Integrand size = 151, antiderivative size = 26 \[ \int \frac {e^{-2 x} \left (\left (-72+48 x-8 x^2+\left (18-12 x+2 x^2+18 x^4-12 x^5+2 x^6\right ) \log \left (\frac {2+2 x^4}{x^4}\right )\right ) \log \left (x \log \left (\frac {2+2 x^4}{x^4}\right )\right )+\left (-24 x+14 x^2-2 x^3-24 x^5+14 x^6-2 x^7\right ) \log \left (\frac {2+2 x^4}{x^4}\right ) \log ^2\left (x \log \left (\frac {2+2 x^4}{x^4}\right )\right )\right )}{\left (x+x^5\right ) \log \left (\frac {2+2 x^4}{x^4}\right )} \, dx=3+e^{-2 x} (-3+x)^2 \log ^2\left (x \log \left (2+\frac {2}{x^4}\right )\right ) \]
\[ \int \frac {e^{-2 x} \left (\left (-72+48 x-8 x^2+\left (18-12 x+2 x^2+18 x^4-12 x^5+2 x^6\right ) \log \left (\frac {2+2 x^4}{x^4}\right )\right ) \log \left (x \log \left (\frac {2+2 x^4}{x^4}\right )\right )+\left (-24 x+14 x^2-2 x^3-24 x^5+14 x^6-2 x^7\right ) \log \left (\frac {2+2 x^4}{x^4}\right ) \log ^2\left (x \log \left (\frac {2+2 x^4}{x^4}\right )\right )\right )}{\left (x+x^5\right ) \log \left (\frac {2+2 x^4}{x^4}\right )} \, dx=\int \frac {e^{-2 x} \left (\left (-72+48 x-8 x^2+\left (18-12 x+2 x^2+18 x^4-12 x^5+2 x^6\right ) \log \left (\frac {2+2 x^4}{x^4}\right )\right ) \log \left (x \log \left (\frac {2+2 x^4}{x^4}\right )\right )+\left (-24 x+14 x^2-2 x^3-24 x^5+14 x^6-2 x^7\right ) \log \left (\frac {2+2 x^4}{x^4}\right ) \log ^2\left (x \log \left (\frac {2+2 x^4}{x^4}\right )\right )\right )}{\left (x+x^5\right ) \log \left (\frac {2+2 x^4}{x^4}\right )} \, dx \]
Integrate[((-72 + 48*x - 8*x^2 + (18 - 12*x + 2*x^2 + 18*x^4 - 12*x^5 + 2* x^6)*Log[(2 + 2*x^4)/x^4])*Log[x*Log[(2 + 2*x^4)/x^4]] + (-24*x + 14*x^2 - 2*x^3 - 24*x^5 + 14*x^6 - 2*x^7)*Log[(2 + 2*x^4)/x^4]*Log[x*Log[(2 + 2*x^ 4)/x^4]]^2)/(E^(2*x)*(x + x^5)*Log[(2 + 2*x^4)/x^4]),x]
Integrate[((-72 + 48*x - 8*x^2 + (18 - 12*x + 2*x^2 + 18*x^4 - 12*x^5 + 2* x^6)*Log[(2 + 2*x^4)/x^4])*Log[x*Log[(2 + 2*x^4)/x^4]] + (-24*x + 14*x^2 - 2*x^3 - 24*x^5 + 14*x^6 - 2*x^7)*Log[(2 + 2*x^4)/x^4]*Log[x*Log[(2 + 2*x^ 4)/x^4]]^2)/(E^(2*x)*(x + x^5)*Log[(2 + 2*x^4)/x^4]), 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.
