Integrand size = 120, antiderivative size = 27 \begin {dmath*} \int \frac {e^{\frac {9+123 x+30 x^2}{10 \log \left (\frac {2+x^2}{x}\right )}} \left (18+246 x+51 x^2-123 x^3-30 x^4+\left (246 x+120 x^2+123 x^3+60 x^4\right ) \log \left (\frac {2+x^2}{x}\right )+\left (20+10 x^2\right ) \log ^2\left (\frac {2+x^2}{x}\right )\right )}{\left (20+10 x^2\right ) \log ^2\left (\frac {2+x^2}{x}\right )} \, dx=e^{\frac {3 \left (x+\left (\frac {1}{10}+x\right ) (3+x)\right )}{\log \left (\frac {2}{x}+x\right )}} x \end {dmath*}
Time = 0.09 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04 \begin {dmath*} \int \frac {e^{\frac {9+123 x+30 x^2}{10 \log \left (\frac {2+x^2}{x}\right )}} \left (18+246 x+51 x^2-123 x^3-30 x^4+\left (246 x+120 x^2+123 x^3+60 x^4\right ) \log \left (\frac {2+x^2}{x}\right )+\left (20+10 x^2\right ) \log ^2\left (\frac {2+x^2}{x}\right )\right )}{\left (20+10 x^2\right ) \log ^2\left (\frac {2+x^2}{x}\right )} \, dx=e^{\frac {3 \left (3+41 x+10 x^2\right )}{10 \log \left (\frac {2}{x}+x\right )}} x \end {dmath*}
Integrate[(E^((9 + 123*x + 30*x^2)/(10*Log[(2 + x^2)/x]))*(18 + 246*x + 51 *x^2 - 123*x^3 - 30*x^4 + (246*x + 120*x^2 + 123*x^3 + 60*x^4)*Log[(2 + x^ 2)/x] + (20 + 10*x^2)*Log[(2 + x^2)/x]^2))/((20 + 10*x^2)*Log[(2 + x^2)/x] ^2),x]
Leaf count is larger than twice the leaf count of optimal. \(164\) vs. \(2(27)=54\).
Time = 1.12 (sec) , antiderivative size = 164, normalized size of antiderivative = 6.07, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.008, Rules used = {2726}
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.
\(\displaystyle \int \frac {e^{\frac {30 x^2+123 x+9}{10 \log \left (\frac {x^2+2}{x}\right )}} \left (-30 x^4-123 x^3+51 x^2+\left (10 x^2+20\right ) \log ^2\left (\frac {x^2+2}{x}\right )+\left (60 x^4+123 x^3+120 x^2+246 x\right ) \log \left (\frac {x^2+2}{x}\right )+246 x+18\right )}{\left (10 x^2+20\right ) \log ^2\left (\frac {x^2+2}{x}\right )} \, dx\) |
\(\Big \downarrow \) 2726 |
\(\displaystyle -\frac {e^{\frac {3 \left (10 x^2+41 x+3\right )}{10 \log \left (\frac {x^2+2}{x}\right )}} \left (-10 x^4-41 x^3+17 x^2+\left (20 x^4+41 x^3+40 x^2+82 x\right ) \log \left (\frac {x^2+2}{x}\right )+82 x+6\right )}{\left (x^2+2\right ) \left (\frac {x \left (10 x^2+41 x+3\right ) \left (2-\frac {x^2+2}{x^2}\right )}{\left (x^2+2\right ) \log ^2\left (\frac {x^2+2}{x}\right )}-\frac {20 x+41}{\log \left (\frac {x^2+2}{x}\right )}\right ) \log ^2\left (\frac {x^2+2}{x}\right )}\) |
Int[(E^((9 + 123*x + 30*x^2)/(10*Log[(2 + x^2)/x]))*(18 + 246*x + 51*x^2 - 123*x^3 - 30*x^4 + (246*x + 120*x^2 + 123*x^3 + 60*x^4)*Log[(2 + x^2)/x] + (20 + 10*x^2)*Log[(2 + x^2)/x]^2))/((20 + 10*x^2)*Log[(2 + x^2)/x]^2),x]
