Integrand size = 81, antiderivative size = 37 \[ \int \frac {e^{2 x} \left (-4+2 x-4 x^2+4 x^3+2 x^5+e^{4 x} \left (-2 x^2+6 x^3\right )+e^{2 x} \left (6 x-8 x^2+2 x^3-8 x^4\right )\right ) \left (3-\log ^2(3)\right )^2}{x^5} \, dx=\frac {e^{2 x} \left (e^{2 x}-\frac {1+x^2}{x}\right )^2 \left (3-\log ^2(3)\right )^2}{x^2} \] Output:
(exp(x)^2-(x^2+1)/x)^2/x^2*exp(ln(-ln(3)^2+3)+x)^2
Time = 2.87 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.86 \[ \int \frac {e^{2 x} \left (-4+2 x-4 x^2+4 x^3+2 x^5+e^{4 x} \left (-2 x^2+6 x^3\right )+e^{2 x} \left (6 x-8 x^2+2 x^3-8 x^4\right )\right ) \left (3-\log ^2(3)\right )^2}{x^5} \, dx=\frac {e^{2 x} \left (1-e^{2 x} x+x^2\right )^2 \left (-3+\log ^2(3)\right )^2}{x^4} \] Input:
Integrate[(E^(2*x)*(-4 + 2*x - 4*x^2 + 4*x^3 + 2*x^5 + E^(4*x)*(-2*x^2 + 6 *x^3) + E^(2*x)*(6*x - 8*x^2 + 2*x^3 - 8*x^4))*(3 - Log[3]^2)^2)/x^5,x]
Output:
(E^(2*x)*(1 - E^(2*x)*x + x^2)^2*(-3 + Log[3]^2)^2)/x^4
Time = 1.30 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.049, Rules used = {27, 27, 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.
\(\displaystyle \int \frac {e^{2 x} \left (2 x^5+4 x^3-4 x^2+e^{4 x} \left (6 x^3-2 x^2\right )+e^{2 x} \left (-8 x^4+2 x^3-8 x^2+6 x\right )+2 x-4\right ) \left (3-\log ^2(3)\right )^2}{x^5} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \left (3-\log ^2(3)\right )^2 \int -\frac {2 e^{2 x} \left (-x^5-2 x^3+2 x^2-x+e^{4 x} \left (x^2-3 x^3\right )-e^{2 x} \left (-4 x^4+x^3-4 x^2+3 x\right )+2\right )}{x^5}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -2 \left (3-\log ^2(3)\right )^2 \int \frac {e^{2 x} \left (-x^5-2 x^3+2 x^2-x+e^{4 x} \left (x^2-3 x^3\right )-e^{2 x} \left (-4 x^4+x^3-4 x^2+3 x\right )+2\right )}{x^5}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -2 \left (3-\log ^2(3)\right )^2 \int \left (-\frac {e^{6 x} (3 x-1)}{x^3}+\frac {e^{4 x} \left (4 x^3-x^2+4 x-3\right )}{x^4}+\frac {e^{2 x} \left (-x^5-2 x^3+2 x^2-x+2\right )}{x^5}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -2 \left (-\frac {e^{2 x}}{2 x^4}+\frac {e^{4 x}}{x^3}-\frac {e^{2 x}}{x^2}-\frac {e^{6 x}}{2 x^2}-\frac {e^{2 x}}{2}+\frac {e^{4 x}}{x}\right ) \left (3-\log ^2(3)\right )^2\) |
Input:
Int[(E^(2*x)*(-4 + 2*x - 4*x^2 + 4*x^3 + 2*x^5 + E^(4*x)*(-2*x^2 + 6*x^3) + E^(2*x)*(6*x - 8*x^2 + 2*x^3 - 8*x^4))*(3 - Log[3]^2)^2)/x^5,x]
Output:
-2*(-1/2*E^(2*x) - E^(2*x)/(2*x^4) + E^(4*x)/x^3 - E^(2*x)/x^2 - E^(6*x)/( 2*x^2) + E^(4*x)/x)*(3 - Log[3]^2)^2
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(71\) vs. \(2(35)=70\).
