Integrand size = 77, antiderivative size = 31 \[ \int \frac {-e^e+e^{2/x} \left (-4+2 x+x^2\right ) \log ^2(2)+e^{\frac {1}{x}} \left (-4+6 x+2 x^2\right ) \log ^2(2) \log (5)+\left (4 x+x^2\right ) \log ^2(2) \log ^2(5)}{8+8 x+2 x^2} \, dx=\frac {e^e+x^2 \log ^2(2) \left (e^{\frac {1}{x}}+\log (5)\right )^2}{2 (2+x)} \] Output:
(exp(exp(1))+(exp(1/x)+ln(5))^2*x^2*ln(2)^2)/(4+2*x)
Time = 0.21 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.94 \[ \int \frac {-e^e+e^{2/x} \left (-4+2 x+x^2\right ) \log ^2(2)+e^{\frac {1}{x}} \left (-4+6 x+2 x^2\right ) \log ^2(2) \log (5)+\left (4 x+x^2\right ) \log ^2(2) \log ^2(5)}{8+8 x+2 x^2} \, dx=\frac {e^e+e^{2/x} x^2 \log ^2(2)+\left (4+2 x+x^2\right ) \log ^2(2) \log ^2(5)+e^{\frac {1}{x}} x^2 \log ^2(2) \log (25)}{2 (2+x)} \] Input:
Integrate[(-E^E + E^(2/x)*(-4 + 2*x + x^2)*Log[2]^2 + E^x^(-1)*(-4 + 6*x + 2*x^2)*Log[2]^2*Log[5] + (4*x + x^2)*Log[2]^2*Log[5]^2)/(8 + 8*x + 2*x^2) ,x]
Output:
(E^E + E^(2/x)*x^2*Log[2]^2 + (4 + 2*x + x^2)*Log[2]^2*Log[5]^2 + E^x^(-1) *x^2*Log[2]^2*Log[25])/(2*(2 + x))
Leaf count is larger than twice the leaf count of optimal. \(128\) vs. \(2(31)=62\).
Time = 1.35 (sec) , antiderivative size = 128, normalized size of antiderivative = 4.13, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.039, Rules used = {2007, 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.
| \(\displaystyle \int \frac {e^{2/x} \left (x^2+2 x-4\right ) \log ^2(2)+\left (x^2+4 x\right ) \log ^2(2) \log ^2(5)+e^{\frac {1}{x}} \left (2 x^2+6 x-4\right ) \log ^2(2) \log (5)-e^e}{2 x^2+8 x+8} \, dx\) |
| \(\Big \downarrow \) 2007 |
| \(\displaystyle \int \frac {e^{2/x} \left (x^2+2 x-4\right ) \log ^2(2)+\left (x^2+4 x\right ) \log ^2(2) \log ^2(5)+e^{\frac {1}{x}} \left (2 x^2+6 x-4\right ) \log ^2(2) \log (5)-e^e}{\left (\sqrt {2} x+2 \sqrt {2}\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {e^{2/x} \left (x^2+2 x-4\right ) \log ^2(2)}{2 (x+2)^2}+\frac {x^2 \log ^2(2) \log ^2(5)+4 x \log ^2(2) \log ^2(5)-e^e}{2 (x+2)^2}+\frac {e^{\frac {1}{x}} \left (x^2+3 x-2\right ) \log ^2(2) \log (5)}{(x+2)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} x \log ^2(2) \log ^2(5)+e^{\frac {1}{x}} x \log ^2(2) \log (5)+\frac {1}{2} e^{2/x} x \log ^2(2)+\frac {e^e+4 \log ^2(2) \log ^2(5)}{2 (x+2)}-2 e^{\frac {1}{x}} \log ^2(2) \log (5)+\frac {4 e^{\frac {1}{x}} \log ^2(2) \log (5)}{x+2}-e^{2/x} \log ^2(2)+\frac {2 e^{2/x} \log ^2(2)}{x+2}\) |
Input:
