Integrand size = 131, antiderivative size = 31 \begin {dmath*} \int \frac {4000-4000 x+e \left (960 x^2-320 x^3-400 x^4+160 x^5\right )+\left (-400 x+400 x^2+e \left (-16 x^4+16 x^5-4 x^6\right )\right ) \log (4)+\left (10 x^2-10 x^3\right ) \log ^2(4)}{1600 x^2-1600 x^3+400 x^4+\left (-160 x^3+160 x^4-40 x^5\right ) \log (4)+\left (4 x^4-4 x^5+x^6\right ) \log ^2(4)} \, dx=\frac {5}{(-2+x) x}-\frac {e (-3-x) x}{5-\frac {1}{4} x \log (4)} \end {dmath*}
Leaf count is larger than twice the leaf count of optimal. \(79\) vs. \(2(31)=62\).
Time = 0.14 (sec) , antiderivative size = 79, normalized size of antiderivative = 2.55 \begin {dmath*} \int \frac {4000-4000 x+e \left (960 x^2-320 x^3-400 x^4+160 x^5\right )+\left (-400 x+400 x^2+e \left (-16 x^4+16 x^5-4 x^6\right )\right ) \log (4)+\left (10 x^2-10 x^3\right ) \log ^2(4)}{1600 x^2-1600 x^3+400 x^4+\left (-160 x^3+160 x^4-40 x^5\right ) \log (4)+\left (4 x^4-4 x^5+x^6\right ) \log ^2(4)} \, dx=-2 \left (\frac {5}{4 x}+\frac {e x \log (16)}{\log ^2(4)}+5 \left (\frac {1}{8-4 x}+\frac {8 e \left (-500 \log ^2(4)-40 \log ^3(4)+3 \log ^4(4)+1000 \log (16)+200 \log (4) \log (16)\right )}{(-10+\log (4))^2 \log ^3(4) (-20+x \log (4))}\right )\right ) \end {dmath*}
Integrate[(4000 - 4000*x + E*(960*x^2 - 320*x^3 - 400*x^4 + 160*x^5) + (-4 00*x + 400*x^2 + E*(-16*x^4 + 16*x^5 - 4*x^6))*Log[4] + (10*x^2 - 10*x^3)* Log[4]^2)/(1600*x^2 - 1600*x^3 + 400*x^4 + (-160*x^3 + 160*x^4 - 40*x^5)*L og[4] + (4*x^4 - 4*x^5 + x^6)*Log[4]^2),x]
-2*(5/(4*x) + (E*x*Log[16])/Log[4]^2 + 5*((8 - 4*x)^(-1) + (8*E*(-500*Log[ 4]^2 - 40*Log[4]^3 + 3*Log[4]^4 + 1000*Log[16] + 200*Log[4]*Log[16]))/((-1 0 + Log[4])^2*Log[4]^3*(-20 + x*Log[4]))))
Time = 0.49 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.52, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {2026, 2462, 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.
\(\displaystyle \int \frac {\left (10 x^2-10 x^3\right ) \log ^2(4)+\left (400 x^2+e \left (-4 x^6+16 x^5-16 x^4\right )-400 x\right ) \log (4)+e \left (160 x^5-400 x^4-320 x^3+960 x^2\right )-4000 x+4000}{400 x^4-1600 x^3+1600 x^2+\left (x^6-4 x^5+4 x^4\right ) \log ^2(4)+\left (-40 x^5+160 x^4-160 x^3\right ) \log (4)} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {\left (10 x^2-10 x^3\right ) \log ^2(4)+\left (400 x^2+e \left (-4 x^6+16 x^5-16 x^4\right )-400 x\right ) \log (4)+e \left (160 x^5-400 x^4-320 x^3+960 x^2\right )-4000 x+4000}{x^2 \left (x^4 \log ^2(4)-4 x^3 \log (4) (10+\log (4))+4 x^2 \left (100+\log ^2(4)+40 \log (4)\right )-160 x (10+\log (4))+1600\right )}dx\) |
\(\Big \downarrow \) 2462 |
