Integrand size = 136, antiderivative size = 36 \[ \int \frac {450-141855 x+93650 x^2-15476 x^3+\left (-11250 x+7470 x^2-1240 x^3\right ) \log \left (\frac {e^{2 x}}{x}\right )+\left (-225 x+150 x^2-25 x^3\right ) \log ^2\left (\frac {e^{2 x}}{x}\right )}{562500 x-372000 x^2+61504 x^3+\left (45000 x-29880 x^2+4960 x^3\right ) \log \left (\frac {e^{2 x}}{x}\right )+\left (900 x-600 x^2+100 x^3\right ) \log ^2\left (\frac {e^{2 x}}{x}\right )} \, dx=\frac {1}{4} \left (-x+\frac {2}{25+\frac {x}{5 (3-x)}+\log \left (\frac {e^{2 x}}{x}\right )}\right ) \]
Time = 0.05 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.44 \[ \int \frac {450-141855 x+93650 x^2-15476 x^3+\left (-11250 x+7470 x^2-1240 x^3\right ) \log \left (\frac {e^{2 x}}{x}\right )+\left (-225 x+150 x^2-25 x^3\right ) \log ^2\left (\frac {e^{2 x}}{x}\right )}{562500 x-372000 x^2+61504 x^3+\left (45000 x-29880 x^2+4960 x^3\right ) \log \left (\frac {e^{2 x}}{x}\right )+\left (900 x-600 x^2+100 x^3\right ) \log ^2\left (\frac {e^{2 x}}{x}\right )} \, dx=-\frac {30-385 x+124 x^2+5 (-3+x) x \log \left (\frac {e^{2 x}}{x}\right )}{4 \left (-375+124 x+5 (-3+x) \log \left (\frac {e^{2 x}}{x}\right )\right )} \]
Integrate[(450 - 141855*x + 93650*x^2 - 15476*x^3 + (-11250*x + 7470*x^2 - 1240*x^3)*Log[E^(2*x)/x] + (-225*x + 150*x^2 - 25*x^3)*Log[E^(2*x)/x]^2)/ (562500*x - 372000*x^2 + 61504*x^3 + (45000*x - 29880*x^2 + 4960*x^3)*Log[ E^(2*x)/x] + (900*x - 600*x^2 + 100*x^3)*Log[E^(2*x)/x]^2),x]
-1/4*(30 - 385*x + 124*x^2 + 5*(-3 + x)*x*Log[E^(2*x)/x])/(-375 + 124*x + 5*(-3 + x)*Log[E^(2*x)/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.
| \(\displaystyle \int \frac {-15476 x^3+93650 x^2+\left (-25 x^3+150 x^2-225 x\right ) \log ^2\left (\frac {e^{2 x}}{x}\right )+\left (-1240 x^3+7470 x^2-11250 x\right ) \log \left (\frac {e^{2 x}}{x}\right )-141855 x+450}{61504 x^3-372000 x^2+\left (100 x^3-600 x^2+900 x\right ) \log ^2\left (\frac {e^{2 x}}{x}\right )+\left (4960 x^3-29880 x^2+45000 x\right ) \log \left (\frac {e^{2 x}}{x}\right )+562500 x} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-15476 x^3+93650 x^2+\left (-25 x^3+150 x^2-225 x\right ) \log ^2\left (\frac {e^{2 x}}{x}\right )+\left (-1240 x^3+7470 x^2-11250 x\right ) \log \left (\frac {e^{2 x}}{x}\right )-141855 x+450}{4 x \left (-124 x-5 x \log \left (\frac {e^{2 x}}{x}\right )+15 \log \left (\frac {e^{2 x}}{x}\right )+375\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \int \frac {-15476 x^3+93650 x^2-141855 x-25 \left (x^3-6 x^2+9 x\right ) \log ^2\left (\frac {e^{2 x}}{x}\right )-10 \left (124 x^3-747 x^2+1125 x\right ) \log \left (\frac {e^{2 x}}{x}\right )+450}{x \left (-5 \log \left (\frac {e^{2 x}}{x}\right ) x-124 x+15 \log \left (\frac {e^{2 x}}{x}\right )+375\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{4} \int \left (-\frac {10 \left (10 x^3-65 x^2+123 x-45\right )}{x \left (5 \log \left (\frac {e^{2 x}}{x}\right ) x+124 x-15 \log \left (\frac {e^{2 x}}{x}\right )-375\right )^2}-1\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{4} \left (-100 \int \frac {x^2}{\left (5 \log \left (\frac {e^{2 x}}{x}\right ) x+124 x-15 \log \left (\frac {e^{2 x}}{x}\right )-375\right )^2}dx-1230 \int \frac {1}{\left (5 \log \left (\frac {e^{2 x}}{x}\right ) x+124 x-15 \log \left (\frac {e^{2 x}}{x}\right )-375\right )^2}dx+450 \int \frac {1}{x \left (5 \log \left (\frac {e^{2 x}}{x}\right ) x+124 x-15 \log \left (\frac {e^{2 x}}{x}\right )-375\right )^2}dx+650 \int \frac {x}{\left (5 \log \left (\frac {e^{2 x}}{x}\right ) x+124 x-15 \log \left (\frac {e^{2 x}}{x}\right )-375\right )^2}dx-x\right )\) |
Int[(450 - 141855*x + 93650*x^2 - 15476*x^3 + (-11250*x + 7470*x^2 - 1240* x^3)*Log[E^(2*x)/x] + (-225*x + 150*x^2 - 25*x^3)*Log[E^(2*x)/x]^2)/(56250 0*x - 372000*x^2 + 61504*x^3 + (45000*x - 29880*x^2 + 4960*x^3)*Log[E^(2*x )/x] + (900*x - 600*x^2 + 100*x^3)*Log[E^(2*x)/x]^2),x]
3.19.95.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(67\) vs. \(2(28)=56\).
