Integrand size = 111, antiderivative size = 33 \[ \int \frac {28080-18720 e^8 x+4320 e^{16} x^2+e^x \left (-18720-18720 x+e^8 \left (12480 x+14880 x^2\right )+e^{16} \left (-2880 x^2-2880 x^3\right )\right )+e^{2 x} \left (3120+6240 x+e^8 \left (-2080 x-4960 x^2\right )+e^{16} \left (480 x^2+960 x^3\right )\right )}{169-156 e^8 x+36 e^{16} x^2} \, dx=\frac {16 \left (-3+e^x\right )^2 x^2}{\frac {6 x}{5}-\frac {x}{3-e^8 x}} \]
Leaf count is larger than twice the leaf count of optimal. \(72\) vs. \(2(33)=66\).
Time = 2.66 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.18 \[ \int \frac {28080-18720 e^8 x+4320 e^{16} x^2+e^x \left (-18720-18720 x+e^8 \left (12480 x+14880 x^2\right )+e^{16} \left (-2880 x^2-2880 x^3\right )\right )+e^{2 x} \left (3120+6240 x+e^8 \left (-2080 x-4960 x^2\right )+e^{16} \left (480 x^2+960 x^3\right )\right )}{169-156 e^8 x+36 e^{16} x^2} \, dx=\frac {80 \left (65+78 e^8 x-72 e^{8+x} x+12 e^{8+2 x} x-36 e^{16} x^2+24 e^{16+x} x^2-4 e^{16+2 x} x^2\right )}{52 e^8-24 e^{16} x} \]
Integrate[(28080 - 18720*E^8*x + 4320*E^16*x^2 + E^x*(-18720 - 18720*x + E ^8*(12480*x + 14880*x^2) + E^16*(-2880*x^2 - 2880*x^3)) + E^(2*x)*(3120 + 6240*x + E^8*(-2080*x - 4960*x^2) + E^16*(480*x^2 + 960*x^3)))/(169 - 156* E^8*x + 36*E^16*x^2),x]
(80*(65 + 78*E^8*x - 72*E^(8 + x)*x + 12*E^(8 + 2*x)*x - 36*E^16*x^2 + 24* E^(16 + x)*x^2 - 4*E^(16 + 2*x)*x^2))/(52*E^8 - 24*E^16*x)
Leaf count is larger than twice the leaf count of optimal. \(127\) vs. \(2(33)=66\).
Time = 1.38 (sec) , antiderivative size = 127, normalized size of antiderivative = 3.85, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {7277, 27, 7292, 7293, 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.
\(\displaystyle \int \frac {4320 e^{16} x^2+e^x \left (e^8 \left (14880 x^2+12480 x\right )+e^{16} \left (-2880 x^3-2880 x^2\right )-18720 x-18720\right )+e^{2 x} \left (e^8 \left (-4960 x^2-2080 x\right )+e^{16} \left (960 x^3+480 x^2\right )+6240 x+3120\right )-18720 e^8 x+28080}{36 e^{16} x^2-156 e^8 x+169} \, dx\) |
\(\Big \downarrow \) 7277 |
\(\displaystyle 144 e^{16} \int \frac {5 \left (54 e^{16} x^2-234 e^8 x-6 e^x \left (39 x-e^8 \left (31 x^2+26 x\right )+6 e^{16} \left (x^3+x^2\right )+39\right )+e^{2 x} \left (78 x-2 e^8 \left (31 x^2+13 x\right )+6 e^{16} \left (2 x^3+x^2\right )+39\right )+351\right )}{9 e^{16} \left (13-6 e^8 x\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 80 \int \frac {54 e^{16} x^2-234 e^8 x-6 e^x \left (39 x-e^8 \left (31 x^2+26 x\right )+6 e^{16} \left (x^3+x^2\right )+39\right )+e^{2 x} \left (78 x-2 e^8 \left (31 x^2+13 x\right )+6 e^{16} \left (2 x^3+x^2\right )+39\right )+351}{\left (13-6 e^8 x\right )^2}dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle 80 \int \frac {\left (3-e^x\right ) \left (-12 e^{x+16} x^3+62 e^{x+8} \left (1-\frac {3 e^8}{31}\right ) x^2+18 e^{16} x^2-78 e^x \left (1-\frac {e^8}{3}\right ) x-78 e^8 x-39 e^x+117\right )}{\left (13-6 e^8 