Integrand size = 208, antiderivative size = 21 \[ \int \frac {-20-10 x+e^x \left (-15 x-5 x^2\right )-5 e^x x \log \left (20 x^2\right )}{27 x+9 x^2+e^x \left (-18 x-6 x^2\right )+e^{2 x} \left (3 x+x^2\right )+\left (9 x-6 e^x x+e^{2 x} x\right ) \log \left (20 x^2\right )+\left (-18 x-6 x^2+e^x \left (6 x+2 x^2\right )+\left (-6 x+2 e^x x\right ) \log \left (20 x^2\right )\right ) \log \left (9+6 x+x^2+(6+2 x) \log \left (20 x^2\right )+\log ^2\left (20 x^2\right )\right )+\left (3 x+x^2+x \log \left (20 x^2\right )\right ) \log ^2\left (9+6 x+x^2+(6+2 x) \log \left (20 x^2\right )+\log ^2\left (20 x^2\right )\right )} \, dx=\frac {5}{-3+e^x+\log \left (\left (3+x+\log \left (20 x^2\right )\right )^2\right )} \]
Time = 0.09 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {-20-10 x+e^x \left (-15 x-5 x^2\right )-5 e^x x \log \left (20 x^2\right )}{27 x+9 x^2+e^x \left (-18 x-6 x^2\right )+e^{2 x} \left (3 x+x^2\right )+\left (9 x-6 e^x x+e^{2 x} x\right ) \log \left (20 x^2\right )+\left (-18 x-6 x^2+e^x \left (6 x+2 x^2\right )+\left (-6 x+2 e^x x\right ) \log \left (20 x^2\right )\right ) \log \left (9+6 x+x^2+(6+2 x) \log \left (20 x^2\right )+\log ^2\left (20 x^2\right )\right )+\left (3 x+x^2+x \log \left (20 x^2\right )\right ) \log ^2\left (9+6 x+x^2+(6+2 x) \log \left (20 x^2\right )+\log ^2\left (20 x^2\right )\right )} \, dx=\frac {5}{-3+e^x+\log \left (\left (3+x+\log \left (20 x^2\right )\right )^2\right )} \]
Integrate[(-20 - 10*x + E^x*(-15*x - 5*x^2) - 5*E^x*x*Log[20*x^2])/(27*x + 9*x^2 + E^x*(-18*x - 6*x^2) + E^(2*x)*(3*x + x^2) + (9*x - 6*E^x*x + E^(2 *x)*x)*Log[20*x^2] + (-18*x - 6*x^2 + E^x*(6*x + 2*x^2) + (-6*x + 2*E^x*x) *Log[20*x^2])*Log[9 + 6*x + x^2 + (6 + 2*x)*Log[20*x^2] + Log[20*x^2]^2] + (3*x + x^2 + x*Log[20*x^2])*Log[9 + 6*x + x^2 + (6 + 2*x)*Log[20*x^2] + L og[20*x^2]^2]^2),x]
Time = 0.84 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {7239, 27, 25, 7237}
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.
| \(\displaystyle \int \frac {e^x \left (-5 x^2-15 x\right )-5 e^x x \log \left (20 x^2\right )-10 x-20}{9 x^2+e^x \left (-6 x^2-18 x\right )+e^{2 x} \left (x^2+3 x\right )+\left (x^2+x \log \left (20 x^2\right )+3 x\right ) \log ^2\left (x^2+\log ^2\left (20 x^2\right )+(2 x+6) \log \left (20 x^2\right )+6 x+9\right )+\left (-6 x^2+e^x \left (2 x^2+6 x\right )+\left (2 e^x x-6 x\right ) \log \left (20 x^2\right )-18 x\right ) \log \left (x^2+\log ^2\left (20 x^2\right )+(2 x+6) \log \left (20 x^2\right )+6 x+9\right )+\left (-6 e^x x+e^{2 x} x+9 x\right ) \log \left (20 x^2\right )+27 x} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {5 \left (-e^x x^2-e^x x \log \left (20 x^2\right )-\left (3 e^x+2\right ) x-4\right )}{x \left (\log \left (20 x^2\right )+x+3\right ) \left (-\log \left (\left (\log \left (20 x^2\right )+x+3\right )^2\right )-e^x+3\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 5 \int -\frac {e^x x^2+\left (2+3 e^x\right ) x+e^x \log \left (20 x^2\right ) x+4}{x \left (x+\log \left (20 x^2\right )+3\right ) \left (-\log \left (\left (x+\log \left (20 x^2\right )+3\right )^2\right )-e^x+3\right )^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -5 \int \frac {e^x x^2+\left (2+3 e^x\right ) x+e^x \log \left (20 x^2\right ) x+4}{x \left (x+\log \left (20 x^2\right )+3\right ) \left (-\log \left (\left (x+\log \left (20 x^2\right )+3\right )^2\right )-e^x+3\right )^2}dx\) |
| \(\Big \downarrow \) 7237 |
