Integrand size = 222, antiderivative size = 29 \[ \int \frac {e^x \left (1280-288 x-144 x^2+32 x^3+16 x^4+e^{20} \left (-32 x+16 x^2\right )+e^{10} \left (192 x-32 x^3\right )-256 \log (2)\right )}{6400+1440 x^2+960 x^3+241 x^4+e^{40} x^4+108 x^5+54 x^6+12 x^7+x^8+e^{30} \left (-12 x^4-4 x^5\right )+e^{20} \left (160 x^2+54 x^4+36 x^5+6 x^6\right )+e^{10} \left (-960 x^2-320 x^3-108 x^4-108 x^5-36 x^6-4 x^7\right )+\left (-2560-288 x^2-32 e^{20} x^2-192 x^3-32 x^4+e^{10} \left (192 x^2+64 x^3\right )\right ) \log (2)+256 \log ^2(2)} \, dx=\frac {e^x}{5+\frac {1}{16} x^2 \left (3-e^{10}+x\right )^2-\log (2)} \]
Time = 5.07 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.31 \[ \int \frac {e^x \left (1280-288 x-144 x^2+32 x^3+16 x^4+e^{20} \left (-32 x+16 x^2\right )+e^{10} \left (192 x-32 x^3\right )-256 \log (2)\right )}{6400+1440 x^2+960 x^3+241 x^4+e^{40} x^4+108 x^5+54 x^6+12 x^7+x^8+e^{30} \left (-12 x^4-4 x^5\right )+e^{20} \left (160 x^2+54 x^4+36 x^5+6 x^6\right )+e^{10} \left (-960 x^2-320 x^3-108 x^4-108 x^5-36 x^6-4 x^7\right )+\left (-2560-288 x^2-32 e^{20} x^2-192 x^3-32 x^4+e^{10} \left (192 x^2+64 x^3\right )\right ) \log (2)+256 \log ^2(2)} \, dx=\frac {16 e^x}{\left (-3+e^{10}\right )^2 x^2-2 \left (-3+e^{10}\right ) x^3+x^4-16 (-5+\log (2))} \]
Integrate[(E^x*(1280 - 288*x - 144*x^2 + 32*x^3 + 16*x^4 + E^20*(-32*x + 1 6*x^2) + E^10*(192*x - 32*x^3) - 256*Log[2]))/(6400 + 1440*x^2 + 960*x^3 + 241*x^4 + E^40*x^4 + 108*x^5 + 54*x^6 + 12*x^7 + x^8 + E^30*(-12*x^4 - 4* x^5) + E^20*(160*x^2 + 54*x^4 + 36*x^5 + 6*x^6) + E^10*(-960*x^2 - 320*x^3 - 108*x^4 - 108*x^5 - 36*x^6 - 4*x^7) + (-2560 - 288*x^2 - 32*E^20*x^2 - 192*x^3 - 32*x^4 + E^10*(192*x^2 + 64*x^3))*Log[2] + 256*Log[2]^2),x]
Time = 1.14 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.52, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.027, Rules used = {6, 2463, 6, 6, 6, 2727}
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.