\(\displaystyle \int \left (\frac {5}{2 x^2}-\frac {5}{2 (x-2)^2}+\frac {80 e (20+\log (64))}{\log (4) (x \log (4)-20)^2}-\frac {4 e}{\log (4)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {5}{2 (2-x)}-\frac {5}{2 x}+\frac {80 e (20+\log (64))}{\log ^2(4) (20-x \log (4))}-\frac {4 e x}{\log (4)}\) |
Int[(4000 - 4000*x + E*(960*x^2 - 320*x^3 - 400*x^4 + 160*x^5) + (-400*x + 400*x^2 + E*(-16*x^4 + 16*x^5 - 4*x^6))*Log[4] + (10*x^2 - 10*x^3)*Log[4] ^2)/(1600*x^2 - 1600*x^3 + 400*x^4 + (-160*x^3 + 160*x^4 - 40*x^5)*Log[4] + (4*x^4 - 4*x^5 + x^6)*Log[4]^2),x]
-5/(2*(2 - x)) - 5/(2*x) - (4*E*x)/Log[4] + (80*E*(20 + Log[64]))/(Log[4]^ 2*(20 - x*Log[4]))
3.2.10.3.1 Defintions of rubi rules used
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_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u*Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && GtQ [Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0 ] && RationalFunctionQ[u, x]
Time = 0.28 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.45
method | result | size |
default | \(-\frac {2 x \,{\mathrm e}}{\ln \left (2\right )}-\frac {5}{2 x}-\frac {20 \,{\mathrm e} \left (3 \ln \left (2\right )+10\right )}{\ln \left (2\right )^{2} \left (x \ln \left (2\right )-10\right )}+\frac {5}{2 \left (-2+x \right )}\) | \(45\) |
norman | \(\frac {-50+\left (-\frac {\ln \left (2\right )^{2}}{4}-2 \,{\mathrm e}\right ) x^{3}+\left (\frac {\ln \left (2\right )^{2}}{2}+\frac {5 \ln \left (2\right )}{2}+12 \,{\mathrm e}\right ) x^{2}-2 x^{4} {\mathrm e}}{x \left (-2+x \right ) \left (x \ln \left (2\right )-10\right )}\) | \(61\) |
gosper | \(-\frac {x^{3} \ln \left (2\right )^{2}+8 x^{4} {\mathrm e}-2 x^{2} \ln \left (2\right )^{2}+8 x^{3} {\mathrm e}-10 x^{2} \ln \left (2\right )-48 x^{2} {\mathrm e}+200}{4 x \left (x^{2} \ln \left (2\right )-2 x \ln \left (2\right )-10 x +20\right )}\) | \(71\) |
parallelrisch | \(-\frac {5 x^{3} \ln \left (2\right )^{2}+1000+40 x^{4} {\mathrm e}-10 x^{2} \ln \left (2\right )^{2}+40 x^{3} {\mathrm e}-50 x^{2} \ln \left (2\right )-240 x^{2} {\mathrm e}}{20 x \left (x^{2} \ln \left (2\right )-2 x \ln \left (2\right )-10 x +20\right )}\) | \(72\) |
risch | \(-\frac {2 x \,{\mathrm e}}{\ln \left (2\right )}+\frac {-\frac {20 \,{\mathrm e} \left (3 \ln \left (2\right )+10\right ) x^{2}}{\ln \left (2\right )}+\frac {5 \left (\ln \left (2\right )^{3}+24 \,{\mathrm e} \ln \left (2\right )+80 \,{\mathrm e}\right ) x}{\ln \left (2\right )}-50 \ln \left (2\right )}{\ln \left (2\right ) x \left (x^{2} \ln \left (2\right )-2 x \ln \left (2\right )-10 x +20\right )}\) | \(81\) |
int((4*(-10*x^3+10*x^2)*ln(2)^2+2*((-4*x^6+16*x^5-16*x^4)*exp(1)+400*x^2-4 00*x)*ln(2)+(160*x^5-400*x^4-320*x^3+960*x^2)*exp(1)-4000*x+4000)/(4*(x^6- 4*x^5+4*x^4)*ln(2)^2+2*(-40*x^5+160*x^4-160*x^3)*ln(2)+400*x^4-1600*x^3+16 00*x^2),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (27) = 54\).