| \(\displaystyle -\frac {5}{-\log \left (\left (\log \left (20 x^2\right )+x+3\right )^2\right )-e^x+3}\) |
Int[(-20 - 10*x + E^x*(-15*x - 5*x^2) - 5*E^x*x*Log[20*x^2])/(27*x + 9*x^2 + E^x*(-18*x - 6*x^2) + E^(2*x)*(3*x + x^2) + (9*x - 6*E^x*x + E^(2*x)*x) *Log[20*x^2] + (-18*x - 6*x^2 + E^x*(6*x + 2*x^2) + (-6*x + 2*E^x*x)*Log[2 0*x^2])*Log[9 + 6*x + x^2 + (6 + 2*x)*Log[20*x^2] + Log[20*x^2]^2] + (3*x + x^2 + x*Log[20*x^2])*Log[9 + 6*x + x^2 + (6 + 2*x)*Log[20*x^2] + Log[20* x^2]^2]^2),x]
3.15.54.3.1 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]]
Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Si mp[q*(y^(m + 1)/(m + 1)), x] /; !FalseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 2.76 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.81
| method | result | size |
| parallelrisch | \(\frac {5}{{\mathrm e}^{x}+\ln \left (\ln \left (20 x^{2}\right )^{2}+\left (2 x +6\right ) \ln \left (20 x^{2}\right )+x^{2}+6 x +9\right )-3}\) | \(38\) |
| risch | \(\frac {10 i}{2 \pi {\operatorname {csgn}\left (i \left (\pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-2 \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+\pi \operatorname {csgn}\left (i x^{2}\right )^{3}+2 i \ln \left (20\right )+2 i x +4 i \ln \left (x \right )+6 i\right )^{2}\right )}^{2}+\pi \operatorname {csgn}\left (-i \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )+2 i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}-i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+2 \ln \left (20\right )+2 x +4 \ln \left (x \right )+6\right )^{2} \operatorname {csgn}\left (i \left (\pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-2 \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+\pi \operatorname {csgn}\left (i x^{2}\right )^{3}+2 i \ln \left (20\right )+2 i x +4 i \ln \left (x \right )+6 i\right )^{2}\right )+2 \pi \,\operatorname {csgn}\left (-i \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )+2 i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}-i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+2 \ln \left (20\right )+2 x +4 \ln \left (x \right )+6\right ) {\operatorname {csgn}\left (i \left (\pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-2 \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+\pi \operatorname {csgn}\left (i x^{2}\right )^{3}+2 i \ln \left (20\right )+2 i x +4 i \ln \left (x \right )+6 i\right )^{2}\right )}^{2}-\pi {\operatorname {csgn}\left (i \left (\pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-2 \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+\pi \operatorname {csgn}\left (i x^{2}\right )^{3}+2 i \ln \left (20\right )+2 i x +4 i \ln \left (x \right )+6 i\right )^{2}\right )}^{3}-2 \pi -4 i \ln \left (2\right )+2 i {\mathrm e}^{x}+4 i \ln \left (\pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-2 \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+\pi \operatorname {csgn}\left (i x^{2}\right )^{3}+2 i \left (2 \ln \left (2\right )+\ln \left (5\right )\right )+2 i x +4 i \ln \left (x \right )+6 i\right )-6 i}\) | \(505\) |
int((-5*x*exp(x)*ln(20*x^2)+(-5*x^2-15*x)*exp(x)-10*x-20)/((x*ln(20*x^2)+x ^2+3*x)*ln(ln(20*x^2)^2+(2*x+6)*ln(20*x^2)+x^2+6*x+9)^2+((2*exp(x)*x-6*x)* ln(20*x^2)+(2*x^2+6*x)*exp(x)-6*x^2-18*x)*ln(ln(20*x^2)^2+(2*x+6)*ln(20*x^ 2)+x^2+6*x+9)+(x*exp(x)^2-6*exp(x)*x+9*x)*ln(20*x^2)+(x^2+3*x)*exp(x)^2+(- 6*x^2-18*x)*exp(x)+9*x^2+27*x),x,method=_RETURNVERBOSE)
