Integrand size = 140, antiderivative size = 22 \[ \int \frac {-16 e^{\frac {16}{\log (x)}}+\left (-3365 x+149462 x^2-2478498 x^3+19518724 x^4\right ) \log ^2(x)+e^{\frac {12}{\log (x)}} \left (96-2256 x+188 x \log ^2(x)\right )+e^{\frac {8}{\log (x)}} \left (-272+9008 x-106032 x^2+\left (-1126 x+26508 x^2\right ) \log ^2(x)\right )+e^{\frac {4}{\log (x)}} \left (288-12752 x+211312 x^2-1661168 x^3+\left (3188 x-105656 x^2+1245876 x^3\right ) \log ^2(x)\right )}{x \log ^2(x)} \, dx=x+\left (5+x+\left (-2+e^{\frac {4}{\log (x)}}+47 x\right )^2\right )^2 \] Output:
(5+x+(exp(4/ln(x))+47*x-2)^2)^2+x
Leaf count is larger than twice the leaf count of optimal. \(87\) vs. \(2(22)=44\).
Time = 0.27 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.95 \[ \int \frac {-16 e^{\frac {16}{\log (x)}}+\left (-3365 x+149462 x^2-2478498 x^3+19518724 x^4\right ) \log ^2(x)+e^{\frac {12}{\log (x)}} \left (96-2256 x+188 x \log ^2(x)\right )+e^{\frac {8}{\log (x)}} \left (-272+9008 x-106032 x^2+\left (-1126 x+26508 x^2\right ) \log ^2(x)\right )+e^{\frac {4}{\log (x)}} \left (288-12752 x+211312 x^2-1661168 x^3+\left (3188 x-105656 x^2+1245876 x^3\right ) \log ^2(x)\right )}{x \log ^2(x)} \, dx=e^{\frac {16}{\log (x)}}-e^{\frac {12}{\log (x)}} (8-188 x)-3365 x+74731 x^2-826166 x^3+4879681 x^4-e^{\frac {8}{\log (x)}} (-34-2 x (-563+6627 x))-e^{\frac {4}{\log (x)}} \left (72-4 x \left (797-13207 x+103823 x^2\right )\right ) \] Input:
Integrate[(-16*E^(16/Log[x]) + (-3365*x + 149462*x^2 - 2478498*x^3 + 19518 724*x^4)*Log[x]^2 + E^(12/Log[x])*(96 - 2256*x + 188*x*Log[x]^2) + E^(8/Lo g[x])*(-272 + 9008*x - 106032*x^2 + (-1126*x + 26508*x^2)*Log[x]^2) + E^(4 /Log[x])*(288 - 12752*x + 211312*x^2 - 1661168*x^3 + (3188*x - 105656*x^2 + 1245876*x^3)*Log[x]^2))/(x*Log[x]^2),x]
Output:
E^(16/Log[x]) - E^(12/Log[x])*(8 - 188*x) - 3365*x + 74731*x^2 - 826166*x^ 3 + 4879681*x^4 - E^(8/Log[x])*(-34 - 2*x*(-563 + 6627*x)) - E^(4/Log[x])* (72 - 4*x*(797 - 13207*x + 103823*x^2))
Leaf count is larger than twice the leaf count of optimal. \(87\) vs. \(2(22)=44\).
Time = 1.50 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.95, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.014, Rules used = {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.
