Integrand size = 79, antiderivative size = 25 \[ \int \frac {\left (-80-256 e^2-192 x\right ) \log (2)}{625 x^2+20480 e^6 x^2+1500 x^3+1200 x^4+320 x^5+e^4 \left (19200 x^2+15360 x^3\right )+e^2 \left (6000 x^2+9600 x^3+3840 x^4\right )} \, dx=e^2+\frac {\log (2)}{5 x \left (\frac {5}{4}+4 e^2+x\right )^2} \] Output:
exp(2)+1/5*ln(2)/(x+5/4+4*exp(2))^2/x
Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {\left (-80-256 e^2-192 x\right ) \log (2)}{625 x^2+20480 e^6 x^2+1500 x^3+1200 x^4+320 x^5+e^4 \left (19200 x^2+15360 x^3\right )+e^2 \left (6000 x^2+9600 x^3+3840 x^4\right )} \, dx=\frac {16 \log (2)}{5 x \left (5+16 e^2+4 x\right )^2} \] Input:
Integrate[((-80 - 256*E^2 - 192*x)*Log[2])/(625*x^2 + 20480*E^6*x^2 + 1500 *x^3 + 1200*x^4 + 320*x^5 + E^4*(19200*x^2 + 15360*x^3) + E^2*(6000*x^2 + 9600*x^3 + 3840*x^4)),x]
Output:
(16*Log[2])/(5*x*(5 + 16*E^2 + 4*x)^2)
Time = 0.46 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.076, Rules used = {6, 27, 27, 2026, 2007, 83}
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.
\(\displaystyle \int \frac {\left (-192 x-256 e^2-80\right ) \log (2)}{320 x^5+1200 x^4+1500 x^3+20480 e^6 x^2+625 x^2+e^4 \left (15360 x^3+19200 x^2\right )+e^2 \left (3840 x^4+9600 x^3+6000 x^2\right )} \, dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {\left (-192 x-256 e^2-80\right ) \log (2)}{320 x^5+1200 x^4+1500 x^3+\left (625+20480 e^6\right ) x^2+e^4 \left (15360 x^3+19200 x^2\right )+e^2 \left (3840 x^4+9600 x^3+6000 x^2\right )}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \log (2) \int -\frac {16 \left (12 x+16 e^2+5\right )}{5 \left (64 x^5+240 x^4+300 x^3+\left (125+4096 e^6\right ) x^2+768 e^4 \left (4 x^3+5 x^2\right )+48 e^2 \left (16 x^4+40 x^3+25 x^2\right )\right )}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {16}{5} \log (2) \int \frac {12 x+16 e^2+5}{64 x^5+240 x^4+300 x^3+\left (125+4096 e^6\right ) x^2+768 e^4 \left (4 x^3+5 x^2\right )+48 e^2 \left (16 x^4+40 x^3+25 x^2\right )}dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle -\frac {16}{5} \log (2) \int \frac {12 x+16 e^2+5}{x^2 \left (64 x^3+48 \left (5+16 e^2\right ) x^2+12 \left (5+16 e^2\right )^2 x+\left (5+16 e^2\right )^3\right )}dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle -\frac {16}{5} \log (2) \int \frac {12 x+16 e^2+5}{x^2 \left (4 x+16 e^2+5\right )^3}dx\) |
\(\Big \downarrow \) 83 |
\(\displaystyle \frac {16 \log (2)}{5 x \left (4 x+16 e^2+5\right )^2}\) |
Input:
Int[((-80 - 256*E^2 - 192*x)*Log[2])/(625*x^2 + 20480*E^6*x^2 + 1500*x^3 + 1200*x^4 + 320*x^5 + E^4*(19200*x^2 + 15360*x^3) + E^2*(6000*x^2 + 9600*x ^3 + 3840*x^4)),x]
Output:
(16*Log[2])/(5*x*(5 + 16*E^2 + 4*x)^2)
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] && EqQ[a*d*f *(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]
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[(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])
Time = 0.19 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76
method | result | size |
norman | \(\frac {16 \ln \left (2\right )}{5 x \left (16 \,{\mathrm e}^{2}+4 x +5\right )^{2}}\) | \(19\) |
risch | \(\frac {16 \ln \left (2\right )}{5 x \left (256 \,{\mathrm e}^{4}+128 \,{\mathrm e}^{2} x +16 x^{2}+160 \,{\mathrm e}^{2}+40 x +25\right )}\) | \(33\) |
gosper | \(\frac {16 \ln \left (2\right )}{5 x \left (256 \,{\mathrm e}^{4}+128 \,{\mathrm e}^{2} x +16 x^{2}+160 \,{\mathrm e}^{2}+40 x +25\right )}\) | \(35\) |
parallelrisch | \(\frac {16 \ln \left (2\right )}{5 x \left (256 \,{\mathrm e}^{4}+128 \,{\mathrm e}^{2} x +16 x^{2}+160 \,{\mathrm e}^{2}+40 x +25\right )}\) | \(35\) |
