Integrand size = 169, antiderivative size = 30 \[ \int \frac {(10-4 x) (i \pi +\log (2))^2+e^{e^{3+x}+e^3 x} \left (-2 e^{3+x} x (i \pi +\log (2))^2+\left (-2-2 e^3 x\right ) (i \pi +\log (2))^2\right )}{-125 x^3+e^{3 e^{3+x}+3 e^3 x} x^3+75 x^4-15 x^5+x^6+e^{2 e^{3+x}+2 e^3 x} \left (-15 x^3+3 x^4\right )+e^{e^{3+x}+e^3 x} \left (75 x^3-30 x^4+3 x^5\right )} \, dx=\frac {(i \pi +\log (2))^2}{x^2 \left (-5+e^{e^3 \left (e^x+x\right )}+x\right )^2} \] Output:
(ln(2)+I*Pi)^2/(exp((exp(x)+x)*exp(3))+x-5)^2/x^2
Time = 0.55 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.10 \[ \int \frac {(10-4 x) (i \pi +\log (2))^2+e^{e^{3+x}+e^3 x} \left (-2 e^{3+x} x (i \pi +\log (2))^2+\left (-2-2 e^3 x\right ) (i \pi +\log (2))^2\right )}{-125 x^3+e^{3 e^{3+x}+3 e^3 x} x^3+75 x^4-15 x^5+x^6+e^{2 e^{3+x}+2 e^3 x} \left (-15 x^3+3 x^4\right )+e^{e^{3+x}+e^3 x} \left (75 x^3-30 x^4+3 x^5\right )} \, dx=-\frac {(\pi -i \log (2))^2}{x^2 \left (-5+e^{e^{3+x}+e^3 x}+x\right )^2} \] Input:
Integrate[((10 - 4*x)*(I*Pi + Log[2])^2 + E^(E^(3 + x) + E^3*x)*(-2*E^(3 + x)*x*(I*Pi + Log[2])^2 + (-2 - 2*E^3*x)*(I*Pi + Log[2])^2))/(-125*x^3 + E ^(3*E^(3 + x) + 3*E^3*x)*x^3 + 75*x^4 - 15*x^5 + x^6 + E^(2*E^(3 + x) + 2* E^3*x)*(-15*x^3 + 3*x^4) + E^(E^(3 + x) + E^3*x)*(75*x^3 - 30*x^4 + 3*x^5) ),x]
Output:
-((Pi - I*Log[2])^2/(x^2*(-5 + E^(E^(3 + x) + E^3*x) + x)^2))
Time = 1.37 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.17, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.018, Rules used = {7239, 27, 7238}
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.
\(\displaystyle \int \frac {(10-4 x) (\log (2)+i \pi )^2+e^{e^3 x+e^{x+3}} \left (\left (-2 e^3 x-2\right ) (\log (2)+i \pi )^2-2 e^{x+3} x (\log (2)+i \pi )^2\right )}{x^6-15 x^5+75 x^4+e^{3 e^3 x+3 e^{x+3}} x^3-125 x^3+e^{2 e^3 x+2 e^{x+3}} \left (3 x^4-15 x^3\right )+e^{e^3 x+e^{x+3}} \left (3 x^5-30 x^4+75 x^3\right )} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {2 \left (-e^{e^3 x+e^{x+3}+3} x-e^{e^3 x+x+e^{x+3}+3} x-2 x-e^{e^3 \left (x+e^x\right )}+5\right ) (\pi -i \log (2))^2}{\left (-x-e^{e^3 \left (x+e^x\right )}+5\right )^3 x^3}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 (\pi -i \log (2))^2 \int \frac {-e^{e^3 x+e^{x+3}+3} x-e^{e^3 x+x+e^{x+3}+3} x-2 x-e^{e^3 \left (x+e^x\right )}+5}{\left (-x-e^{e^3 \left (x+e^x\right )}+5\right )^3 x^3}dx\) |