\(\displaystyle \int \frac {e^x \left (16 x^4+32 x^3+e^{10} \left (192 x-32 x^3\right )-144 x^2+e^{20} \left (16 x^2-32 x\right )-288 x+1280-256 \log (2)\right )}{x^8+12 x^7+54 x^6+108 x^5+e^{40} x^4+241 x^4+960 x^3+1440 x^2+e^{30} \left (-4 x^5-12 x^4\right )+\left (-32 x^4-192 x^3-32 e^{20} x^2-288 x^2+e^{10} \left (64 x^3+192 x^2\right )-2560\right ) \log (2)+e^{20} \left (6 x^6+36 x^5+54 x^4+160 x^2\right )+e^{10} \left (-4 x^7-36 x^6-108 x^5-108 x^4-320 x^3-960 x^2\right )+6400+256 \log ^2(2)} \, dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {e^x \left (16 x^4+32 x^3+e^{10} \left (192 x-32 x^3\right )-144 x^2+e^{20} \left (16 x^2-32 x\right )-288 x+1280-256 \log (2)\right )}{x^8+12 x^7+54 x^6+108 x^5+\left (241+e^{40}\right ) x^4+960 x^3+1440 x^2+e^{30} \left (-4 x^5-12 x^4\right )+\left (-32 x^4-192 x^3-32 e^{20} x^2-288 x^2+e^{10} \left (64 x^3+192 x^2\right )-2560\right ) \log (2)+e^{20} \left (6 x^6+36 x^5+54 x^4+160 x^2\right )+e^{10} \left (-4 x^7-36 x^6-108 x^5-108 x^4-320 x^3-960 x^2\right )+6400+256 \log ^2(2)}dx\) |
\(\Big \downarrow \) 2463 |
\(\displaystyle \int \frac {e^x \left (16 x^4+32 x^3+e^{10} \left (192 x-32 x^3\right )-144 x^2+e^{20} \left (16 x^2-32 x\right )-288 x+1280-256 \log (2)\right )}{\left (x^4-2 e^{10} x^3+6 x^3+e^{20} x^2-6 e^{10} x^2+9 x^2+80-16 \log (2)\right )^2}dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {e^x \left (16 x^4+32 x^3+e^{10} \left (192 x-32 x^3\right )-144 x^2+e^{20} \left (16 x^2-32 x\right )-288 x+1280-256 \log (2)\right )}{\left (x^4-2 e^{10} x^3+6 x^3+\left (9-6 e^{10}\right ) x^2+e^{20} x^2+80-16 \log (2)\right )^2}dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {e^x \left (16 x^4+32 x^3+e^{10} \left (192 x-32 x^3\right )-144 x^2+e^{20} \left (16 x^2-32 x\right )-288 x+1280-256 \log (2)\right )}{\left (x^4-2 e^{10} x^3+6 x^3+\left (9-6 e^{10}+e^{20}\right ) x^2+80-16 \log (2)\right )^2}dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {e^x \left (16 x^4+32 x^3+e^{10} \left (192 x-32 x^3\right )-144 x^2+e^{20} \left (16 x^2-32 x\right )-288 x+1280-256 \log (2)\right )}{\left (x^4+\left (6-2 e^{10}\right ) x^3+\left (9-6 e^{10}+e^{20}\right ) x^2+80-16 \log (2)\right )^2}dx\) |
\(\Big \downarrow \) 2727 |
\(\displaystyle \frac {16 e^x}{x^4+2 \left (3-e^{10}\right ) x^3+\left (3-e^{10}\right )^2 x^2+16 (5-\log (2))}\) |
Int[(E^x*(1280 - 288*x - 144*x^2 + 32*x^3 + 16*x^4 + E^20*(-32*x + 16*x^2) + E^10*(192*x - 32*x^3) - 256*Log[2]))/(6400 + 1440*x^2 + 960*x^3 + 241*x ^4 + E^40*x^4 + 108*x^5 + 54*x^6 + 12*x^7 + x^8 + E^30*(-12*x^4 - 4*x^5) + E^20*(160*x^2 + 54*x^4 + 36*x^5 + 6*x^6) + E^10*(-960*x^2 - 320*x^3 - 108 *x^4 - 108*x^5 - 36*x^6 - 4*x^7) + (-2560 - 288*x^2 - 32*E^20*x^2 - 192*x^ 3 - 32*x^4 + E^10*(192*x^2 + 64*x^3))*Log[2] + 256*Log[2]^2),x]
