Integrand size = 117, antiderivative size = 34 \[ \int \frac {-6 x^3-2 x^4-36 x^6-30 x^7-6 x^8+(-18-6 x) \log ^2(2)+e^{2 x} \left (-2 x^3-4 x^6-2 x^7+(-2+2 x) \log ^2(2)\right )+e^x \left (8 x^3+2 x^4+24 x^6+16 x^7+2 x^8+\left (12-4 x-2 x^2\right ) \log ^2(2)\right )}{x^3} \, dx=\left (-\frac {e^x}{x}+\frac {3+x}{x}\right )^2 \left (-x^2-x^6+\log ^2(2)\right ) \]
Leaf count is larger than twice the leaf count of optimal. \(80\) vs. \(2(34)=68\).
Time = 0.74 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.35 \[ \int \frac {-6 x^3-2 x^4-36 x^6-30 x^7-6 x^8+(-18-6 x) \log ^2(2)+e^{2 x} \left (-2 x^3-4 x^6-2 x^7+(-2+2 x) \log ^2(2)\right )+e^x \left (8 x^3+2 x^4+24 x^6+16 x^7+2 x^8+\left (12-4 x-2 x^2\right ) \log ^2(2)\right )}{x^3} \, dx=-\frac {6 x^3+x^4+9 x^6+6 x^7+x^8-9 \log ^2(2)-6 x \log ^2(2)+e^{2 x} \left (x^2+x^6-\log ^2(2)\right )-2 e^x (3+x) \left (x^2+x^6-\log ^2(2)\right )}{x^2} \]
Integrate[(-6*x^3 - 2*x^4 - 36*x^6 - 30*x^7 - 6*x^8 + (-18 - 6*x)*Log[2]^2 + E^(2*x)*(-2*x^3 - 4*x^6 - 2*x^7 + (-2 + 2*x)*Log[2]^2) + E^x*(8*x^3 + 2 *x^4 + 24*x^6 + 16*x^7 + 2*x^8 + (12 - 4*x - 2*x^2)*Log[2]^2))/x^3,x]
-((6*x^3 + x^4 + 9*x^6 + 6*x^7 + x^8 - 9*Log[2]^2 - 6*x*Log[2]^2 + E^(2*x) *(x^2 + x^6 - Log[2]^2) - 2*E^x*(3 + x)*(x^2 + x^6 - Log[2]^2))/x^2)
Leaf count is larger than twice the leaf count of optimal. \(123\) vs. \(2(34)=68\).
Time = 0.68 (sec) , antiderivative size = 123, normalized size of antiderivative = 3.62, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.017, 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.
\(\displaystyle \int \frac {-6 x^8-30 x^7-36 x^6-2 x^4-6 x^3+e^{2 x} \left (-2 x^7-4 x^6-2 x^3+(2 x-2) \log ^2(2)\right )+e^x \left (2 x^8+16 x^7+24 x^6+2 x^4+8 x^3+\left (-2 x^2-4 x+12\right ) \log ^2(2)\right )+(-6 x-18) \log ^2(2)}{x^3} \, dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \int \left (-\frac {2 (x+3) \left (3 x^7+6 x^6+x^3+3 \log ^2(2)\right )}{x^3}-\frac {2 e^{2 x} \left (x^7+2 x^6+x^3-x \log ^2(2)+\log ^2(2)\right )}{x^3}+\frac {2 e^x \left (x^8+8 x^7+12 x^6+x^4+4 x^3-x^2 \log ^2(2)-2 x \log ^2(2)+6 \log ^2(2)\right )}{x^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -x^6+2 e^x x^5-6 x^5+6 e^x x^4-e^{2 x} x^4-9 x^4-x^2-\frac {6 e^x \log ^2(2)}{x^2}+\frac {e^{2 x} \log ^2(2)}{x^2}+\frac {9 \log ^2(2)}{x^2}+2 e^x x-6 x+6 e^x-e^{2 x}-\frac {2 e^x \log ^2(2)}{x}+\frac {6 \log ^2(2)}{x}\) |
Int[(-6*x^3 - 2*x^4 - 36*x^6 - 30*x^7 - 6*x^8 + (-18 - 6*x)*Log[2]^2 + E^( 2*x)*(-2*x^3 - 4*x^6 - 2*x^7 + (-2 + 2*x)*Log[2]^2) + E^x*(8*x^3 + 2*x^4 + 24*x^6 + 16*x^7 + 2*x^8 + (12 - 4*x - 2*x^2)*Log[2]^2))/x^3,x]
6*E^x - E^(2*x) - 6*x + 2*E^x*x - x^2 - 9*x^4 + 6*E^x*x^4 - E^(2*x)*x^4 - 6*x^5 + 2*E^x*x^5 - x^6 + (9*Log[2]^2)/x^2 - (6*E^x*Log[2]^2)/x^2 + (E^(2* x)*Log[2]^2)/x^2 + (6*Log[2]^2)/x - (2*E^x*Log[2]^2)/x
3.21.71.3.1 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. \(105\) vs. \(2(33)=66\).