x\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle 80 \int \left (\frac {9 \left (6 e^{16} x^2-26 e^8 x+39\right )}{\left (6 e^8 x-13\right )^2}+\frac {6 e^x \left (-6 e^{16} x^3+e^8 \left (31-6 e^8\right ) x^2-13 \left (3-2 e^8\right ) x-39\right )}{\left (13-6 e^8 x\right )^2}+\frac {e^{2 x} \left (12 e^{16} x^3-2 e^8 \left (31-3 e^8\right ) x^2+26 \left (3-e^8\right ) x+39\right )}{\left (13-6 e^8 x\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 80 \left (-e^x x+\frac {1}{6} e^{2 x} x+\frac {3 x}{2}+e^x-\frac {e^{2 x}}{12}-\frac {65 e^{x-8}}{6 \left (13-6 e^8 x\right )}+\frac {65 e^{2 x-8}}{36 \left (13-6 e^8 x\right )}+\frac {65}{4 e^8 \left (13-6 e^8 x\right )}-\frac {1}{36} \left (5-3 e^8\right ) e^{2 x-8}+\frac {1}{6} \left (5-6 e^8\right ) e^{x-8}\right )\) |
Int[(28080 - 18720*E^8*x + 4320*E^16*x^2 + E^x*(-18720 - 18720*x + E^8*(12 480*x + 14880*x^2) + E^16*(-2880*x^2 - 2880*x^3)) + E^(2*x)*(3120 + 6240*x + E^8*(-2080*x - 4960*x^2) + E^16*(480*x^2 + 960*x^3)))/(169 - 156*E^8*x + 36*E^16*x^2),x]
80*(E^x - E^(2*x)/12 + (E^(-8 + x)*(5 - 6*E^8))/6 - (E^(-8 + 2*x)*(5 - 3*E ^8))/36 + (3*x)/2 - E^x*x + (E^(2*x)*x)/6 + 65/(4*E^8*(13 - 6*E^8*x)) - (6 5*E^(-8 + x))/(6*(13 - 6*E^8*x)) + (65*E^(-8 + 2*x))/(36*(13 - 6*E^8*x)))
3.5.59.3.1 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[(u_)*((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_.), x_Symbol] :> Simp[1/(4^p*c^p) Int[u*(b + 2*c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n} , x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p] && !AlgebraicFu nctionQ[u, x]
Time = 0.72 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.79
method | result | size |
risch | \(\frac {80 x^{2} {\mathrm e}^{2 x +8}-480 x^{2} {\mathrm e}^{x +8}+720 x^{2} {\mathrm e}^{8}-240 x \,{\mathrm e}^{2 x}+1440 \,{\mathrm e}^{x} x -1300 \,{\mathrm e}^{-8}-1560 x}{6 x \,{\mathrm e}^{8}-13}\) | \(59\) |
norman | \(\frac {-2160 x -240 x \,{\mathrm e}^{2 x}+720 x^{2} {\mathrm e}^{8}+1440 \,{\mathrm e}^{x} x -480 \,{\mathrm e}^{8} {\mathrm e}^{x} x^{2}+80 \,{\mathrm e}^{8} {\mathrm e}^{2 x} x^{2}}{6 x \,{\mathrm e}^{8}-13}\) | \(62\) |
parallelrisch | \(\frac {1040 \,{\mathrm e}^{8} {\mathrm e}^{2 x} x^{2}-6240 \,{\mathrm e}^{8} {\mathrm e}^{x} x^{2}+9360 x^{2} {\mathrm e}^{8}-3120 x \,{\mathrm e}^{2 x}+18720 \,{\mathrm e}^{x} x -28080 x}{78 x \,{\mathrm e}^{8}-169}\) | \(63\) |
parts | \(\text {Expression too large to display}\) | \(815\) |
default | \(\text {Expression too large to display}\) | \(829\) |
int((((960*x^3+480*x^2)*exp(4)^4+(-4960*x^2-2080*x)*exp(4)^2+6240*x+3120)* exp(x)^2+((-2880*x^3-2880*x^2)*exp(4)^4+(14880*x^2+12480*x)*exp(4)^2-18720 *x-18720)*exp(x)+4320*x^2*exp(4)^4-18720*x*exp(4)^2+28080)/(36*x^2*exp(4)^ 4-156*x*exp(4)^2+169),x,method=_RETURNVERBOSE)
20*(4*x^2*exp(2*x+8)-24*x^2*exp(x+8)+36*x^2*exp(8)-12*x*exp(2*x)+72*exp(x) *x-65*exp(-8)-78*x)/(6*x*exp(8)-13)
Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (28) = 56\).
