Integrand size = 108, antiderivative size = 30 \[ \int \frac {-64-648 x-1522 x^2-216 x^3+e^x \left (40 x^3-40 x^4-310 x^5-40 x^6+\left (-80 x^2-230 x^3+270 x^4+40 x^5\right ) \log (4)\right )+e^{2 x} \left (50 x^6+50 x^7+\left (-50 x^5-100 x^6\right ) \log (4)+50 x^5 \log ^2(4)\right )}{x^5} \, dx=\frac {\left (-2-\frac {4}{x}+x-5 \left (5+x+e^x x (-x+\log (4))\right )\right )^2}{x^2} \]
Leaf count is larger than twice the leaf count of optimal. \(95\) vs. \(2(30)=60\).
Time = 0.15 (sec) , antiderivative size = 95, normalized size of antiderivative = 3.17 \[ \int \frac {-64-648 x-1522 x^2-216 x^3+e^x \left (40 x^3-40 x^4-310 x^5-40 x^6+\left (-80 x^2-230 x^3+270 x^4+40 x^5\right ) \log (4)\right )+e^{2 x} \left (50 x^6+50 x^7+\left (-50 x^5-100 x^6\right ) \log (4)+50 x^5 \log ^2(4)\right )}{x^5} \, dx=2 \left (\frac {8}{x^4}+\frac {108}{x^3}+\frac {761}{2 x^2}+\frac {108}{x}+e^{2 x} \left (\frac {25 x^2}{2}-\frac {25}{2} x \log (16)+\frac {25}{4} \left (-2 \log (4)+2 \log ^2(4)+\log (16)\right )\right )+e^x \left (-20 x+\frac {20 \log (4)}{x^2}+\frac {5 (-4+27 \log (4))}{x}+5 (-27+\log (256))\right )\right ) \]
Integrate[(-64 - 648*x - 1522*x^2 - 216*x^3 + E^x*(40*x^3 - 40*x^4 - 310*x ^5 - 40*x^6 + (-80*x^2 - 230*x^3 + 270*x^4 + 40*x^5)*Log[4]) + E^(2*x)*(50 *x^6 + 50*x^7 + (-50*x^5 - 100*x^6)*Log[4] + 50*x^5*Log[4]^2))/x^5,x]
2*(8/x^4 + 108/x^3 + 761/(2*x^2) + 108/x + E^(2*x)*((25*x^2)/2 - (25*x*Log [16])/2 + (25*(-2*Log[4] + 2*Log[4]^2 + Log[16]))/4) + E^x*(-20*x + (20*Lo g[4])/x^2 + (5*(-4 + 27*Log[4]))/x + 5*(-27 + Log[256])))
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.51 (sec) , antiderivative size = 174, normalized size of antiderivative = 5.80, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {2010, 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.
\(\displaystyle \int \frac {-216 x^3-1522 x^2+e^{2 x} \left (50 x^7+50 x^6+50 x^5 \log ^2(4)+\left (-100 x^6-50 x^5\right ) \log (4)\right )+e^x \left (-40 x^6-310 x^5-40 x^4+40 x^3+\left (40 x^5+270 x^4-230 x^3-80 x^2\right ) \log (4)\right )-648 x-64}{x^5} \, dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \int \left (-\frac {2 \left (108 x^3+761 x^2+324 x+32\right )}{x^5}+\frac {10 e^x \left (-4 x^4-x^3 (31-4 \log (4))-x^2 (4-27 \log (4))+x (4-23 \log (4))-8 \log (4)\right )}{x^3}+50 e^{2 x} (x-\log (4)) (x+1-\log (4))\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -40 \log (4) \operatorname {ExpIntegralEi}(x)+10 (4-23 \log (4)) \operatorname {ExpIntegralEi}(x)-10 (4-27 \log (4)) \operatorname {ExpIntegralEi}(x)+\frac {16}{x^4}+\frac {216}{x^3}+25 e^{2 x} x^2+\frac {761}{x^2}+\frac {40 e^x \log (4)}{x^2}-40 e^x x-25 e^{2 x} x+40 e^x+\frac {25 e^{2 x}}{2}+\frac {216}{x}+25 e^{2 x} x (1-2 \log (4))-\frac {25}{2} e^{2 x} (1-\log (16))-25 e^{2 x} (1-\log (4)) \log (4)-10 e^x (31-8 \log (2))+\frac {40 e^x \log (4)}{x}-\frac {10 e^x (4-23 \log (4))}{x}\) |