Time = 0.36 (sec) , antiderivative size = 106, normalized size of antiderivative = 3.12
method | result | size |
risch | \(-x^{6}-6 x^{5}-9 x^{4}-x^{2}-6 x +\frac {6 x \ln \left (2\right )^{2}+9 \ln \left (2\right )^{2}}{x^{2}}+\frac {\left (\ln \left (2\right )^{2}-x^{2}-x^{6}\right ) {\mathrm e}^{2 x}}{x^{2}}-\frac {2 \left (-x^{7}-3 x^{6}+x \ln \left (2\right )^{2}-x^{3}+3 \ln \left (2\right )^{2}-3 x^{2}\right ) {\mathrm e}^{x}}{x^{2}}\) | \(106\) |
parallelrisch | \(-\frac {-2 x^{7} {\mathrm e}^{x}+x^{8}-6 x^{6} {\mathrm e}^{x}+{\mathrm e}^{2 x} x^{6}+6 x^{7}+9 x^{6}+2 x \ln \left (2\right )^{2} {\mathrm e}^{x}-2 \,{\mathrm e}^{x} x^{3}+x^{4}+6 \ln \left (2\right )^{2} {\mathrm e}^{x}-6 \,{\mathrm e}^{x} x^{2}-\ln \left (2\right )^{2} {\mathrm e}^{2 x}-6 x \ln \left (2\right )^{2}+{\mathrm e}^{2 x} x^{2}+6 x^{3}-9 \ln \left (2\right )^{2}}{x^{2}}\) | \(112\) |
parts | \(-x^{6}-6 x^{5}-9 x^{4}-x^{2}-6 x +\frac {9 \ln \left (2\right )^{2}}{x^{2}}+\frac {6 \ln \left (2\right )^{2}}{x}-{\mathrm e}^{2 x}-{\mathrm e}^{2 x} x^{4}+\frac {\ln \left (2\right )^{2} {\mathrm e}^{2 x}}{x^{2}}+2 x^{5} {\mathrm e}^{x}+6 \,{\mathrm e}^{x} x^{4}+2 \,{\mathrm e}^{x} x +6 \,{\mathrm e}^{x}-\frac {6 \ln \left (2\right )^{2} {\mathrm e}^{x}}{x^{2}}-\frac {2 \,{\mathrm e}^{x} \ln \left (2\right )^{2}}{x}\) | \(115\) |
norman | \(\frac {\ln \left (2\right )^{2} {\mathrm e}^{2 x}-6 x^{3}-x^{4}-9 x^{6}-6 x^{7}-x^{8}+9 \ln \left (2\right )^{2}+6 x \ln \left (2\right )^{2}+6 x^{6} {\mathrm e}^{x}-{\mathrm e}^{2 x} x^{6}+2 x^{7} {\mathrm e}^{x}+6 \,{\mathrm e}^{x} x^{2}+2 \,{\mathrm e}^{x} x^{3}-{\mathrm e}^{2 x} x^{2}-6 \ln \left (2\right )^{2} {\mathrm e}^{x}-2 x \ln \left (2\right )^{2} {\mathrm e}^{x}}{x^{2}}\) | \(116\) |
default | \(-x^{6}-6 x^{5}-9 x^{4}-x^{2}-6 x -{\mathrm e}^{2 x}+\frac {9 \ln \left (2\right )^{2}}{x^{2}}+2 x^{5} {\mathrm e}^{x}+6 \,{\mathrm e}^{x} x^{4}+2 \,{\mathrm e}^{x} x +6 \,{\mathrm e}^{x}-{\mathrm e}^{2 x} x^{4}+\frac {6 \ln \left (2\right )^{2}}{x}+12 \ln \left (2\right )^{2} \left (-\frac {{\mathrm