3.21.26.3.1 Defintions of rubi rules used
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[(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[(F_)^(u_)*(v_)^(n_.)*(w_), x_Symbol] :> With[{z = Log[F]*v*D[u, x] + (n + 1)*D[v, x]}, Simp[(Coefficient[w, x, Exponent[w, x]]/Coefficient[z, x, E xponent[z, x]])*F^u*v^(n + 1), x] /; EqQ[Exponent[w, x], Exponent[z, x]] && EqQ[w*Coefficient[z, x, Exponent[z, x]], z*Coefficient[w, x, Exponent[w, x ]]]] /; FreeQ[{F, n}, x] && PolynomialQ[u, x] && PolynomialQ[v, x] && Polyn omialQ[w, x]
Time = 0.53 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.76
method | result | size |
gosper | \(-\frac {16 \,{\mathrm e}^{x}}{-x^{2} {\mathrm e}^{20}+2 x^{3} {\mathrm e}^{10}-x^{4}+6 x^{2} {\mathrm e}^{10}-6 x^{3}-9 x^{2}+16 \ln \left (2\right )-80}\) | \(51\) |
norman | \(-\frac {16 \,{\mathrm e}^{x}}{-x^{2} {\mathrm e}^{20}+2 x^{3} {\mathrm e}^{10}-x^{4}+6 x^{2} {\mathrm e}^{10}-6 x^{3}-9 x^{2}+16 \ln \left (2\right )-80}\) | \(51\) |
parallelrisch | \(-\frac {16 \,{\mathrm e}^{x}}{-x^{2} {\mathrm e}^{20}+2 x^{3} {\mathrm e}^{10}-x^{4}+6 x^{2} {\mathrm e}^{10}-6 x^{3}-9 x^{2}+16 \ln \left (2\right )-80}\) | \(51\) |
default | \(\text {Expression too large to display}\) | \(5691\) |
int((-256*ln(2)+(16*x^2-32*x)*exp(10)^2+(-32*x^3+192*x)*exp(10)+16*x^4+32* x^3-144*x^2-288*x+1280)*exp(x)/(256*ln(2)^2+(-32*x^2*exp(10)^2+(64*x^3+192 *x^2)*exp(10)-32*x^4-192*x^3-288*x^2-2560)*ln(2)+x^4*exp(10)^4+(-4*x^5-12* x^4)*exp(10)^3+(6*x^6+36*x^5+54*x^4+160*x^2)*exp(10)^2+(-4*x^7-36*x^6-108* x^5-108*x^4-320*x^3-960*x^2)*exp(10)+x^8+12*x^7+54*x^6+108*x^5+241*x^4+960 *x^3+1440*x^2+6400),x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.52 \[ \int \frac {e^x \left (1280-288 x-144 x^2+32 x^3+16 x^4+e^{20} \left (-32 x+16 x^2\right )+e^{10} \left (192 x-32 x^3\right )-256 \log (2)\right )}{6400+1440 x^2+960 x^3+241 x^4+e^{40} x^4+108 x^5+54 x^6+12 x^7+x^8+e^{30} \left (-12 x^4-4 x^5\right )+e^{20} \left (160 x^2+54 x^4+36 x^5+6 x^6\right )+e^{10} \left (-960 x^2-320 x^3-108 x^4-108 x^5-36 x^6-4 x^7\right )+\left (-2560-288 x^2-32 e^{20} x^2-192 x^3-32 x^4+e^{10} \left (192 x^2+64 x^3\right )\right ) \log (2)+256 \log ^2(2)} \, dx=\frac {16 \, e^{x}}{x^{4} + 6 \, x^{3} + x^{2} e^{20} + 9 \, x^{2} - 2 \, {\left (x^{3} + 3 \, x^{2}\right )} e^{10} - 16 \, \log \left (2\right ) + 80} \]
integrate((-256*log(2)+(16*x^2-32*x)*exp(10)^2+(-32*x^3+192*x)*exp(10)+16* x^4+32*x^3-144*x^2-288*x+1280)*exp(x)/(256*log(2)^2+(-32*x^2*exp(10)^2+(64 *x^3+192*x^2)*exp(10)-32*x^4-192*x^3-288*x^2-2560)*log(2)+x^4*exp(10)^4+(- 4*x^5-12*x^4)*exp(10)^3+(6*x^6+36*x^5+54*x^4+160*x^2)*exp(10)^2+(-4*x^7-36 *x^6-108*x^5-108*x^4-320*x^3-960*x^2)*exp(10)+x^8+12*x^7+54*x^6+108*x^5+24 1*x^4+960*x^3+1440*x^2+6400),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (20) = 40\).
