Integrand size = 125, antiderivative size = 25 \[ \int \frac {-75 x^3-75 x^4+\left (300 e^5 x^3-300 x^4-300 x^3 \log (x)\right ) \log \left (-e^5+x+\log (x)\right )}{-9 e^5+9 x+9 \log (x)+\left (-30 e^5 x^4+30 x^5+30 x^4 \log (x)\right ) \log \left (-e^5+x+\log (x)\right )+\left (-25 e^5 x^8+25 x^9+25 x^8 \log (x)\right ) \log ^2\left (-e^5+x+\log (x)\right )} \, dx=3 e+\frac {15}{3+5 x^4 \log \left (-e^5+x+\log (x)\right )} \] Output:
3*exp(1)+15/(5*ln(ln(x)-exp(5)+x)*x^4+3)
Time = 0.03 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {-75 x^3-75 x^4+\left (300 e^5 x^3-300 x^4-300 x^3 \log (x)\right ) \log \left (-e^5+x+\log (x)\right )}{-9 e^5+9 x+9 \log (x)+\left (-30 e^5 x^4+30 x^5+30 x^4 \log (x)\right ) \log \left (-e^5+x+\log (x)\right )+\left (-25 e^5 x^8+25 x^9+25 x^8 \log (x)\right ) \log ^2\left (-e^5+x+\log (x)\right )} \, dx=\frac {75}{15+25 x^4 \log \left (-e^5+x+\log (x)\right )} \] Input:
Integrate[(-75*x^3 - 75*x^4 + (300*E^5*x^3 - 300*x^4 - 300*x^3*Log[x])*Log [-E^5 + x + Log[x]])/(-9*E^5 + 9*x + 9*Log[x] + (-30*E^5*x^4 + 30*x^5 + 30 *x^4*Log[x])*Log[-E^5 + x + Log[x]] + (-25*E^5*x^8 + 25*x^9 + 25*x^8*Log[x ])*Log[-E^5 + x + Log[x]]^2),x]
Output:
75/(15 + 25*x^4*Log[-E^5 + x + Log[x]])
Time = 0.62 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {7239, 27, 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.
\(\displaystyle \int \frac {-75 x^4-75 x^3+\left (-300 x^4+300 e^5 x^3-300 x^3 \log (x)\right ) \log \left (x+\log (x)-e^5\right )}{\left (25 x^9-25 e^5 x^8+25 x^8 \log (x)\right ) \log ^2\left (x+\log (x)-e^5\right )+\left (30 x^5-30 e^5 x^4+30 x^4 \log (x)\right ) \log \left (x+\log (x)-e^5\right )+9 x+9 \log (x)-9 e^5} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {75 x^3 \left (x+4 \left (x+\log (x)-e^5\right ) \log \left (x+\log (x)-e^5\right )+1\right )}{\left (-x-\log (x)+e^5\right ) \left (5 x^4 \log \left (x+\log (x)-e^5\right )+3\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 75 \int \frac {x^3 \left (x-4 \left (-x-\log (x)+e^5\right ) \log \left (x+\log (x)-e^5\right )+1\right )}{\left (-x-\log (x)+e^5\right ) \left (5 \log \left (x+\log (x)-e^5\right ) x^4+3\right )^2}dx\) |
\(\Big \downarrow \) 7237 |
\(\displaystyle \frac {15}{5 x^4 \log \left (x+\log (x)-e^5\right )+3}\) |
Input:
Int[(-75*x^3 - 75*x^4 + (300*E^5*x^3 - 300*x^4 - 300*x^3*Log[x])*Log[-E^5 + x + Log[x]])/(-9*E^5 + 9*x + 9*Log[x] + (-30*E^5*x^4 + 30*x^5 + 30*x^4*L og[x])*Log[-E^5 + x + Log[x]] + (-25*E^5*x^8 + 25*x^9 + 25*x^8*Log[x])*Log [-E^5 + x + Log[x]]^2),x]
Output:
15/(3 + 5*x^4*Log[-E^5 + x + Log[x]])
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 = 14.37 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84
method | result | size |
default | \(\frac {15}{5 \ln \left (\ln \left (x \right )-{\mathrm e}^{5}+x \right ) x^{4}+3}\) | \(21\) |
risch | \(\frac {15}{5 \ln \left (\ln \left (x \right )-{\mathrm e}^{5}+x \right ) x^{4}+3}\) | \(21\) |
parallelrisch | \(\frac {15}{5 \ln \left (\ln \left (x \right )-{\mathrm e}^{5}+x \right ) x^{4}+3}\) | \(21\) |
Input:
int(((-300*x^3*ln(x)+300*x^3*exp(5)-300*x^4)*ln(ln(x)-exp(5)+x)-75*x^4-75* x^3)/((25*x^8*ln(x)-25*x^8*exp(5)+25*x^9)*ln(ln(x)-exp(5)+x)^2+(30*x^4*ln( x)-30*x^4*exp(5)+30*x^5)*ln(ln(x)-exp(5)+x)+9*ln(x)-9*exp(5)+9*x),x,method =_RETURNVERBOSE)
Output:
15/(5*ln(ln(x)-exp(5)+x)*x^4+3)
Time = 0.07 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {-75 x^3-75 x^4+\left (300 e^5 x^3-300 x^4-300 x^3 \log (x)\right ) \log \left (-e^5+x+\log (x)\right )}{-9 e^5+9 x+9 \log (x)+\left (-30 e^5 x^4+30 x^5+30 x^4 \log (x)\right ) \log \left (-e^5+x+\log (x)\right )+\left (-25 e^5 x^8+25 x^9+25 x^8 \log (x)\right ) \log ^2\left (-e^5+x+\log (x)\right )} \, dx=\frac {15}{5 \, x^{4} \log \left (x - e^{5} + \log \left (x\right )\right ) + 3} \] Input:
integrate(((-300*x^3*log(x)+300*x^3*exp(5)-300*x^4)*log(log(x)-exp(5)+x)-7 5*x^4-75*x^3)/((25*x^8*log(x)-25*x^8*exp(5)+25*x^9)*log(log(x)-exp(5)+x)^2 +(30*x^4*log(x)-30*x^4*exp(5)+30*x^5)*log(log(x)-exp(5)+x)+9*log(x)-9*exp( 5)+9*x),x, algorithm="fricas")
Output:
15/(5*x^4*log(x - e^5 + log(x)) + 3)
Time = 0.14 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.68 \[ \int \frac {-75 x^3-75 x^4+\left (300 e^5 x^3-300 x^4-300 x^3 \log (x)\right ) \log \left (-e^5+x+\log (x)\right )}{-9 e^5+9 x+9 \log (x)+\left (-30 e^5 x^4+30 x^5+30 x^4 \log (x)\right ) \log \left (-e^5+x+\log (x)\right )+\left (-25 e^5 x^8+25 x^9+25 x^8 \log (x)\right ) \log ^2\left (-e^5+x+\log (x)\right )} \, dx=\frac {15}{5 x^{4} \log {\left (x + \log {\left (x \right )} - e^{5} \right )} + 3} \] Input:
integrate(((-300*x**3*ln(x)+300*x**3*exp(5)-300*x**4)*ln(ln(x)-exp(5)+x)-7 5*x**4-75*x**3)/((25*x**8*ln(x)-25*x**8*exp(5)+25*x**9)*ln(ln(x)-exp(5)+x) **2+(30*x**4*ln(x)-30*x**4*exp(5)+30*x**5)*ln(ln(x)-exp(5)+x)+9*ln(x)-9*ex p(5)+9*x),x)
Output:
15/(5*x**4*log(x + log(x) - exp(5)) + 3)
Time = 0.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {-75 x^3-75 x^4+\left (300 e^5 x^3-300 x^4-300 x^3 \log (x)\right ) \log \left (-e^5+x+\log (x)\right )}{-9 e^5+9 x+9 \log (x)+\left (-30 e^5 x^4+30 x^5+30 x^4 \log (x)\right ) \log \left (-e^5+x+\log (x)\right )+\left (-25 e^5 x^8+25 x^9+25 x^8 \log (x)\right ) \log ^2\left (-e^5+x+\log (x)\right )} \, dx=\frac {15}{5 \, x^{4} \log \left (x - e^{5} + \log \left (x\right )\right ) + 3} \] Input:
integrate(((-300*x^3*log(x)+300*x^3*exp(5)-300*x^4)*log(log(x)-exp(5)+x)-7 5*x^4-75*x^3)/((25*x^8*log(x)-25*x^8*exp(5)+25*x^9)*log(log(x)-exp(5)+x)^2 +(30*x^4*log(x)-30*x^4*exp(5)+30*x^5)*log(log(x)-exp(5)+x)+9*log(x)-9*exp( 5)+9*x),x, algorithm="maxima")
Output:
15/(5*x^4*log(x - e^5 + log(x)) + 3)
Time = 0.19 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {-75 x^3-75 x^4+\left (300 e^5 x^3-300 x^4-300 x^3 \log (x)\right ) \log \left (-e^5+x+\log (x)\right )}{-9 e^5+9 x+9 \log (x)+\left (-30 e^5 x^4+30 x^5+30 x^4 \log (x)\right ) \log \left (-e^5+x+\log (x)\right )+\left (-25 e^5 x^8+25 x^9+25 x^8 \log (x)\right ) \log ^2\left (-e^5+x+\log (x)\right )} \, dx=\frac {15}{5 \, x^{4} \log \left (x - e^{5} + \log \left (x\right )\right ) + 3} \] Input:
integrate(((-300*x^3*log(x)+300*x^3*exp(5)-300*x^4)*log(log(x)-exp(5)+x)-7 5*x^4-75*x^3)/((25*x^8*log(x)-25*x^8*exp(5)+25*x^9)*log(log(x)-exp(5)+x)^2 +(30*x^4*log(x)-30*x^4*exp(5)+30*x^5)*log(log(x)-exp(5)+x)+9*log(x)-9*exp( 5)+9*x),x, algorithm="giac")
Output:
15/(5*x^4*log(x - e^5 + log(x)) + 3)
Time = 0.64 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {-75 x^3-75 x^4+\left (300 e^5 x^3-300 x^4-300 x^3 \log (x)\right ) \log \left (-e^5+x+\log (x)\right )}{-9 e^5+9 x+9 \log (x)+\left (-30 e^5 x^4+30 x^5+30 x^4 \log (x)\right ) \log \left (-e^5+x+\log (x)\right )+\left (-25 e^5 x^8+25 x^9+25 x^8 \log (x)\right ) \log ^2\left (-e^5+x+\log (x)\right )} \, dx=\frac {15}{5\,x^4\,\ln \left (x-{\mathrm {e}}^5+\ln \left (x\right )\right )+3} \] Input:
int(-(log(x - exp(5) + log(x))*(300*x^3*log(x) - 300*x^3*exp(5) + 300*x^4) + 75*x^3 + 75*x^4)/(9*x - 9*exp(5) + 9*log(x) + log(x - exp(5) + log(x))* (30*x^4*log(x) - 30*x^4*exp(5) + 30*x^5) + log(x - exp(5) + log(x))^2*(25* x^8*log(x) - 25*x^8*exp(5) + 25*x^9)),x)
Output:
15/(5*x^4*log(x - exp(5) + log(x)) + 3)
\[ \int \frac {-75 x^3-75 x^4+\left (300 e^5 x^3-300 x^4-300 x^3 \log (x)\right ) \log \left (-e^5+x+\log (x)\right )}{-9 e^5+9 x+9 \log (x)+\left (-30 e^5 x^4+30 x^5+30 x^4 \log (x)\right ) \log \left (-e^5+x+\log (x)\right )+\left (-25 e^5 x^8+25 x^9+25 x^8 \log (x)\right ) \log ^2\left (-e^5+x+\log (x)\right )} \, dx =\text {Too large to display} \] Input:
int(((-300*x^3*log(x)+300*x^3*exp(5)-300*x^4)*log(log(x)-exp(5)+x)-75*x^4- 75*x^3)/((25*x^8*log(x)-25*x^8*exp(5)+25*x^9)*log(log(x)-exp(5)+x)^2+(30*x ^4*log(x)-30*x^4*exp(5)+30*x^5)*log(log(x)-exp(5)+x)+9*log(x)-9*exp(5)+9*x ),x)
Output:
75*( - int(x**4/(25*log(log(x) - e**5 + x)**2*log(x)*x**8 - 25*log(log(x) - e**5 + x)**2*e**5*x**8 + 25*log(log(x) - e**5 + x)**2*x**9 + 30*log(log( x) - e**5 + x)*log(x)*x**4 - 30*log(log(x) - e**5 + x)*e**5*x**4 + 30*log( log(x) - e**5 + x)*x**5 + 9*log(x) - 9*e**5 + 9*x),x) - int(x**3/(25*log(l og(x) - e**5 + x)**2*log(x)*x**8 - 25*log(log(x) - e**5 + x)**2*e**5*x**8 + 25*log(log(x) - e**5 + x)**2*x**9 + 30*log(log(x) - e**5 + x)*log(x)*x** 4 - 30*log(log(x) - e**5 + x)*e**5*x**4 + 30*log(log(x) - e**5 + x)*x**5 + 9*log(x) - 9*e**5 + 9*x),x) - 4*int((log(log(x) - e**5 + x)*x**4)/(25*log (log(x) - e**5 + x)**2*log(x)*x**8 - 25*log(log(x) - e**5 + x)**2*e**5*x** 8 + 25*log(log(x) - e**5 + x)**2*x**9 + 30*log(log(x) - e**5 + x)*log(x)*x **4 - 30*log(log(x) - e**5 + x)*e**5*x**4 + 30*log(log(x) - e**5 + x)*x**5 + 9*log(x) - 9*e**5 + 9*x),x) + 4*int((log(log(x) - e**5 + x)*x**3)/(25*l og(log(x) - e**5 + x)**2*log(x)*x**8 - 25*log(log(x) - e**5 + x)**2*e**5*x **8 + 25*log(log(x) - e**5 + x)**2*x**9 + 30*log(log(x) - e**5 + x)*log(x) *x**4 - 30*log(log(x) - e**5 + x)*e**5*x**4 + 30*log(log(x) - e**5 + x)*x* *5 + 9*log(x) - 9*e**5 + 9*x),x)*e**5 - 4*int((log(log(x) - e**5 + x)*log( x)*x**3)/(25*log(log(x) - e**5 + x)**2*log(x)*x**8 - 25*log(log(x) - e**5 + x)**2*e**5*x**8 + 25*log(log(x) - e**5 + x)**2*x**9 + 30*log(log(x) - e* *5 + x)*log(x)*x**4 - 30*log(log(x) - e**5 + x)*e**5*x**4 + 30*log(log(x) - e**5 + x)*x**5 + 9*log(x) - 9*e**5 + 9*x),x))