| \(\displaystyle \int \frac {e^{\frac {8}{\log (x)}} \left (-106032 x^2+\left (26508 x^2-1126 x\right ) \log ^2(x)+9008 x-272\right )+e^{\frac {4}{\log (x)}} \left (-1661168 x^3+211312 x^2+\left (1245876 x^3-105656 x^2+3188 x\right ) \log ^2(x)-12752 x+288\right )+\left (19518724 x^4-2478498 x^3+149462 x^2-3365 x\right ) \log ^2(x)+e^{\frac {12}{\log (x)}} \left (-2256 x+188 x \log ^2(x)+96\right )-16 e^{\frac {16}{\log (x)}}}{x \log ^2(x)} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (19518724 x^3-2478498 x^2+\frac {2 e^{\frac {8}{\log (x)}} \left (-53016 x^2+13254 x^2 \log ^2(x)+4504 x-563 x \log ^2(x)-136\right )}{x \log ^2(x)}+\frac {4 e^{\frac {4}{\log (x)}} \left (-415292 x^3+311469 x^3 \log ^2(x)+52828 x^2-26414 x^2 \log ^2(x)-3188 x+797 x \log ^2(x)+72\right )}{x \log ^2(x)}+149462 x+\frac {4 e^{\frac {12}{\log (x)}} \left (-564 x+47 x \log ^2(x)+24\right )}{x \log ^2(x)}-\frac {16 e^{\frac {16}{\log (x)}}}{x \log ^2(x)}-3365\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 4879681 x^4-826166 x^3+74731 x^2+2 \left (6627 x^2-563 x+17\right ) e^{\frac {8}{\log (x)}}-4 \left (-103823 x^3+13207 x^2-797 x+18\right ) e^{\frac {4}{\log (x)}}-3365 x+e^{\frac {16}{\log (x)}}-4 (2-47 x) e^{\frac {12}{\log (x)}}\) |
Input:
Int[(-16*E^(16/Log[x]) + (-3365*x + 149462*x^2 - 2478498*x^3 + 19518724*x^ 4)*Log[x]^2 + E^(12/Log[x])*(96 - 2256*x + 188*x*Log[x]^2) + E^(8/Log[x])* (-272 + 9008*x - 106032*x^2 + (-1126*x + 26508*x^2)*Log[x]^2) + E^(4/Log[x ])*(288 - 12752*x + 211312*x^2 - 1661168*x^3 + (3188*x - 105656*x^2 + 1245 876*x^3)*Log[x]^2))/(x*Log[x]^2),x]
Output:
E^(16/Log[x]) - 4*E^(12/Log[x])*(2 - 47*x) - 3365*x + 74731*x^2 - 826166*x ^3 + 4879681*x^4 + 2*E^(8/Log[x])*(17 - 563*x + 6627*x^2) - 4*E^(4/Log[x]) *(18 - 797*x + 13207*x^2 - 103823*x^3)
Leaf count of result is larger than twice the leaf count of optimal. \(80\) vs. \(2(21)=42\).
Time = 0.26 (sec) , antiderivative size = 81, normalized size of antiderivative = 3.68
| method | result | size |
| risch | \(4879681 x^{4}+{\mathrm e}^{\frac {16}{\ln \left (x \right )}}-826166 x^{3}+74731 x^{2}-3365 x +\left (-8+188 x \right ) {\mathrm e}^{\frac {12}{\ln \left (x \right )}}+\left (13254 x^{2}-1126 x +34\right ) {\mathrm e}^{\frac {8}{\ln \left (x \right )}}+\left (415292 x^{3}-52828 x^{2}+3188 x -72\right ) {\mathrm e}^{\frac {4}{\ln \left (x \right )}}\) | \(81\) |
| parallelrisch | \(415292 \,{\mathrm e}^{\frac {4}{\ln \left (x \right )}} x^{3}+4879681 x^{4}+13254 \,{\mathrm e}^{\frac {8}{\ln \left (x \right )}} x^{2}-52828 \,{\mathrm e}^{\frac {4}{\ln \left (x \right )}} x^{2}-826166 x^{3}+188 \,{\mathrm e}^{\frac {12}{\ln \left (x \right )}} x -1126 \,{\mathrm e}^{\frac {8}{\ln \left (x \right )}} x +3188 \,{\mathrm e}^{\frac {4}{\ln \left (x \right )}} x +74731 x^{2}+{\mathrm e}^{\frac {16}{\ln \left (x \right )}}-8 \,{\mathrm e}^{\frac {12}{\ln \left (x \right )}}+34 \,{\mathrm e}^{\frac {8}{\ln \left (x \right )}}-72 \,{\mathrm e}^{\frac {4}{\ln \left (x \right )}}-3365 x\) | \(132\) |
Input:
int((-16*exp(4/ln(x))^4+(188*x*ln(x)^2-2256*x+96)*exp(4/ln(x))^3+((26508*x ^2-1126*x)*ln(x)^2-106032*x^2+9008*x-272)*exp(4/ln(x))^2+((1245876*x^3-105 656*x^2+3188*x)*ln(x)^2-1661168*x^3+211312*x^2-12752*x+288)*exp(4/ln(x))+( 19518724*x^4-2478498*x^3+149462*x^2-3365*x)*ln(x)^2)/x/ln(x)^2,x,method=_R ETURNVERBOSE)
Output:
4879681*x^4+exp(16/ln(x))-826166*x^3+74731*x^2-3365*x+(-8+188*x)*exp(12/ln (x))+(13254*x^2-1126*x+34)*exp(8/ln(x))+(415292*x^3-52828*x^2+3188*x-72)*e xp(4/ln(x))
Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (21) = 42\).