Int[(-64 - 648*x - 1522*x^2 - 216*x^3 + E^x*(40*x^3 - 40*x^4 - 310*x^5 - 4 0*x^6 + (-80*x^2 - 230*x^3 + 270*x^4 + 40*x^5)*Log[4]) + E^(2*x)*(50*x^6 + 50*x^7 + (-50*x^5 - 100*x^6)*Log[4] + 50*x^5*Log[4]^2))/x^5,x]
40*E^x + (25*E^(2*x))/2 + 16/x^4 + 216/x^3 + 761/x^2 + 216/x - 40*E^x*x - 25*E^(2*x)*x + 25*E^(2*x)*x^2 - 10*E^x*(31 - 8*Log[2]) - 10*ExpIntegralEi[ x]*(4 - 27*Log[4]) - (10*E^x*(4 - 23*Log[4]))/x + 10*ExpIntegralEi[x]*(4 - 23*Log[4]) + 25*E^(2*x)*x*(1 - 2*Log[4]) + (40*E^x*Log[4])/x^2 + (40*E^x* Log[4])/x - 40*ExpIntegralEi[x]*Log[4] - 25*E^(2*x)*(1 - Log[4])*Log[4] - (25*E^(2*x)*(1 - Log[16]))/2
3.6.95.3.1 Defintions of rubi rules used
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] , x] /; FreeQ[{c, m}, x] && SumQ[u] && !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
Leaf count of result is larger than twice the leaf count of optimal. \(79\) vs. \(2(29)=58\).
Time = 0.85 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.67
method | result | size |
risch | \(\frac {216 x^{3}+761 x^{2}+216 x +16}{x^{4}}+\left (100 \ln \left (2\right )^{2}-100 x \ln \left (2\right )+25 x^{2}\right ) {\mathrm e}^{2 x}+\frac {10 \left (8 x^{2} \ln \left (2\right )-4 x^{3}+54 x \ln \left (2\right )-27 x^{2}+8 \ln \left (2\right )-4 x \right ) {\mathrm e}^{x}}{x^{2}}\) | \(80\) |
parts | \(25 \,{\mathrm e}^{2 x} x^{2}+100 \ln \left (2\right )^{2} {\mathrm e}^{2 x}-100 x \ln \left (2\right ) {\mathrm e}^{2 x}+\frac {16}{x^{4}}+\frac {216}{x^{3}}+\frac {761}{x^{2}}+\frac {216}{x}-\frac {40 \,{\mathrm e}^{x}}{x}-40 \,{\mathrm e}^{x} x -270 \,{\mathrm e}^{x}+80 \,{\mathrm e}^{x} \ln \left (2\right )+\frac {80 \ln \left (2\right ) {\mathrm e}^{x}}{x^{2}}+\frac {540 \ln \left (2\right ) {\mathrm e}^{x}}{x}\) | \(90\) |
norman | \(\frac {16+\left (-270+80 \ln \left (2\right )\right ) x^{4} {\mathrm e}^{x}+\left (540 \ln \left (2\right )-40\right ) x^{3} {\mathrm e}^{x}+216 x +761 x^{2}+216 x^{3}-40 x^{5} {\mathrm e}^{x}+25 \,{\mathrm e}^{2 x} x^{6}+80 x^{2} \ln \left (2\right ) {\mathrm e}^{x}+100 \ln \left (2\right )^{2} {\mathrm e}^{2 x} x^{4}-100 \ln \left (2\right ) {\mathrm e}^{2 x} x^{5}}{x^{4}}\) | \(93\) |