Time = 0.74 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.89
| method | result | size |
| parallelrisch | \(\frac {5475-25 \ln \left (\frac {{\mathrm e}^{2 x}}{x}\right ) x^{2}-620 x^{2}+65 x +225 \ln \left (\frac {{\mathrm e}^{2 x}}{x}\right )}{100 \ln \left (\frac {{\mathrm e}^{2 x}}{x}\right ) x -300 \ln \left (\frac {{\mathrm e}^{2 x}}{x}\right )+2480 x -7500}\) | \(68\) |
| default | \(-\frac {13 x +\left (-124-5 \ln \left (\frac {{\mathrm e}^{2 x}}{x}\right )-5 \ln \left (x \right )+10 x \right ) x^{2}-10 x^{3}+5 x^{2} \ln \left (x \right )+1095+45 \ln \left (\frac {{\mathrm e}^{2 x}}{x}\right )}{4 \left (5 x \ln \left (x \right )-10 x^{2}-5 x \left (\ln \left (\frac {{\mathrm e}^{2 x}}{x}\right )-2 \ln \left ({\mathrm e}^{x}\right )+\ln \left (x \right )\right )-10 x \left (\ln \left ({\mathrm e}^{x}\right )-x \right )-124 x +15 \ln \left (\frac {{\mathrm e}^{2 x}}{x}\right )+375\right )}\) | \(113\) |
| risch | \(-\frac {x}{4}+\frac {5 i \left (-3+x \right )}{5 \pi x \operatorname {csgn}\left (i {\mathrm e}^{2 x}\right )^{3}-10 \pi x \operatorname {csgn}\left (i {\mathrm e}^{2 x}\right )^{2} \operatorname {csgn}\left (i {\mathrm e}^{x}\right )+5 \pi x \,\operatorname {csgn}\left (i {\mathrm e}^{2 x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{x}\right )^{2}-5 \pi x \,\operatorname {csgn}\left (i {\mathrm e}^{2 x}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 x}}{x}\right )^{2}+5 \pi x \,\operatorname {csgn}\left (i {\mathrm e}^{2 x}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 x}}{x}\right ) \operatorname {csgn}\left (\frac {i}{x}\right )+5 \pi x \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 x}}{x}\right )^{3}-5 \pi x \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 x}}{x}\right )^{2} \operatorname {csgn}\left (\frac {i}{x}\right )-15 \pi \operatorname {csgn}\left (i {\mathrm e}^{2 x}\right )^{3}+30 \pi \operatorname {csgn}\left (i {\mathrm e}^{2 x}\right )^{2} \operatorname {csgn}\left (i {\mathrm e}^{x}\right )-15 \pi \,\operatorname {csgn}\left (i {\mathrm e}^{2 x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{x}\right )^{2}+15 \pi \,\operatorname {csgn}\left (i {\mathrm e}^{2 x}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 x}}{x}\right )^{2}-15 \pi \,\operatorname {csgn}\left (i {\mathrm e}^{2 x}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 x}}{x}\right ) \operatorname {csgn}\left (\frac {i}{x}\right )-15 \pi \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 x}}{x}\right )^{3}+15 \pi \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 x}}{x}\right )^{2} \operatorname {csgn}\left (\frac {i}{x}\right )+20 i x \ln \left ({\mathrm e}^{x}\right )-60 i \ln \left ({\mathrm e}^{x}\right )+30 i \ln \left (x \right )+248 i x -10 i x \ln \left (x \right )-750 i}\) | \(337\) |