Time = 0.22 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.66 \[ \int \frac {e^x \left (1280-288 x-144 x^2+32 x^3+16 x^4+e^{20} \left (-32 x+16 x^2\right )+e^{10} \left (192 x-32 x^3\right )-256 \log (2)\right )}{6400+1440 x^2+960 x^3+241 x^4+e^{40} x^4+108 x^5+54 x^6+12 x^7+x^8+e^{30} \left (-12 x^4-4 x^5\right )+e^{20} \left (160 x^2+54 x^4+36 x^5+6 x^6\right )+e^{10} \left (-960 x^2-320 x^3-108 x^4-108 x^5-36 x^6-4 x^7\right )+\left (-2560-288 x^2-32 e^{20} x^2-192 x^3-32 x^4+e^{10} \left (192 x^2+64 x^3\right )\right ) \log (2)+256 \log ^2(2)} \, dx=\frac {16 e^{x}}{x^{4} - 2 x^{3} e^{10} + 6 x^{3} - 6 x^{2} e^{10} + 9 x^{2} + x^{2} e^{20} - 16 \log {\left (2 \right )} + 80} \]
integrate((-256*ln(2)+(16*x**2-32*x)*exp(10)**2+(-32*x**3+192*x)*exp(10)+1 6*x**4+32*x**3-144*x**2-288*x+1280)*exp(x)/(256*ln(2)**2+(-32*x**2*exp(10) **2+(64*x**3+192*x**2)*exp(10)-32*x**4-192*x**3-288*x**2-2560)*ln(2)+x**4* exp(10)**4+(-4*x**5-12*x**4)*exp(10)**3+(6*x**6+36*x**5+54*x**4+160*x**2)* exp(10)**2+(-4*x**7-36*x**6-108*x**5-108*x**4-320*x**3-960*x**2)*exp(10)+x **8+12*x**7+54*x**6+108*x**5+241*x**4+960*x**3+1440*x**2+6400),x)
16*exp(x)/(x**4 - 2*x**3*exp(10) + 6*x**3 - 6*x**2*exp(10) + 9*x**2 + x**2 *exp(20) - 16*log(2) + 80)
Time = 0.35 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.24 \[ \int \frac {e^x \left (1280-288 x-144 x^2+32 x^3+16 x^4+e^{20} \left (-32 x+16 x^2\right )+e^{10} \left (192 x-32 x^3\right )-256 \log (2)\right )}{6400+1440 x^2+960 x^3+241 x^4+e^{40} x^4+108 x^5+54 x^6+12 x^7+x^8+e^{30} \left (-12 x^4-4 x^5\right )+e^{20} \left (160 x^2+54 x^4+36 x^5+6 x^6\right )+e^{10} \left (-960 x^2-320 x^3-108 x^4-108 x^5-36 x^6-4 x^7\right )+\left (-2560-288 x^2-32 e^{20} x^2-192 x^3-32 x^4+e^{10} \left (192 x^2+64 x^3\right )\right ) \log (2)+256 \log ^2(2)} \, dx=\frac {16 \, e^{x}}{x^{4} - 2 \, x^{3} {\left (e^{10} - 3\right )} + x^{2} {\left (e^{20} - 6 \, e^{10} + 9\right )} - 16 \, \log \left (2\right ) + 80} \]
integrate((-256*log(2)+(16*x^2-32*x)*exp(10)^2+(-32*x^3+192*x)*exp(10)+16* x^4+32*x^3-144*x^2-288*x+1280)*exp(x)/(256*log(2)^2+(-32*x^2*exp(10)^2+(64 *x^3+192*x^2)*exp(10)-32*x^4-192*x^3-288*x^2-2560)*log(2)+x^4*exp(10)^4+(- 4*x^5-12*x^4)*exp(10)^3+(6*x^6+36*x^5+54*x^4+160*x^2)*exp(10)^2+(-4*x^7-36 *x^6-108*x^5-108*x^4-320*x^3-960*x^2)*exp(10)+x^8+12*x^7+54*x^6+108*x^5+24 1*x^4+960*x^3+1440*x^2+6400),x, algorithm=\