parallelrisch | \(\frac {100 \ln \left (2\right )^{2} {\mathrm e}^{2 x} x^{4}-100 \ln \left (2\right ) {\mathrm e}^{2 x} x^{5}+25 \,{\mathrm e}^{2 x} x^{6}+80 \ln \left (2\right ) {\mathrm e}^{x} x^{4}-40 x^{5} {\mathrm e}^{x}+540 x^{3} \ln \left (2\right ) {\mathrm e}^{x}-270 \,{\mathrm e}^{x} x^{4}+80 x^{2} \ln \left (2\right ) {\mathrm e}^{x}-40 \,{\mathrm e}^{x} x^{3}+216 x^{3}+761 x^{2}+216 x +16}{x^{4}}\) | \(101\) |
default | \(\frac {16}{x^{4}}+\frac {216}{x^{3}}+\frac {761}{x^{2}}+\frac {216}{x}-\frac {40 \,{\mathrm e}^{x}}{x}-40 \,{\mathrm e}^{x} x -270 \,{\mathrm e}^{x}+80 \,{\mathrm e}^{x} \ln \left (2\right )+25 \,{\mathrm e}^{2 x} x^{2}+100 \ln \left (2\right )^{2} {\mathrm e}^{2 x}-160 \ln \left (2\right ) \left (-\frac {{\mathrm e}^{x}}{2 x^{2}}-\frac {{\mathrm e}^{x}}{2 x}-\frac {\operatorname {Ei}_{1}\left (-x \right )}{2}\right )-460 \ln \left (2\right ) \left (-\frac {{\mathrm e}^{x}}{x}-\operatorname {Ei}_{1}\left (-x \right )\right )-100 x \ln \left (2\right ) {\mathrm e}^{2 x}-540 \ln \left (2\right ) \operatorname {Ei}_{1}\left (-x \right )\) | \(126\) |
int(((200*x^5*ln(2)^2+2*(-100*x^6-50*x^5)*ln(2)+50*x^7+50*x^6)*exp(x)^2+(2 *(40*x^5+270*x^4-230*x^3-80*x^2)*ln(2)-40*x^6-310*x^5-40*x^4+40*x^3)*exp(x )-216*x^3-1522*x^2-648*x-64)/x^5,x,method=_RETURNVERBOSE)
(216*x^3+761*x^2+216*x+16)/x^4+(100*ln(2)^2-100*x*ln(2)+25*x^2)*exp(x)^2+1 0*(8*x^2*ln(2)-4*x^3+54*x*ln(2)-27*x^2+8*ln(2)-4*x)/x^2*exp(x)
Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (27) = 54\).
Time = 0.26 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.83 \[ \int \frac {-64-648 x-1522 x^2-216 x^3+e^x \left (40 x^3-40 x^4-310 x^5-40 x^6+\left (-80 x^2-230 x^3+270 x^4+40 x^5\right ) \log (4)\right )+e^{2 x} \left (50 x^6+50 x^7+\left (-50 x^5-100 x^6\right ) \log (4)+50 x^5 \log ^2(4)\right )}{x^5} \, dx=\frac {216 \, x^{3} + 761 \, x^{2} + 25 \, {\left (x^{6} - 4 \, x^{5} \log \left (2\right ) + 4 \, x^{4} \log \left (2\right )^{2}\right )} e^{\left (2 \, x\right )} - 10 \, {\left (4 \, x^{5} + 27 \, x^{4} + 4 \, x^{3} - 2 \, {\left (4 \, x^{4} + 27 \, x^{3} + 4 \, x^{2}\right )} \log \left (2\right )\right )} e^{x} + 216 \, x + 16}{x^{4}} \]
integrate(((200*x^5*log(2)^2+2*(-100*x^6-50*x^5)*log(2)+50*x^7+50*x^6)*exp (x)^2+(2*(40*x^5+270*x^4-230*x^3-80*x^2)*log(2)-40*x^6-310*x^5-40*x^4+40*x ^3)*exp(x)-216*x^3-1522*x^2-648*x-64)/x^5,x, algorithm=\
(216*x^3 + 761*x^2 + 25*(x^6 - 4*x^5*log(2) + 4*x^4*log(2)^2)*e^(2*x) - 10 *(4*x^5 + 27*x^4 + 4*x^3 - 2*(4*x^4 + 27*x^3 + 4*x^2)*log(2))*e^x + 216*x + 16)/x^4
Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (27) = 54\).
