Integrand size = 152, antiderivative size = 22 \[ \int \frac {4 e^{15 e x}-x+\left (4 e^{30 e x}-30 e^{1+15 e x} x\right ) \log \left (\frac {4}{x^2}\right )-30 e^{1+30 e x} x \log ^2\left (\frac {4}{x^2}\right )}{4 x+4 x^2+x^3+e^{15 e x} \left (8 x+4 x^2\right ) \log \left (\frac {4}{x^2}\right )+e^{30 e x} \left (8 x+2 x^2\right ) \log ^2\left (\frac {4}{x^2}\right )+4 e^{45 e x} x \log ^3\left (\frac {4}{x^2}\right )+e^{60 e x} x \log ^4\left (\frac {4}{x^2}\right )} \, dx=\frac {1}{1+x+\left (1+e^{15 e x} \log \left (\frac {4}{x^2}\right )\right )^2} \] Output:
1/(x+(1+exp(15*x*exp(1))*ln(4/x^2))^2+1)
Time = 0.07 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.55 \[ \int \frac {4 e^{15 e x}-x+\left (4 e^{30 e x}-30 e^{1+15 e x} x\right ) \log \left (\frac {4}{x^2}\right )-30 e^{1+30 e x} x \log ^2\left (\frac {4}{x^2}\right )}{4 x+4 x^2+x^3+e^{15 e x} \left (8 x+4 x^2\right ) \log \left (\frac {4}{x^2}\right )+e^{30 e x} \left (8 x+2 x^2\right ) \log ^2\left (\frac {4}{x^2}\right )+4 e^{45 e x} x \log ^3\left (\frac {4}{x^2}\right )+e^{60 e x} x \log ^4\left (\frac {4}{x^2}\right )} \, dx=\frac {1}{2+x+2 e^{15 e x} \log \left (\frac {4}{x^2}\right )+e^{30 e x} \log ^2\left (\frac {4}{x^2}\right )} \] Input:
Integrate[(4*E^(15*E*x) - x + (4*E^(30*E*x) - 30*E^(1 + 15*E*x)*x)*Log[4/x ^2] - 30*E^(1 + 30*E*x)*x*Log[4/x^2]^2)/(4*x + 4*x^2 + x^3 + E^(15*E*x)*(8 *x + 4*x^2)*Log[4/x^2] + E^(30*E*x)*(8*x + 2*x^2)*Log[4/x^2]^2 + 4*E^(45*E *x)*x*Log[4/x^2]^3 + E^(60*E*x)*x*Log[4/x^2]^4),x]
Output:
(2 + x + 2*E^(15*E*x)*Log[4/x^2] + E^(30*E*x)*Log[4/x^2]^2)^(-1)
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.
| \(\displaystyle \int \frac {-30 e^{30 e x+1} x \log ^2\left (\frac {4}{x^2}\right )+\left (4 e^{30 e x}-30 e^{15 e x+1} x\right ) \log \left (\frac {4}{x^2}\right )+4 e^{15 e x}-x}{x^3+4 x^2+e^{60 e x} x \log ^4\left (\frac {4}{x^2}\right )+4 e^{45 e x} x \log ^3\left (\frac {4}{x^2}\right )+e^{30 e x} \left (2 x^2+8 x\right ) \log ^2\left (\frac {4}{x^2}\right )+e^{15 e x} \left (4 x^2+8 x\right ) \log \left (\frac {4}{x^2}\right )+4 x} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-30 e^{30 e x+1} x \log ^2\left (\frac {4}{x^2}\right )+\left (4 e^{30 e x}-30 e^{15 e x+1} x\right ) \log \left (\frac {4}{x^2}\right )+4 e^{15 e x}-x}{x \left (e^{30 e x} \log ^2\left (\frac {4}{x^2}\right )+2 e^{15 e x} \log \left (\frac {4}{x^2}\right )+x+2\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {30 