e}^{x}}{2 x^{2}}-\frac {{\mathrm e}^{x}}{2 x}-\frac {\operatorname {Ei}_{1}\left (-x \right )}{2}\right )-2 \ln \left (2\right )^{2} \left (-\frac {{\mathrm e}^{2 x}}{2 x^{2}}-\frac {{\mathrm e}^{2 x}}{x}-2 \,\operatorname {Ei}_{1}\left (-2 x \right )\right )-4 \ln \left (2\right )^{2} \left (-\frac {{\mathrm e}^{x}}{x}-\operatorname {Ei}_{1}\left (-x \right )\right )+2 \ln \left (2\right )^{2} \left (-\frac {{\mathrm e}^{2 x}}{x}-2 \,\operatorname {Ei}_{1}\left (-2 x \right )\right )+2 \ln \left (2\right )^{2} \operatorname {Ei}_{1}\left (-x \right )\) | \(196\) |
int((((-2+2*x)*ln(2)^2-2*x^7-4*x^6-2*x^3)*exp(x)^2+((-2*x^2-4*x+12)*ln(2)^ 2+2*x^8+16*x^7+24*x^6+2*x^4+8*x^3)*exp(x)+(-6*x-18)*ln(2)^2-6*x^8-30*x^7-3 6*x^6-2*x^4-6*x^3)/x^3,x,method=_RETURNVERBOSE)
-x^6-6*x^5-9*x^4-x^2-6*x+(6*x*ln(2)^2+9*ln(2)^2)/x^2+(ln(2)^2-x^2-x^6)/x^2 *exp(x)^2-2*(-x^7-3*x^6+x*ln(2)^2-x^3+3*ln(2)^2-3*x^2)/x^2*exp(x)
Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (32) = 64\).
Time = 0.26 (sec) , antiderivative size = 86, normalized size of antiderivative = 2.53 \[ \int \frac {-6 x^3-2 x^4-36 x^6-30 x^7-6 x^8+(-18-6 x) \log ^2(2)+e^{2 x} \left (-2 x^3-4 x^6-2 x^7+(-2+2 x) \log ^2(2)\right )+e^x \left (8 x^3+2 x^4+24 x^6+16 x^7+2 x^8+\left (12-4 x-2 x^2\right ) \log ^2(2)\right )}{x^3} \, dx=-\frac {x^{8} + 6 \, x^{7} + 9 \, x^{6} + x^{4} + 6 \, x^{3} - 3 \, {\left (2 \, x + 3\right )} \log \left (2\right )^{2} + {\left (x^{6} + x^{2} - \log \left (2\right )^{2}\right )} e^{\left (2 \, x\right )} - 2 \, {\left (x^{7} + 3 \, x^{6} + x^{3} - {\left (x + 3\right )} \log \left (2\right )^{2} + 3 \, x^{2}\right )} e^{x}}{x^{2}} \]
integrate((((-2+2*x)*log(2)^2-2*x^7-4*x^6-2*x^3)*exp(x)^2+((-2*x^2-4*x+12) *log(2)^2+2*x^8+16*x^7+24*x^6+2*x^4+8*x^3)*exp(x)+(-6*x-18)*log(2)^2-6*x^8 -30*x^7-36*x^6-2*x^4-6*x^3)/x^3,x, algorithm=\
-(x^8 + 6*x^7 + 9*x^6 + x^4 + 6*x^3 - 3*(2*x + 3)*log(2)^2 + (x^6 + x^2 - log(2)^2)*e^(2*x) - 2*(x^7 + 3*x^6 + x^3 - (x + 3)*log(2)^2 + 3*x^2)*e^x)/ x^2
Leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (22) = 44\).