Int[(-E^E + E^(2/x)*(-4 + 2*x + x^2)*Log[2]^2 + E^x^(-1)*(-4 + 6*x + 2*x^2 )*Log[2]^2*Log[5] + (4*x + x^2)*Log[2]^2*Log[5]^2)/(8 + 8*x + 2*x^2),x]
Output:
-(E^(2/x)*Log[2]^2) + (E^(2/x)*x*Log[2]^2)/2 + (2*E^(2/x)*Log[2]^2)/(2 + x ) - 2*E^x^(-1)*Log[2]^2*Log[5] + E^x^(-1)*x*Log[2]^2*Log[5] + (4*E^x^(-1)* Log[2]^2*Log[5])/(2 + x) + (x*Log[2]^2*Log[5]^2)/2 + (E^E + 4*Log[2]^2*Log [5]^2)/(2*(2 + x))
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^(Ex pon[Px, x]*p), x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; IntegerQ[p] && Pol yQ[Px, x] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 0]
Time = 0.51 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.71
| method | result | size |
| parallelrisch | \(\frac {\ln \left (2\right )^{2} \ln \left (5\right )^{2} x^{2}+2 \ln \left (2\right )^{2} {\mathrm e}^{\frac {1}{x}} \ln \left (5\right ) x^{2}+\ln \left (2\right )^{2} x^{2} {\mathrm e}^{\frac {2}{x}}+{\mathrm e}^{{\mathrm e}}}{4+2 x}\) | \(53\) |
| norman | \(\frac {\ln \left (2\right )^{2} {\mathrm e}^{\frac {1}{x}} \ln \left (5\right ) x^{2}+\frac {\ln \left (2\right )^{2} x^{2} {\mathrm e}^{\frac {2}{x}}}{2}+\frac {\ln \left (2\right )^{2} \ln \left (5\right )^{2} x^{2}}{2}+\frac {{\mathrm e}^{{\mathrm e}}}{2}}{2+x}\) | \(55\) |
| risch | \(\frac {\ln \left (2\right )^{2} \ln \left (5\right )^{2} x}{2}+\frac {2 \ln \left (2\right )^{2} \ln \left (5\right )^{2}}{2+x}+\frac {{\mathrm e}^{{\mathrm e}}}{4+2 x}+\frac {\ln \left (2\right )^{2} x^{2} {\mathrm e}^{\frac {2}{x}}}{4+2 x}+\frac {\ln \left (2\right )^{2} \ln \left (5\right ) x^{2} {\mathrm e}^{\frac {1}{x}}}{2+x}\) | \(77\) |
| parts | \(\ln \left (2\right )^{2} \ln \left (5\right ) \left (x \,{\mathrm e}^{\frac {1}{x}}-\frac {{\mathrm e}^{\frac {1}{x}}}{\frac {1}{x}+\frac {1}{2}}\right )+\frac {\ln \left (2\right )^{2} \ln \left (5\right )^{2} x}{2}-\frac {-4 \ln \left (2\right )^{2} \ln \left (5\right )^{2}-{\mathrm e}^{{\mathrm e}}}{2 \left (2+x \right )}+\frac {\ln \left (2\right )^{2} \left (x \,{\mathrm e}^{\frac {2}{x}}-\frac {{\mathrm e}^{\frac {2}{x}}}{\frac {1}{x}+\frac {1}{2}}\right )}{2}\) | \(93\) |
