\(\int \frac {e^{2 \log ^2(x)} (-216+27 x+90 x^3+216 x^4-27 x^5-12 x^7-72 x^8+9 x^9-6 x^{11}+8 x^{12}-x^{13}+(-864-108 x-72 x^3+864 x^4+108 x^5+48 x^7-288 x^8-36 x^9-8 x^{11}+32 x^{12}+4 x^{13}) \log (x))}{-13824-5184 x-648 x^2-3483 x^3+12960 x^4+5130 x^5+360 x^6+2295 x^7-4032 x^8-1700 x^9-120 x^{10}-381 x^{11}+416 x^{12}+186 x^{13}+24 x^{14}+x^{15}} \, dx\) [3393]

Optimal result

Integrand size = 187, antiderivative size = 26 \[ \int \frac {e^{2 \log ^2(x)} \left (-216+27 x+90 x^3+216 x^4-27 x^5-12 x^7-72 x^8+9 x^9-6 x^{11}+8 x^{12}-x^{13}+\left (-864-108 x-72 x^3+864 x^4+108 x^5+48 x^7-288 x^8-36 x^9-8 x^{11}+32 x^{12}+4 x^{13}\right ) \log (x)\right )}{-13824-5184 x-648 x^2-3483 x^3+12960 x^4+5130 x^5+360 x^6+2295 x^7-4032 x^8-1700 x^9-120 x^{10}-381 x^{11}+416 x^{12}+186 x^{13}+24 x^{14}+x^{15}} \, dx=\frac {e^{2 \log ^2(x)} x}{\left (8+x-\frac {2}{-\frac {3}{x^3}+x}\right )^2} \]

[Out]

x/(8+x-2/(x-3/x^3))^2*exp(ln(x)^2)^2

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(137\) vs. \(2(26)=52\).

Time = 0.36 (sec) , antiderivative size = 137, normalized size of antiderivative = 5.27, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.005, Rules used = {2326} \[ \int \frac {e^{2 \log ^2(x)} \left (-216+27 x+90 x^3+216 x^4-27 x^5-12 x^7-72 x^8+9 x^9-6 x^{11}+8 x^{12}-x^{13}+\left (-864-108 x-72 x^3+864 x^4+108 x^5+48 x^7-288 x^8-36 x^9-8 x^{11}+32 x^{12}+4 x^{13}\right ) \log (x)\right )}{-13824-5184 x-648 x^2-3483 x^3+12960 x^4+5130 x^5+360 x^6+2295 x^7-4032 x^8-1700 x^9-120 x^{10}-381 x^{11}+416 x^{12}+186 x^{13}+24 x^{14}+x^{15}} \, dx=\frac {x \left (-x^{13}-8 x^{12}+2 x^{11}+9 x^9+72 x^8-12 x^7-27 x^5-216 x^4+18 x^3+27 x+216\right ) e^{2 \log ^2(x)}}{-x^{15}-24 x^{14}-186 x^{13}-416 x^{12}+381 x^{11}+120 x^{10}+1700 x^9+4032 x^8-2295 x^7-360 x^6-5130 x^5-12960 x^4+3483 x^3+648 x^2+5184 x+13824} \]

[In]

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^1
3 + (-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 - 1
20*x^10 - 381*x^11 + 416*x^12 + 186*x^13 + 24*x^14 + x^15),x]

[Out]

(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))
/(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)

Rule 2326

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]

Rubi steps \begin{align*} \text {integral}& = \frac {e^{2 \log ^2(x)} x \left (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}\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}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.42 \[ \int \frac {e^{2 \log ^2(x)} \left (-216+27 x+90 x^3+216 x^4-27 x^5-12 x^7-72 x^8+9 x^9-6 x^{11}+8 x^{12}-x^{13}+\left (-864-108 x-72 x^3+864 x^4+108 x^5+48 x^7-288 x^8-36 x^9-8 x^{11}+32 x^{12}+4 x^{13}\right ) \log (x)\right )}{-13824-5184 x-648 x^2-3483 x^3+12960 x^4+5130 x^5+360 x^6+2295 x^7-4032 x^8-1700 x^9-120 x^{10}-381 x^{11}+416 x^{12}+186 x^{13}+24 x^{14}+x^{15}} \, dx=\frac {e^{2 \log ^2(x)} x \left (-3+x^4\right )^2}{\left (-24-3 x-2 x^3+8 x^4+x^5\right )^2} \]

[In]

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)*
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]

[Out]

(E^(2*Log[x]^2)*x*(-3 + x^4)^2)/(-24 - 3*x - 2*x^3 + 8*x^4 + x^5)^2

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(69\) vs. \(2(25)=50\).

