Integrand size = 127, antiderivative size = 25 \[ \int \frac {-5 e+5 e^{e^5 x^2+x^3}+5 x+\left (5 e-10 x+e^{e^5 x^2+x^3} \left (-5-10 e^5 x^2-15 x^3\right )\right ) \log (x)}{e^2 x^2+e^{2 e^5 x^2+2 x^3} x^2-2 e x^3+x^4+e^{e^5 x^2+x^3} \left (-2 e x^2+2 x^3\right )} \, dx=\frac {5 \log (x)}{x \left (-e+e^{x^2 \left (e^5+x\right )}+x\right )} \]
Time = 0.12 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {-5 e+5 e^{e^5 x^2+x^3}+5 x+\left (5 e-10 x+e^{e^5 x^2+x^3} \left (-5-10 e^5 x^2-15 x^3\right )\right ) \log (x)}{e^2 x^2+e^{2 e^5 x^2+2 x^3} x^2-2 e x^3+x^4+e^{e^5 x^2+x^3} \left (-2 e x^2+2 x^3\right )} \, dx=\frac {5 \log (x)}{x \left (-e+e^{e^5 x^2+x^3}+x\right )} \]
Integrate[(-5*E + 5*E^(E^5*x^2 + x^3) + 5*x + (5*E - 10*x + E^(E^5*x^2 + x ^3)*(-5 - 10*E^5*x^2 - 15*x^3))*Log[x])/(E^2*x^2 + E^(2*E^5*x^2 + 2*x^3)*x ^2 - 2*E*x^3 + x^4 + E^(E^5*x^2 + x^3)*(-2*E*x^2 + 2*x^3)),x]
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.
\(\displaystyle \int \frac {5 e^{x^3+e^5 x^2}+\left (e^{x^3+e^5 x^2} \left (-15 x^3-10 e^5 x^2-5\right )-10 x+5 e\right ) \log (x)+5 x-5 e}{x^4-2 e x^3+e^2 x^2+e^{2 x^3+2 e^5 x^2} x^2+e^{x^3+e^5 x^2} \left (2 x^3-2 e x^2\right )} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {5 e^{x^3+e^5 x^2}+\left (e^{x^3+e^5 x^2} \left (-15 x^3-10 e^5 x^2-5\right )-10 x+5 e\right ) \log (x)+5 x-5 e}{\left (-e^{x^2 \left (x+e^5\right )}-x+e\right )^2 x^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {5 e}{\left (-e^{x^2 \left (x+e^5\right )}-x+e\right )^2 x^2}+\frac {5}{x \left (e^{x^2 \left (x+e^5\right )}+x-e\right )^2}+\frac {5 e \log (x)}{\left (-e^{x^2 \left (x+e^5\right )}-x+e\right )^2 x^2}-\frac {10 \log (x)}{x \left (e^{x^2 \left (x+e^5\right )}+x-e\right )^2}+\frac {5 e^{x^3+e^5 x^2}}{\left (-e^{x^2 \left (x+e^5\right )}-x+e\right )^2 x^2}-\frac {15 e^{x^3+e^5 x^2} x \log (x)}{\left (-e^{x^2 \left (x+e^5\right )}-x+e\right )^2}-\frac {10 e^{x^3+e^5 x^2+5} \log (x)}{\left (-e^{x^2 \left (x+e^5\right )}-x+e\right )^2}-\frac {5 e^{x^3+e^5 x^2} \log (x)}{\left (-e^{x^2 \left (x+e^5\right )}-x+e\right )^2 x^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -5 e \int \frac {1}{\left (-x-e^{x^2 \left (x+e^5\right )}+e\right )^2 x^2}dx+5 \int \frac {1}{x \left (x+e^{x^2 \left (x+e^5\right )}-e\right )^2}dx-5 e \int \frac {\int \frac {1}{x^2 \left (x+e^{x^2 \left (x+e^5\right )}-e\right )^2}dx}{x}dx+10 \int \frac {\int \frac {1}{x \left (x+e^{x^2 \left (x+e^5\right )}-e\right )^2}dx}{x}dx+5 e \log (x) \int \frac {1}{\left (-x-e^{x^2 \left (x+e^5\right )}+e\right )^2 x^2}dx-10 \log (x) \int \frac {1}{x \left (x+e^{x^2 \left (x+e^5\right )}-e\right )^2}dx+5 \int \frac {e^{x^3+e^5 x^2}}{\left (-x-e^{x^2 \left (x+e^5\right )}+e\right )^2 x^2}dx+10 \int \frac {\int \frac {e^{x^3+e^5 x^2+5}}{\left (x+e^{x^2 \left (x+e^5\right )}-e\right )^2}dx}{x}dx+5 \int \frac {\int \frac {e^{x^3+e^5 x^2}}{x^2 \left (x+e^{x^2 \left (x+e^5\right )}-e\right )^2}dx}{x}dx+15 \int \frac {\int \frac {e^{x^3+e^5 x^2} x}{\left (x+e^{x^2 \left (x+e^5\right )}-e\right )^2}dx}{x}dx-10 \log (x) \int \frac {e^{x^3+e^5 x^2+5}}{\left (-x-e^{x^2 \left (x+e^5\right )}+e\right )^2}dx-5 \log (x) \int \frac {e^{x^3+e^5 x^2}}{\left (-x-e^{x^2 \left (x+e^5\right )}+e\right )^2 x^2}dx-15 \log (x) \int \frac {e^{x^3+e^5 x^2} x}{\left (-x-e^{x^2 \left (x+e^5\right )}+e\right )^2}dx\) |