| derivativedivides | \(-\frac {\ln \left (2\right )^{2} \left (2 \,\operatorname {expIntegral}_{1}\left (-\frac {2}{x}\right )-\frac {{\mathrm e}^{\frac {2}{x}}}{\frac {1}{x}+\frac {1}{2}}-6 \,{\mathrm e}^{-1} \operatorname {expIntegral}_{1}\left (-\frac {2}{x}-1\right )-x \,{\mathrm e}^{\frac {2}{x}}\right )}{2}-\frac {\ln \left (2\right )^{2} \ln \left (5\right )^{2} \left (-x -4 \ln \left (\frac {1}{x}\right )-\frac {2}{\frac {2}{x}+1}+4 \ln \left (\frac {2}{x}+1\right )\right )}{2}-\frac {{\mathrm e}^{{\mathrm e}}}{4 \left (\frac {2}{x}+1\right )}-\ln \left (2\right )^{2} \left (-\operatorname {expIntegral}_{1}\left (-\frac {2}{x}\right )+\frac {{\mathrm e}^{\frac {2}{x}}}{\frac {2}{x}+1}+2 \,{\mathrm e}^{-1} \operatorname {expIntegral}_{1}\left (-\frac {2}{x}-1\right )\right )-2 \ln \left (2\right )^{2} \ln \left (5\right )^{2} \left (\ln \left (\frac {1}{x}\right )+\frac {1}{\frac {2}{x}+1}-\ln \left (\frac {2}{x}+1\right )\right )-\frac {\ln \left (2\right )^{2} {\mathrm e}^{\frac {2}{x}}}{2 \left (\frac {1}{x}+\frac {1}{2}\right )}-\ln \left (2\right )^{2} {\mathrm e}^{-1} \operatorname {expIntegral}_{1}\left (-\frac {2}{x}-1\right )-\ln \left (2\right )^{2} \ln \left (5\right ) \left (3 \,\operatorname {expIntegral}_{1}\left (-\frac {1}{x}\right )-\frac {{\mathrm e}^{\frac {1}{x}}}{\frac {1}{x}+\frac {1}{2}}-5 \,{\mathrm e}^{-\frac {1}{2}} \operatorname {expIntegral}_{1}\left (-\frac {1}{x}-\frac {1}{2}\right )-x \,{\mathrm e}^{\frac {1}{x}}\right )-3 \ln \left (2\right )^{2} \ln \left (5\right ) \left (-\operatorname {expIntegral}_{1}\left (-\frac {1}{x}\right )+\frac {{\mathrm e}^{\frac {1}{x}}}{\frac {2}{x}+1}+\frac {3 \,{\mathrm e}^{-\frac {1}{2}} \operatorname {expIntegral}_{1}\left (-\frac {1}{x}-\frac {1}{2}\right )}{2}\right )+2 \ln \left (2\right )^{2} \ln \left (5\right ) \left (-\frac {{\mathrm e}^{\frac {1}{x}}}{4 \left (\frac {1}{x}+\frac {1}{2}\right )}-\frac {{\mathrm e}^{-\frac {1}{2}} \operatorname {expIntegral}_{1}\left (-\frac {1}{x}-\frac {1}{2}\right )}{4}\right )\) | \(354\) |
| default | \(-\frac {\ln \left (2\right )^{2} \left (2 \,\operatorname {expIntegral}_{1}\left (-\frac {2}{x}\right )-\frac {{\mathrm e}^{\frac {2}{x}}}{\frac {1}{x}+\frac {1}{2}}-6 \,{\mathrm e}^{-1} \operatorname {expIntegral}_{1}\left (-\frac {2}{x}-1\right )-x \,{\mathrm e}^{\frac {2}{x}}\right )}{2}-\frac {\ln \left (2\right )^{2} \ln \left (5\right )^{2} \left (-x -4 \ln \left (\frac {1}{x}\right )-\frac {2}{\frac {2}{x}+1}+4 \ln \left (\frac {2}{x}+1\right )\right )}{2}-\frac {{\mathrm e}^{{\mathrm e}}}{4 \left (\frac {2}{x}+1\right )}-\ln \left (2\right )^{2} \left (-\operatorname {expIntegral}_{1}\left (-\frac {2}{x}\right )+\frac {{\mathrm e}^{\frac {2}{x}}}{\frac {2}{x}+1}+2 \,{\mathrm e}^{-1} \operatorname {expIntegral}_{1}\left (-\frac {2}{x}-1\right )\right )-2 \ln \left (2\right )^{2} \ln \left (5\right )^{2} \left (\ln \left (\frac {1}{x}\right )+\frac {1}{\frac {2}{x}+1}-\ln \left (\frac {2}{x}+1\right )\right )-\frac {\ln \left (2\right )^{2} {\mathrm e}^{\frac {2}{x}}}{2 \left (\frac {1}{x}+\frac {1}{2}\right )}-\ln \left (2\right )^{2} {\mathrm e}^{-1} \operatorname {expIntegral}_{1}\left (-\frac {2}{x}-1\right )-\ln \left (2\right )^{2} \ln \left (5\right ) \left (3 \,\operatorname {expIntegral}_{1}\left (-\frac {1}{x}\right )-\frac {{\mathrm e}^{\frac {1}{x}}}{\frac {1}{x}+\frac {1}{2}}-5 \,{\mathrm e}^{-\frac {1}{2}} \operatorname {expIntegral}_{1}\left (-\frac {1}{x}-\frac {1}{2}\right )-x \,{\mathrm e}^{\frac {1}{x}}\right )-3 \ln \left (2\right )^{2} \ln \left (5\right ) \left (-\operatorname {expIntegral}_{1}\left (-\frac {1}{x}\right )+\frac {{\mathrm e}^{\frac {1}{x}}}{\frac {2}{x}+1}+\frac {3 \,{\mathrm e}^{-\frac {1}{2}} \operatorname {expIntegral}_{1}\left (-\frac {1}{x}-\frac {1}{2}\right )}{2}\right )+2 \ln \left (2\right )^{2} \ln \left (5\right ) \left (-\frac {{\mathrm e}^{\frac {1}{x}}}{4 \left (\frac {1}{x}+\frac {1}{2}\right )}-\frac {{\mathrm e}^{-\frac {1}{2}} \operatorname {expIntegral}_{1}\left (-\frac {1}{x}-\frac {1}{2}\right )}{4}\right )\) | \(354\) |
Input:
int((-exp(exp(1))+(x^2+2*x-4)*ln(2)^2*exp(1/x)^2+(2*x^2+6*x-4)*ln(2)^2*ln( 5)*exp(1/x)+(x^2+4*x)*ln(2)^2*ln(5)^2)/(2*x^2+8*x+8),x,method=_RETURNVERBO SE)
Output:
1/2*(ln(2)^2*ln(5)^2*x^2+2*ln(2)^2*exp(1/x)*ln(5)*x^2+ln(2)^2*exp(1/x)^2*x ^2+exp(exp(1)))/(2+x)
Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (28) = 56\).
Time = 0.09 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.84 \[ \int \frac {-e^e+e^{2/x} \left (-4+2 x+x^2\right ) \log ^2(2)+e^{\frac {1}{x}} \left (-4+6 x+2 x^2\right ) \log ^2(2) \log (5)+\left (4 x+x^2\right ) \log ^2(2) \log ^2(5)}{8+8 x+2 x^2} \, dx=\frac {2 \, x^{2} e^{\frac {1}{x}} \log \left (5\right ) \log \left (2\right )^{2} + x^{2} e^{\frac {2}{x}} \log \left (2\right )^{2} + {\left (x^{2} + 2 \, x + 4\right )} \log \left (5\right )^{2} \log \left (2\right )^{2} + e^{e}}{2 \, {\left (x + 2\right )}} \] Input:
integrate((-exp(exp(1))+(x^2+2*x-4)*log(2)^2*exp(1/x)^2+(2*x^2+6*x-4)*log( 2)^2*log(5)*exp(1/x)+(x^2+4*x)*log(2)^2*log(5)^2)/(2*x^2+8*x+8),x, algorit hm="fricas")
Output:
1/2*(2*x^2*e^(1/x)*log(5)*log(2)^2 + x^2*e^(2/x)*log(2)^2 + (x^2 + 2*x + 4 )*log(5)^2*log(2)^2 + e^e)/(x + 2)
Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (27) = 54\).