Time = 0.11 (sec) , antiderivative size = 83, normalized size of antiderivative = 3.77 \[ \int \frac {-16 e^{\frac {16}{\log (x)}}+\left (-3365 x+149462 x^2-2478498 x^3+19518724 x^4\right ) \log ^2(x)+e^{\frac {12}{\log (x)}} \left (96-2256 x+188 x \log ^2(x)\right )+e^{\frac {8}{\log (x)}} \left (-272+9008 x-106032 x^2+\left (-1126 x+26508 x^2\right ) \log ^2(x)\right )+e^{\frac {4}{\log (x)}} \left (288-12752 x+211312 x^2-1661168 x^3+\left (3188 x-105656 x^2+1245876 x^3\right ) \log ^2(x)\right )}{x \log ^2(x)} \, dx=4879681 \, x^{4} - 826166 \, x^{3} + 74731 \, x^{2} + 4 \, {\left (47 \, x - 2\right )} e^{\frac {12}{\log \left (x\right )}} + 2 \, {\left (6627 \, x^{2} - 563 \, x + 17\right )} e^{\frac {8}{\log \left (x\right )}} + 4 \, {\left (103823 \, x^{3} - 13207 \, x^{2} + 797 \, x - 18\right )} e^{\frac {4}{\log \left (x\right )}} - 3365 \, x + e^{\frac {16}{\log \left (x\right )}} \] Input:
integrate((-16*exp(4/log(x))^4+(188*x*log(x)^2-2256*x+96)*exp(4/log(x))^3+ ((26508*x^2-1126*x)*log(x)^2-106032*x^2+9008*x-272)*exp(4/log(x))^2+((1245 876*x^3-105656*x^2+3188*x)*log(x)^2-1661168*x^3+211312*x^2-12752*x+288)*ex p(4/log(x))+(19518724*x^4-2478498*x^3+149462*x^2-3365*x)*log(x)^2)/x/log(x )^2,x, algorithm="fricas")
Output:
4879681*x^4 - 826166*x^3 + 74731*x^2 + 4*(47*x - 2)*e^(12/log(x)) + 2*(662 7*x^2 - 563*x + 17)*e^(8/log(x)) + 4*(103823*x^3 - 13207*x^2 + 797*x - 18) *e^(4/log(x)) - 3365*x + e^(16/log(x))
Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (19) = 38\).