| \(\displaystyle \int \frac {e^{-2 x} \left (\left (-8 x^2+\left (2 x^6-12 x^5+18 x^4+2 x^2-12 x+18\right ) \log \left (\frac {2 x^4+2}{x^4}\right )+48 x-72\right ) \log \left (x \log \left (\frac {2 x^4+2}{x^4}\right )\right )+\left (-2 x^7+14 x^6-24 x^5-2 x^3+14 x^2-24 x\right ) \log \left (\frac {2 x^4+2}{x^4}\right ) \log ^2\left (x \log \left (\frac {2 x^4+2}{x^4}\right )\right )\right )}{\left (x^5+x\right ) \log \left (\frac {2 x^4+2}{x^4}\right )} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {e^{-2 x} \left (\left (-8 x^2+\left (2 x^6-12 x^5+18 x^4+2 x^2-12 x+18\right ) \log \left (\frac {2 x^4+2}{x^4}\right )+48 x-72\right ) \log \left (x \log \left (\frac {2 x^4+2}{x^4}\right )\right )+\left (-2 x^7+14 x^6-24 x^5-2 x^3+14 x^2-24 x\right ) \log \left (\frac {2 x^4+2}{x^4}\right ) \log ^2\left (x \log \left (\frac {2 x^4+2}{x^4}\right )\right )\right )}{x \left (x^4+1\right ) \log \left (\frac {2 x^4+2}{x^4}\right )}dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {2 e^{-2 x} (3-x) \log \left (x \log \left (\frac {2}{x^4}+2\right )\right ) \left (\left (x^4+1\right ) \log \left (\frac {2}{x^4}+2\right ) \left ((x-4) x \log \left (x \log \left (\frac {2}{x^4}+2\right )\right )-x+3\right )+4 (x-3)\right )}{x \left (x^4+1\right ) \log \left (\frac {2}{x^4}+2\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int -\frac {e^{-2 x} (3-x) \log \left (x \log \left (2+\frac {2}{x^4}\right )\right ) \left (4 (3-x)-\left (x^4+1\right ) \log \left (2+\frac {2}{x^4}\right ) \left (-\left ((4-x) \log \left (x \log \left (2+\frac {2}{x^4}\right )\right ) x\right )-x+3\right )\right )}{x \left (x^4+1\right ) \log \left (2+\frac {2}{x^4}\right )}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -2 \int \frac {e^{-2 x} (3-x) \log \left (x \log \left (2+\frac {2}{x^4}\right )\right ) \left (4 (3-x)-\left (x^4+1\right ) \log \left (2+\frac {2}{x^4}\right ) \left (-\left ((4-x) \log \left (x \log \left (2+\frac {2}{x^4}\right )\right ) x\right )-x+3\right )\right )}{x \left (x^4+1\right ) \log \left (2+\frac {2}{x^4}\right )}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -2 \int \left (e^{-2 x} (x-4) (x-3) \log ^2\left (x \log \left (2+\frac {2}{x^4}\right )\right )-\frac {e^{-2 x} (x-3)^2 \left (\log \left (2+\frac {2}{x^4}\right ) x^4+\log \left (2+\frac {2}{x^4}\right )-4\right ) \log \left (x \log \left (2+\frac {2}{x^4}\right )\right )}{x \left (x^4+1\right ) \log \left (2+\frac {2}{x^4}\right )}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle -2 \int \left (e^{-2 x} (x-4) (x-3) \log ^2\left (x \log \left (2+\frac {2}{x^4}\right )\right )-\frac {e^{-2 x} (x-3)^2 \left (\log \left (2+\frac {2}{x^4}\right ) x^4+\log \left (2+\frac {2}{x^4}\right )-4\right ) \log \left (x \log \left (2+\frac {2}{x^4}\right )\right )}{x \left (x^4+1\right ) \log \left (2+\frac {2}{x^4}\right )}\right )dx\) |
Int[((-72 + 48*x - 8*x^2 + (18 - 12*x + 2*x^2 + 18*x^4 - 12*x^5 + 2*x^6)*L og[(2 + 2*x^4)/x^4])*Log[x*Log[(2 + 2*x^4)/x^4]] + (-24*x + 14*x^2 - 2*x^3 - 24*x^5 + 14*x^6 - 2*x^7)*Log[(2 + 2*x^4)/x^4]*Log[x*Log[(2 + 2*x^4)/x^4 ]]^2)/(E^(2*x)*(x + x^5)*Log[(2 + 2*x^4)/x^4]),x]
3.8.24.3.1 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[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(65\) vs. \(2(25)=50\).