Time = 0.19 (sec) , antiderivative size = 100, normalized size of antiderivative = 3.23 \[ \int \frac {-e^e+e^{2/x} \left (-4+2 x+x^2\right ) \log ^2(2)+e^{\frac {1}{x}} \left (-4+6 x+2 x^2\right ) \log ^2(2) \log (5)+\left (4 x+x^2\right ) \log ^2(2) \log ^2(5)}{8+8 x+2 x^2} \, dx=\frac {x \log {\left (2 \right )}^{2} \log {\left (5 \right )}^{2}}{2} + \frac {\left (x^{3} \log {\left (2 \right )}^{2} + 2 x^{2} \log {\left (2 \right )}^{2}\right ) e^{\frac {2}{x}} + \left (2 x^{3} \log {\left (2 \right )}^{2} \log {\left (5 \right )} + 4 x^{2} \log {\left (2 \right )}^{2} \log {\left (5 \right )}\right ) e^{\frac {1}{x}}}{2 x^{2} + 8 x + 8} + \frac {4 \log {\left (2 \right )}^{2} \log {\left (5 \right )}^{2} + e^{e}}{2 x + 4} \] Input:
integrate((-exp(exp(1))+(x**2+2*x-4)*ln(2)**2*exp(1/x)**2+(2*x**2+6*x-4)*l n(2)**2*ln(5)*exp(1/x)+(x**2+4*x)*ln(2)**2*ln(5)**2)/(2*x**2+8*x+8),x)
Output:
x*log(2)**2*log(5)**2/2 + ((x**3*log(2)**2 + 2*x**2*log(2)**2)*exp(2/x) + (2*x**3*log(2)**2*log(5) + 4*x**2*log(2)**2*log(5))*exp(1/x))/(2*x**2 + 8* x + 8) + (4*log(2)**2*log(5)**2 + exp(E))/(2*x + 4)
\[ \int \frac {-e^e+e^{2/x} \left (-4+2 x+x^2\right ) \log ^2(2)+e^{\frac {1}{x}} \left (-4+6 x+2 x^2\right ) \log ^2(2) \log (5)+\left (4 x+x^2\right ) \log ^2(2) \log ^2(5)}{8+8 x+2 x^2} \, dx=\int { \frac {2 \, {\left (x^{2} + 3 \, x - 2\right )} e^{\frac {1}{x}} \log \left (5\right ) \log \left (2\right )^{2} + {\left (x^{2} + 4 \, x\right )} \log \left (5\right )^{2} \log \left (2\right )^{2} + {\left (x^{2} + 2 \, x - 4\right )} e^{\frac {2}{x}} \log \left (2\right )^{2} - e^{e}}{2 \, {\left (x^{2} + 4 \, x + 4\right )}} \,d x } \] Input:
integrate((-exp(exp(1))+(x^2+2*x-4)*log(2)^2*exp(1/x)^2+(2*x^2+6*x-4)*log( 2)^2*log(5)*exp(1/x)+(x^2+4*x)*log(2)^2*log(5)^2)/(2*x^2+8*x+8),x, algorit hm="maxima")
Output:
1/2*(x - 4/(x + 2) - 4*log(x + 2))*log(5)^2*log(2)^2 + 2*(2/(x + 2) + log( x + 2))*log(5)^2*log(2)^2 - 4*integrate(e^(2/x)/(x^3 + 4*x^2 + 4*x), x)*lo g(2)^2 + 1/2*((x^3*log(2)^2 - 4*x*log(2)^2)*e^(2/x) + 2*(x^3*log(5)*log(2) ^2 - 6*x*log(5)*log(2)^2)*e^(1/x))/(x^2 + 4*x + 4) + 1/2*e^e/(x + 2) + 1/2 *integrate(4*(x*log(5)*log(2)^2 - 6*log(5)*log(2)^2)*e^(1/x)/(x^4 + 6*x^3 + 12*x^2 + 8*x), x)