Time = 2.16 (sec) , antiderivative size = 75, normalized size of antiderivative = 3.41 \[ \int \frac {-16 e^{\frac {16}{\log (x)}}+\left (-3365 x+149462 x^2-2478498 x^3+19518724 x^4\right ) \log ^2(x)+e^{\frac {12}{\log (x)}} \left (96-2256 x+188 x \log ^2(x)\right )+e^{\frac {8}{\log (x)}} \left (-272+9008 x-106032 x^2+\left (-1126 x+26508 x^2\right ) \log ^2(x)\right )+e^{\frac {4}{\log (x)}} \left (288-12752 x+211312 x^2-1661168 x^3+\left (3188 x-105656 x^2+1245876 x^3\right ) \log ^2(x)\right )}{x \log ^2(x)} \, dx=4879681 x^{4} - 826166 x^{3} + 74731 x^{2} - 3365 x + \left (188 x - 8\right ) e^{\frac {12}{\log {\left (x \right )}}} + \left (13254 x^{2} - 1126 x + 34\right ) e^{\frac {8}{\log {\left (x \right )}}} + \left (415292 x^{3} - 52828 x^{2} + 3188 x - 72\right ) e^{\frac {4}{\log {\left (x \right )}}} + e^{\frac {16}{\log {\left (x \right )}}} \] Input:
integrate((-16*exp(4/ln(x))**4+(188*x*ln(x)**2-2256*x+96)*exp(4/ln(x))**3+ ((26508*x**2-1126*x)*ln(x)**2-106032*x**2+9008*x-272)*exp(4/ln(x))**2+((12 45876*x**3-105656*x**2+3188*x)*ln(x)**2-1661168*x**3+211312*x**2-12752*x+2 88)*exp(4/ln(x))+(19518724*x**4-2478498*x**3+149462*x**2-3365*x)*ln(x)**2) /x/ln(x)**2,x)
Output:
4879681*x**4 - 826166*x**3 + 74731*x**2 - 3365*x + (188*x - 8)*exp(12/log( x)) + (13254*x**2 - 1126*x + 34)*exp(8/log(x)) + (415292*x**3 - 52828*x**2 + 3188*x - 72)*exp(4/log(x)) + exp(16/log(x))
\[ \int \frac {-16 e^{\frac {16}{\log (x)}}+\left (-3365 x+149462 x^2-2478498 x^3+19518724 x^4\right ) \log ^2(x)+e^{\frac {12}{\log (x)}} \left (96-2256 x+188 x \log ^2(x)\right )+e^{\frac {8}{\log (x)}} \left (-272+9008 x-106032 x^2+\left (-1126 x+26508 x^2\right ) \log ^2(x)\right )+e^{\frac {4}{\log (x)}} \left (288-12752 x+211312 x^2-1661168 x^3+\left (3188 x-105656 x^2+1245876 x^3\right ) \log ^2(x)\right )}{x \log ^2(x)} \, dx=\int { \frac {{\left (19518724 \, x^{4} - 2478498 \, x^{3} + 149462 \, x^{2} - 3365 \, x\right )} \log \left (x\right )^{2} + 4 \, {\left (47 \, x \log \left (x\right )^{2} - 564 \, x + 24\right )} e^{\frac {12}{\log \left (x\right )}} + 2 \, {\left ({\left (13254 \, x^{2} - 563 \, x\right )} \log \left (x\right )^{2} - 53016 \, x^{2} + 4504 \, x - 136\right )} e^{\frac {8}{\log \left (x\right )}} - 4 \, {\left (415292 \, x^{3} - {\left (311469 \, x^{3} - 26414 \, x^{2} + 797 \, x\right )} \log \left (x\right )^{2} - 52828 \, x^{2} + 3188 \, x - 72\right )} e^{\frac {4}{\log \left (x\right )}} - 16 \, e^{\frac {16}{\log \left (x\right )}}}{x \log \left (x\right )^{2}} \,d x } \] Input:
integrate((-16*exp(4/log(x))^4+(188*x*log(x)^2-2256*x+96)*exp(4/log(x))^3+ ((26508*x^2-1126*x)*log(x)^2-106032*x^2+9008*x-272)*exp(4/log(x))^2+((1245 876*x^3-105656*x^2+3188*x)*log(x)^2-1661168*x^3+211312*x^2-12752*x+288)*ex p(4/log(x))+(19518724*x^4-2478498*x^3+149462*x^2-3365*x)*log(x)^2)/x/log(x )^2,x, algorithm="maxima")
Output:
4879681*x^4 - 826166*x^3 + 74731*x^2 - 3365*x + e^(16/log(x)) + integrate( 4*(47*x*log(x)^2 - 564*x + 24)*e^(12/log(x))/(x*log(x)^2), x) + integrate( 2*((13254*x^2 - 563*x)*log(x)^2 - 53016*x^2 + 4504*x - 136)*e^(8/log(x))/( x*log(x)^2), x) + integrate(-4*(415292*x^3 - (311469*x^3 - 26414*x^2 + 797 *x)*log(x)^2 - 52828*x^2 + 3188*x - 72)*e^(4/log(x))/(x*log(x)^2), x)
Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (21) = 42\).