-((E^((3*(3 + 41*x + 10*x^2))/(10*Log[(2 + x^2)/x]))*(6 + 82*x + 17*x^2 - 41*x^3 - 10*x^4 + (82*x + 40*x^2 + 41*x^3 + 20*x^4)*Log[(2 + x^2)/x]))/((2 + x^2)*((x*(3 + 41*x + 10*x^2)*(2 - (2 + x^2)/x^2))/((2 + x^2)*Log[(2 + x ^2)/x]^2) - (41 + 20*x)/Log[(2 + x^2)/x])*Log[(2 + x^2)/x]^2))
3.6.41.3.1 Defintions of rubi rules used
Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, x]))}, Simp[F^u*z, x] /; EqQ[D[z, x], w*y]] /; FreeQ[F, x]
Time = 1.27 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04
method | result | size |
risch | \({\mathrm e}^{\frac {3 x^{2}+\frac {123}{10} x +\frac {9}{10}}{\ln \left (\frac {x^{2}+2}{x}\right )}} x\) | \(28\) |
parallelrisch | \({\mathrm e}^{\frac {3 x^{2}+\frac {123}{10} x +\frac {9}{10}}{\ln \left (\frac {x^{2}+2}{x}\right )}} x\) | \(28\) |
int(((10*x^2+20)*ln((x^2+2)/x)^2+(60*x^4+123*x^3+120*x^2+246*x)*ln((x^2+2) /x)-30*x^4-123*x^3+51*x^2+246*x+18)*exp(1/10*(30*x^2+123*x+9)/ln((x^2+2)/x ))/(10*x^2+20)/ln((x^2+2)/x)^2,x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \begin {dmath*} \int \frac {e^{\frac {9+123 x+30 x^2}{10 \log \left (\frac {2+x^2}{x}\right )}} \left (18+246 x+51 x^2-123 x^3-30 x^4+\left (246 x+120 x^2+123 x^3+60 x^4\right ) \log \left (\frac {2+x^2}{x}\right )+\left (20+10 x^2\right ) \log ^2\left (\frac {2+x^2}{x}\right )\right )}{\left (20+10 x^2\right ) \log ^2\left (\frac {2+x^2}{x}\right )} \, dx=x e^{\left (\frac {3 \, {\left (10 \, x^{2} + 41 \, x + 3\right )}}{10 \, \log \left (\frac {x^{2} + 2}{x}\right )}\right )} \end {dmath*}
integrate(((10*x^2+20)*log((x^2+2)/x)^2+(60*x^4+123*x^3+120*x^2+246*x)*log ((x^2+2)/x)-30*x^4-123*x^3+51*x^2+246*x+18)*exp(1/10*(30*x^2+123*x+9)/log( (x^2+2)/x))/(10*x^2+20)/log((x^2+2)/x)^2,x, algorithm=\
Exception generated. \begin {dmath*} \int \frac {e^{\frac {9+123 x+30 x^2}{10 \log \left (\frac {2+x^2}{x}\right )}} \left (18+246 x+51 x^2-123 x^3-30 x^4+\left (246 x+120 x^2+123 x^3+60 x^4\right ) \log \left (\frac {2+x^2}{x}\right )+\left (20+10 x^2\right ) \log ^2\left (\frac {2+x^2}{x}\right )\right )}{\left (20+10 x^2\right ) \log ^2\left (\frac {2+x^2}{x}\right )} \, dx=\text {Exception raised: TypeError} \end {dmath*}
integrate(((10*x**2+20)*ln((x**2+2)/x)**2+(60*x**4+123*x**3+120*x**2+246*x )*ln((x**2+2)/x)-30*x**4-123*x**3+51*x**2+246*x+18)*exp(1/10*(30*x**2+123* x+9)/ln((x**2+2)/x))/(10*x**2+20)/ln((x**2+2)/x)**2,x)
Exception generated. \begin {dmath*} \int \frac {e^{\frac {9+123 x+30 x^2}{10 \log \left (\frac {2+x^2}{x}\right )}} \left (18+246 x+51 x^2-123 x^3-30 x^4+\left (246 x+120 x^2+123 x^3+60 x^4\right ) \log \left (\frac {2+x^2}{x}\right )+\left (20+10 x^2\right ) \log ^2\left (\frac {2+x^2}{x}\right )\right )}{\left (20+10 x^2\right ) \log ^2\left (\frac {2+x^2}{x}\right )} \, dx=\text {Exception raised: RuntimeError} \end {dmath*}