Time = 0.14 (sec) , antiderivative size = 107, normalized size of antiderivative = 3.15 \[ \int \frac {-6 x^3-2 x^4-36 x^6-30 x^7-6 x^8+(-18-6 x) \log ^2(2)+e^{2 x} \left (-2 x^3-4 x^6-2 x^7+(-2+2 x) \log ^2(2)\right )+e^x \left (8 x^3+2 x^4+24 x^6+16 x^7+2 x^8+\left (12-4 x-2 x^2\right ) \log ^2(2)\right )}{x^3} \, dx=- x^{6} - 6 x^{5} - 9 x^{4} - x^{2} - 6 x - \frac {- 6 x \log {\left (2 \right )}^{2} - 9 \log {\left (2 \right )}^{2}}{x^{2}} + \frac {\left (- x^{8} - x^{4} + x^{2} \log {\left (2 \right )}^{2}\right ) e^{2 x} + \left (2 x^{9} + 6 x^{8} + 2 x^{5} + 6 x^{4} - 2 x^{3} \log {\left (2 \right )}^{2} - 6 x^{2} \log {\left (2 \right )}^{2}\right ) e^{x}}{x^{4}} \]
integrate((((-2+2*x)*ln(2)**2-2*x**7-4*x**6-2*x**3)*exp(x)**2+((-2*x**2-4* x+12)*ln(2)**2+2*x**8+16*x**7+24*x**6+2*x**4+8*x**3)*exp(x)+(-6*x-18)*ln(2 )**2-6*x**8-30*x**7-36*x**6-2*x**4-6*x**3)/x**3,x)
-x**6 - 6*x**5 - 9*x**4 - x**2 - 6*x - (-6*x*log(2)**2 - 9*log(2)**2)/x**2 + ((-x**8 - x**4 + x**2*log(2)**2)*exp(2*x) + (2*x**9 + 6*x**8 + 2*x**5 + 6*x**4 - 2*x**3*log(2)**2 - 6*x**2*log(2)**2)*exp(x))/x**4
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.24 (sec) , antiderivative size = 224, normalized size of antiderivative = 6.59 \[ \int \frac {-6 x^3-2 x^4-36 x^6-30 x^7-6 x^8+(-18-6 x) \log ^2(2)+e^{2 x} \left (-2 x^3-4 x^6-2 x^7+(-2+2 x) \log ^2(2)\right )+e^x \left (8 x^3+2 x^4+24 x^6+16 x^7+2 x^8+\left (12-4 x-2 x^2\right ) \log ^2(2)\right )}{x^3} \, dx=-x^{6} - 6 \, x^{5} - 9 \, x^{4} - 2 \, {\rm Ei}\left (x\right ) \log \left (2\right )^{2} - 4 \, \Gamma \left (-1, -x\right ) \log \left (2\right )^{2} + 4 \, \Gamma \left (-1, -2 \, x\right ) \log \left (2\right )^{2} - 12 \, \Gamma \left (-2, -x\right ) \log \left (2\right )^{2} + 8 \, \Gamma \left (-2, -2 \, x\right ) \log \left (2\right )^{2} - x^{2} - \frac {1}{2} \, {\left (2 \, x^{4} - 4 \, x^{3} + 6 \, x^{2} - 6 \, x + 3\right )} e^{\left (2 \, x\right )} - \frac {1}{2} \, {\left (4 \, x^{3} - 6 \, x^{2} + 6 \, x - 3\right )} e^{\left (2 \, x\right )} + 2 \, {\left (x^{5} - 5 \, x^{4} + 20 \, x^{3} - 60 \, x^{2} + 120 \, x - 120\right )} e^{x} + 16 \, {\left (x^{4} - 4 \, x^{3} + 12 \, x^{2} - 24 \, x + 24\right )} e^{x} + 24 \, {\left (x^{3} - 3 \, x^{2} + 6 \, x - 6\right )} e^{x} + 2 \, {\left (x - 1\right )} e^{x} - 6 \, x + \frac {6 \, \log \left (2\right )^{2}}{x} + \frac {9 \, \log \left (2\right )^{2}}{x^{2}} - e^{\left (2 \, x\right )} + 8 \, e^{x} \]
integrate((((-2+2*x)*log(2)^2-2*x^7-4*x^6-2*x^3)*exp(x)^2+((-2*x^2-4*x+12) *log(2)^2+2*x^8+16*x^7+24*x^6+2*x^4+8*x^3)*exp(x)+(-6*x-18)*log(2)^2-6*x^8 -30*x^7-36*x^6-2*x^4-6*x^3)/x^3,x, algorithm=\
-x^6 - 6*x^5 - 9*x^4 - 2*Ei(x)*log(2)^2 - 4*gamma(-1, -x)*log(2)^2 + 4*gam ma(-1, -2*x)*log(2)^2 - 12*gamma(-2, -x)*log(2)^2 + 8*gamma(-2, -2*x)*log( 2)^2 - x^2 - 1/2*(2*x^4 - 4*x^3 + 6*x^2 - 6*x + 3)*e^(2*x) - 1/2*(4*x^3 - 6*x^2 + 6*x - 3)*e^(2*x) + 2*(x^5 - 5*x^4 + 20*x^3 - 60*x^2 + 120*x - 120) *e^x + 16*(x^4 - 4*x^3 + 12*x^2 - 24*x + 24)*e^x + 24*(x^3 - 3*x^2 + 6*x - 6)*e^x + 2*(x - 1)*e^x - 6*x + 6*log(2)^2/x + 9*log(2)^2/x^2 - e^(2*x) + 8*e^x
Leaf count of result is larger than twice the leaf count of optimal. 111 vs. \(2 (32) = 64\).