default | \(16 \ln \left (2\right ) \left (-\frac {-65536 \,{\mathrm e}^{8}-81920 \,{\mathrm e}^{6}-38400 \,{\mathrm e}^{4}-8000 \,{\mathrm e}^{2}-625}{5 \left (4096 \,{\mathrm e}^{6}+3840 \,{\mathrm e}^{4}+1200 \,{\mathrm e}^{2}+125\right )^{2} x}+\frac {4 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (64 \textit {\_Z}^{3}+\left (768 \,{\mathrm e}^{2}+240\right ) \textit {\_Z}^{2}+\left (3072 \,{\mathrm e}^{4}+1920 \,{\mathrm e}^{2}+300\right ) \textit {\_Z} +4096 \,{\mathrm e}^{6}+3840 \,{\mathrm e}^{4}+1200 \,{\mathrm e}^{2}+125\right )}{\sum }\frac {\left (9375+262144 \textit {\_R} \,{\mathrm e}^{8}+327680 \textit {\_R} \,{\mathrm e}^{6}+153600 \textit {\_R} \,{\mathrm e}^{4}+32000 \,{\mathrm e}^{2} \textit {\_R} +4915200 \,{\mathrm e}^{8}+3145728 \,{\mathrm e}^{10}+3072000 \,{\mathrm e}^{6}+960000 \,{\mathrm e}^{4}+150000 \,{\mathrm e}^{2}+2500 \textit {\_R} \right ) \ln \left (x -\textit {\_R} \right )}{25+256 \,{\mathrm e}^{4}+128 \,{\mathrm e}^{2} \textit {\_R} +16 \textit {\_R}^{2}+160 \,{\mathrm e}^{2}+40 \textit {\_R}}\right )}{15 \left (4096 \,{\mathrm e}^{6}+3840 \,{\mathrm e}^{4}+1200 \,{\mathrm e}^{2}+125\right )^{2}}\right )\) | \(185\) |
Input:
int((-256*exp(2)-192*x-80)*ln(2)/(20480*x^2*exp(2)^3+(15360*x^3+19200*x^2) *exp(2)^2+(3840*x^4+9600*x^3+6000*x^2)*exp(2)+320*x^5+1200*x^4+1500*x^3+62 5*x^2),x,method=_RETURNVERBOSE)
Output:
16/5*ln(2)/x/(16*exp(2)+4*x+5)^2
Time = 0.08 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.52 \[ \int \frac {\left (-80-256 e^2-192 x\right ) \log (2)}{625 x^2+20480 e^6 x^2+1500 x^3+1200 x^4+320 x^5+e^4 \left (19200 x^2+15360 x^3\right )+e^2 \left (6000 x^2+9600 x^3+3840 x^4\right )} \, dx=\frac {16 \, \log \left (2\right )}{5 \, {\left (16 \, x^{3} + 40 \, x^{2} + 256 \, x e^{4} + 32 \, {\left (4 \, x^{2} + 5 \, x\right )} e^{2} + 25 \, x\right )}} \] Input:
integrate((-256*exp(2)-192*x-80)*log(2)/(20480*x^2*exp(2)^3+(15360*x^3+192 00*x^2)*exp(2)^2+(3840*x^4+9600*x^3+6000*x^2)*exp(2)+320*x^5+1200*x^4+1500 *x^3+625*x^2),x, algorithm="fricas")
Output:
16/5*log(2)/(16*x^3 + 40*x^2 + 256*x*e^4 + 32*(4*x^2 + 5*x)*e^2 + 25*x)
Time = 0.41 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28 \[ \int \frac {\left (-80-256 e^2-192 x\right ) \log (2)}{625 x^2+20480 e^6 x^2+1500 x^3+1200 x^4+320 x^5+e^4 \left (19200 x^2+15360 x^3\right )+e^2 \left (6000 x^2+9600 x^3+3840 x^4\right )} \, dx=\frac {16 \log {\left (2 \right )}}{80 x^{3} + x^{2} \cdot \left (200 + 640 e^{2}\right ) + x \left (125 + 800 e^{2} + 1280 e^{4}\right )} \] Input:
integrate((-256*exp(2)-192*x-80)*ln(2)/(20480*x**2*exp(2)**3+(15360*x**3+1 9200*x**2)*exp(2)**2+(3840*x**4+9600*x**3+6000*x**2)*exp(2)+320*x**5+1200* x**4+1500*x**3+625*x**2),x)
Output:
16*log(2)/(80*x**3 + x**2*(200 + 640*exp(2)) + x*(125 + 800*exp(2) + 1280* exp(4)))
Time = 0.05 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.40 \[ \int \frac {\left (-80-256 e^2-192 x\right ) \log (2)}{625 x^2+20480 e^6 x^2+1500 x^3+1200 x^4+320 x^5+e^4 \left (19200 x^2+15360 x^3\right )+e^2 \left (6000 x^2+9600 x^3+3840 x^4\right )} \, dx=\frac {16 \, \log \left (2\right )}{5 \, {\left (16 \, x^{3} + 8 \, x^{2} {\left (16 \, e^{2} + 5\right )} + x {\left (256 \, e^{4} + 160 \, e^{2} + 25\right )}\right )}} \] Input:
integrate((-256*exp(2)-192*x-80)*log(2)/(20480*x^2*exp(2)^3+(15360*x^3+192 00*x^2)*exp(2)^2+(3840*x^4+9600*x^3+6000*x^2)*exp(2)+320*x^5+1200*x^4+1500 *x^3+625*x^2),x, algorithm="maxima")
Output:
16/5*log(2)/(16*x^3 + 8*x^2*(16*e^2 + 5) + x*(256*e^4 + 160*e^2 + 25))
Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (21) = 42\).