e^{15 e x+1} x \log ^2\left (\frac {4}{x^2}\right )+30 e x^2 \log \left (\frac {4}{x^2}\right )-(1-60 e) x \log \left (\frac {4}{x^2}\right )-4 e^{15 e x} \log \left (\frac {4}{x^2}\right )-4 x-8}{x \log \left (\frac {4}{x^2}\right ) \left (e^{30 e x} \log ^2\left (\frac {4}{x^2}\right )+2 e^{15 e x} \log \left (\frac {4}{x^2}\right )+x+2\right )^2}-\frac {2 \left (15 e x \log \left (\frac {4}{x^2}\right )-2\right )}{x \log \left (\frac {4}{x^2}\right ) \left (e^{30 e x} \log ^2\left (\frac {4}{x^2}\right )+2 e^{15 e x} \log \left (\frac {4}{x^2}\right )+x+2\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\left ((1-60 e) \int \frac {1}{\left (e^{30 e x} \log ^2\left (\frac {4}{x^2}\right )+2 e^{15 e x} \log \left (\frac {4}{x^2}\right )+x+2\right )^2}dx\right )-4 \int \frac {e^{15 e x}}{x \left (e^{30 e x} \log ^2\left (\frac {4}{x^2}\right )+2 e^{15 e x} \log \left (\frac {4}{x^2}\right )+x+2\right )^2}dx+30 e \int \frac {x}{\left (e^{30 e x} \log ^2\left (\frac {4}{x^2}\right )+2 e^{15 e x} \log \left (\frac {4}{x^2}\right )+x+2\right )^2}dx-4 \int \frac {1}{\log \left (\frac {4}{x^2}\right ) \left (e^{30 e x} \log ^2\left (\frac {4}{x^2}\right )+2 e^{15 e x} \log \left (\frac {4}{x^2}\right )+x+2\right )^2}dx-8 \int \frac {1}{x \log \left (\frac {4}{x^2}\right ) \left (e^{30 e x} \log ^2\left (\frac {4}{x^2}\right )+2 e^{15 e x} \log \left (\frac {4}{x^2}\right )+x+2\right )^2}dx+30 \int \frac {e^{15 e x+1} \log \left (\frac {4}{x^2}\right )}{\left (e^{30 e x} \log ^2\left (\frac {4}{x^2}\right )+2 e^{15 e x} \log \left (\frac {4}{x^2}\right )+x+2\right )^2}dx-30 e \int \frac {1}{e^{30 e x} \log ^2\left (\frac {4}{x^2}\right )+2 e^{15 e x} \log \left (\frac {4}{x^2}\right )+x+2}dx+4 \int \frac {1}{x \log \left (\frac {4}{x^2}\right ) \left (e^{30 e x} \log ^2\left (\frac {4}{x^2}\right )+2 e^{15 e x} \log \left (\frac {4}{x^2}\right )+x+2\right )}dx\) |
Input:
Int[(4*E^(15*E*x) - x + (4*E^(30*E*x) - 30*E^(1 + 15*E*x)*x)*Log[4/x^2] - 30*E^(1 + 30*E*x)*x*Log[4/x^2]^2)/(4*x + 4*x^2 + x^3 + E^(15*E*x)*(8*x + 4 *x^2)*Log[4/x^2] + E^(30*E*x)*(8*x + 2*x^2)*Log[4/x^2]^2 + 4*E^(45*E*x)*x* Log[4/x^2]^3 + E^(60*E*x)*x*Log[4/x^2]^4),x]
Output:
$Aborted
Time = 5.28 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.68
| method | result | size |
| parallelrisch | \(\frac {1}{{\mathrm e}^{30 x \,{\mathrm e}} \ln \left (\frac {4}{x^{2}}\right )^{2}+2 \,{\mathrm e}^{15 x \,{\mathrm e}} \ln \left (\frac {4}{x^{2}}\right )+x +2}\) | \(37\) |