Time = 0.27 (sec) , antiderivative size = 111, normalized size of antiderivative = 3.26 \[ \int \frac {-6 x^3-2 x^4-36 x^6-30 x^7-6 x^8+(-18-6 x) \log ^2(2)+e^{2 x} \left (-2 x^3-4 x^6-2 x^7+(-2+2 x) \log ^2(2)\right )+e^x \left (8 x^3+2 x^4+24 x^6+16 x^7+2 x^8+\left (12-4 x-2 x^2\right ) \log ^2(2)\right )}{x^3} \, dx=-\frac {x^{8} - 2 \, x^{7} e^{x} + 6 \, x^{7} + x^{6} e^{\left (2 \, x\right )} - 6 \, x^{6} e^{x} + 9 \, x^{6} + x^{4} - 2 \, x^{3} e^{x} + 2 \, x e^{x} \log \left (2\right )^{2} + 6 \, x^{3} + x^{2} e^{\left (2 \, x\right )} - 6 \, x^{2} e^{x} - 6 \, x \log \left (2\right )^{2} - e^{\left (2 \, x\right )} \log \left (2\right )^{2} + 6 \, e^{x} \log \left (2\right )^{2} - 9 \, \log \left (2\right )^{2}}{x^{2}} \]
integrate((((-2+2*x)*log(2)^2-2*x^7-4*x^6-2*x^3)*exp(x)^2+((-2*x^2-4*x+12) *log(2)^2+2*x^8+16*x^7+24*x^6+2*x^4+8*x^3)*exp(x)+(-6*x-18)*log(2)^2-6*x^8 -30*x^7-36*x^6-2*x^4-6*x^3)/x^3,x, algorithm=\
-(x^8 - 2*x^7*e^x + 6*x^7 + x^6*e^(2*x) - 6*x^6*e^x + 9*x^6 + x^4 - 2*x^3* e^x + 2*x*e^x*log(2)^2 + 6*x^3 + x^2*e^(2*x) - 6*x^2*e^x - 6*x*log(2)^2 - e^(2*x)*log(2)^2 + 6*e^x*log(2)^2 - 9*log(2)^2)/x^2
Time = 0.19 (sec) , antiderivative size = 101, normalized size of antiderivative = 2.97 \[ \int \frac {-6 x^3-2 x^4-36 x^6-30 x^7-6 x^8+(-18-6 x) \log ^2(2)+e^{2 x} \left (-2 x^3-4 x^6-2 x^7+(-2+2 x) \log ^2(2)\right )+e^x \left (8 x^3+2 x^4+24 x^6+16 x^7+2 x^8+\left (12-4 x-2 x^2\right ) \log ^2(2)\right )}{x^3} \, dx=6\,{\mathrm {e}}^x-{\mathrm {e}}^{2\,x}-x^4\,\left ({\mathrm {e}}^{2\,x}-6\,{\mathrm {e}}^x+9\right )+x\,\left (2\,{\mathrm {e}}^x-6\right )-\frac {6\,{\mathrm {e}}^x\,{\ln \left (2\right )}^2-{\mathrm {e}}^{2\,x}\,{\ln \left (2\right )}^2+x\,\left (2\,{\mathrm {e}}^x\,{\ln \left (2\right )}^2-6\,{\ln \left (2\right )}^2\right )-9\,{\ln \left (2\right )}^2}{x^2}+x^5\,\left (2\,{\mathrm {e}}^x-6\right )-x^2-x^6 \]
int(-(exp(2*x)*(2*x^3 - log(2)^2*(2*x - 2) + 4*x^6 + 2*x^7) - exp(x)*(8*x^ 3 - log(2)^2*(4*x + 2*x^2 - 12) + 2*x^4 + 24*x^6 + 16*x^7 + 2*x^8) + log(2 )^2*(6*x + 18) + 6*x^3 + 2*x^4 + 36*x^6 + 30*x^7 + 6*x^8)/x^3,x)