Time = 0.13 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.93 \[ \int \frac {-64-648 x-1522 x^2-216 x^3+e^x \left (40 x^3-40 x^4-310 x^5-40 x^6+\left (-80 x^2-230 x^3+270 x^4+40 x^5\right ) \log (4)\right )+e^{2 x} \left (50 x^6+50 x^7+\left (-50 x^5-100 x^6\right ) \log (4)+50 x^5 \log ^2(4)\right )}{x^5} \, dx=\frac {\left (25 x^{4} - 100 x^{3} \log {\left (2 \right )} + 100 x^{2} \log {\left (2 \right )}^{2}\right ) e^{2 x} + \left (- 40 x^{3} - 270 x^{2} + 80 x^{2} \log {\left (2 \right )} - 40 x + 540 x \log {\left (2 \right )} + 80 \log {\left (2 \right )}\right ) e^{x}}{x^{2}} - \frac {- 216 x^{3} - 761 x^{2} - 216 x - 16}{x^{4}} \]
integrate(((200*x**5*ln(2)**2+2*(-100*x**6-50*x**5)*ln(2)+50*x**7+50*x**6) *exp(x)**2+(2*(40*x**5+270*x**4-230*x**3-80*x**2)*ln(2)-40*x**6-310*x**5-4 0*x**4+40*x**3)*exp(x)-216*x**3-1522*x**2-648*x-64)/x**5,x)
((25*x**4 - 100*x**3*log(2) + 100*x**2*log(2)**2)*exp(2*x) + (-40*x**3 - 2 70*x**2 + 80*x**2*log(2) - 40*x + 540*x*log(2) + 80*log(2))*exp(x))/x**2 - (-216*x**3 - 761*x**2 - 216*x - 16)/x**4
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.24 (sec) , antiderivative size = 131, normalized size of antiderivative = 4.37 \[ \int \frac {-64-648 x-1522 x^2-216 x^3+e^x \left (40 x^3-40 x^4-310 x^5-40 x^6+\left (-80 x^2-230 x^3+270 x^4+40 x^5\right ) \log (4)\right )+e^{2 x} \left (50 x^6+50 x^7+\left (-50 x^5-100 x^6\right ) \log (4)+50 x^5 \log ^2(4)\right )}{x^5} \, dx=-50 \, {\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} \log \left (2\right ) + 100 \, e^{\left (2 \, x\right )} \log \left (2\right )^{2} + \frac {25}{2} \, {\left (2 \, x^{2} - 2 \, x + 1\right )} e^{\left (2 \, x\right )} + \frac {25}{2} \, {\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} - 40 \, {\left (x - 1\right )} e^{x} + 540 \, {\rm Ei}\left (x\right ) \log \left (2\right ) - 50 \, e^{\left (2 \, x\right )} \log \left (2\right ) + 80 \, e^{x} \log \left (2\right ) - 460 \, \Gamma \left (-1, -x\right ) \log \left (2\right ) + 160 \, \Gamma \left (-2, -x\right ) \log \left (2\right ) + \frac {216}{x} + \frac {761}{x^{2}} + \frac {216}{x^{3}} + \frac {16}{x^{4}} - 40 \, {\rm Ei}\left (x\right ) - 310 \, e^{x} + 40 \, \Gamma \left (-1, -x\right ) \]
integrate(((200*x^5*log(2)^2+2*(-100*x^6-50*x^5)*log(2)+50*x^7+50*x^6)*exp (x)^2+(2*(40*x^5+270*x^4-230*x^3-80*x^2)*log(2)-40*x^6-310*x^5-40*x^4+40*x ^3)*exp(x)-216*x^3-1522*x^2-648*x-64)/x^5,x, algorithm=\
-50*(2*x - 1)*e^(2*x)*log(2) + 100*e^(2*x)*log(2)^2 + 25/2*(2*x^2 - 2*x + 1)*e^(2*x) + 25/2*(2*x - 1)*e^(2*x) - 40*(x - 1)*e^x + 540*Ei(x)*log(2) - 50*e^(2*x)*log(2) + 80*e^x*log(2) - 460*gamma(-1, -x)*log(2) + 160*gamma(- 2, -x)*log(2) + 216/x + 761/x^2 + 216/x^3 + 16/x^4 - 40*Ei(x) - 310*e^x + 40*gamma(-1, -x)
Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (27) = 54\).
