Integrand size = 134, antiderivative size = 20 \[ \int \frac {-4398046511104 e^{-20460-20 x}-17179869184 e^{-15345-15 x} x-25165824 e^{-10230-10 x} x^2-16384 e^{-5115-5 x} x^3-4 x^4+\left (-21990232555520 e^{-20460-20 x} x+4 x^4+1073741824 e^{-15345-15 x} \left (4 x-60 x^2\right )+1048576 e^{-10230-10 x} \left (12 x^2-60 x^3\right )+1024 e^{-5115-5 x} \left (12 x^3-20 x^4\right )\right ) \log (x)}{x \log ^5(x)} \, dx=\frac {\left (e^{5 (-1023-x+\log (4))}+x\right )^4}{\log ^4(x)} \] Output:
(x+exp(10*ln(2)-5*x-5115))^4/ln(x)^4
Time = 5.06 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15 \[ \int \frac {-4398046511104 e^{-20460-20 x}-17179869184 e^{-15345-15 x} x-25165824 e^{-10230-10 x} x^2-16384 e^{-5115-5 x} x^3-4 x^4+\left (-21990232555520 e^{-20460-20 x} x+4 x^4+1073741824 e^{-15345-15 x} \left (4 x-60 x^2\right )+1048576 e^{-10230-10 x} \left (12 x^2-60 x^3\right )+1024 e^{-5115-5 x} \left (12 x^3-20 x^4\right )\right ) \log (x)}{x \log ^5(x)} \, dx=\frac {\left (1024 e^{-5 x}+e^{5115} x\right )^4}{e^{20460} \log ^4(x)} \] Input:
Integrate[(-4398046511104*E^(-20460 - 20*x) - 17179869184*E^(-15345 - 15*x )*x - 25165824*E^(-10230 - 10*x)*x^2 - 16384*E^(-5115 - 5*x)*x^3 - 4*x^4 + (-21990232555520*E^(-20460 - 20*x)*x + 4*x^4 + 1073741824*E^(-15345 - 15* x)*(4*x - 60*x^2) + 1048576*E^(-10230 - 10*x)*(12*x^2 - 60*x^3) + 1024*E^( -5115 - 5*x)*(12*x^3 - 20*x^4))*Log[x])/(x*Log[x]^5),x]
Output:
(1024/E^(5*x) + E^5115*x)^4/(E^20460*Log[x]^4)
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 2.37 (sec) , antiderivative size = 203, normalized size of antiderivative = 10.15, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {7239, 27, 25, 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.
| \(\displaystyle \int \frac {-4 x^4-16384 e^{-5 x-5115} x^3-25165824 e^{-10 x-10230} x^2+\left (4 x^4+1073741824 e^{-15 x-15345} \left (4 x-60 x^2\right )+1024 e^{-5 x-5115} \left (12 x^3-20 x^4\right )+1048576 e^{-10 x-10230} \left (12 x^2-60 x^3\right )-21990232555520 e^{-20 x-20460} x\right ) \log (x)-17179869184 e^{-15 x-15345} x-4398046511104 e^{-20 x-20460}}{x \log ^5(x)} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {4 e^{-20 x-20460} \left (e^{5 x+5115} x+1024\right )^3 \left (-e^{5 x+5115} x+\left (e^{5 x+5115}-5120\right ) x \log (x)-1024\right )}{x \log ^5(x)}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 4 \int -\frac {e^{-20 x-20460} \left (e^{5 x+5115} x+1024\right )^3 \left (e^{5 x+5115} x+\left (5120-e^{5 x+5115}\right ) \log (x) x+1024\right )}{x \log ^5(x)}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -4 \int \frac {e^{-20 x-20460} \left (e^{5 x+5115} x+1024\right )^3 \left (e^{5 x+5115} x+\left (5120-e^{5 x+5115}\right ) \log (x) x+1024\right )}{x \log ^5(x)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -4 \int \left (-\frac {(\log (x)-1) x^3}{\log ^5(x)}+\frac {1024 e^{-5 x-5115} (5 x \log (x)-3 \log (x)+4) x^2}{\log ^5(x)}+\frac {3145728 e^{-10 x-10230} (5 x \log (x)-\log (x)+2) x}{\log ^5(x)}+\frac {1073741824 e^{-15 x-15345} (15 x \log (x)-\log (x)+4)}{\log ^5(x)}+\frac {1099511627776 e^{-20 x-20460} (5 x \log (x)+1)}{\log ^5(x) x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -4 \left (-\frac {32}{3} \operatorname {ExpIntegralEi}(4 \log (x))+\frac {32}{3} (1-\log (x)) \operatorname {ExpIntegralEi}(4 \log (x))+\frac {32}{3} \log (x) \operatorname {ExpIntegralEi}(4 \log (x))-\frac {8 x^4}{3}-\frac {x^4 (1-\log (x))}{4 \log ^4(x)}-\frac {x^4 (1-\log (x))}{3 \log ^3(x)}+\frac {x^4}{12 \log ^3(x)}-\frac {2 x^4 (1-\log (x))}{3 \log ^2(x)}+\frac {x^4}{3 \log ^2(x)}-\frac {8 x^4 (1-\log (x))}{3 \log (x)}+\frac {2 x^4}{\log (x)}-\frac {1024 e^{-5 x-5115} x^3}{\log ^4(x)}-\frac {1572864 e^{-10 x-10230} x^2}{\log ^4(x)}-\frac {1073741824 e^{-15 x-15345} x}{\log ^4(x)}-\frac {274877906944 e^{-20 x-20460}}{\log ^4(x)}\right )\) |
Input:
Int[(-4398046511104*E^(-20460 - 20*x) - 17179869184*E^(-15345 - 15*x)*x - 25165824*E^(-10230 - 10*x)*x^2 - 16384*E^(-5115 - 5*x)*x^3 - 4*x^4 + (-219 90232555520*E^(-20460 - 20*x)*x + 4*x^4 + 1073741824*E^(-15345 - 15*x)*(4* x - 60*x^2) + 1048576*E^(-10230 - 10*x)*(12*x^2 - 60*x^3) + 1024*E^(-5115 - 5*x)*(12*x^3 - 20*x^4))*Log[x])/(x*Log[x]^5),x]
Output:
-4*((-8*x^4)/3 - (32*ExpIntegralEi[4*Log[x]])/3 + (32*ExpIntegralEi[4*Log[ x]]*(1 - Log[x]))/3 - (274877906944*E^(-20460 - 20*x))/Log[x]^4 - (1073741 824*E^(-15345 - 15*x)*x)/Log[x]^4 - (1572864*E^(-10230 - 10*x)*x^2)/Log[x] ^4 - (1024*E^(-5115 - 5*x)*x^3)/Log[x]^4 - (x^4*(1 - Log[x]))/(4*Log[x]^4) + x^4/(12*Log[x]^3) - (x^4*(1 - Log[x]))/(3*Log[x]^3) + x^4/(3*Log[x]^2) - (2*x^4*(1 - Log[x]))/(3*Log[x]^2) + (2*x^4)/Log[x] - (8*x^4*(1 - Log[x]) )/(3*Log[x]) + (32*ExpIntegralEi[4*Log[x]]*Log[x])/3)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(48\) vs. \(2(19)=38\).
Time = 381.94 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.45
| method | result | size |
| risch | \(\frac {x^{4}+4096 \,{\mathrm e}^{-5115-5 x} x^{3}+6291456 \,{\mathrm e}^{-10230-10 x} x^{2}+4294967296 \,{\mathrm e}^{-15345-15 x} x +1099511627776 \,{\mathrm e}^{-20460-20 x}}{\ln \left (x \right )^{4}}\) | \(49\) |
| parallelrisch | \(-\frac {-x^{4}-4 x^{3} {\mathrm e}^{10 \ln \left (2\right )-5 x -5115}-6291456 \,{\mathrm e}^{-10230-10 x} x^{2}-4294967296 \,{\mathrm e}^{-15345-15 x} x -1099511627776 \,{\mathrm e}^{-20460-20 x}}{\ln \left (x \right )^{4}}\) | \(74\) |
Input:
int(((-20*x*exp(10*ln(2)-5*x-5115)^4+(-60*x^2+4*x)*exp(10*ln(2)-5*x-5115)^ 3+(-60*x^3+12*x^2)*exp(10*ln(2)-5*x-5115)^2+(-20*x^4+12*x^3)*exp(10*ln(2)- 5*x-5115)+4*x^4)*ln(x)-4*exp(10*ln(2)-5*x-5115)^4-16*x*exp(10*ln(2)-5*x-51 15)^3-24*x^2*exp(10*ln(2)-5*x-5115)^2-16*x^3*exp(10*ln(2)-5*x-5115)-4*x^4) /x/ln(x)^5,x,method=_RETURNVERBOSE)
Output:
(x^4+4096*exp(-5115-5*x)*x^3+6291456*exp(-10230-10*x)*x^2+4294967296*exp(- 15345-15*x)*x+1099511627776*exp(-20460-20*x))/ln(x)^4
Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (19) = 38\).