Time = 0.25 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.88 \[ \int \frac {28080-18720 e^8 x+4320 e^{16} x^2+e^x \left (-18720-18720 x+e^8 \left (12480 x+14880 x^2\right )+e^{16} \left (-2880 x^2-2880 x^3\right )\right )+e^{2 x} \left (3120+6240 x+e^8 \left (-2080 x-4960 x^2\right )+e^{16} \left (480 x^2+960 x^3\right )\right )}{169-156 e^8 x+36 e^{16} x^2} \, dx=\frac {20 \, {\left (36 \, x^{2} e^{16} - 78 \, x e^{8} + 4 \, {\left (x^{2} e^{16} - 3 \, x e^{8}\right )} e^{\left (2 \, x\right )} - 24 \, {\left (x^{2} e^{16} - 3 \, x e^{8}\right )} e^{x} - 65\right )}}{6 \, x e^{16} - 13 \, e^{8}} \]
integrate((((960*x^3+480*x^2)*exp(4)^4+(-4960*x^2-2080*x)*exp(4)^2+6240*x+ 3120)*exp(x)^2+((-2880*x^3-2880*x^2)*exp(4)^4+(14880*x^2+12480*x)*exp(4)^2 -18720*x-18720)*exp(x)+4320*x^2*exp(4)^4-18720*x*exp(4)^2+28080)/(36*x^2*e xp(4)^4-156*x*exp(4)^2+169),x, algorithm=\
20*(36*x^2*e^16 - 78*x*e^8 + 4*(x^2*e^16 - 3*x*e^8)*e^(2*x) - 24*(x^2*e^16 - 3*x*e^8)*e^x - 65)/(6*x*e^16 - 13*e^8)
Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (24) = 48\).
Time = 0.20 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.48 \[ \int \frac {28080-18720 e^8 x+4320 e^{16} x^2+e^x \left (-18720-18720 x+e^8 \left (12480 x+14880 x^2\right )+e^{16} \left (-2880 x^2-2880 x^3\right )\right )+e^{2 x} \left (3120+6240 x+e^8 \left (-2080 x-4960 x^2\right )+e^{16} \left (480 x^2+960 x^3\right )\right )}{169-156 e^8 x+36 e^{16} x^2} \, dx=120 x + \frac {\left (- 2880 x^{3} e^{16} + 14880 x^{2} e^{8} - 18720 x\right ) e^{x} + \left (480 x^{3} e^{16} - 2480 x^{2} e^{8} + 3120 x\right ) e^{2 x}}{36 x^{2} e^{16} - 156 x e^{8} + 169} - \frac {1300}{6 x e^{16} - 13 e^{8}} \]
integrate((((960*x**3+480*x**2)*exp(4)**4+(-4960*x**2-2080*x)*exp(4)**2+62 40*x+3120)*exp(x)**2+((-2880*x**3-2880*x**2)*exp(4)**4+(14880*x**2+12480*x )*exp(4)**2-18720*x-18720)*exp(x)+4320*x**2*exp(4)**4-18720*x*exp(4)**2+28 080)/(36*x**2*exp(4)**4-156*x*exp(4)**2+169),x)
120*x + ((-2880*x**3*exp(16) + 14880*x**2*exp(8) - 18720*x)*exp(x) + (480* x**3*exp(16) - 2480*x**2*exp(8) + 3120*x)*exp(2*x))/(36*x**2*exp(16) - 156 *x*exp(8) + 169) - 1300/(6*x*exp(16) - 13*exp(8))
\[ \int \frac {28080-18720 e^8 x+4320 e^{16} x^2+e^x \left (-18720-18720 x+e^8 \left (12480 x+14880 x^2\right )+e^{16} \left (-2880 x^2-2880 x^3\right )\right )+e^{2 x} \left (3120+6240 x+e^8 \left (-2080 x-4960 x^2\right )+e^{16} \left (480 x^2+960 x^3\right )\right )}{169-156 e^8 x+36 e^{16} x^2} \, dx=\int { \frac {80 \, {\left (54 \, x^{2} e^{16} - 234 \, x e^{8} + {\left (6 \, {\left (2 \, x^{3} + x^{2}\right )} e^{16} - 2 \, {\left (31 \, x^{2} + 13 \, x\right )} e^{8} + 78 \, x + 39\right )} e^{\left (2 \, x\right )} - 6 \, {\left (6 \, {\left (x^{3} + x^{2}\right )} e^{16} - {\left (31 \, x^{2} + 26 \, x\right )} e^{8} + 39 \, x + 39\right )} e^{x} + 351\right )}}{36 \, x^{2} e^{16} - 156 \, x e^{8} + 169} \,d x } \]
integrate((((960*x^3+480*x^2)*exp(4)^4+(-4960*x^2-2080*x)*exp(4)^2+6240*x+ 3120)*exp(x)^2+((-2880*x^3-2880*x^2)*exp(4)^4+(14880*x^2+12480*x)*exp(4)^2 -18720*x-18720)*exp(x)+4320*x^2*exp(4)^4-18720*x*exp(4)^2+28080)/(36*x^2*e xp(4)^4-156*x*exp(4)^2+169),x, algorithm=\
20*(6*x*e^(-16) + 26*e^(-24)*log(6*x*e^8 - 13) - 169/(6*x*e^32 - 13*e^24)) *e^16 - 520*(e^(-16)*log(6*x*e^8 - 13) - 13/(6*x*e^24 - 13*e^16))*e^8 + 31 20*e^(13/6*e^(-8) - 8)*exp_integral_e(2, -1/6*(6*x*e^8 - 13)*e^(-8))/(6*x* e^8 - 13) + 80*((x^2*e^8 - 3*x)*e^(2*x) - 6*(x^2*e^8 - 3*x)*e^x)/(6*x*e^8 - 13) - 4680/(6*x*e^16 - 13*e^8) + 18720*integrate(e^x/(36*x^2*e^16 - 156* x*e^8 + 169), x)
Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (28) = 56\).