Time = 39.04 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.54
| method | result | size |
| parallelrisch | \(\frac {\left (2 {\ln \left (x \ln \left (\frac {2 x^{4}+2}{x^{4}}\right )\right )}^{2} x^{2}-12 {\ln \left (x \ln \left (\frac {2 x^{4}+2}{x^{4}}\right )\right )}^{2} x +18 {\ln \left (x \ln \left (\frac {2 x^{4}+2}{x^{4}}\right )\right )}^{2}\right ) {\mathrm e}^{-2 x}}{2}\) | \(66\) |
| risch | \(\text {Expression too large to display}\) | \(110816\) |
int(((-2*x^7+14*x^6-24*x^5-2*x^3+14*x^2-24*x)*ln((2*x^4+2)/x^4)*ln(x*ln((2 *x^4+2)/x^4))^2+((2*x^6-12*x^5+18*x^4+2*x^2-12*x+18)*ln((2*x^4+2)/x^4)-8*x ^2+48*x-72)*ln(x*ln((2*x^4+2)/x^4)))/(x^5+x)/exp(x)^2/ln((2*x^4+2)/x^4),x, method=_RETURNVERBOSE)
1/2*(2*ln(x*ln(2*(x^4+1)/x^4))^2*x^2-12*ln(x*ln(2*(x^4+1)/x^4))^2*x+18*ln( x*ln(2*(x^4+1)/x^4))^2)/exp(x)^2
Time = 0.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {e^{-2 x} \left (\left (-72+48 x-8 x^2+\left (18-12 x+2 x^2+18 x^4-12 x^5+2 x^6\right ) \log \left (\frac {2+2 x^4}{x^4}\right )\right ) \log \left (x \log \left (\frac {2+2 x^4}{x^4}\right )\right )+\left (-24 x+14 x^2-2 x^3-24 x^5+14 x^6-2 x^7\right ) \log \left (\frac {2+2 x^4}{x^4}\right ) \log ^2\left (x \log \left (\frac {2+2 x^4}{x^4}\right )\right )\right )}{\left (x+x^5\right ) \log \left (\frac {2+2 x^4}{x^4}\right )} \, dx={\left (x^{2} - 6 \, x + 9\right )} e^{\left (-2 \, x\right )} \log \left (x \log \left (\frac {2 \, {\left (x^{4} + 1\right )}}{x^{4}}\right )\right )^{2} \]
integrate(((-2*x^7+14*x^6-24*x^5-2*x^3+14*x^2-24*x)*log((2*x^4+2)/x^4)*log (x*log((2*x^4+2)/x^4))^2+((2*x^6-12*x^5+18*x^4+2*x^2-12*x+18)*log((2*x^4+2 )/x^4)-8*x^2+48*x-72)*log(x*log((2*x^4+2)/x^4)))/(x^5+x)/exp(x)^2/log((2*x ^4+2)/x^4),x, algorithm=\
Timed out. \[ \int \frac {e^{-2 x} \left (\left (-72+48 x-8 x^2+\left (18-12 x+2 x^2+18 x^4-12 x^5+2 x^6\right ) \log \left (\frac {2+2 x^4}{x^4}\right )\right ) \log \left (x \log \left (\frac {2+2 x^4}{x^4}\right )\right )+\left (-24 x+14 x^2-2 x^3-24 x^5+14 x^6-2 x^7\right ) \log \left (\frac {2+2 x^4}{x^4}\right ) \log ^2\left (x \log \left (\frac {2+2 x^4}{x^4}\right )\right )\right )}{\left (x+x^5\right ) \log \left (\frac {2+2 x^4}{x^4}\right )} \, dx=\text {Timed out} \]
integrate(((-2*x**7+14*x**6-24*x**5-2*x**3+14*x**2-24*x)*ln((2*x**4+2)/x** 4)*ln(x*ln((2*x**4+2)/x**4))**2+((2*x**6-12*x**5+18*x**4+2*x**2-12*x+18)*l n((2*x**4+2)/x**4)-8*x**2+48*x-72)*ln(x*ln((2*x**4+2)/x**4)))/(x**5+x)/exp (x)**2/ln((2*x**4+2)/x**4),x)
Leaf count of result is larger than twice the leaf count of optimal. 77 vs. \(2 (25) = 50\).