Time = 0.24 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.71 \[ \int \frac {-20-10 x+e^x \left (-15 x-5 x^2\right )-5 e^x x \log \left (20 x^2\right )}{27 x+9 x^2+e^x \left (-18 x-6 x^2\right )+e^{2 x} \left (3 x+x^2\right )+\left (9 x-6 e^x x+e^{2 x} x\right ) \log \left (20 x^2\right )+\left (-18 x-6 x^2+e^x \left (6 x+2 x^2\right )+\left (-6 x+2 e^x x\right ) \log \left (20 x^2\right )\right ) \log \left (9+6 x+x^2+(6+2 x) \log \left (20 x^2\right )+\log ^2\left (20 x^2\right )\right )+\left (3 x+x^2+x \log \left (20 x^2\right )\right ) \log ^2\left (9+6 x+x^2+(6+2 x) \log \left (20 x^2\right )+\log ^2\left (20 x^2\right )\right )} \, dx=\frac {5}{e^{x} + \log \left (x^{2} + 2 \, {\left (x + 3\right )} \log \left (20 \, x^{2}\right ) + \log \left (20 \, x^{2}\right )^{2} + 6 \, x + 9\right ) - 3} \]
integrate((-5*x*exp(x)*log(20*x^2)+(-5*x^2-15*x)*exp(x)-10*x-20)/((x*log(2 0*x^2)+x^2+3*x)*log(log(20*x^2)^2+(2*x+6)*log(20*x^2)+x^2+6*x+9)^2+((2*exp (x)*x-6*x)*log(20*x^2)+(2*x^2+6*x)*exp(x)-6*x^2-18*x)*log(log(20*x^2)^2+(2 *x+6)*log(20*x^2)+x^2+6*x+9)+(x*exp(x)^2-6*exp(x)*x+9*x)*log(20*x^2)+(x^2+ 3*x)*exp(x)^2+(-6*x^2-18*x)*exp(x)+9*x^2+27*x),x, algorithm=\
Time = 0.27 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.71 \[ \int \frac {-20-10 x+e^x \left (-15 x-5 x^2\right )-5 e^x x \log \left (20 x^2\right )}{27 x+9 x^2+e^x \left (-18 x-6 x^2\right )+e^{2 x} \left (3 x+x^2\right )+\left (9 x-6 e^x x+e^{2 x} x\right ) \log \left (20 x^2\right )+\left (-18 x-6 x^2+e^x \left (6 x+2 x^2\right )+\left (-6 x+2 e^x x\right ) \log \left (20 x^2\right )\right ) \log \left (9+6 x+x^2+(6+2 x) \log \left (20 x^2\right )+\log ^2\left (20 x^2\right )\right )+\left (3 x+x^2+x \log \left (20 x^2\right )\right ) \log ^2\left (9+6 x+x^2+(6+2 x) \log \left (20 x^2\right )+\log ^2\left (20 x^2\right )\right )} \, dx=\frac {5}{e^{x} + \log {\left (x^{2} + 6 x + \left (2 x + 6\right ) \log {\left (20 x^{2} \right )} + \log {\left (20 x^{2} \right )}^{2} + 9 \right )} - 3} \]
integrate((-5*x*exp(x)*ln(20*x**2)+(-5*x**2-15*x)*exp(x)-10*x-20)/((x*ln(2 0*x**2)+x**2+3*x)*ln(ln(20*x**2)**2+(2*x+6)*ln(20*x**2)+x**2+6*x+9)**2+((2 *exp(x)*x-6*x)*ln(20*x**2)+(2*x**2+6*x)*exp(x)-6*x**2-18*x)*ln(ln(20*x**2) **2+(2*x+6)*ln(20*x**2)+x**2+6*x+9)+(x*exp(x)**2-6*exp(x)*x+9*x)*ln(20*x** 2)+(x**2+3*x)*exp(x)**2+(-6*x**2-18*x)*exp(x)+9*x**2+27*x),x)
Time = 0.38 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14 \[ \int \frac {-20-10 x+e^x \left (-15 x-5 x^2\right )-5 e^x x \log \left (20 x^2\right )}{27 x+9 x^2+e^x \left (-18 x-6 x^2\right )+e^{2 x} \left (3 x+x^2\right )+\left (9 x-6 e^x x+e^{2 x} x\right ) \log \left (20 x^2\right )+\left (-18 x-6 x^2+e^x \left (6 x+2 x^2\right )+\left (-6 x+2 e^x x\right ) \log \left (20 x^2\right )\right ) \log \left (9+6 x+x^2+(6+2 x) \log \left (20 x^2\right )+\log ^2\left (20 x^2\right )\right )+\left (3 x+x^2+x \log \left (20 x^2\right )\right ) \log ^2\left (9+6 x+x^2+(6+2 x) \log \left (20 x^2\right )+\log ^2\left (20 x^2\right )\right )} \, dx=\frac {5}{e^{x} + 2 \, \log \left (x + \log \left (5\right ) + 2 \, \log \left (2\right ) + 2 \, \log \left (x\right ) + 3\right ) - 3} \]
integrate((-5*x*exp(x)*log(20*x^2)+(-5*x^2-15*x)*exp(x)-10*x-20)/((x*log(2 0*x^2)+x^2+3*x)*log(log(20*x^2)^2+(2*x+6)*log(20*x^2)+x^2+6*x+9)^2+((2*exp (x)*x-6*x)*log(20*x^2)+(2*x^2+6*x)*exp(x)-6*x^2-18*x)*log(log(20*x^2)^2+(2 *x+6)*log(20*x^2)+x^2+6*x+9)+(x*exp(x)^2-6*exp(x)*x+9*x)*log(20*x^2)+(x^2+ 3*x)*exp(x)^2+(-6*x^2-18*x)*exp(x)+9*x^2+27*x),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (20) = 40\).