int(((-25*x^3+150*x^2-225*x)*ln(exp(x)^2/x)^2+(-1240*x^3+7470*x^2-11250*x) *ln(exp(x)^2/x)-15476*x^3+93650*x^2-141855*x+450)/((100*x^3-600*x^2+900*x) *ln(exp(x)^2/x)^2+(4960*x^3-29880*x^2+45000*x)*ln(exp(x)^2/x)+61504*x^3-37 2000*x^2+562500*x),x,method=_RETURNVERBOSE)
1/20*(5475-25*ln(exp(x)^2/x)*x^2-620*x^2+65*x+225*ln(exp(x)^2/x))/(5*ln(ex p(x)^2/x)*x-15*ln(exp(x)^2/x)+124*x-375)
Time = 0.25 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.42 \[ \int \frac {450-141855 x+93650 x^2-15476 x^3+\left (-11250 x+7470 x^2-1240 x^3\right ) \log \left (\frac {e^{2 x}}{x}\right )+\left (-225 x+150 x^2-25 x^3\right ) \log ^2\left (\frac {e^{2 x}}{x}\right )}{562500 x-372000 x^2+61504 x^3+\left (45000 x-29880 x^2+4960 x^3\right ) \log \left (\frac {e^{2 x}}{x}\right )+\left (900 x-600 x^2+100 x^3\right ) \log ^2\left (\frac {e^{2 x}}{x}\right )} \, dx=-\frac {124 \, x^{2} + 5 \, {\left (x^{2} - 3 \, x\right )} \log \left (\frac {e^{\left (2 \, x\right )}}{x}\right ) - 385 \, x + 30}{4 \, {\left (5 \, {\left (x - 3\right )} \log \left (\frac {e^{\left (2 \, x\right )}}{x}\right ) + 124 \, x - 375\right )}} \]
integrate(((-25*x^3+150*x^2-225*x)*log(exp(x)^2/x)^2+(-1240*x^3+7470*x^2-1 1250*x)*log(exp(x)^2/x)-15476*x^3+93650*x^2-141855*x+450)/((100*x^3-600*x^ 2+900*x)*log(exp(x)^2/x)^2+(4960*x^3-29880*x^2+45000*x)*log(exp(x)^2/x)+61 504*x^3-372000*x^2+562500*x),x, algorithm=\
-1/4*(124*x^2 + 5*(x^2 - 3*x)*log(e^(2*x)/x) - 385*x + 30)/(5*(x - 3)*log( e^(2*x)/x) + 124*x - 375)
Time = 0.14 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.72 \[ \int \frac {450-141855 x+93650 x^2-15476 x^3+\left (-11250 x+7470 x^2-1240 x^3\right ) \log \left (\frac {e^{2 x}}{x}\right )+\left (-225 x+150 x^2-25 x^3\right ) \log ^2\left (\frac {e^{2 x}}{x}\right )}{562500 x-372000 x^2+61504 x^3+\left (45000 x-29880 x^2+4960 x^3\right ) \log \left (\frac {e^{2 x}}{x}\right )+\left (900 x-600 x^2+100 x^3\right ) \log ^2\left (\frac {e^{2 x}}{x}\right )} \, dx=- \frac {x}{4} + \frac {5 x - 15}{248 x + \left (10 x - 30\right ) \log {\left (\frac {e^{2 x}}{x} \right )} - 750} \]
integrate(((-25*x**3+150*x**2-225*x)*ln(exp(x)**2/x)**2+(-1240*x**3+7470*x **2-11250*x)*ln(exp(x)**2/x)-15476*x**3+93650*x**2-141855*x+450)/((100*x** 3-600*x**2+900*x)*ln(exp(x)**2/x)**2+(4960*x**3-29880*x**2+45000*x)*ln(exp (x)**2/x)+61504*x**3-372000*x**2+562500*x),x)
Time = 0.24 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.31 \[ \int \frac {450-141855 x+93650 x^2-15476 x^3+\left (-11250 x+7470 x^2-1240 x^3\right ) \log \left (\frac {e^{2 x}}{x}\right )+\left (-225 x+150 x^2-25 x^3\right ) \log ^2\left (\frac {e^{2 x}}{x}\right )}{562500 x-372000 x^2+61504 x^3+\left (45000 x-29880 x^2+4960 x^3\right ) \log \left (\frac {e^{2 x}}{x}\right )+\left (900 x-600 x^2+100 x^3\right ) \log ^2\left (\frac {e^{2 x}}{x}\right )} \, dx=-\frac {10 \, x^{3} + 94 \, x^{2} - 5 \, {\left (x^{2} - 3 \, x\right )} \log \left (x\right ) - 385 \, x + 30}{4 \, {\left (10 \, x^{2} - 5 \, {\left (x - 3\right )} \log \left (x\right ) + 94 \, x - 375\right )}} \]