Time = 0.25 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.84 \begin {dmath*} \int \frac {4000-4000 x+e \left (960 x^2-320 x^3-400 x^4+160 x^5\right )+\left (-400 x+400 x^2+e \left (-16 x^4+16 x^5-4 x^6\right )\right ) \log (4)+\left (10 x^2-10 x^3\right ) \log ^2(4)}{1600 x^2-1600 x^3+400 x^4+\left (-160 x^3+160 x^4-40 x^5\right ) \log (4)+\left (4 x^4-4 x^5+x^6\right ) \log ^2(4)} \, dx=\frac {5 \, x \log \left (2\right )^{3} + 20 \, {\left (x^{3} - 5 \, x^{2} + 6 \, x\right )} e \log \left (2\right ) - 2 \, {\left ({\left (x^{4} - 2 \, x^{3}\right )} e + 25\right )} \log \left (2\right )^{2} - 200 \, {\left (x^{2} - 2 \, x\right )} e}{{\left (x^{3} - 2 \, x^{2}\right )} \log \left (2\right )^{3} - 10 \, {\left (x^{2} - 2 \, x\right )} \log \left (2\right )^{2}} \end {dmath*}
integrate((4*(-10*x^3+10*x^2)*log(2)^2+2*((-4*x^6+16*x^5-16*x^4)*exp(1)+40 0*x^2-400*x)*log(2)+(160*x^5-400*x^4-320*x^3+960*x^2)*exp(1)-4000*x+4000)/ (4*(x^6-4*x^5+4*x^4)*log(2)^2+2*(-40*x^5+160*x^4-160*x^3)*log(2)+400*x^4-1 600*x^3+1600*x^2),x, algorithm=\
(5*x*log(2)^3 + 20*(x^3 - 5*x^2 + 6*x)*e*log(2) - 2*((x^4 - 2*x^3)*e + 25) *log(2)^2 - 200*(x^2 - 2*x)*e)/((x^3 - 2*x^2)*log(2)^3 - 10*(x^2 - 2*x)*lo g(2)^2)
Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (24) = 48\).
Time = 3.60 (sec) , antiderivative size = 94, normalized size of antiderivative = 3.03 \begin {dmath*} \int \frac {4000-4000 x+e \left (960 x^2-320 x^3-400 x^4+160 x^5\right )+\left (-400 x+400 x^2+e \left (-16 x^4+16 x^5-4 x^6\right )\right ) \log (4)+\left (10 x^2-10 x^3\right ) \log ^2(4)}{1600 x^2-1600 x^3+400 x^4+\left (-160 x^3+160 x^4-40 x^5\right ) \log (4)+\left (4 x^4-4 x^5+x^6\right ) \log ^2(4)} \, dx=- \frac {2 e x}{\log {\left (2 \right )}} - \frac {x^{2} \cdot \left (60 e \log {\left (2 \right )} + 200 e\right ) + x \left (- 400 e - 120 e \log {\left (2 \right )} - 5 \log {\left (2 \right )}^{3}\right ) + 50 \log {\left (2 \right )}^{2}}{x^{3} \log {\left (2 \right )}^{3} + x^{2} \left (- 10 \log {\left (2 \right )}^{2} - 2 \log {\left (2 \right )}^{3}\right ) + 20 x \log {\left (2 \right )}^{2}} \end {dmath*}
integrate((4*(-10*x**3+10*x**2)*ln(2)**2+2*((-4*x**6+16*x**5-16*x**4)*exp( 1)+400*x**2-400*x)*ln(2)+(160*x**5-400*x**4-320*x**3+960*x**2)*exp(1)-4000 *x+4000)/(4*(x**6-4*x**5+4*x**4)*ln(2)**2+2*(-40*x**5+160*x**4-160*x**3)*l n(2)+400*x**4-1600*x**3+1600*x**2),x)
-2*E*x/log(2) - (x**2*(60*E*log(2) + 200*E) + x*(-400*E - 120*E*log(2) - 5 *log(2)**3) + 50*log(2)**2)/(x**3*log(2)**3 + x**2*(-10*log(2)**2 - 2*log( 2)**3) + 20*x*log(2)**2)
Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (27) = 54\).
Time = 0.19 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.81 \begin {dmath*} \int \frac {4000-4000 x+e \left (960 x^2-320 x^3-400 x^4+160 x^5\right )+\left (-400 x+400 x^2+e \left (-16 x^4+16 x^5-4 x^6\right )\right ) \log (4)+\left (10 x^2-10 x^3\right ) \log ^2(4)}{1600 x^2-1600 x^3+400 x^4+\left (-160 x^3+160 x^4-40 x^5\right ) \log (4)+\left (4 x^4-4 x^5+x^6\right ) \log ^2(4)} \, dx=-\frac {2 \, x e}{\log \left (2\right )} - \frac {5 \, {\left (4 \, {\left (3 \, e \log \left (2\right ) + 10 \, e\right )} x^{2} - {\left (\log \left (2\right )^{3} + 24 \, e \log \left (2\right ) + 80 \, e\right )} x + 10 \, \log \left (2\right )^{2}\right )}}{x^{3} \log \left (2\right )^{3} - 2 \, {\left (\log \left (2\right )^{3} + 5 \, \log \left (2\right )^{2}\right )} x^{2} + 20 \, x \log \left (2\right )^{2}} \end {dmath*}
integrate((4*(-10*x^3+10*x^2)*log(2)^2+2*((-4*x^6+16*x^5-16*x^4)*exp(1)+40 0*x^2-400*x)*log(2)+(160*x^5-400*x^4-320*x^3+960*x^2)*exp(1)-4000*x+4000)/ (4*(x^6-4*x^5+4*x^4)*log(2)^2+2*(-40*x^5+160*x^4-160*x^3)*log(2)+400*x^4-1 600*x^3+1600*x^2),x, algorithm=\
-2*x*e/log(2) - 5*(4*(3*e*log(2) + 10*e)*x^2 - (log(2)^3 + 24*e*log(2) + 8 0*e)*x + 10*log(2)^2)/(x^3*log(2)^3 - 2*(log(2)^3 + 5*log(2)^2)*x^2 + 20*x *log(2)^2)
Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (27) = 54\).