Int[(-5*E + 5*E^(E^5*x^2 + x^3) + 5*x + (5*E - 10*x + E^(E^5*x^2 + x^3)*(- 5 - 10*E^5*x^2 - 15*x^3))*Log[x])/(E^2*x^2 + E^(2*E^5*x^2 + 2*x^3)*x^2 - 2 *E*x^3 + x^4 + E^(E^5*x^2 + x^3)*(-2*E*x^2 + 2*x^3)),x]
3.17.20.3.1 Defintions of rubi rules used
Time = 0.51 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08
method | result | size |
risch | \(-\frac {5 \ln \left (x \right )}{x \left ({\mathrm e}-x -{\mathrm e}^{x^{2} \left ({\mathrm e}^{5}+x \right )}\right )}\) | \(27\) |
parallelrisch | \(-\frac {5 \ln \left (x \right )}{x \left ({\mathrm e}-x -{\mathrm e}^{x^{2} \left ({\mathrm e}^{5}+x \right )}\right )}\) | \(27\) |
int((((-10*x^2*exp(5)-15*x^3-5)*exp(x^2*exp(5)+x^3)+5*exp(1)-10*x)*ln(x)+5 *exp(x^2*exp(5)+x^3)-5*exp(1)+5*x)/(x^2*exp(x^2*exp(5)+x^3)^2+(-2*x^2*exp( 1)+2*x^3)*exp(x^2*exp(5)+x^3)+x^2*exp(1)^2-2*x^3*exp(1)+x^4),x,method=_RET URNVERBOSE)
Time = 0.24 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {-5 e+5 e^{e^5 x^2+x^3}+5 x+\left (5 e-10 x+e^{e^5 x^2+x^3} \left (-5-10 e^5 x^2-15 x^3\right )\right ) \log (x)}{e^2 x^2+e^{2 e^5 x^2+2 x^3} x^2-2 e x^3+x^4+e^{e^5 x^2+x^3} \left (-2 e x^2+2 x^3\right )} \, dx=\frac {5 \, \log \left (x\right )}{x^{2} - x e + x e^{\left (x^{3} + x^{2} e^{5}\right )}} \]
integrate((((-10*x^2*exp(5)-15*x^3-5)*exp(x^2*exp(5)+x^3)+5*exp(1)-10*x)*l og(x)+5*exp(x^2*exp(5)+x^3)-5*exp(1)+5*x)/(x^2*exp(x^2*exp(5)+x^3)^2+(-2*x ^2*exp(1)+2*x^3)*exp(x^2*exp(5)+x^3)+x^2*exp(1)^2-2*x^3*exp(1)+x^4),x, alg orithm=\
Time = 0.09 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04 \[ \int \frac {-5 e+5 e^{e^5 x^2+x^3}+5 x+\left (5 e-10 x+e^{e^5 x^2+x^3} \left (-5-10 e^5 x^2-15 x^3\right )\right ) \log (x)}{e^2 x^2+e^{2 e^5 x^2+2 x^3} x^2-2 e x^3+x^4+e^{e^5 x^2+x^3} \left (-2 e x^2+2 x^3\right )} \, dx=\frac {5 \log {\left (x \right )}}{x^{2} + x e^{x^{3} + x^{2} e^{5}} - e x} \]
integrate((((-10*x**2*exp(5)-15*x**3-5)*exp(x**2*exp(5)+x**3)+5*exp(1)-10* x)*ln(x)+5*exp(x**2*exp(5)+x**3)-5*exp(1)+5*x)/(x**2*exp(x**2*exp(5)+x**3) **2+(-2*x**2*exp(1)+2*x**3)*exp(x**2*exp(5)+x**3)+x**2*exp(1)**2-2*x**3*ex p(1)+x**4),x)
Time = 0.26 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {-5 e+5 e^{e^5 x^2+x^3}+5 x+\left (5 e-10 x+e^{e^5 x^2+x^3} \left (-5-10 e^5 x^2-15 x^3\right )\right ) \log (x)}{e^2 x^2+e^{2 e^5 x^2+2 x^3} x^2-2 e x^3+x^4+e^{e^5 x^2+x^3} \left (-2 e x^2+2 x^3\right )} \, dx=\frac {5 \, \log \left (x\right )}{x^{2} - x e + x e^{\left (x^{3} + x^{2} e^{5}\right )}} \]
integrate((((-10*x^2*exp(5)-15*x^3-5)*exp(x^2*exp(5)+x^3)+5*exp(1)-10*x)*l og(x)+5*exp(x^2*exp(5)+x^3)-5*exp(1)+5*x)/(x^2*exp(x^2*exp(5)+x^3)^2+(-2*x ^2*exp(1)+2*x^3)*exp(x^2*exp(5)+x^3)+x^2*exp(1)^2-2*x^3*exp(1)+x^4),x, alg orithm=\
Leaf count of result is larger than twice the leaf count of optimal. 172 vs. \(2 (24) = 48\).