Time = 1.03 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.95
method | result | size |
risch | \(\frac {\left (-\ln \left (3\right )^{2}+3\right )^{2} {\mathrm e}^{6 x}}{x^{2}}-\frac {2 \left (-\ln \left (3\right )^{2}+3\right )^{2} \left (x^{2}+1\right ) {\mathrm e}^{4 x}}{x^{3}}+\frac {\left (-\ln \left (3\right )^{2}+3\right )^{2} \left (x^{4}+2 x^{2}+1\right ) {\mathrm e}^{2 x}}{x^{4}}\) | \(72\) |
parallelrisch | \(\frac {x^{2} {\mathrm e}^{4 x} \left (\ln \left (3\right )^{2}-3\right )^{2} {\mathrm e}^{2 x}-2 \left (\ln \left (3\right )^{2}-3\right )^{2} \left ({\mathrm e}^{2 x}\right )^{2} x^{3}+\left (\ln \left (3\right )^{2}-3\right )^{2} {\mathrm e}^{2 x} x^{4}-2 \left (\ln \left (3\right )^{2}-3\right )^{2} \left ({\mathrm e}^{2 x}\right )^{2} x +2 x^{2} \left (\ln \left (3\right )^{2}-3\right )^{2} {\mathrm e}^{2 x}+\left (\ln \left (3\right )^{2}-3\right )^{2} {\mathrm e}^{2 x}}{x^{4}}\) | \(123\) |
parts | \(\left (\ln \left (3\right )^{2}-3\right )^{2} {\mathrm e}^{2 x}+\frac {2 \left (\ln \left (3\right )^{2}-3\right )^{2} {\mathrm e}^{2 x}}{x^{2}}+\frac {\left (\ln \left (3\right )^{2}-3\right )^{2} {\mathrm e}^{2 x}}{x^{4}}-\frac {18 \,{\mathrm e}^{4 x}}{x^{3}}-\frac {18 \,{\mathrm e}^{4 x}}{x}+\frac {12 \,{\mathrm e}^{4 x} \ln \left (3\right )^{2}}{x^{3}}+\frac {12 \,{\mathrm e}^{4 x} \ln \left (3\right )^{2}}{x}-\frac {2 \,{\mathrm e}^{4 x} \ln \left (3\right )^{4}}{x^{3}}-\frac {2 \,{\mathrm e}^{4 x} \ln \left (3\right )^{4}}{x}+\frac {9 \,{\mathrm e}^{6 x}}{x^{2}}-\frac {6 \,{\mathrm e}^{6 x} \ln \left (3\right )^{2}}{x^{2}}+\frac {{\mathrm e}^{6 x} \ln \left (3\right )^{4}}{x^{2}}\) | \(157\) |
default | \(\text {Expression too large to display}\) | \(688\) |
orering | \(\text {Expression too large to display}\) | \(1186\) |
Input:
int(((6*x^3-2*x^2)*exp(x)^4+(-8*x^4+2*x^3-8*x^2+6*x)*exp(x)^2+2*x^5+4*x^3- 4*x^2+2*x-4)*exp(ln(-ln(3)^2+3)+x)^2/x^5,x,method=_RETURNVERBOSE)
Output:
(-ln(3)^2+3)^2/x^2*exp(6*x)-2*(-ln(3)^2+3)^2*(x^2+1)/x^3*exp(4*x)+(-ln(3)^ 2+3)^2*(x^4+2*x^2+1)/x^4*exp(2*x)
Leaf count of result is larger than twice the leaf count of optimal. 205 vs. \(2 (38) = 76\).