Time = 354.49 (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	\(\frac {864 x \,{\mathrm e}^{2 \ln \left (x \right )^{2}}-576 \,{\mathrm e}^{2 \ln \left (x \right )^{2}} x^{5}+96 \,{\mathrm e}^{2 \ln \left (x \right )^{2}} x^{9}}{96 x^{10}+1536 x^{9}+5760 x^{8}-3072 x^{7}-192 x^{6}-9216 x^{5}-35712 x^{4}+9216 x^{3}+864 x^{2}+13824 x +55296}\)	\(88\)

[In]

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^3-108*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,method=_RETURNVER
BOSE)

[Out]

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)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (25) = 50\).

Time = 0.25 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.69 \[ \int \frac {e^{2 \log ^2(x)} \left (-216+27 x+90 x^3+216 x^4-27 x^5-12 x^7-72 x^8+9 x^9-6 x^{11}+8 x^{12}-x^{13}+\left (-864-108 x-72 x^3+864 x^4+108 x^5+48 x^7-288 x^8-36 x^9-8 x^{11}+32 x^{12}+4 x^{13}\right ) \log (x)\right )}{-13824-5184 x-648 x^2-3483 x^3+12960 x^4+5130 x^5+360 x^6+2295 x^7-4032 x^8-1700 x^9-120 x^{10}-381 x^{11}+416 x^{12}+186 x^{13}+24 x^{14}+x^{15}} \, dx=\frac {{\left (x^{9} - 6 \, x^{5} + 9 \, x\right )} e^{\left (2 \, \log \left (x\right )^{2}\right )}}{x^{10} + 16 \, x^{9} + 60 \, x^{8} - 32 \, x^{7} - 2 \, x^{6} - 96 \, x^{5} - 372 \, x^{4} + 96 \, x^{3} + 9 \, x^{2} + 144 \, x + 576} \]

[In]

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, algorith
m="fricas")

[Out]

(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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (22) = 44\).

Time = 0.21 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.62 \[ \int \frac {e^{2 \log ^2(x)} \left (-216+27 x+90 x^3+216 x^4-27 x^5-12 x^7-72 x^8+9 x^9-6 x^{11}+8 x^{12}-x^{13}+\left (-864-108 x-72 x^3+864 x^4+108 x^5+48 x^7-288 x^8-36 x^9-8 x^{11}+32 x^{12}+4 x^{13}\right ) \log (x)\right )}{-13824-5184 x-648 x^2-3483 x^3+12960 x^4+5130 x^5+360 x^6+2295 x^7-4032 x^8-1700 x^9-120 x^{10}-381 x^{11}+416 x^{12}+186 x^{13}+24 x^{14}+x^{15}} \, dx=\frac {\left (x^{9} - 6 x^{5} + 9 x\right ) e^{2 \log {\left (x \right )}^{2}}}{x^{10} + 16 x^{9} + 60 x^{8} - 32 x^{7} - 2 x^{6} - 96 x^{5} - 372 x^{4} + 96 x^{3} + 9 x^{2} + 144 x + 576} \]

[In]

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+18
6*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**3-64
8*x**2-5184*x-13824),x)

[Out]

(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)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (25) = 50\).