| risch | \(-\frac {4}{-8-4 x +16 i \pi \ln \left (2\right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2} {\mathrm e}^{30 x \,{\mathrm e}}+{\mathrm e}^{30 x \,{\mathrm e}} \pi ^{2} \operatorname {csgn}\left (i x \right )^{4} \operatorname {csgn}\left (i x^{2}\right )^{2}-4 \,{\mathrm e}^{30 x \,{\mathrm e}} \pi ^{2} \operatorname {csgn}\left (i x \right )^{3} \operatorname {csgn}\left (i x^{2}\right )^{3}+6 \,{\mathrm e}^{30 x \,{\mathrm e}} \pi ^{2} \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )^{4}-4 \,{\mathrm e}^{30 x \,{\mathrm e}} \pi ^{2} \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{5}+16 \,{\mathrm e}^{15 x \,{\mathrm e}} \ln \left (x \right )-16 \,{\mathrm e}^{30 x \,{\mathrm e}} \ln \left (x \right )^{2}-16 \,{\mathrm e}^{30 x \,{\mathrm e}} \ln \left (2\right )^{2}-16 \ln \left (2\right ) {\mathrm e}^{15 x \,{\mathrm e}}+32 \,{\mathrm e}^{30 x \,{\mathrm e}} \ln \left (2\right ) \ln \left (x \right )+8 i \pi \operatorname {csgn}\left (i x^{2}\right )^{3} {\mathrm e}^{30 x \,{\mathrm e}} \ln \left (x \right )-16 i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2} {\mathrm e}^{30 x \,{\mathrm e}} \ln \left (x \right )+8 i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2} {\mathrm e}^{15 x \,{\mathrm e}}-4 i \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right ) {\mathrm e}^{15 x \,{\mathrm e}}+8 i \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right ) {\mathrm e}^{30 x \,{\mathrm e}} \ln \left (x \right )-8 i \pi \ln \left (2\right ) \operatorname {csgn}\left (i x^{2}\right )^{3} {\mathrm e}^{30 x \,{\mathrm e}}-4 i \pi \operatorname {csgn}\left (i x^{2}\right )^{3} {\mathrm e}^{15 x \,{\mathrm e}}-8 i \pi \ln \left (2\right ) \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right ) {\mathrm e}^{30 x \,{\mathrm e}}+{\mathrm e}^{30 x \,{\mathrm e}} \pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{6}}\) | \(403\) |
Input:
int((-30*x*exp(1)*exp(15*x*exp(1))^2*ln(4/x^2)^2+(4*exp(15*x*exp(1))^2-30* x*exp(1)*exp(15*x*exp(1)))*ln(4/x^2)+4*exp(15*x*exp(1))-x)/(x*exp(15*x*exp (1))^4*ln(4/x^2)^4+4*x*exp(15*x*exp(1))^3*ln(4/x^2)^3+(2*x^2+8*x)*exp(15*x *exp(1))^2*ln(4/x^2)^2+(4*x^2+8*x)*exp(15*x*exp(1))*ln(4/x^2)+x^3+4*x^2+4* x),x,method=_RETURNVERBOSE)
Output:
1/(exp(15*x*exp(1))^2*ln(4/x^2)^2+2*exp(15*x*exp(1))*ln(4/x^2)+x+2)
Leaf count of result is larger than twice the leaf count of optimal. 45 vs. \(2 (22) = 44\).