Time = 0.38 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.96 \[ \int \frac {e^{-2 x} \left (\left (-72+48 x-8 x^2+\left (18-12 x+2 x^2+18 x^4-12 x^5+2 x^6\right ) \log \left (\frac {2+2 x^4}{x^4}\right )\right ) \log \left (x \log \left (\frac {2+2 x^4}{x^4}\right )\right )+\left (-24 x+14 x^2-2 x^3-24 x^5+14 x^6-2 x^7\right ) \log \left (\frac {2+2 x^4}{x^4}\right ) \log ^2\left (x \log \left (\frac {2+2 x^4}{x^4}\right )\right )\right )}{\left (x+x^5\right ) \log \left (\frac {2+2 x^4}{x^4}\right )} \, dx={\left (x^{2} - 6 \, x + 9\right )} e^{\left (-2 \, x\right )} \log \left (x\right )^{2} + 2 \, {\left (x^{2} - 6 \, x + 9\right )} e^{\left (-2 \, x\right )} \log \left (x\right ) \log \left (\log \left (2\right ) + \log \left (x^{4} + 1\right ) - 4 \, \log \left (x\right )\right ) + {\left (x^{2} - 6 \, x + 9\right )} e^{\left (-2 \, x\right )} \log \left (\log \left (2\right ) + \log \left (x^{4} + 1\right ) - 4 \, \log \left (x\right )\right )^{2} \]
integrate(((-2*x^7+14*x^6-24*x^5-2*x^3+14*x^2-24*x)*log((2*x^4+2)/x^4)*log (x*log((2*x^4+2)/x^4))^2+((2*x^6-12*x^5+18*x^4+2*x^2-12*x+18)*log((2*x^4+2 )/x^4)-8*x^2+48*x-72)*log(x*log((2*x^4+2)/x^4)))/(x^5+x)/exp(x)^2/log((2*x ^4+2)/x^4),x, algorithm=\
(x^2 - 6*x + 9)*e^(-2*x)*log(x)^2 + 2*(x^2 - 6*x + 9)*e^(-2*x)*log(x)*log( log(2) + log(x^4 + 1) - 4*log(x)) + (x^2 - 6*x + 9)*e^(-2*x)*log(log(2) + log(x^4 + 1) - 4*log(x))^2
Timed out. \[ \int \frac {e^{-2 x} \left (\left (-72+48 x-8 x^2+\left (18-12 x+2 x^2+18 x^4-12 x^5+2 x^6\right ) \log \left (\frac {2+2 x^4}{x^4}\right )\right ) \log \left (x \log \left (\frac {2+2 x^4}{x^4}\right )\right )+\left (-24 x+14 x^2-2 x^3-24 x^5+14 x^6-2 x^7\right ) \log \left (\frac {2+2 x^4}{x^4}\right ) \log ^2\left (x \log \left (\frac {2+2 x^4}{x^4}\right )\right )\right )}{\left (x+x^5\right ) \log \left (\frac {2+2 x^4}{x^4}\right )} \, dx=\text {Timed out} \]
integrate(((-2*x^7+14*x^6-24*x^5-2*x^3+14*x^2-24*x)*log((2*x^4+2)/x^4)*log (x*log((2*x^4+2)/x^4))^2+((2*x^6-12*x^5+18*x^4+2*x^2-12*x+18)*log((2*x^4+2 )/x^4)-8*x^2+48*x-72)*log(x*log((2*x^4+2)/x^4)))/(x^5+x)/exp(x)^2/log((2*x ^4+2)/x^4),x, algorithm=\
Time = 12.67 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-2 x} \left (\left (-72+48 x-8 x^2+\left (18-12 x+2 x^2+18 x^4-12 x^5+2 x^6\right ) \log \left (\frac {2+2 x^4}{x^4}\right )\right ) \log \left (x \log \left (\frac {2+2 x^4}{x^4}\right )\right )+\left (-24 x+14 x^2-2 x^3-24 x^5+14 x^6-2 x^7\right ) \log \left (\frac {2+2 x^4}{x^4}\right ) \log ^2\left (x \log \left (\frac {2+2 x^4}{x^4}\right )\right )\right )}{\left (x+x^5\right ) \log \left (\frac {2+2 x^4}{x^4}\right )} \, dx={\ln \left (x\,\ln \left (\frac {2\,\left (x^4+1\right )}{x^4}\right )\right )}^2\,{\mathrm {e}}^{-2\,x}\,{\left (x-3\right )}^2 \]