\(\Big \downarrow \) 7238 |
\(\displaystyle -\frac {(\pi -i \log (2))^2}{\left (-x-e^{e^3 \left (x+e^x\right )}+5\right )^2 x^2}\) |
Input:
Int[((10 - 4*x)*(I*Pi + Log[2])^2 + E^(E^(3 + x) + E^3*x)*(-2*E^(3 + x)*x* (I*Pi + Log[2])^2 + (-2 - 2*E^3*x)*(I*Pi + Log[2])^2))/(-125*x^3 + E^(3*E^ (3 + x) + 3*E^3*x)*x^3 + 75*x^4 - 15*x^5 + x^6 + E^(2*E^(3 + x) + 2*E^3*x) *(-15*x^3 + 3*x^4) + E^(E^(3 + x) + E^3*x)*(75*x^3 - 30*x^4 + 3*x^5)),x]
Output:
-((Pi - I*Log[2])^2/((5 - E^(E^3*(E^x + x)) - x)^2*x^2))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*(y_)^(m_.)*(z_)^(n_.), x_Symbol] :> With[{q = DerivativeDivides[y* z, u*z^(n - m), x]}, Simp[q*y^(m + 1)*(z^(m + 1)/(m + 1)), x] /; !FalseQ[q ]] /; FreeQ[{m, n}, x] && NeQ[m, -1]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.84 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.20
method | result | size |
risch | \(\frac {\ln \left (2\right )^{2}+2 i \ln \left (2\right ) \pi -\pi ^{2}}{x^{2} \left ({\mathrm e}^{{\mathrm e}^{3+x}+x \,{\mathrm e}^{3}}+x -5\right )^{2}}\) | \(36\) |
parallelrisch | \(\frac {\ln \left (2\right )^{2}+2 i \ln \left (2\right ) \pi -\pi ^{2}}{x^{2} \left (x^{2}+2 \,{\mathrm e}^{\left ({\mathrm e}^{x}+x \right ) {\mathrm e}^{3}} x +{\mathrm e}^{2 \left ({\mathrm e}^{x}+x \right ) {\mathrm e}^{3}}-10 x -10 \,{\mathrm e}^{\left ({\mathrm e}^{x}+x \right ) {\mathrm e}^{3}}+25\right )}\) | \(62\) |
Input:
int(((-2*x*exp(3)*(ln(2)+I*Pi)^2*exp(x)+(-2*x*exp(3)-2)*(ln(2)+I*Pi)^2)*ex p(exp(x)*exp(3)+x*exp(3))+(10-4*x)*(ln(2)+I*Pi)^2)/(x^3*exp(exp(x)*exp(3)+ x*exp(3))^3+(3*x^4-15*x^3)*exp(exp(x)*exp(3)+x*exp(3))^2+(3*x^5-30*x^4+75* x^3)*exp(exp(x)*exp(3)+x*exp(3))+x^6-15*x^5+75*x^4-125*x^3),x,method=_RETU RNVERBOSE)
Output:
(ln(2)^2+2*I*ln(2)*Pi-Pi^2)/x^2/(exp(exp(3+x)+x*exp(3))+x-5)^2
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (25) = 50\).