integrate(((-25*x^3+150*x^2-225*x)*log(exp(x)^2/x)^2+(-1240*x^3+7470*x^2-1 1250*x)*log(exp(x)^2/x)-15476*x^3+93650*x^2-141855*x+450)/((100*x^3-600*x^ 2+900*x)*log(exp(x)^2/x)^2+(4960*x^3-29880*x^2+45000*x)*log(exp(x)^2/x)+61 504*x^3-372000*x^2+562500*x),x, algorithm=\
-1/4*(10*x^3 + 94*x^2 - 5*(x^2 - 3*x)*log(x) - 385*x + 30)/(10*x^2 - 5*(x - 3)*log(x) + 94*x - 375)
Time = 0.29 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.83 \[ \int \frac {450-141855 x+93650 x^2-15476 x^3+\left (-11250 x+7470 x^2-1240 x^3\right ) \log \left (\frac {e^{2 x}}{x}\right )+\left (-225 x+150 x^2-25 x^3\right ) \log ^2\left (\frac {e^{2 x}}{x}\right )}{562500 x-372000 x^2+61504 x^3+\left (45000 x-29880 x^2+4960 x^3\right ) \log \left (\frac {e^{2 x}}{x}\right )+\left (900 x-600 x^2+100 x^3\right ) \log ^2\left (\frac {e^{2 x}}{x}\right )} \, dx=-\frac {1}{4} \, x + \frac {5 \, {\left (x - 3\right )}}{2 \, {\left (10 \, x^{2} - 5 \, x \log \left (x\right ) + 94 \, x + 15 \, \log \left (x\right ) - 375\right )}} \]
integrate(((-25*x^3+150*x^2-225*x)*log(exp(x)^2/x)^2+(-1240*x^3+7470*x^2-1 1250*x)*log(exp(x)^2/x)-15476*x^3+93650*x^2-141855*x+450)/((100*x^3-600*x^ 2+900*x)*log(exp(x)^2/x)^2+(4960*x^3-29880*x^2+45000*x)*log(exp(x)^2/x)+61 504*x^3-372000*x^2+562500*x),x, algorithm=\
Timed out. \[ \int \frac {450-141855 x+93650 x^2-15476 x^3+\left (-11250 x+7470 x^2-1240 x^3\right ) \log \left (\frac {e^{2 x}}{x}\right )+\left (-225 x+150 x^2-25 x^3\right ) \log ^2\left (\frac {e^{2 x}}{x}\right )}{562500 x-372000 x^2+61504 x^3+\left (45000 x-29880 x^2+4960 x^3\right ) \log \left (\frac {e^{2 x}}{x}\right )+\left (900 x-600 x^2+100 x^3\right ) \log ^2\left (\frac {e^{2 x}}{x}\right )} \, dx=\int -\frac {141855\,x+{\ln \left (\frac {{\mathrm {e}}^{2\,x}}{x}\right )}^2\,\left (25\,x^3-150\,x^2+225\,x\right )-93650\,x^2+15476\,x^3+\ln \left (\frac {{\mathrm {e}}^{2\,x}}{x}\right )\,\left (1240\,x^3-7470\,x^2+11250\,x\right )-450}{562500\,x+{\ln \left (\frac {{\mathrm {e}}^{2\,x}}{x}\right )}^2\,\left (100\,x^3-600\,x^2+900\,x\right )-372000\,x^2+61504\,x^3+\ln \left (\frac {{\mathrm {e}}^{2\,x}}{x}\right )\,\left (4960\,x^3-29880\,x^2+45000\,x\right )} \,d x \]
int(-(141855*x + log(exp(2*x)/x)^2*(225*x - 150*x^2 + 25*x^3) - 93650*x^2 + 15476*x^3 + log(exp(2*x)/x)*(11250*x - 7470*x^2 + 1240*x^3) - 450)/(5625 00*x + log(exp(2*x)/x)^2*(900*x - 600*x^2 + 100*x^3) - 372000*x^2 + 61504* x^3 + log(exp(2*x)/x)*(45000*x - 29880*x^2 + 4960*x^3)),x)