Time = 0.10 (sec) , antiderivative size = 62, normalized size of antiderivative = 3.10 \[ \int \frac {-4398046511104 e^{-20460-20 x}-17179869184 e^{-15345-15 x} x-25165824 e^{-10230-10 x} x^2-16384 e^{-5115-5 x} x^3-4 x^4+\left (-21990232555520 e^{-20460-20 x} x+4 x^4+1073741824 e^{-15345-15 x} \left (4 x-60 x^2\right )+1048576 e^{-10230-10 x} \left (12 x^2-60 x^3\right )+1024 e^{-5115-5 x} \left (12 x^3-20 x^4\right )\right ) \log (x)}{x \log ^5(x)} \, dx=\frac {x^{4} + 4 \, x^{3} e^{\left (-5 \, x + 10 \, \log \left (2\right ) - 5115\right )} + 6 \, x^{2} e^{\left (-10 \, x + 20 \, \log \left (2\right ) - 10230\right )} + 4 \, x e^{\left (-15 \, x + 30 \, \log \left (2\right ) - 15345\right )} + e^{\left (-20 \, x + 40 \, \log \left (2\right ) - 20460\right )}}{\log \left (x\right )^{4}} \] Input:
integrate(((-20*x*exp(10*log(2)-5*x-5115)^4+(-60*x^2+4*x)*exp(10*log(2)-5* x-5115)^3+(-60*x^3+12*x^2)*exp(10*log(2)-5*x-5115)^2+(-20*x^4+12*x^3)*exp( 10*log(2)-5*x-5115)+4*x^4)*log(x)-4*exp(10*log(2)-5*x-5115)^4-16*x*exp(10* log(2)-5*x-5115)^3-24*x^2*exp(10*log(2)-5*x-5115)^2-16*x^3*exp(10*log(2)-5 *x-5115)-4*x^4)/x/log(x)^5,x, algorithm="fricas")
Output:
(x^4 + 4*x^3*e^(-5*x + 10*log(2) - 5115) + 6*x^2*e^(-10*x + 20*log(2) - 10 230) + 4*x*e^(-15*x + 30*log(2) - 15345) + e^(-20*x + 40*log(2) - 20460))/ log(x)^4
Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (17) = 34\).
Time = 0.18 (sec) , antiderivative size = 82, normalized size of antiderivative = 4.10 \[ \int \frac {-4398046511104 e^{-20460-20 x}-17179869184 e^{-15345-15 x} x-25165824 e^{-10230-10 x} x^2-16384 e^{-5115-5 x} x^3-4 x^4+\left (-21990232555520 e^{-20460-20 x} x+4 x^4+1073741824 e^{-15345-15 x} \left (4 x-60 x^2\right )+1048576 e^{-10230-10 x} \left (12 x^2-60 x^3\right )+1024 e^{-5115-5 x} \left (12 x^3-20 x^4\right )\right ) \log (x)}{x \log ^5(x)} \, dx=\frac {x^{4}}{\log {\left (x \right )}^{4}} + \frac {4096 x^{3} e^{- 5 x - 5115} \log {\left (x \right )}^{12} + 6291456 x^{2} e^{- 10 x - 10230} \log {\left (x \right )}^{12} + 4294967296 x e^{- 15 x - 15345} \log {\left (x \right )}^{12} + 1099511627776 e^{- 20 x - 20460} \log {\left (x \right )}^{12}}{\log {\left (x \right )}^{16}} \] Input:
integrate(((-20*x*exp(10*ln(2)-5*x-5115)**4+(-60*x**2+4*x)*exp(10*ln(2)-5* x-5115)**3+(-60*x**3+12*x**2)*exp(10*ln(2)-5*x-5115)**2+(-20*x**4+12*x**3) *exp(10*ln(2)-5*x-5115)+4*x**4)*ln(x)-4*exp(10*ln(2)-5*x-5115)**4-16*x*exp (10*ln(2)-5*x-5115)**3-24*x**2*exp(10*ln(2)-5*x-5115)**2-16*x**3*exp(10*ln (2)-5*x-5115)-4*x**4)/x/ln(x)**5,x)
Output:
x**4/log(x)**4 + (4096*x**3*exp(-5*x - 5115)*log(x)**12 + 6291456*x**2*exp (-10*x - 10230)*log(x)**12 + 4294967296*x*exp(-15*x - 15345)*log(x)**12 + 1099511627776*exp(-20*x - 20460)*log(x)**12)/log(x)**16