integrate(((10*x^2+20)*log((x^2+2)/x)^2+(60*x^4+123*x^3+120*x^2+246*x)*log ((x^2+2)/x)-30*x^4-123*x^3+51*x^2+246*x+18)*exp(1/10*(30*x^2+123*x+9)/log( (x^2+2)/x))/(10*x^2+20)/log((x^2+2)/x)^2,x, algorithm=\
\begin {dmath*} \int \frac {e^{\frac {9+123 x+30 x^2}{10 \log \left (\frac {2+x^2}{x}\right )}} \left (18+246 x+51 x^2-123 x^3-30 x^4+\left (246 x+120 x^2+123 x^3+60 x^4\right ) \log \left (\frac {2+x^2}{x}\right )+\left (20+10 x^2\right ) \log ^2\left (\frac {2+x^2}{x}\right )\right )}{\left (20+10 x^2\right ) \log ^2\left (\frac {2+x^2}{x}\right )} \, dx=\int { -\frac {{\left (30 \, x^{4} + 123 \, x^{3} - 10 \, {\left (x^{2} + 2\right )} \log \left (\frac {x^{2} + 2}{x}\right )^{2} - 51 \, x^{2} - 3 \, {\left (20 \, x^{4} + 41 \, x^{3} + 40 \, x^{2} + 82 \, x\right )} \log \left (\frac {x^{2} + 2}{x}\right ) - 246 \, x - 18\right )} e^{\left (\frac {3 \, {\left (10 \, x^{2} + 41 \, x + 3\right )}}{10 \, \log \left (\frac {x^{2} + 2}{x}\right )}\right )}}{10 \, {\left (x^{2} + 2\right )} \log \left (\frac {x^{2} + 2}{x}\right )^{2}} \,d x } \end {dmath*}
integrate(((10*x^2+20)*log((x^2+2)/x)^2+(60*x^4+123*x^3+120*x^2+246*x)*log ((x^2+2)/x)-30*x^4-123*x^3+51*x^2+246*x+18)*exp(1/10*(30*x^2+123*x+9)/log( (x^2+2)/x))/(10*x^2+20)/log((x^2+2)/x)^2,x, algorithm=\
integrate(-1/10*(30*x^4 + 123*x^3 - 10*(x^2 + 2)*log((x^2 + 2)/x)^2 - 51*x ^2 - 3*(20*x^4 + 41*x^3 + 40*x^2 + 82*x)*log((x^2 + 2)/x) - 246*x - 18)*e^ (3/10*(10*x^2 + 41*x + 3)/log((x^2 + 2)/x))/((x^2 + 2)*log((x^2 + 2)/x)^2) , x)
Time = 15.67 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.30 \begin {dmath*} \int \frac {e^{\frac {9+123 x+30 x^2}{10 \log \left (\frac {2+x^2}{x}\right )}} \left (18+246 x+51 x^2-123 x^3-30 x^4+\left (246 x+120 x^2+123 x^3+60 x^4\right ) \log \left (\frac {2+x^2}{x}\right )+\left (20+10 x^2\right ) \log ^2\left (\frac {2+x^2}{x}\right )\right )}{\left (20+10 x^2\right ) \log ^2\left (\frac {2+x^2}{x}\right )} \, dx=x\,{\mathrm {e}}^{\frac {3\,x^2}{\ln \left (\frac {1}{x}\right )+\ln \left (x^2+2\right )}}\,{\mathrm {e}}^{\frac {9}{10\,\left (\ln \left (\frac {1}{x}\right )+\ln \left (x^2+2\right )\right )}}\,{\mathrm {e}}^{\frac {123\,x}{10\,\left (\ln \left (\frac {1}{x}\right )+\ln \left (x^2+2\right )\right )}} \end {dmath*}
int((exp(((123*x)/10 + 3*x^2 + 9/10)/log((x^2 + 2)/x))*(246*x + log((x^2 + 2)/x)*(246*x + 120*x^2 + 123*x^3 + 60*x^4) + log((x^2 + 2)/x)^2*(10*x^2 + 20) + 51*x^2 - 123*x^3 - 30*x^4 + 18))/(log((x^2 + 2)/x)^2*(10*x^2 + 20)) ,x)