Time = 0.36 (sec) , antiderivative size = 172, normalized size of antiderivative = 6.88 \[ \int \frac {-5 e+5 e^{e^5 x^2+x^3}+5 x+\left (5 e-10 x+e^{e^5 x^2+x^3} \left (-5-10 e^5 x^2-15 x^3\right )\right ) \log (x)}{e^2 x^2+e^{2 e^5 x^2+2 x^3} x^2-2 e x^3+x^4+e^{e^5 x^2+x^3} \left (-2 e x^2+2 x^3\right )} \, dx=\frac {5 \, {\left (3 \, x^{3} \log \left (x\right ) + 2 \, x^{2} e^{5} \log \left (x\right ) - 3 \, x^{2} e \log \left (x\right ) - 2 \, x e^{6} \log \left (x\right ) - \log \left (x\right )\right )}}{3 \, x^{5} + 2 \, x^{4} e^{5} - 6 \, x^{4} e + 3 \, x^{4} e^{\left (x^{3} + x^{2} e^{5}\right )} - 4 \, x^{3} e^{6} + 3 \, x^{3} e^{2} + 2 \, x^{3} e^{\left (x^{3} + x^{2} e^{5} + 5\right )} - 3 \, x^{3} e^{\left (x^{3} + x^{2} e^{5} + 1\right )} + 2 \, x^{2} e^{7} - 2 \, x^{2} e^{\left (x^{3} + x^{2} e^{5} + 6\right )} - x^{2} + x e - x e^{\left (x^{3} + x^{2} e^{5}\right )}} \]
integrate((((-10*x^2*exp(5)-15*x^3-5)*exp(x^2*exp(5)+x^3)+5*exp(1)-10*x)*l og(x)+5*exp(x^2*exp(5)+x^3)-5*exp(1)+5*x)/(x^2*exp(x^2*exp(5)+x^3)^2+(-2*x ^2*exp(1)+2*x^3)*exp(x^2*exp(5)+x^3)+x^2*exp(1)^2-2*x^3*exp(1)+x^4),x, alg orithm=\
5*(3*x^3*log(x) + 2*x^2*e^5*log(x) - 3*x^2*e*log(x) - 2*x*e^6*log(x) - log (x))/(3*x^5 + 2*x^4*e^5 - 6*x^4*e + 3*x^4*e^(x^3 + x^2*e^5) - 4*x^3*e^6 + 3*x^3*e^2 + 2*x^3*e^(x^3 + x^2*e^5 + 5) - 3*x^3*e^(x^3 + x^2*e^5 + 1) + 2* x^2*e^7 - 2*x^2*e^(x^3 + x^2*e^5 + 6) - x^2 + x*e - x*e^(x^3 + x^2*e^5))
Time = 10.69 (sec) , antiderivative size = 90, normalized size of antiderivative = 3.60 \[ \int \frac {-5 e+5 e^{e^5 x^2+x^3}+5 x+\left (5 e-10 x+e^{e^5 x^2+x^3} \left (-5-10 e^5 x^2-15 x^3\right )\right ) \log (x)}{e^2 x^2+e^{2 e^5 x^2+2 x^3} x^2-2 e x^3+x^4+e^{e^5 x^2+x^3} \left (-2 e x^2+2 x^3\right )} \, dx=\frac {5\,x\,\ln \left (x\right )-15\,x^4\,\ln \left (x\right )+x^3\,\left (15\,\mathrm {e}\,\ln \left (x\right )-10\,{\mathrm {e}}^5\,\ln \left (x\right )\right )+10\,x^2\,{\mathrm {e}}^6\,\ln \left (x\right )}{x^2\,\left (x+{\mathrm {e}}^{x^3+{\mathrm {e}}^5\,x^2}-\mathrm {e}\right )\,\left (2\,x\,{\mathrm {e}}^6+3\,x^2\,\mathrm {e}-2\,x^2\,{\mathrm {e}}^5-3\,x^3+1\right )} \]
int((5*x + 5*exp(x^2*exp(5) + x^3) - 5*exp(1) - log(x)*(10*x - 5*exp(1) + exp(x^2*exp(5) + x^3)*(10*x^2*exp(5) + 15*x^3 + 5)))/(x^2*exp(2) - 2*x^3*e xp(1) - exp(x^2*exp(5) + x^3)*(2*x^2*exp(1) - 2*x^3) + x^2*exp(2*x^2*exp(5 ) + 2*x^3) + x^4),x)