Time = 0.11 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.37 \[ \int \frac {(10-4 x) (i \pi +\log (2))^2+e^{e^{3+x}+e^3 x} \left (-2 e^{3+x} x (i \pi +\log (2))^2+\left (-2-2 e^3 x\right ) (i \pi +\log (2))^2\right )}{-125 x^3+e^{3 e^{3+x}+3 e^3 x} x^3+75 x^4-15 x^5+x^6+e^{2 e^{3+x}+2 e^3 x} \left (-15 x^3+3 x^4\right )+e^{e^{3+x}+e^3 x} \left (75 x^3-30 x^4+3 x^5\right )} \, dx=-\frac {\pi ^{2} - 2 i \, \pi \log \left (2\right ) - \log \left (2\right )^{2}}{x^{4} - 10 \, x^{3} + x^{2} e^{\left (2 \, x e^{3} + 2 \, e^{\left (x + 3\right )}\right )} + 25 \, x^{2} + 2 \, {\left (x^{3} - 5 \, x^{2}\right )} e^{\left (x e^{3} + e^{\left (x + 3\right )}\right )}} \] Input:
integrate(((-2*x*exp(3)*(log(2)+I*pi)^2*exp(x)+(-2*x*exp(3)-2)*(log(2)+I*p i)^2)*exp(exp(x)*exp(3)+x*exp(3))+(10-4*x)*(log(2)+I*pi)^2)/(x^3*exp(exp(x )*exp(3)+x*exp(3))^3+(3*x^4-15*x^3)*exp(exp(x)*exp(3)+x*exp(3))^2+(3*x^5-3 0*x^4+75*x^3)*exp(exp(x)*exp(3)+x*exp(3))+x^6-15*x^5+75*x^4-125*x^3),x, al gorithm="fricas")
Output:
-(pi^2 - 2*I*pi*log(2) - log(2)^2)/(x^4 - 10*x^3 + x^2*e^(2*x*e^3 + 2*e^(x + 3)) + 25*x^2 + 2*(x^3 - 5*x^2)*e^(x*e^3 + e^(x + 3)))
Timed out. \[ \int \frac {(10-4 x) (i \pi +\log (2))^2+e^{e^{3+x}+e^3 x} \left (-2 e^{3+x} x (i \pi +\log (2))^2+\left (-2-2 e^3 x\right ) (i \pi +\log (2))^2\right )}{-125 x^3+e^{3 e^{3+x}+3 e^3 x} x^3+75 x^4-15 x^5+x^6+e^{2 e^{3+x}+2 e^3 x} \left (-15 x^3+3 x^4\right )+e^{e^{3+x}+e^3 x} \left (75 x^3-30 x^4+3 x^5\right )} \, dx=\text {Timed out} \] Input:
integrate(((-2*x*exp(3)*(ln(2)+I*pi)**2*exp(x)+(-2*x*exp(3)-2)*(ln(2)+I*pi )**2)*exp(exp(x)*exp(3)+x*exp(3))+(10-4*x)*(ln(2)+I*pi)**2)/(x**3*exp(exp( x)*exp(3)+x*exp(3))**3+(3*x**4-15*x**3)*exp(exp(x)*exp(3)+x*exp(3))**2+(3* x**5-30*x**4+75*x**3)*exp(exp(x)*exp(3)+x*exp(3))+x**6-15*x**5+75*x**4-125 *x**3),x)
Output:
Timed out
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (25) = 50\).
Time = 3.87 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.37 \[ \int \frac {(10-4 x) (i \pi +\log (2))^2+e^{e^{3+x}+e^3 x} \left (-2 e^{3+x} x (i \pi +\log (2))^2+\left (-2-2 e^3 x\right ) (i \pi +\log (2))^2\right )}{-125 x^3+e^{3 e^{3+x}+3 e^3 x} x^3+75 x^4-15 x^5+x^6+e^{2 e^{3+x}+2 e^3 x} \left (-15 x^3+3 x^4\right )+e^{e^{3+x}+e^3 x} \left (75 x^3-30 x^4+3 x^5\right )} \, dx=-\frac {\pi ^{2} - 2 i \, \pi \log \left (2\right ) - \log \left (2\right )^{2}}{x^{4} - 10 \, x^{3} + x^{2} e^{\left (2 \, x e^{3} + 2 \, e^{\left (x + 3\right )}\right )} + 25 \, x^{2} + 2 \, {\left (x^{3} - 5 \, x^{2}\right )} e^{\left (x e^{3} + e^{\left (x + 3\right )}\right )}} \] Input:
integrate(((-2*x*exp(3)*(log(2)+I*pi)^2*exp(x)+(-2*x*exp(3)-2)*(log(2)+I*p i)^2)*exp(exp(x)*exp(3)+x*exp(3))+(10-4*x)*(log(2)+I*pi)^2)/(x^3*exp(exp(x )*exp(3)+x*exp(3))^3+(3*x^4-15*x^3)*exp(exp(x)*exp(3)+x*exp(3))^2+(3*x^5-3 0*x^4+75*x^3)*exp(exp(x)*exp(3)+x*exp(3))+x^6-15*x^5+75*x^4-125*x^3),x, al gorithm="maxima")
Output:
-(pi^2 - 2*I*pi*log(2) - log(2)^2)/(x^4 - 10*x^3 + x^2*e^(2*x*e^3 + 2*e^(x + 3)) + 25*x^2 + 2*(x^3 - 5*x^2)*e^(x*e^3 + e^(x + 3)))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (25) = 50\).