Time = 0.12 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.24 \[ \int \frac {\left (-80-256 e^2-192 x\right ) \log (2)}{625 x^2+20480 e^6 x^2+1500 x^3+1200 x^4+320 x^5+e^4 \left (19200 x^2+15360 x^3\right )+e^2 \left (6000 x^2+9600 x^3+3840 x^4\right )} \, dx=-\frac {16}{5} \, {\left (\frac {8 \, {\left (2 \, x + 16 \, e^{2} + 5\right )}}{{\left (4 \, x + 16 \, e^{2} + 5\right )}^{2} {\left (256 \, e^{4} + 160 \, e^{2} + 25\right )}} - \frac {1}{x {\left (256 \, e^{4} + 160 \, e^{2} + 25\right )}}\right )} \log \left (2\right ) \] Input:
integrate((-256*exp(2)-192*x-80)*log(2)/(20480*x^2*exp(2)^3+(15360*x^3+192 00*x^2)*exp(2)^2+(3840*x^4+9600*x^3+6000*x^2)*exp(2)+320*x^5+1200*x^4+1500 *x^3+625*x^2),x, algorithm="giac")
Output:
-16/5*(8*(2*x + 16*e^2 + 5)/((4*x + 16*e^2 + 5)^2*(256*e^4 + 160*e^2 + 25) ) - 1/(x*(256*e^4 + 160*e^2 + 25)))*log(2)
Time = 2.64 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.72 \[ \int \frac {\left (-80-256 e^2-192 x\right ) \log (2)}{625 x^2+20480 e^6 x^2+1500 x^3+1200 x^4+320 x^5+e^4 \left (19200 x^2+15360 x^3\right )+e^2 \left (6000 x^2+9600 x^3+3840 x^4\right )} \, dx=\frac {16\,\ln \left (2\right )}{5\,x\,{\left (4\,x+16\,{\mathrm {e}}^2+5\right )}^2} \] Input:
int(-(log(2)*(192*x + 256*exp(2) + 80))/(exp(4)*(19200*x^2 + 15360*x^3) + 20480*x^2*exp(6) + exp(2)*(6000*x^2 + 9600*x^3 + 3840*x^4) + 625*x^2 + 150 0*x^3 + 1200*x^4 + 320*x^5),x)
Output:
(16*log(2))/(5*x*(4*x + 16*exp(2) + 5)^2)
Time = 0.20 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.40 \[ \int \frac {\left (-80-256 e^2-192 x\right ) \log (2)}{625 x^2+20480 e^6 x^2+1500 x^3+1200 x^4+320 x^5+e^4 \left (19200 x^2+15360 x^3\right )+e^2 \left (6000 x^2+9600 x^3+3840 x^4\right )} \, dx=\frac {16 \,\mathrm {log}\left (2\right )}{5 x \left (256 e^{4}+128 e^{2} x +160 e^{2}+16 x^{2}+40 x +25\right )} \] Input:
int((-256*exp(2)-192*x-80)*log(2)/(20480*x^2*exp(2)^3+(15360*x^3+19200*x^2 )*exp(2)^2+(3840*x^4+9600*x^3+6000*x^2)*exp(2)+320*x^5+1200*x^4+1500*x^3+6 25*x^2),x)
Output:
(16*log(2))/(5*x*(256*e**4 + 128*e**2*x + 160*e**2 + 16*x**2 + 40*x + 25))