\[ \int \frac {-4398046511104 e^{-20460-20 x}-17179869184 e^{-15345-15 x} x-25165824 e^{-10230-10 x} x^2-16384 e^{-5115-5 x} x^3-4 x^4+\left (-21990232555520 e^{-20460-20 x} x+4 x^4+1073741824 e^{-15345-15 x} \left (4 x-60 x^2\right )+1048576 e^{-10230-10 x} \left (12 x^2-60 x^3\right )+1024 e^{-5115-5 x} \left (12 x^3-20 x^4\right )\right ) \log (x)}{x \log ^5(x)} \, dx=\int { -\frac {4 \, {\left (x^{4} + 4 \, x^{3} e^{\left (-5 \, x + 10 \, \log \left (2\right ) - 5115\right )} + 6 \, x^{2} e^{\left (-10 \, x + 20 \, \log \left (2\right ) - 10230\right )} + 4 \, x e^{\left (-15 \, x + 30 \, \log \left (2\right ) - 15345\right )} - {\left (x^{4} - {\left (5 \, x^{4} - 3 \, x^{3}\right )} e^{\left (-5 \, x + 10 \, \log \left (2\right ) - 5115\right )} - 3 \, {\left (5 \, x^{3} - x^{2}\right )} e^{\left (-10 \, x + 20 \, \log \left (2\right ) - 10230\right )} - {\left (15 \, x^{2} - x\right )} e^{\left (-15 \, x + 30 \, \log \left (2\right ) - 15345\right )} - 5 \, x e^{\left (-20 \, x + 40 \, \log \left (2\right ) - 20460\right )}\right )} \log \left (x\right ) + e^{\left (-20 \, x + 40 \, \log \left (2\right ) - 20460\right )}\right )}}{x \log \left (x\right )^{5}} \,d x } \] Input:
integrate(((-20*x*exp(10*log(2)-5*x-5115)^4+(-60*x^2+4*x)*exp(10*log(2)-5* x-5115)^3+(-60*x^3+12*x^2)*exp(10*log(2)-5*x-5115)^2+(-20*x^4+12*x^3)*exp( 10*log(2)-5*x-5115)+4*x^4)*log(x)-4*exp(10*log(2)-5*x-5115)^4-16*x*exp(10* log(2)-5*x-5115)^3-24*x^2*exp(10*log(2)-5*x-5115)^2-16*x^3*exp(10*log(2)-5 *x-5115)-4*x^4)/x/log(x)^5,x, algorithm="maxima")
Output:
-4/3*(8*x^4*e^20460*log(x)^3 + 2*x^4*e^20460*log(x)^2 + x^4*e^20460*log(x) - 3072*x^3*e^(-5*x + 15345) - 4718592*x^2*e^(-10*x + 10230) - 3221225472* x*e^(-15*x + 5115) - 824633720832*e^(-20*x))*e^(-20460)/log(x)^4 + 1024*ga mma(-4, -4*log(x)) + 128/3*integrate(x^3/log(x), x)
Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (19) = 38\).
Time = 0.13 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.55 \[ \int \frac {-4398046511104 e^{-20460-20 x}-17179869184 e^{-15345-15 x} x-25165824 e^{-10230-10 x} x^2-16384 e^{-5115-5 x} x^3-4 x^4+\left (-21990232555520 e^{-20460-20 x} x+4 x^4+1073741824 e^{-15345-15 x} \left (4 x-60 x^2\right )+1048576 e^{-10230-10 x} \left (12 x^2-60 x^3\right )+1024 e^{-5115-5 x} \left (12 x^3-20 x^4\right )\right ) \log (x)}{x \log ^5(x)} \, dx=\frac {{\left (x^{4} e^{20460} + 4096 \, x^{3} e^{\left (-5 \, x + 15345\right )} + 6291456 \, x^{2} e^{\left (-10 \, x + 10230\right )} + 4294967296 \, x e^{\left (-15 \, x + 5115\right )} + 1099511627776 \, e^{\left (-20 \, x\right )}\right )} e^{\left (-20460\right )}}{\log \left (x\right )^{4}} \] Input:
integrate(((-20*x*exp(10*log(2)-5*x-5115)^4+(-60*x^2+4*x)*exp(10*log(2)-5* x-5115)^3+(-60*x^3+12*x^2)*exp(10*log(2)-5*x-5115)^2+(-20*x^4+12*x^3)*exp( 10*log(2)-5*x-5115)+4*x^4)*log(x)-4*exp(10*log(2)-5*x-5115)^4-16*x*exp(10* log(2)-5*x-5115)^3-24*x^2*exp(10*log(2)-5*x-5115)^2-16*x^3*exp(10*log(2)-5 *x-5115)-4*x^4)/x/log(x)^5,x, algorithm="giac")