Timed out. \[ \int \frac {e^x \left (1280-288 x-144 x^2+32 x^3+16 x^4+e^{20} \left (-32 x+16 x^2\right )+e^{10} \left (192 x-32 x^3\right )-256 \log (2)\right )}{6400+1440 x^2+960 x^3+241 x^4+e^{40} x^4+108 x^5+54 x^6+12 x^7+x^8+e^{30} \left (-12 x^4-4 x^5\right )+e^{20} \left (160 x^2+54 x^4+36 x^5+6 x^6\right )+e^{10} \left (-960 x^2-320 x^3-108 x^4-108 x^5-36 x^6-4 x^7\right )+\left (-2560-288 x^2-32 e^{20} x^2-192 x^3-32 x^4+e^{10} \left (192 x^2+64 x^3\right )\right ) \log (2)+256 \log ^2(2)} \, dx=\text {Timed out} \]
integrate((-256*log(2)+(16*x^2-32*x)*exp(10)^2+(-32*x^3+192*x)*exp(10)+16* x^4+32*x^3-144*x^2-288*x+1280)*exp(x)/(256*log(2)^2+(-32*x^2*exp(10)^2+(64 *x^3+192*x^2)*exp(10)-32*x^4-192*x^3-288*x^2-2560)*log(2)+x^4*exp(10)^4+(- 4*x^5-12*x^4)*exp(10)^3+(6*x^6+36*x^5+54*x^4+160*x^2)*exp(10)^2+(-4*x^7-36 *x^6-108*x^5-108*x^4-320*x^3-960*x^2)*exp(10)+x^8+12*x^7+54*x^6+108*x^5+24 1*x^4+960*x^3+1440*x^2+6400),x, algorithm=\
Time = 14.18 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.24 \[ \int \frac {e^x \left (1280-288 x-144 x^2+32 x^3+16 x^4+e^{20} \left (-32 x+16 x^2\right )+e^{10} \left (192 x-32 x^3\right )-256 \log (2)\right )}{6400+1440 x^2+960 x^3+241 x^4+e^{40} x^4+108 x^5+54 x^6+12 x^7+x^8+e^{30} \left (-12 x^4-4 x^5\right )+e^{20} \left (160 x^2+54 x^4+36 x^5+6 x^6\right )+e^{10} \left (-960 x^2-320 x^3-108 x^4-108 x^5-36 x^6-4 x^7\right )+\left (-2560-288 x^2-32 e^{20} x^2-192 x^3-32 x^4+e^{10} \left (192 x^2+64 x^3\right )\right ) \log (2)+256 \log ^2(2)} \, dx=\frac {16\,{\mathrm {e}}^x}{x^4+\left (6-2\,{\mathrm {e}}^{10}\right )\,x^3+{\left ({\mathrm {e}}^{10}-3\right )}^2\,x^2-16\,\ln \left (2\right )+80} \]
int(-(exp(x)*(288*x + 256*log(2) + exp(20)*(32*x - 16*x^2) - exp(10)*(192* x - 32*x^3) + 144*x^2 - 32*x^3 - 16*x^4 - 1280))/(x^4*exp(40) - exp(30)*(1 2*x^4 + 4*x^5) - exp(10)*(960*x^2 + 320*x^3 + 108*x^4 + 108*x^5 + 36*x^6 + 4*x^7) + 256*log(2)^2 + 1440*x^2 + 960*x^3 + 241*x^4 + 108*x^5 + 54*x^6 + 12*x^7 + x^8 - log(2)*(32*x^2*exp(20) - exp(10)*(192*x^2 + 64*x^3) + 288* x^2 + 192*x^3 + 32*x^4 + 2560) + exp(20)*(160*x^2 + 54*x^4 + 36*x^5 + 6*x^ 6) + 6400),x)