Time = 1.39 (sec) , antiderivative size = 42, normalized size of antiderivative = 2.00 \[ \int \frac {-20-10 x+e^x \left (-15 x-5 x^2\right )-5 e^x x \log \left (20 x^2\right )}{27 x+9 x^2+e^x \left (-18 x-6 x^2\right )+e^{2 x} \left (3 x+x^2\right )+\left (9 x-6 e^x x+e^{2 x} x\right ) \log \left (20 x^2\right )+\left (-18 x-6 x^2+e^x \left (6 x+2 x^2\right )+\left (-6 x+2 e^x x\right ) \log \left (20 x^2\right )\right ) \log \left (9+6 x+x^2+(6+2 x) \log \left (20 x^2\right )+\log ^2\left (20 x^2\right )\right )+\left (3 x+x^2+x \log \left (20 x^2\right )\right ) \log ^2\left (9+6 x+x^2+(6+2 x) \log \left (20 x^2\right )+\log ^2\left (20 x^2\right )\right )} \, dx=\frac {5}{e^{x} + \log \left (x^{2} + 2 \, x \log \left (20 \, x^{2}\right ) + \log \left (20 \, x^{2}\right )^{2} + 6 \, x + 6 \, \log \left (20 \, x^{2}\right ) + 9\right ) - 3} \]
integrate((-5*x*exp(x)*log(20*x^2)+(-5*x^2-15*x)*exp(x)-10*x-20)/((x*log(2 0*x^2)+x^2+3*x)*log(log(20*x^2)^2+(2*x+6)*log(20*x^2)+x^2+6*x+9)^2+((2*exp (x)*x-6*x)*log(20*x^2)+(2*x^2+6*x)*exp(x)-6*x^2-18*x)*log(log(20*x^2)^2+(2 *x+6)*log(20*x^2)+x^2+6*x+9)+(x*exp(x)^2-6*exp(x)*x+9*x)*log(20*x^2)+(x^2+ 3*x)*exp(x)^2+(-6*x^2-18*x)*exp(x)+9*x^2+27*x),x, algorithm=\
Time = 10.19 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.76 \[ \int \frac {-20-10 x+e^x \left (-15 x-5 x^2\right )-5 e^x x \log \left (20 x^2\right )}{27 x+9 x^2+e^x \left (-18 x-6 x^2\right )+e^{2 x} \left (3 x+x^2\right )+\left (9 x-6 e^x x+e^{2 x} x\right ) \log \left (20 x^2\right )+\left (-18 x-6 x^2+e^x \left (6 x+2 x^2\right )+\left (-6 x+2 e^x x\right ) \log \left (20 x^2\right )\right ) \log \left (9+6 x+x^2+(6+2 x) \log \left (20 x^2\right )+\log ^2\left (20 x^2\right )\right )+\left (3 x+x^2+x \log \left (20 x^2\right )\right ) \log ^2\left (9+6 x+x^2+(6+2 x) \log \left (20 x^2\right )+\log ^2\left (20 x^2\right )\right )} \, dx=\frac {5}{\ln \left (6\,x+{\ln \left (20\,x^2\right )}^2+\ln \left (20\,x^2\right )\,\left (2\,x+6\right )+x^2+9\right )+{\mathrm {e}}^x-3} \]
int(-(10*x + exp(x)*(15*x + 5*x^2) + 5*x*exp(x)*log(20*x^2) + 20)/(27*x + log(20*x^2)*(9*x + x*exp(2*x) - 6*x*exp(x)) - log(6*x + log(20*x^2)^2 + lo g(20*x^2)*(2*x + 6) + x^2 + 9)*(18*x - exp(x)*(6*x + 2*x^2) + 6*x^2 + log( 20*x^2)*(6*x - 2*x*exp(x))) + exp(2*x)*(3*x + x^2) - exp(x)*(18*x + 6*x^2) + 9*x^2 + log(6*x + log(20*x^2)^2 + log(20*x^2)*(2*x + 6) + x^2 + 9)^2*(3 *x + x*log(20*x^2) + x^2)),x)