Time = 0.29 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.67 \[ \int \frac {(10-4 x) (i \pi +\log (2))^2+e^{e^{3+x}+e^3 x} \left (-2 e^{3+x} x (i \pi +\log (2))^2+\left (-2-2 e^3 x\right ) (i \pi +\log (2))^2\right )}{-125 x^3+e^{3 e^{3+x}+3 e^3 x} x^3+75 x^4-15 x^5+x^6+e^{2 e^{3+x}+2 e^3 x} \left (-15 x^3+3 x^4\right )+e^{e^{3+x}+e^3 x} \left (75 x^3-30 x^4+3 x^5\right )} \, dx=-\frac {\pi ^{2} - 2 i \, \pi \log \left (2\right ) - \log \left (2\right )^{2}}{x^{4} + 2 \, x^{3} e^{\left (x e^{3} + e^{\left (x + 3\right )}\right )} - 10 \, x^{3} + x^{2} e^{\left (2 \, x e^{3} + 2 \, e^{\left (x + 3\right )}\right )} - 10 \, x^{2} e^{\left (x e^{3} + e^{\left (x + 3\right )}\right )} + 25 \, x^{2}} \] Input:
integrate(((-2*x*exp(3)*(log(2)+I*pi)^2*exp(x)+(-2*x*exp(3)-2)*(log(2)+I*p i)^2)*exp(exp(x)*exp(3)+x*exp(3))+(10-4*x)*(log(2)+I*pi)^2)/(x^3*exp(exp(x )*exp(3)+x*exp(3))^3+(3*x^4-15*x^3)*exp(exp(x)*exp(3)+x*exp(3))^2+(3*x^5-3 0*x^4+75*x^3)*exp(exp(x)*exp(3)+x*exp(3))+x^6-15*x^5+75*x^4-125*x^3),x, al gorithm="giac")
Output:
-(pi^2 - 2*I*pi*log(2) - log(2)^2)/(x^4 + 2*x^3*e^(x*e^3 + e^(x + 3)) - 10 *x^3 + x^2*e^(2*x*e^3 + 2*e^(x + 3)) - 10*x^2*e^(x*e^3 + e^(x + 3)) + 25*x ^2)
Timed out. \[ \int \frac {(10-4 x) (i \pi +\log (2))^2+e^{e^{3+x}+e^3 x} \left (-2 e^{3+x} x (i \pi +\log (2))^2+\left (-2-2 e^3 x\right ) (i \pi +\log (2))^2\right )}{-125 x^3+e^{3 e^{3+x}+3 e^3 x} x^3+75 x^4-15 x^5+x^6+e^{2 e^{3+x}+2 e^3 x} \left (-15 x^3+3 x^4\right )+e^{e^{3+x}+e^3 x} \left (75 x^3-30 x^4+3 x^5\right )} \, dx=\int -\frac {{\mathrm {e}}^{x\,{\mathrm {e}}^3+{\mathrm {e}}^3\,{\mathrm {e}}^x}\,\left ({\left (\ln \left (2\right )+\Pi \,1{}\mathrm {i}\right )}^2\,\left (2\,x\,{\mathrm {e}}^3+2\right )+2\,x\,{\mathrm {e}}^3\,{\mathrm {e}}^x\,{\left (\ln \left (2\right )+\Pi \,1{}\mathrm {i}\right )}^2\right )+\left (4\,x-10\right )\,{\left (\ln \left (2\right )+\Pi \,1{}\mathrm {i}\right )}^2}{{\mathrm {e}}^{x\,{\mathrm {e}}^3+{\mathrm {e}}^3\,{\mathrm {e}}^x}\,\left (3\,x^5-30\,x^4+75\,x^3\right )-{\mathrm {e}}^{2\,x\,{\mathrm {e}}^3+2\,{\mathrm {e}}^3\,{\mathrm {e}}^x}\,\left (15\,x^3-3\,x^4\right )-125\,x^3+75\,x^4-15\,x^5+x^6+x^3\,{\mathrm {e}}^{3\,x\,{\mathrm {e}}^3+3\,{\mathrm {e}}^3\,{\mathrm {e}}^x}} \,d x \] Input:
int(-(exp(x*exp(3) + exp(3)*exp(x))*((Pi*1i + log(2))^2*(2*x*exp(3) + 2) + 2*x*exp(3)*exp(x)*(Pi*1i + log(2))^2) + (4*x - 10)*(Pi*1i + log(2))^2)/(e xp(x*exp(3) + exp(3)*exp(x))*(75*x^3 - 30*x^4 + 3*x^5) - exp(2*x*exp(3) + 2*exp(3)*exp(x))*(15*x^3 - 3*x^4) - 125*x^3 + 75*x^4 - 15*x^5 + x^6 + x^3* exp(3*x*exp(3) + 3*exp(3)*exp(x))),x)
Output:
int(-(exp(x*exp(3) + exp(3)*exp(x))*((Pi*1i + log(2))^2*(2*x*exp(3) + 2) + 2*x*exp(3)*exp(x)*(Pi*1i + log(2))^2) + (4*x - 10)*(Pi*1i + log(2))^2)/(e xp(x*exp(3) + exp(3)*exp(x))*(75*x^3 - 30*x^4 + 3*x^5) - exp(2*x*exp(3) + 2*exp(3)*exp(x))*(15*x^3 - 3*x^4) - 125*x^3 + 75*x^4 - 15*x^5 + x^6 + x^3* exp(3*x*exp(3) + 3*exp(3)*exp(x))), x)
Time = 0.16 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.73 \[ \int \frac {(10-4 x) (i \pi +\log (2))^2+e^{e^{3+x}+e^3 x} \left (-2 e^{3+x} x (i \pi +\log (2))^2+\left (-2-2 e^3 x\right ) (i \pi +\log (2))^2\right )}{-125 x^3+e^{3 e^{3+x}+3 e^3 x} x^3+75 x^4-15 x^5+x^6+e^{2 e^{3+x}+2 e^3 x} \left (-15 x^3+3 x^4\right )+e^{e^{3+x}+e^3 x} \left (75 x^3-30 x^4+3 x^5\right )} \, dx=\frac {\mathrm {log}\left (2\right )^{2}+2 \,\mathrm {log}\left (2\right ) i \pi -\pi ^{2}}{x^{2} \left (e^{2 e^{x} e^{3}+2 e^{3} x}+2 e^{e^{x} e^{3}+e^{3} x} x -10 e^{e^{x} e^{3}+e^{3} x}+x^{2}-10 x +25\right )} \] Input:
int(((-2*x*exp(3)*(log(2)+I*Pi)^2*exp(x)+(-2*x*exp(3)-2)*(log(2)+I*Pi)^2)* exp(exp(x)*exp(3)+x*exp(3))+(10-4*x)*(log(2)+I*Pi)^2)/(x^3*exp(exp(x)*exp( 3)+x*exp(3))^3+(3*x^4-15*x^3)*exp(exp(x)*exp(3)+x*exp(3))^2+(3*x^5-30*x^4+ 75*x^3)*exp(exp(x)*exp(3)+x*exp(3))+x^6-15*x^5+75*x^4-125*x^3),x)
Output:
(log(2)**2 + 2*log(2)*i*pi - pi**2)/(x**2*(e**(2*e**x*e**3 + 2*e**3*x) + 2 *e**(e**x*e**3 + e**3*x)*x - 10*e**(e**x*e**3 + e**3*x) + x**2 - 10*x + 25 ))