Time = 0.31 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.69 \[ \int \frac {e^{2 \log ^2(x)} \left (-216+27 x+90 x^3+216 x^4-27 x^5-12 x^7-72 x^8+9 x^9-6 x^{11}+8 x^{12}-x^{13}+\left (-864-108 x-72 x^3+864 x^4+108 x^5+48 x^7-288 x^8-36 x^9-8 x^{11}+32 x^{12}+4 x^{13}\right ) \log (x)\right )}{-13824-5184 x-648 x^2-3483 x^3+12960 x^4+5130 x^5+360 x^6+2295 x^7-4032 x^8-1700 x^9-120 x^{10}-381 x^{11}+416 x^{12}+186 x^{13}+24 x^{14}+x^{15}} \, dx=\frac {{\left (x^{9} - 6 \, x^{5} + 9 \, x\right )} e^{\left (2 \, \log \left (x\right )^{2}\right )}}{x^{10} + 16 \, x^{9} + 60 \, x^{8} - 32 \, x^{7} - 2 \, x^{6} - 96 \, x^{5} - 372 \, x^{4} + 96 \, x^{3} + 9 \, x^{2} + 144 \, x + 576} \]

[In]

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, algorith
m="maxima")

[Out]

(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)

Giac [F]

\[ \int \frac {e^{2 \log ^2(x)} \left (-216+27 x+90 x^3+216 x^4-27 x^5-12 x^7-72 x^8+9 x^9-6 x^{11}+8 x^{12}-x^{13}+\left (-864-108 x-72 x^3+864 x^4+108 x^5+48 x^7-288 x^8-36 x^9-8 x^{11}+32 x^{12}+4 x^{13}\right ) \log (x)\right )}{-13824-5184 x-648 x^2-3483 x^3+12960 x^4+5130 x^5+360 x^6+2295 x^7-4032 x^8-1700 x^9-120 x^{10}-381 x^{11}+416 x^{12}+186 x^{13}+24 x^{14}+x^{15}} \, dx=\int { -\frac {{\left (x^{13} - 8 \, x^{12} + 6 \, x^{11} - 9 \, x^{9} + 72 \, x^{8} + 12 \, x^{7} + 27 \, x^{5} - 216 \, x^{4} - 90 \, x^{3} - 4 \, {\left (x^{13} + 8 \, x^{12} - 2 \, x^{11} - 9 \, x^{9} - 72 \, x^{8} + 12 \, x^{7} + 27 \, x^{5} + 216 \, x^{4} - 18 \, x^{3} - 27 \, x - 216\right )} \log \left (x\right ) - 27 \, x + 216\right )} e^{\left (2 \, \log \left (x\right )^{2}\right )}}{x^{15} + 24 \, x^{14} + 186 \, x^{13} + 416 \, x^{12} - 381 \, x^{11} - 120 \, x^{10} - 1700 \, x^{9} - 4032 \, x^{8} + 2295 \, x^{7} + 360 \, x^{6} + 5130 \, x^{5} + 12960 \, x^{4} - 3483 \, x^{3} - 648 \, x^{2} - 5184 \, x - 13824} \,d x } \]

[In]

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, algorith
m="giac")

[Out]

integrate(-(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*(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 - 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)

Mupad [B] (verification not implemented)

Time = 8.82 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.46 \[ \int \frac {e^{2 \log ^2(x)} \left (-216+27 x+90 x^3+216 x^4-27 x^5-12 x^7-72 x^8+9 x^9-6 x^{11}+8 x^{12}-x^{13}+\left (-864-108 x-72 x^3+864 x^4+108 x^5+48 x^7-288 x^8-36 x^9-8 x^{11}+32 x^{12}+4 x^{13}\right ) \log (x)\right )}{-13824-5184 x-648 x^2-3483 x^3+12960 x^4+5130 x^5+360 x^6+2295 x^7-4032 x^8-1700 x^9-120 x^{10}-381 x^{11}+416 x^{12}+186 x^{13}+24 x^{14}+x^{15}} \, dx=\frac {x\,{\mathrm {e}}^{2\,{\ln \left (x\right )}^2}\,{\left (x^4-3\right )}^2}{{\left (-x^5-8\,x^4+2\,x^3+3\,x+24\right )}^2} \]

[In]

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^1
2 - 4*x^13 + 864) - 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 + 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)

[Out]

(x*exp(2*log(x)^2)*(x^4 - 3)^2)/(3*x + 2*x^3 - 8*x^4 - x^5 + 24)^2