Time = 0.12 (sec) , antiderivative size = 119, normalized size of antiderivative = 5.41 \[ \int \frac {-16 e^{\frac {16}{\log (x)}}+\left (-3365 x+149462 x^2-2478498 x^3+19518724 x^4\right ) \log ^2(x)+e^{\frac {12}{\log (x)}} \left (96-2256 x+188 x \log ^2(x)\right )+e^{\frac {8}{\log (x)}} \left (-272+9008 x-106032 x^2+\left (-1126 x+26508 x^2\right ) \log ^2(x)\right )+e^{\frac {4}{\log (x)}} \left (288-12752 x+211312 x^2-1661168 x^3+\left (3188 x-105656 x^2+1245876 x^3\right ) \log ^2(x)\right )}{x \log ^2(x)} \, dx=4879681 \, x^{4} + 415292 \, x^{3} e^{\frac {4}{\log \left (x\right )}} - 826166 \, x^{3} + 13254 \, x^{2} e^{\frac {8}{\log \left (x\right )}} - 52828 \, x^{2} e^{\frac {4}{\log \left (x\right )}} + 74731 \, x^{2} + 188 \, x e^{\frac {12}{\log \left (x\right )}} - 1126 \, x e^{\frac {8}{\log \left (x\right )}} + 3188 \, x e^{\frac {4}{\log \left (x\right )}} - 3365 \, x + e^{\frac {16}{\log \left (x\right )}} - 8 \, e^{\frac {12}{\log \left (x\right )}} + 34 \, e^{\frac {8}{\log \left (x\right )}} - 72 \, e^{\frac {4}{\log \left (x\right )}} \] Input:
integrate((-16*exp(4/log(x))^4+(188*x*log(x)^2-2256*x+96)*exp(4/log(x))^3+ ((26508*x^2-1126*x)*log(x)^2-106032*x^2+9008*x-272)*exp(4/log(x))^2+((1245 876*x^3-105656*x^2+3188*x)*log(x)^2-1661168*x^3+211312*x^2-12752*x+288)*ex p(4/log(x))+(19518724*x^4-2478498*x^3+149462*x^2-3365*x)*log(x)^2)/x/log(x )^2,x, algorithm="giac")
Output:
4879681*x^4 + 415292*x^3*e^(4/log(x)) - 826166*x^3 + 13254*x^2*e^(8/log(x) ) - 52828*x^2*e^(4/log(x)) + 74731*x^2 + 188*x*e^(12/log(x)) - 1126*x*e^(8 /log(x)) + 3188*x*e^(4/log(x)) - 3365*x + e^(16/log(x)) - 8*e^(12/log(x)) + 34*e^(8/log(x)) - 72*e^(4/log(x))
Timed out. \[ \int \frac {-16 e^{\frac {16}{\log (x)}}+\left (-3365 x+149462 x^2-2478498 x^3+19518724 x^4\right ) \log ^2(x)+e^{\frac {12}{\log (x)}} \left (96-2256 x+188 x \log ^2(x)\right )+e^{\frac {8}{\log (x)}} \left (-272+9008 x-106032 x^2+\left (-1126 x+26508 x^2\right ) \log ^2(x)\right )+e^{\frac {4}{\log (x)}} \left (288-12752 x+211312 x^2-1661168 x^3+\left (3188 x-105656 x^2+1245876 x^3\right ) \log ^2(x)\right )}{x \log ^2(x)} \, dx=-\int \frac {16\,{\mathrm {e}}^{\frac {16}{\ln \left (x\right )}}-{\mathrm {e}}^{\frac {4}{\ln \left (x\right )}}\,\left ({\ln \left (x\right )}^2\,\left (1245876\,x^3-105656\,x^2+3188\,x\right )-12752\,x+211312\,x^2-1661168\,x^3+288\right )+{\ln \left (x\right )}^2\,\left (-19518724\,x^4+2478498\,x^3-149462\,x^2+3365\,x\right )-{\mathrm {e}}^{\frac {12}{\ln \left (x\right )}}\,\left (188\,x\,{\ln \left (x\right )}^2-2256\,x+96\right )+{\mathrm {e}}^{\frac {8}{\ln \left (x\right )}}\,\left ({\ln \left (x\right )}^2\,\left (1126\,x-26508\,x^2\right )-9008\,x+106032\,x^2+272\right )}{x\,{\ln \left (x\right )}^2} \,d x \] Input:
int(-(16*exp(16/log(x)) - exp(4/log(x))*(log(x)^2*(3188*x - 105656*x^2 + 1 245876*x^3) - 12752*x + 211312*x^2 - 1661168*x^3 + 288) + log(x)^2*(3365*x - 149462*x^2 + 2478498*x^3 - 19518724*x^4) - exp(12/log(x))*(188*x*log(x) ^2 - 2256*x + 96) + exp(8/log(x))*(log(x)^2*(1126*x - 26508*x^2) - 9008*x + 106032*x^2 + 272))/(x*log(x)^2),x)
Output:
-int((16*exp(16/log(x)) - exp(4/log(x))*(log(x)^2*(3188*x - 105656*x^2 + 1 245876*x^3) - 12752*x + 211312*x^2 - 1661168*x^3 + 288) + log(x)^2*(3365*x - 149462*x^2 + 2478498*x^3 - 19518724*x^4) - exp(12/log(x))*(188*x*log(x) ^2 - 2256*x + 96) + exp(8/log(x))*(log(x)^2*(1126*x - 26508*x^2) - 9008*x + 106032*x^2 + 272))/(x*log(x)^2), x)
Time = 0.16 (sec) , antiderivative size = 129, normalized size of antiderivative = 5.86 \[ \int \frac {-16 e^{\frac {16}{\log (x)}}+\left (-3365 x+149462 x^2-2478498 x^3+19518724 x^4\right ) \log ^2(x)+e^{\frac {12}{\log (x)}} \left (96-2256 x+188 x \log ^2(x)\right )+e^{\frac {8}{\log (x)}} \left (-272+9008 x-106032 x^2+\left (-1126 x+26508 x^2\right ) \log ^2(x)\right )+e^{\frac {4}{\log (x)}} \left (288-12752 x+211312 x^2-1661168 x^3+\left (3188 x-105656 x^2+1245876 x^3\right ) \log ^2(x)\right )}{x \log ^2(x)} \, dx=e^{\frac {16}{\mathrm {log}\left (x \right )}}+188 e^{\frac {12}{\mathrm {log}\left (x \right )}} x -8 e^{\frac {12}{\mathrm {log}\left (x \right )}}+13254 e^{\frac {8}{\mathrm {log}\left (x \right )}} x^{2}-1126 e^{\frac {8}{\mathrm {log}\left (x \right )}} x +34 e^{\frac {8}{\mathrm {log}\left (x \right )}}+415292 e^{\frac {4}{\mathrm {log}\left (x \right )}} x^{3}-52828 e^{\frac {4}{\mathrm {log}\left (x \right )}} x^{2}+3188 e^{\frac {4}{\mathrm {log}\left (x \right )}} x -72 e^{\frac {4}{\mathrm {log}\left (x \right )}}+4879681 x^{4}-826166 x^{3}+74731 x^{2}-3365 x \] Input:
int((-16*exp(4/log(x))^4+(188*x*log(x)^2-2256*x+96)*exp(4/log(x))^3+((2650 8*x^2-1126*x)*log(x)^2-106032*x^2+9008*x-272)*exp(4/log(x))^2+((1245876*x^ 3-105656*x^2+3188*x)*log(x)^2-1661168*x^3+211312*x^2-12752*x+288)*exp(4/lo g(x))+(19518724*x^4-2478498*x^3+149462*x^2-3365*x)*log(x)^2)/x/log(x)^2,x)
Output:
e**(16/log(x)) + 188*e**(12/log(x))*x - 8*e**(12/log(x)) + 13254*e**(8/log (x))*x**2 - 1126*e**(8/log(x))*x + 34*e**(8/log(x)) + 415292*e**(4/log(x)) *x**3 - 52828*e**(4/log(x))*x**2 + 3188*e**(4/log(x))*x - 72*e**(4/log(x)) + 4879681*x**4 - 826166*x**3 + 74731*x**2 - 3365*x