Time = 0.28 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.03 \[ \int \frac {28080-18720 e^8 x+4320 e^{16} x^2+e^x \left (-18720-18720 x+e^8 \left (12480 x+14880 x^2\right )+e^{16} \left (-2880 x^2-2880 x^3\right )\right )+e^{2 x} \left (3120+6240 x+e^8 \left (-2080 x-4960 x^2\right )+e^{16} \left (480 x^2+960 x^3\right )\right )}{169-156 e^8 x+36 e^{16} x^2} \, dx=\frac {20 \, {\left (36 \, x^{2} e^{40} + 4 \, x^{2} e^{\left (2 \, x + 40\right )} - 24 \, x^{2} e^{\left (x + 40\right )} - 78 \, x e^{32} - 12 \, x e^{\left (2 \, x + 32\right )} + 72 \, x e^{\left (x + 32\right )} - 65 \, e^{24}\right )}}{6 \, x e^{40} - 13 \, e^{32}} \]
integrate((((960*x^3+480*x^2)*exp(4)^4+(-4960*x^2-2080*x)*exp(4)^2+6240*x+ 3120)*exp(x)^2+((-2880*x^3-2880*x^2)*exp(4)^4+(14880*x^2+12480*x)*exp(4)^2 -18720*x-18720)*exp(x)+4320*x^2*exp(4)^4-18720*x*exp(4)^2+28080)/(36*x^2*e xp(4)^4-156*x*exp(4)^2+169),x, algorithm=\
20*(36*x^2*e^40 + 4*x^2*e^(2*x + 40) - 24*x^2*e^(x + 40) - 78*x*e^32 - 12* x*e^(2*x + 32) + 72*x*e^(x + 32) - 65*e^24)/(6*x*e^40 - 13*e^32)
Time = 0.70 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.94 \[ \int \frac {28080-18720 e^8 x+4320 e^{16} x^2+e^x \left (-18720-18720 x+e^8 \left (12480 x+14880 x^2\right )+e^{16} \left (-2880 x^2-2880 x^3\right )\right )+e^{2 x} \left (3120+6240 x+e^8 \left (-2080 x-4960 x^2\right )+e^{16} \left (480 x^2+960 x^3\right )\right )}{169-156 e^8 x+36 e^{16} x^2} \, dx=120\,x-\frac {1300\,{\mathrm {e}}^{-8}}{6\,x\,{\mathrm {e}}^8-13}+\frac {{\mathrm {e}}^x\,\left (240\,x\,{\mathrm {e}}^{-8}-80\,x^2\right )}{x-\frac {13\,{\mathrm {e}}^{-8}}{6}}-\frac {{\mathrm {e}}^{2\,x}\,\left (40\,x\,{\mathrm {e}}^{-8}-\frac {40\,x^2}{3}\right )}{x-\frac {13\,{\mathrm {e}}^{-8}}{6}} \]
int((exp(2*x)*(6240*x - exp(8)*(2080*x + 4960*x^2) + exp(16)*(480*x^2 + 96 0*x^3) + 3120) - 18720*x*exp(8) - exp(x)*(18720*x - exp(8)*(12480*x + 1488 0*x^2) + exp(16)*(2880*x^2 + 2880*x^3) + 18720) + 4320*x^2*exp(16) + 28080 )/(36*x^2*exp(16) - 156*x*exp(8) + 169),x)