Time = 0.10 (sec) , antiderivative size = 45, normalized size of antiderivative = 2.05 \[ \int \frac {4 e^{15 e x}-x+\left (4 e^{30 e x}-30 e^{1+15 e x} x\right ) \log \left (\frac {4}{x^2}\right )-30 e^{1+30 e x} x \log ^2\left (\frac {4}{x^2}\right )}{4 x+4 x^2+x^3+e^{15 e x} \left (8 x+4 x^2\right ) \log \left (\frac {4}{x^2}\right )+e^{30 e x} \left (8 x+2 x^2\right ) \log ^2\left (\frac {4}{x^2}\right )+4 e^{45 e x} x \log ^3\left (\frac {4}{x^2}\right )+e^{60 e x} x \log ^4\left (\frac {4}{x^2}\right )} \, dx=\frac {e^{2}}{e^{\left (30 \, x e + 2\right )} \log \left (\frac {4}{x^{2}}\right )^{2} + {\left (x + 2\right )} e^{2} + 2 \, e^{\left (15 \, x e + 2\right )} \log \left (\frac {4}{x^{2}}\right )} \] Input:
integrate((-30*x*exp(1)*exp(15*exp(1)*x)^2*log(4/x^2)^2+(4*exp(15*exp(1)*x )^2-30*x*exp(1)*exp(15*exp(1)*x))*log(4/x^2)+4*exp(15*exp(1)*x)-x)/(x*exp( 15*exp(1)*x)^4*log(4/x^2)^4+4*x*exp(15*exp(1)*x)^3*log(4/x^2)^3+(2*x^2+8*x )*exp(15*exp(1)*x)^2*log(4/x^2)^2+(4*x^2+8*x)*exp(15*exp(1)*x)*log(4/x^2)+ x^3+4*x^2+4*x),x, algorithm="fricas")
Output:
e^2/(e^(30*x*e + 2)*log(4/x^2)^2 + (x + 2)*e^2 + 2*e^(15*x*e + 2)*log(4/x^ 2))
Time = 0.16 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.68 \[ \int \frac {4 e^{15 e x}-x+\left (4 e^{30 e x}-30 e^{1+15 e x} x\right ) \log \left (\frac {4}{x^2}\right )-30 e^{1+30 e x} x \log ^2\left (\frac {4}{x^2}\right )}{4 x+4 x^2+x^3+e^{15 e x} \left (8 x+4 x^2\right ) \log \left (\frac {4}{x^2}\right )+e^{30 e x} \left (8 x+2 x^2\right ) \log ^2\left (\frac {4}{x^2}\right )+4 e^{45 e x} x \log ^3\left (\frac {4}{x^2}\right )+e^{60 e x} x \log ^4\left (\frac {4}{x^2}\right )} \, dx=\frac {1}{x + e^{30 e x} \log {\left (\frac {4}{x^{2}} \right )}^{2} + 2 e^{15 e x} \log {\left (\frac {4}{x^{2}} \right )} + 2} \] Input:
integrate((-30*x*exp(1)*exp(15*exp(1)*x)**2*ln(4/x**2)**2+(4*exp(15*exp(1) *x)**2-30*x*exp(1)*exp(15*exp(1)*x))*ln(4/x**2)+4*exp(15*exp(1)*x)-x)/(x*e xp(15*exp(1)*x)**4*ln(4/x**2)**4+4*x*exp(15*exp(1)*x)**3*ln(4/x**2)**3+(2* x**2+8*x)*exp(15*exp(1)*x)**2*ln(4/x**2)**2+(4*x**2+8*x)*exp(15*exp(1)*x)* ln(4/x**2)+x**3+4*x**2+4*x),x)
Output:
1/(x + exp(30*E*x)*log(4/x**2)**2 + 2*exp(15*E*x)*log(4/x**2) + 2)