Time = 0.27 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.65 \begin {dmath*} \int \frac {4000-4000 x+e \left (960 x^2-320 x^3-400 x^4+160 x^5\right )+\left (-400 x+400 x^2+e \left (-16 x^4+16 x^5-4 x^6\right )\right ) \log (4)+\left (10 x^2-10 x^3\right ) \log ^2(4)}{1600 x^2-1600 x^3+400 x^4+\left (-160 x^3+160 x^4-40 x^5\right ) \log (4)+\left (4 x^4-4 x^5+x^6\right ) \log ^2(4)} \, dx=-\frac {2 \, x e}{\log \left (2\right )} - \frac {5 \, {\left (12 \, x^{2} e \log \left (2\right ) - x \log \left (2\right )^{3} + 40 \, x^{2} e - 24 \, x e \log \left (2\right ) - 80 \, x e + 10 \, \log \left (2\right )^{2}\right )}}{{\left (x^{3} \log \left (2\right ) - 2 \, x^{2} \log \left (2\right ) - 10 \, x^{2} + 20 \, x\right )} \log \left (2\right )^{2}} \end {dmath*}
integrate((4*(-10*x^3+10*x^2)*log(2)^2+2*((-4*x^6+16*x^5-16*x^4)*exp(1)+40 0*x^2-400*x)*log(2)+(160*x^5-400*x^4-320*x^3+960*x^2)*exp(1)-4000*x+4000)/ (4*(x^6-4*x^5+4*x^4)*log(2)^2+2*(-40*x^5+160*x^4-160*x^3)*log(2)+400*x^4-1 600*x^3+1600*x^2),x, algorithm=\
-2*x*e/log(2) - 5*(12*x^2*e*log(2) - x*log(2)^3 + 40*x^2*e - 24*x*e*log(2) - 80*x*e + 10*log(2)^2)/((x^3*log(2) - 2*x^2*log(2) - 10*x^2 + 20*x)*log( 2)^2)
Time = 13.66 (sec) , antiderivative size = 91, normalized size of antiderivative = 2.94 \begin {dmath*} \int \frac {4000-4000 x+e \left (960 x^2-320 x^3-400 x^4+160 x^5\right )+\left (-400 x+400 x^2+e \left (-16 x^4+16 x^5-4 x^6\right )\right ) \log (4)+\left (10 x^2-10 x^3\right ) \log ^2(4)}{1600 x^2-1600 x^3+400 x^4+\left (-160 x^3+160 x^4-40 x^5\right ) \log (4)+\left (4 x^4-4 x^5+x^6\right ) \log ^2(4)} \, dx=-\frac {\frac {20\,\left (10\,\mathrm {e}+3\,\mathrm {e}\,\ln \left (2\right )\right )\,x^2}{\ln \left (2\right )}-\frac {5\,\left (80\,\mathrm {e}+24\,\mathrm {e}\,\ln \left (2\right )+{\ln \left (2\right )}^3\right )\,x}{\ln \left (2\right )}+50\,\ln \left (2\right )}{{\ln \left (2\right )}^2\,x^3+\left (-10\,\ln \left (2\right )-2\,{\ln \left (2\right )}^2\right )\,x^2+20\,\ln \left (2\right )\,x}-\frac {2\,x\,\mathrm {e}}{\ln \left (2\right )} \end {dmath*}
int((4*log(2)^2*(10*x^2 - 10*x^3) - 2*log(2)*(400*x + exp(1)*(16*x^4 - 16* x^5 + 4*x^6) - 400*x^2) - 4000*x + exp(1)*(960*x^2 - 320*x^3 - 400*x^4 + 1 60*x^5) + 4000)/(4*log(2)^2*(4*x^4 - 4*x^5 + x^6) - 2*log(2)*(160*x^3 - 16 0*x^4 + 40*x^5) + 1600*x^2 - 1600*x^3 + 400*x^4),x)