Output:
(x^4*e^20460 + 4096*x^3*e^(-5*x + 15345) + 6291456*x^2*e^(-10*x + 10230) + 4294967296*x*e^(-15*x + 5115) + 1099511627776*e^(-20*x))*e^(-20460)/log(x )^4
Time = 2.81 (sec) , antiderivative size = 64, normalized size of antiderivative = 3.20 \[ \int \frac {-4398046511104 e^{-20460-20 x}-17179869184 e^{-15345-15 x} x-25165824 e^{-10230-10 x} x^2-16384 e^{-5115-5 x} x^3-4 x^4+\left (-21990232555520 e^{-20460-20 x} x+4 x^4+1073741824 e^{-15345-15 x} \left (4 x-60 x^2\right )+1048576 e^{-10230-10 x} \left (12 x^2-60 x^3\right )+1024 e^{-5115-5 x} \left (12 x^3-20 x^4\right )\right ) \log (x)}{x \log ^5(x)} \, dx=\frac {1099511627776\,{\mathrm {e}}^{-20\,x-20460}}{{\ln \left (x\right )}^4}+\frac {x^4}{{\ln \left (x\right )}^4}+\frac {4096\,x^3\,{\mathrm {e}}^{-5\,x-5115}}{{\ln \left (x\right )}^4}+\frac {6291456\,x^2\,{\mathrm {e}}^{-10\,x-10230}}{{\ln \left (x\right )}^4}+\frac {4294967296\,x\,{\mathrm {e}}^{-15\,x-15345}}{{\ln \left (x\right )}^4} \] Input:
int(-(4*exp(40*log(2) - 20*x - 20460) + 16*x*exp(30*log(2) - 15*x - 15345) - log(x)*(exp(30*log(2) - 15*x - 15345)*(4*x - 60*x^2) - 20*x*exp(40*log( 2) - 20*x - 20460) + exp(10*log(2) - 5*x - 5115)*(12*x^3 - 20*x^4) + exp(2 0*log(2) - 10*x - 10230)*(12*x^2 - 60*x^3) + 4*x^4) + 16*x^3*exp(10*log(2) - 5*x - 5115) + 24*x^2*exp(20*log(2) - 10*x - 10230) + 4*x^4)/(x*log(x)^5 ),x)
Output:
(1099511627776*exp(- 20*x - 20460))/log(x)^4 + x^4/log(x)^4 + (4096*x^3*ex p(- 5*x - 5115))/log(x)^4 + (6291456*x^2*exp(- 10*x - 10230))/log(x)^4 + ( 4294967296*x*exp(- 15*x - 15345))/log(x)^4
Time = 0.19 (sec) , antiderivative size = 66, normalized size of antiderivative = 3.30 \[ \int \frac {-4398046511104 e^{-20460-20 x}-17179869184 e^{-15345-15 x} x-25165824 e^{-10230-10 x} x^2-16384 e^{-5115-5 x} x^3-4 x^4+\left (-21990232555520 e^{-20460-20 x} x+4 x^4+1073741824 e^{-15345-15 x} \left (4 x-60 x^2\right )+1048576 e^{-10230-10 x} \left (12 x^2-60 x^3\right )+1024 e^{-5115-5 x} \left (12 x^3-20 x^4\right )\right ) \log (x)}{x \log ^5(x)} \, dx=\frac {e^{20 x} e^{20460} x^{4}+4096 e^{15 x} e^{15345} x^{3}+6291456 e^{10 x} e^{10230} x^{2}+4294967296 e^{5 x} e^{5115} x +1099511627776}{e^{20 x} \mathrm {log}\left (x \right )^{4} e^{20460}} \] Input:
int(((-20*x*exp(10*log(2)-5*x-5115)^4+(-60*x^2+4*x)*exp(10*log(2)-5*x-5115 )^3+(-60*x^3+12*x^2)*exp(10*log(2)-5*x-5115)^2+(-20*x^4+12*x^3)*exp(10*log (2)-5*x-5115)+4*x^4)*log(x)-4*exp(10*log(2)-5*x-5115)^4-16*x*exp(10*log(2) -5*x-5115)^3-24*x^2*exp(10*log(2)-5*x-5115)^2-16*x^3*exp(10*log(2)-5*x-511 5)-4*x^4)/x/log(x)^5,x)
Output:
(e**(20*x)*e**20460*x**4 + 4096*e**(15*x)*e**15345*x**3 + 6291456*e**(10*x )*e**10230*x**2 + 4294967296*e**(5*x)*e**5115*x + 1099511627776)/(e**(20*x )*log(x)**4*e**20460)