Time = 0.42 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.95 \[ \int \frac {4 e^{15 e x}-x+\left (4 e^{30 e x}-30 e^{1+15 e x} x\right ) \log \left (\frac {4}{x^2}\right )-30 e^{1+30 e x} x \log ^2\left (\frac {4}{x^2}\right )}{4 x+4 x^2+x^3+e^{15 e x} \left (8 x+4 x^2\right ) \log \left (\frac {4}{x^2}\right )+e^{30 e x} \left (8 x+2 x^2\right ) \log ^2\left (\frac {4}{x^2}\right )+4 e^{45 e x} x \log ^3\left (\frac {4}{x^2}\right )+e^{60 e x} x \log ^4\left (\frac {4}{x^2}\right )} \, dx=\frac {1}{4 \, {\left (\log \left (2\right )^{2} - 2 \, \log \left (2\right ) \log \left (x\right ) + \log \left (x\right )^{2}\right )} e^{\left (30 \, x e\right )} + 4 \, {\left (\log \left (2\right ) - \log \left (x\right )\right )} e^{\left (15 \, x e\right )} + x + 2} \] Input:
integrate((-30*x*exp(1)*exp(15*exp(1)*x)^2*log(4/x^2)^2+(4*exp(15*exp(1)*x )^2-30*x*exp(1)*exp(15*exp(1)*x))*log(4/x^2)+4*exp(15*exp(1)*x)-x)/(x*exp( 15*exp(1)*x)^4*log(4/x^2)^4+4*x*exp(15*exp(1)*x)^3*log(4/x^2)^3+(2*x^2+8*x )*exp(15*exp(1)*x)^2*log(4/x^2)^2+(4*x^2+8*x)*exp(15*exp(1)*x)*log(4/x^2)+ x^3+4*x^2+4*x),x, algorithm="maxima")
Output:
1/(4*(log(2)^2 - 2*log(2)*log(x) + log(x)^2)*e^(30*x*e) + 4*(log(2) - log( x))*e^(15*x*e) + x + 2)
Timed out. \[ \int \frac {4 e^{15 e x}-x+\left (4 e^{30 e x}-30 e^{1+15 e x} x\right ) \log \left (\frac {4}{x^2}\right )-30 e^{1+30 e x} x \log ^2\left (\frac {4}{x^2}\right )}{4 x+4 x^2+x^3+e^{15 e x} \left (8 x+4 x^2\right ) \log \left (\frac {4}{x^2}\right )+e^{30 e x} \left (8 x+2 x^2\right ) \log ^2\left (\frac {4}{x^2}\right )+4 e^{45 e x} x \log ^3\left (\frac {4}{x^2}\right )+e^{60 e x} x \log ^4\left (\frac {4}{x^2}\right )} \, dx=\text {Timed out} \] Input:
integrate((-30*x*exp(1)*exp(15*exp(1)*x)^2*log(4/x^2)^2+(4*exp(15*exp(1)*x )^2-30*x*exp(1)*exp(15*exp(1)*x))*log(4/x^2)+4*exp(15*exp(1)*x)-x)/(x*exp( 15*exp(1)*x)^4*log(4/x^2)^4+4*x*exp(15*exp(1)*x)^3*log(4/x^2)^3+(2*x^2+8*x )*exp(15*exp(1)*x)^2*log(4/x^2)^2+(4*x^2+8*x)*exp(15*exp(1)*x)*log(4/x^2)+ x^3+4*x^2+4*x),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {4 e^{15 e x}-x+\left (4 e^{30 e x}-30 e^{1+15 e x} x\right ) \log \left (\frac {4}{x^2}\right )-30 e^{1+30 e x} x \log ^2\left (\frac {4}{x^2}\right )}{4 x+4 x^2+x^3+e^{15 e x} \left (8 x+4 x^2\right ) \log \left (\frac {4}{x^2}\right )+e^{30 e x} \left (8 x+2 x^2\right ) \log ^2\left (\frac {4}{x^2}\right )+4 e^{45 e x} x \log ^3\left (\frac {4}{x^2}\right )+e^{60 e x} x \log ^4\left (\frac {4}{x^2}\right )} \, dx=\int -\frac {30\,x\,\mathrm {e}\,{\mathrm {e}}^{30\,x\,\mathrm {e}}\,{\ln \left (\frac {4}{x^2}\right )}^2+\left (30\,x\,\mathrm {e}\,{\mathrm {e}}^{15\,x\,\mathrm {e}}-4\,{\mathrm {e}}^{30\,x\,\mathrm {e}}\right )\,\ln \left (\frac {4}{x^2}\right )+x-4\,{\mathrm {e}}^{15\,x\,\mathrm {e}}}{4\,x+4\,x^2+x^3+{\mathrm {e}}^{15\,x\,\mathrm {e}}\,\ln \left (\frac {4}{x^2}\right )\,\left (4\,x^2+8\,x\right )+{\mathrm {e}}^{30\,x\,\mathrm {e}}\,{\ln \left (\frac {4}{x^2}\right )}^2\,\left (2\,x^2+8\,x\right )+4\,x\,{\mathrm {e}}^{45\,x\,\mathrm {e}}\,{\ln \left (\frac {4}{x^2}\right )}^3+x\,{\mathrm {e}}^{60\,x\,\mathrm {e}}\,{\ln \left (\frac {4}{x^2}\right )}^4} \,d x \] Input:
int(-(x - 4*exp(15*x*exp(1)) - log(4/x^2)*(4*exp(30*x*exp(1)) - 30*x*exp(1 )*exp(15*x*exp(1))) + 30*x*exp(1)*exp(30*x*exp(1))*log(4/x^2)^2)/(4*x + 4* x^2 + x^3 + exp(15*x*exp(1))*log(4/x^2)*(8*x + 4*x^2) + exp(30*x*exp(1))*l og(4/x^2)^2*(8*x + 2*x^2) + 4*x*exp(45*x*exp(1))*log(4/x^2)^3 + x*exp(60*x *exp(1))*log(4/x^2)^4),x)
Output:
int(-(x - 4*exp(15*x*exp(1)) - log(4/x^2)*(4*exp(30*x*exp(1)) - 30*x*exp(1 )*exp(15*x*exp(1))) + 30*x*exp(1)*exp(30*x*exp(1))*log(4/x^2)^2)/(4*x + 4* x^2 + x^3 + exp(15*x*exp(1))*log(4/x^2)*(8*x + 4*x^2) + exp(30*x*exp(1))*l og(4/x^2)^2*(8*x + 2*x^2) + 4*x*exp(45*x*exp(1))*log(4/x^2)^3 + x*exp(60*x *exp(1))*log(4/x^2)^4), x)
Time = 0.22 (sec) , antiderivative size = 72, normalized size of antiderivative = 3.27 \[ \int \frac {4 e^{15 e x}-x+\left (4 e^{30 e x}-30 e^{1+15 e x} x\right ) \log \left (\frac {4}{x^2}\right )-30 e^{1+30 e x} x \log ^2\left (\frac {4}{x^2}\right )}{4 x+4 x^2+x^3+e^{15 e x} \left (8 x+4 x^2\right ) \log \left (\frac {4}{x^2}\right )+e^{30 e x} \left (8 x+2 x^2\right ) \log ^2\left (\frac {4}{x^2}\right )+4 e^{45 e x} x \log ^3\left (\frac {4}{x^2}\right )+e^{60 e x} x \log ^4\left (\frac {4}{x^2}\right )} \, dx=\frac {-e^{30 e x} \mathrm {log}\left (\frac {4}{x^{2}}\right )^{2}-2 e^{15 e x} \mathrm {log}\left (\frac {4}{x^{2}}\right )-x}{2 e^{30 e x} \mathrm {log}\left (\frac {4}{x^{2}}\right )^{2}+4 e^{15 e x} \mathrm {log}\left (\frac {4}{x^{2}}\right )+2 x +4} \] Input:
int((-30*x*exp(1)*exp(15*exp(1)*x)^2*log(4/x^2)^2+(4*exp(15*exp(1)*x)^2-30 *x*exp(1)*exp(15*exp(1)*x))*log(4/x^2)+4*exp(15*exp(1)*x)-x)/(x*exp(15*exp (1)*x)^4*log(4/x^2)^4+4*x*exp(15*exp(1)*x)^3*log(4/x^2)^3+(2*x^2+8*x)*exp( 15*exp(1)*x)^2*log(4/x^2)^2+(4*x^2+8*x)*exp(15*exp(1)*x)*log(4/x^2)+x^3+4* x^2+4*x),x)
Output:
( - e**(30*e*x)*log(4/x**2)**2 - 2*e**(15*e*x)*log(4/x**2) - x)/(2*(e**(30 *e*x)*log(4/x**2)**2 + 2*e**(15*e*x)*log(4/x**2) + x + 2))