Time = 0.09 (sec) , antiderivative size = 205, normalized size of antiderivative = 5.54 \[ \int \frac {e^{2 x} \left (-4+2 x-4 x^2+4 x^3+2 x^5+e^{4 x} \left (-2 x^2+6 x^3\right )+e^{2 x} \left (6 x-8 x^2+2 x^3-8 x^4\right )\right ) \left (3-\log ^2(3)\right )^2}{x^5} \, dx=\frac {x^{2} e^{\left (6 \, x + 6 \, \log \left (-\log \left (3\right )^{2} + 3\right )\right )} - 2 \, {\left ({\left (x^{3} + x\right )} \log \left (3\right )^{4} + 9 \, x^{3} - 6 \, {\left (x^{3} + x\right )} \log \left (3\right )^{2} + 9 \, x\right )} e^{\left (4 \, x + 4 \, \log \left (-\log \left (3\right )^{2} + 3\right )\right )} + {\left ({\left (x^{4} + 2 \, x^{2} + 1\right )} \log \left (3\right )^{8} - 12 \, {\left (x^{4} + 2 \, x^{2} + 1\right )} \log \left (3\right )^{6} + 54 \, {\left (x^{4} + 2 \, x^{2} + 1\right )} \log \left (3\right )^{4} + 81 \, x^{4} - 108 \, {\left (x^{4} + 2 \, x^{2} + 1\right )} \log \left (3\right )^{2} + 162 \, x^{2} + 81\right )} e^{\left (2 \, x + 2 \, \log \left (-\log \left (3\right )^{2} + 3\right )\right )}}{x^{4} \log \left (3\right )^{8} - 12 \, x^{4} \log \left (3\right )^{6} + 54 \, x^{4} \log \left (3\right )^{4} - 108 \, x^{4} \log \left (3\right )^{2} + 81 \, x^{4}} \] Input:
integrate(((6*x^3-2*x^2)*exp(x)^4+(-8*x^4+2*x^3-8*x^2+6*x)*exp(x)^2+2*x^5+ 4*x^3-4*x^2+2*x-4)*exp(log(-log(3)^2+3)+x)^2/x^5,x, algorithm="fricas")
Output:
(x^2*e^(6*x + 6*log(-log(3)^2 + 3)) - 2*((x^3 + x)*log(3)^4 + 9*x^3 - 6*(x ^3 + x)*log(3)^2 + 9*x)*e^(4*x + 4*log(-log(3)^2 + 3)) + ((x^4 + 2*x^2 + 1 )*log(3)^8 - 12*(x^4 + 2*x^2 + 1)*log(3)^6 + 54*(x^4 + 2*x^2 + 1)*log(3)^4 + 81*x^4 - 108*(x^4 + 2*x^2 + 1)*log(3)^2 + 162*x^2 + 81)*e^(2*x + 2*log( -log(3)^2 + 3)))/(x^4*log(3)^8 - 12*x^4*log(3)^6 + 54*x^4*log(3)^4 - 108*x ^4*log(3)^2 + 81*x^4)
Leaf count of result is larger than twice the leaf count of optimal. 165 vs. \(2 (29) = 58\).
Time = 0.18 (sec) , antiderivative size = 165, normalized size of antiderivative = 4.46 \[ \int \frac {e^{2 x} \left (-4+2 x-4 x^2+4 x^3+2 x^5+e^{4 x} \left (-2 x^2+6 x^3\right )+e^{2 x} \left (6 x-8 x^2+2 x^3-8 x^4\right )\right ) \left (3-\log ^2(3)\right )^2}{x^5} \, dx=\frac {\left (- 6 x^{7} \log {\left (3 \right )}^{2} + x^{7} \log {\left (3 \right )}^{4} + 9 x^{7}\right ) e^{6 x} + \left (- 18 x^{8} - 2 x^{8} \log {\left (3 \right )}^{4} + 12 x^{8} \log {\left (3 \right )}^{2} - 18 x^{6} - 2 x^{6} \log {\left (3 \right )}^{4} + 12 x^{6} \log {\left (3 \right )}^{2}\right ) e^{4 x} + \left (- 6 x^{9} \log {\left (3 \right )}^{2} + x^{9} \log {\left (3 \right )}^{4} + 9 x^{9} - 12 x^{7} \log {\left (3 \right )}^{2} + 2 x^{7} \log {\left (3 \right )}^{4} + 18 x^{7} - 6 x^{5} \log {\left (3 \right )}^{2} + x^{5} \log {\left (3 \right )}^{4} + 9 x^{5}\right ) e^{2 x}}{x^{9}} \] Input:
integrate(((6*x**3-2*x**2)*exp(x)**4+(-8*x**4+2*x**3-8*x**2+6*x)*exp(x)**2 +2*x**5+4*x**3-4*x**2+2*x-4)*exp(ln(-ln(3)**2+3)+x)**2/x**5,x)
Output:
((-6*x**7*log(3)**2 + x**7*log(3)**4 + 9*x**7)*exp(6*x) + (-18*x**8 - 2*x* *8*log(3)**4 + 12*x**8*log(3)**2 - 18*x**6 - 2*x**6*log(3)**4 + 12*x**6*lo g(3)**2)*exp(4*x) + (-6*x**9*log(3)**2 + x**9*log(3)**4 + 9*x**9 - 12*x**7 *log(3)**2 + 2*x**7*log(3)**4 + 18*x**7 - 6*x**5*log(3)**2 + x**5*log(3)** 4 + 9*x**5)*exp(2*x))/x**9
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.09 (sec) , antiderivative size = 86, normalized size of antiderivative = 2.32 \[ \int \frac {e^{2 x} \left (-4+2 x-4 x^2+4 x^3+2 x^5+e^{4 x} \left (-2 x^2+6 x^3\right )+e^{2 x} \left (6 x-8 x^2+2 x^3-8 x^4\right )\right ) \left (3-\log ^2(3)\right )^2}{x^5} \, dx=-{\left (\log \left (3\right )^{2} - 3\right )}^{2} {\left (8 \, {\rm Ei}\left (4 \, x\right ) - e^{\left (2 \, x\right )} - 8 \, \Gamma \left (-1, -2 \, x\right ) - 8 \, \Gamma \left (-1, -4 \, x\right ) - 36 \, \Gamma \left (-1, -6 \, x\right ) - 16 \, \Gamma \left (-2, -2 \, x\right ) - 128 \, \Gamma \left (-2, -4 \, x\right ) - 72 \, \Gamma \left (-2, -6 \, x\right ) - 16 \, \Gamma \left (-3, -2 \, x\right ) - 384 \, \Gamma \left (-3, -4 \, x\right ) - 64 \, \Gamma \left (-4, -2 \, x\right )\right )} \] Input:
integrate(((6*x^3-2*x^2)*exp(x)^4+(-8*x^4+2*x^3-8*x^2+6*x)*exp(x)^2+2*x^5+ 4*x^3-4*x^2+2*x-4)*exp(log(-log(3)^2+3)+x)^2/x^5,x, algorithm="maxima")
Output:
-(log(3)^2 - 3)^2*(8*Ei(4*x) - e^(2*x) - 8*gamma(-1, -2*x) - 8*gamma(-1, - 4*x) - 36*gamma(-1, -6*x) - 16*gamma(-2, -2*x) - 128*gamma(-2, -4*x) - 72* gamma(-2, -6*x) - 16*gamma(-3, -2*x) - 384*gamma(-3, -4*x) - 64*gamma(-4, -2*x))
Leaf count of result is larger than twice the leaf count of optimal. 197 vs. \(2 (38) = 76\).
Time = 0.12 (sec) , antiderivative size = 197, normalized size of antiderivative = 5.32 \[ \int \frac {e^{2 x} \left (-4+2 x-4 x^2+4 x^3+2 x^5+e^{4 x} \left (-2 x^2+6 x^3\right )+e^{2 x} \left (6 x-8 x^2+2 x^3-8 x^4\right )\right ) \left (3-\log ^2(3)\right )^2}{x^5} \, dx=\frac {x^{4} e^{\left (2 \, x\right )} \log \left (3\right )^{4} - 2 \, x^{3} e^{\left (4 \, x\right )} \log \left (3\right )^{4} - 6 \, x^{4} e^{\left (2 \, x\right )} \log \left (3\right )^{2} + x^{2} e^{\left (6 \, x\right )} \log \left (3\right )^{4} + 2 \, x^{2} e^{\left (2 \, x\right )} \log \left (3\right )^{4} + 12 \, x^{3} e^{\left (4 \, x\right )} \log \left (3\right )^{2} - 2 \, x e^{\left (4 \, x\right )} \log \left (3\right )^{4} + 9 \, x^{4} e^{\left (2 \, x\right )} - 6 \, x^{2} e^{\left (6 \, x\right )} \log \left (3\right )^{2} - 12 \, x^{2} e^{\left (2 \, x\right )} \log \left (3\right )^{2} + e^{\left (2 \, x\right )} \log \left (3\right )^{4} - 18 \, x^{3} e^{\left (4 \, x\right )} + 12 \, x e^{\left (4 \, x\right )} \log \left (3\right )^{2} + 9 \, x^{2} e^{\left (6 \, x\right )} + 18 \, x^{2} e^{\left (2 \, x\right )} - 6 \, e^{\left (2 \, x\right )} \log \left (3\right )^{2} - 18 \, x e^{\left (4 \, x\right )} + 9 \, e^{\left (2 \, x\right )}}{x^{4}} \] Input:
integrate(((6*x^3-2*x^2)*exp(x)^4+(-8*x^4+2*x^3-8*x^2+6*x)*exp(x)^2+2*x^5+ 4*x^3-4*x^2+2*x-4)*exp(log(-log(3)^2+3)+x)^2/x^5,x, algorithm="giac")
Output:
(x^4*e^(2*x)*log(3)^4 - 2*x^3*e^(4*x)*log(3)^4 - 6*x^4*e^(2*x)*log(3)^2 + x^2*e^(6*x)*log(3)^4 + 2*x^2*e^(2*x)*log(3)^4 + 12*x^3*e^(4*x)*log(3)^2 - 2*x*e^(4*x)*log(3)^4 + 9*x^4*e^(2*x) - 6*x^2*e^(6*x)*log(3)^2 - 12*x^2*e^( 2*x)*log(3)^2 + e^(2*x)*log(3)^4 - 18*x^3*e^(4*x) + 12*x*e^(4*x)*log(3)^2 + 9*x^2*e^(6*x) + 18*x^2*e^(2*x) - 6*e^(2*x)*log(3)^2 - 18*x*e^(4*x) + 9*e ^(2*x))/x^4
Time = 0.17 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.81 \[ \int \frac {e^{2 x} \left (-4+2 x-4 x^2+4 x^3+2 x^5+e^{4 x} \left (-2 x^2+6 x^3\right )+e^{2 x} \left (6 x-8 x^2+2 x^3-8 x^4\right )\right ) \left (3-\log ^2(3)\right )^2}{x^5} \, dx=\frac {{\mathrm {e}}^{2\,x}\,{\left ({\ln \left (3\right )}^2-3\right )}^2\,{\left (x^2-x\,{\mathrm {e}}^{2\,x}+1\right )}^2}{x^4} \] Input:
int((exp(2*x + 2*log(3 - log(3)^2))*(2*x - exp(4*x)*(2*x^2 - 6*x^3) + exp( 2*x)*(6*x - 8*x^2 + 2*x^3 - 8*x^4) - 4*x^2 + 4*x^3 + 2*x^5 - 4))/x^5,x)
Output:
(exp(2*x)*(log(3)^2 - 3)^2*(x^2 - x*exp(2*x) + 1)^2)/x^4
Time = 0.16 (sec) , antiderivative size = 173, normalized size of antiderivative = 4.68 \[ \int \frac {e^{2 x} \left (-4+2 x-4 x^2+4 x^3+2 x^5+e^{4 x} \left (-2 x^2+6 x^3\right )+e^{2 x} \left (6 x-8 x^2+2 x^3-8 x^4\right )\right ) \left (3-\log ^2(3)\right )^2}{x^5} \, dx=\frac {e^{2 x} \left (e^{4 x} \mathrm {log}\left (3\right )^{4} x^{2}-6 e^{4 x} \mathrm {log}\left (3\right )^{2} x^{2}+9 e^{4 x} x^{2}-2 e^{2 x} \mathrm {log}\left (3\right )^{4} x^{3}-2 e^{2 x} \mathrm {log}\left (3\right )^{4} x +12 e^{2 x} \mathrm {log}\left (3\right )^{2} x^{3}+12 e^{2 x} \mathrm {log}\left (3\right )^{2} x -18 e^{2 x} x^{3}-18 e^{2 x} x +\mathrm {log}\left (3\right )^{4} x^{4}+2 \mathrm {log}\left (3\right )^{4} x^{2}+\mathrm {log}\left (3\right )^{4}-6 \mathrm {log}\left (3\right )^{2} x^{4}-12 \mathrm {log}\left (3\right )^{2} x^{2}-6 \mathrm {log}\left (3\right )^{2}+9 x^{4}+18 x^{2}+9\right )}{x^{4}} \] Input:
int(((6*x^3-2*x^2)*exp(x)^4+(-8*x^4+2*x^3-8*x^2+6*x)*exp(x)^2+2*x^5+4*x^3- 4*x^2+2*x-4)*exp(log(-log(3)^2+3)+x)^2/x^5,x)
Output:
(e**(2*x)*(e**(4*x)*log(3)**4*x**2 - 6*e**(4*x)*log(3)**2*x**2 + 9*e**(4*x )*x**2 - 2*e**(2*x)*log(3)**4*x**3 - 2*e**(2*x)*log(3)**4*x + 12*e**(2*x)* log(3)**2*x**3 + 12*e**(2*x)*log(3)**2*x - 18*e**(2*x)*x**3 - 18*e**(2*x)* x + log(3)**4*x**4 + 2*log(3)**4*x**2 + log(3)**4 - 6*log(3)**2*x**4 - 12* log(3)**2*x**2 - 6*log(3)**2 + 9*x**4 + 18*x**2 + 9))/x**4