Time = 0.13 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.81 \[ \int \frac {-e^e+e^{2/x} \left (-4+2 x+x^2\right ) \log ^2(2)+e^{\frac {1}{x}} \left (-4+6 x+2 x^2\right ) \log ^2(2) \log (5)+\left (4 x+x^2\right ) \log ^2(2) \log ^2(5)}{8+8 x+2 x^2} \, dx=\frac {4 \, e^{\frac {1}{x}} \log \left (5\right ) \log \left (2\right )^{2} + 2 \, \log \left (5\right )^{2} \log \left (2\right )^{2} + 2 \, e^{\frac {2}{x}} \log \left (2\right )^{2} - \frac {e^{e}}{x}}{4 \, {\left (\frac {1}{x} + \frac {2}{x^{2}}\right )}} \] Input:
integrate((-exp(exp(1))+(x^2+2*x-4)*log(2)^2*exp(1/x)^2+(2*x^2+6*x-4)*log( 2)^2*log(5)*exp(1/x)+(x^2+4*x)*log(2)^2*log(5)^2)/(2*x^2+8*x+8),x, algorit hm="giac")
Output:
1/4*(4*e^(1/x)*log(5)*log(2)^2 + 2*log(5)^2*log(2)^2 + 2*e^(2/x)*log(2)^2 - e^e/x)/(1/x + 2/x^2)
Time = 2.97 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.87 \[ \int \frac {-e^e+e^{2/x} \left (-4+2 x+x^2\right ) \log ^2(2)+e^{\frac {1}{x}} \left (-4+6 x+2 x^2\right ) \log ^2(2) \log (5)+\left (4 x+x^2\right ) \log ^2(2) \log ^2(5)}{8+8 x+2 x^2} \, dx=\frac {2\,x^2\,{\mathrm {e}}^{2/x}\,{\ln \left (2\right )}^2-x\,{\mathrm {e}}^{\mathrm {e}}+2\,x^2\,{\ln \left (2\right )}^2\,{\ln \left (5\right )}^2+4\,x^2\,{\mathrm {e}}^{1/x}\,{\ln \left (2\right )}^2\,\ln \left (5\right )}{4\,x+8} \] Input:
int((log(2)^2*log(5)^2*(4*x + x^2) - exp(exp(1)) + exp(2/x)*log(2)^2*(2*x + x^2 - 4) + exp(1/x)*log(2)^2*log(5)*(6*x + 2*x^2 - 4))/(8*x + 2*x^2 + 8) ,x)
Output:
(2*x^2*exp(2/x)*log(2)^2 - x*exp(exp(1)) + 2*x^2*log(2)^2*log(5)^2 + 4*x^2 *exp(1/x)*log(2)^2*log(5))/(4*x + 8)
Time = 0.17 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.74 \[ \int \frac {-e^e+e^{2/x} \left (-4+2 x+x^2\right ) \log ^2(2)+e^{\frac {1}{x}} \left (-4+6 x+2 x^2\right ) \log ^2(2) \log (5)+\left (4 x+x^2\right ) \log ^2(2) \log ^2(5)}{8+8 x+2 x^2} \, dx=\frac {x \left (2 e^{\frac {2}{x}} \mathrm {log}\left (2\right )^{2} x +4 e^{\frac {1}{x}} \mathrm {log}\left (5\right ) \mathrm {log}\left (2\right )^{2} x -e^{e}+2 \mathrm {log}\left (5\right )^{2} \mathrm {log}\left (2\right )^{2} x \right )}{4 x +8} \] Input:
int((-exp(exp(1))+(x^2+2*x-4)*log(2)^2*exp(1/x)^2+(2*x^2+6*x-4)*log(2)^2*l og(5)*exp(1/x)+(x^2+4*x)*log(2)^2*log(5)^2)/(2*x^2+8*x+8),x)
Output:
(x*(2*e**(2/x)*log(2)**2*x + 4*e**(1/x)*log(5)*log(2)**2*x - e**e + 2*log( 5)**2*log(2)**2*x))/(4*(x + 2))