Time = 0.27 (sec) , antiderivative size = 100, normalized size of antiderivative = 3.33 \[ \int \frac {-64-648 x-1522 x^2-216 x^3+e^x \left (40 x^3-40 x^4-310 x^5-40 x^6+\left (-80 x^2-230 x^3+270 x^4+40 x^5\right ) \log (4)\right )+e^{2 x} \left (50 x^6+50 x^7+\left (-50 x^5-100 x^6\right ) \log (4)+50 x^5 \log ^2(4)\right )}{x^5} \, dx=\frac {25 \, x^{6} e^{\left (2 \, x\right )} - 100 \, x^{5} e^{\left (2 \, x\right )} \log \left (2\right ) + 100 \, x^{4} e^{\left (2 \, x\right )} \log \left (2\right )^{2} - 40 \, x^{5} e^{x} + 80 \, x^{4} e^{x} \log \left (2\right ) - 270 \, x^{4} e^{x} + 540 \, x^{3} e^{x} \log \left (2\right ) - 40 \, x^{3} e^{x} + 80 \, x^{2} e^{x} \log \left (2\right ) + 216 \, x^{3} + 761 \, x^{2} + 216 \, x + 16}{x^{4}} \]
integrate(((200*x^5*log(2)^2+2*(-100*x^6-50*x^5)*log(2)+50*x^7+50*x^6)*exp (x)^2+(2*(40*x^5+270*x^4-230*x^3-80*x^2)*log(2)-40*x^6-310*x^5-40*x^4+40*x ^3)*exp(x)-216*x^3-1522*x^2-648*x-64)/x^5,x, algorithm=\
(25*x^6*e^(2*x) - 100*x^5*e^(2*x)*log(2) + 100*x^4*e^(2*x)*log(2)^2 - 40*x ^5*e^x + 80*x^4*e^x*log(2) - 270*x^4*e^x + 540*x^3*e^x*log(2) - 40*x^3*e^x + 80*x^2*e^x*log(2) + 216*x^3 + 761*x^2 + 216*x + 16)/x^4
Time = 9.03 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.70 \[ \int \frac {-64-648 x-1522 x^2-216 x^3+e^x \left (40 x^3-40 x^4-310 x^5-40 x^6+\left (-80 x^2-230 x^3+270 x^4+40 x^5\right ) \log (4)\right )+e^{2 x} \left (50 x^6+50 x^7+\left (-50 x^5-100 x^6\right ) \log (4)+50 x^5 \log ^2(4)\right )}{x^5} \, dx={\mathrm {e}}^x\,\left (80\,\ln \left (2\right )-270\right )+100\,{\mathrm {e}}^{2\,x}\,{\ln \left (2\right )}^2+25\,x^2\,{\mathrm {e}}^{2\,x}-x\,\left (40\,{\mathrm {e}}^x+100\,{\mathrm {e}}^{2\,x}\,\ln \left (2\right )\right )+\frac {216\,x+x^2\,\left (80\,{\mathrm {e}}^x\,\ln \left (2\right )+761\right )+x^3\,\left ({\mathrm {e}}^x\,\left (540\,\ln \left (2\right )-40\right )+216\right )+16}{x^4} \]
int(-(648*x - exp(2*x)*(200*x^5*log(2)^2 - 2*log(2)*(50*x^5 + 100*x^6) + 5 0*x^6 + 50*x^7) + exp(x)*(2*log(2)*(80*x^2 + 230*x^3 - 270*x^4 - 40*x^5) - 40*x^3 + 40*x^4 + 310*x^5 + 40*x^6) + 1522*x^2 + 216*x^3 + 64)/x^5,x)