 - 12960*x^4 - 5130*x^5 - 360*x^6 - 2295*x^7 + 4032*x^8 + 1700*x^9 + 120*x^10 + 381*x^11 - 416*x^12 - 186*x^13 - 24*x^14 - x^15 + 13824),x)
Output:
(x*exp(2*log(x)^2)*(x^4 - 3)^2)/(3*x + 2*x^3 - 8*x^4 - x^5 + 24)^2
Time = 0.22 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.69 \[ \int \frac {e^{2 \log ^2(x)} \left (-216+27 x+90 x^3+216 x^4-27 x^5-12 x^7-72 x^8+9 x^9-6 x^{11}+8 x^{12}-x^{13}+\left (-864-108 x-72 x^3+864 x^4+108 x^5+48 x^7-288 x^8-36 x^9-8 x^{11}+32 x^{12}+4 x^{13}\right ) \log (x)\right )}{-13824-5184 x-648 x^2-3483 x^3+12960 x^4+5130 x^5+360 x^6+2295 x^7-4032 x^8-1700 x^9-120 x^{10}-381 x^{11}+416 x^{12}+186 x^{13}+24 x^{14}+x^{15}} \, dx=\frac {e^{2 \mathrm {log}\left (x \right )^{2}} x \left (x^{8}-6 x^{4}+9\right )}{x^{10}+16 x^{9}+60 x^{8}-32 x^{7}-2 x^{6}-96 x^{5}-372 x^{4}+96 x^{3}+9 x^{2}+144 x +576} \] Input:
int(((4*x^13+32*x^12-8*x^11-36*x^9-288*x^8+48*x^7+108*x^5+864*x^4-72*x^3-1 08*x-864)*log(x)-x^13+8*x^12-6*x^11+9*x^9-72*x^8-12*x^7-27*x^5+216*x^4+90* x^3+27*x-216)*exp(log(x)^2)^2/(x^15+24*x^14+186*x^13+416*x^12-381*x^11-120 *x^10-1700*x^9-4032*x^8+2295*x^7+360*x^6+5130*x^5+12960*x^4-3483*x^3-648*x ^2-5184*x-13824),x)
Output:
(e**(2*log(x)**2)*x*(x**8 - 6*x**4 + 9))/(x**10 + 16*x**9 + 60*x**8 - 32*x **7 - 2*x**6 - 96*x**5 - 372*x**4 + 96*x**3 + 9*x**2 + 144*x + 576)