Integrand size = 173, antiderivative size = 26 \[ \int \frac {\left (-36 x^3-24 x^4-4 x^5\right ) \log ^2(x)+e^{\frac {1}{\left (3 x^3+x^4\right ) \log (x)}} \left (3+x+(9+4 x) \log (x)+\left (9 x^3+6 x^4+x^5\right ) \log ^2(x)\right )}{e^{\frac {1}{\left (3 x^3+x^4\right ) \log (x)}} \left (-72 x^3-48 x^4-8 x^5\right ) \log ^2(x)+e^{\frac {2}{\left (3 x^3+x^4\right ) \log (x)}} \left (9 x^3+6 x^4+x^5\right ) \log ^2(x)+\left (144 x^3+96 x^4+16 x^5\right ) \log ^2(x)} \, dx=2-\frac {x}{4-e^{\frac {1}{x^3 (3+x) \log (x)}}} \]
Time = 0.17 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.81 \[ \int \frac {\left (-36 x^3-24 x^4-4 x^5\right ) \log ^2(x)+e^{\frac {1}{\left (3 x^3+x^4\right ) \log (x)}} \left (3+x+(9+4 x) \log (x)+\left (9 x^3+6 x^4+x^5\right ) \log ^2(x)\right )}{e^{\frac {1}{\left (3 x^3+x^4\right ) \log (x)}} \left (-72 x^3-48 x^4-8 x^5\right ) \log ^2(x)+e^{\frac {2}{\left (3 x^3+x^4\right ) \log (x)}} \left (9 x^3+6 x^4+x^5\right ) \log ^2(x)+\left (144 x^3+96 x^4+16 x^5\right ) \log ^2(x)} \, dx=\frac {x}{-4+e^{\frac {1}{x^3 (3+x) \log (x)}}} \]
Integrate[((-36*x^3 - 24*x^4 - 4*x^5)*Log[x]^2 + E^(1/((3*x^3 + x^4)*Log[x ]))*(3 + x + (9 + 4*x)*Log[x] + (9*x^3 + 6*x^4 + x^5)*Log[x]^2))/(E^(1/((3 *x^3 + x^4)*Log[x]))*(-72*x^3 - 48*x^4 - 8*x^5)*Log[x]^2 + E^(2/((3*x^3 + x^4)*Log[x]))*(9*x^3 + 6*x^4 + x^5)*Log[x]^2 + (144*x^3 + 96*x^4 + 16*x^5) *Log[x]^2),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.
\(\displaystyle \int \frac {\left (-4 x^5-24 x^4-36 x^3\right ) \log ^2(x)+e^{\frac {1}{\left (x^4+3 x^3\right ) \log (x)}} \left (\left (x^5+6 x^4+9 x^3\right ) \log ^2(x)+x+(4 x+9) \log (x)+3\right )}{\left (-8 x^5-48 x^4-72 x^3\right ) e^{\frac {1}{\left (x^4+3 x^3\right ) \log (x)}} \log ^2(x)+\left (x^5+6 x^4+9 x^3\right ) e^{\frac {2}{\left (x^4+3 x^3\right ) \log (x)}} \log ^2(x)+\left (16 x^5+96 x^4+144 x^3\right ) \log ^2(x)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {\left (-4 x^5-24 x^4-36 x^3\right ) \log ^2(x)+e^{\frac {1}{\left (x^4+3 x^3\right ) \log (x)}} \left (\left (x^5+6 x^4+9 x^3\right ) \log ^2(x)+x+(4 x+9) \log (x)+3\right )}{x^3 (x+3)^2 \left (4-e^{\frac {1}{x^3 (x+3) \log (x)}}\right )^2 \log ^2(x)}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {4 (x+4 x \log (x)+9 \log (x)+3)}{x^3 (x+3)^2 \left (e^{\frac {1}{x^3 (x+3) \log (x)}}-4\right )^2 \log ^2(x)}+\frac {x^5 \log ^2(x)+6 x^4 \log ^2(x)+9 x^3 \log ^2(x)+x+4 x \log (x)+9 \log (x)+3}{x^3 (x+3)^2 \left (e^{\frac {1}{x^3 (x+3) \log (x)}}-4\right ) \log ^2(x)}\right )dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \left (\frac {4 (x+4 x \log (x)+9 \log (x)+3)}{x^3 (x+3)^2 \left (e^{\frac {1}{x^3 (x+3) \log (x)}}-4\right )^2 \log ^2(x)}+\frac {x^5 \log ^2(x)+6 x^4 \log ^2(x)+9 x^3 \log ^2(x)+x+4 x \log (x)+9 \log (x)+3}{x^3 (x+3)^2 \left (e^{\frac {1}{x^3 (x+3) \log (x)}}-4\right ) \log ^2(x)}\right )dx\) |
Int[((-36*x^3 - 24*x^4 - 4*x^5)*Log[x]^2 + E^(1/((3*x^3 + x^4)*Log[x]))*(3 + x + (9 + 4*x)*Log[x] + (9*x^3 + 6*x^4 + x^5)*Log[x]^2))/(E^(1/((3*x^3 + x^4)*Log[x]))*(-72*x^3 - 48*x^4 - 8*x^5)*Log[x]^2 + E^(2/((3*x^3 + x^4)*L og[x]))*(9*x^3 + 6*x^4 + x^5)*Log[x]^2 + (144*x^3 + 96*x^4 + 16*x^5)*Log[x ]^2),x]
3.7.84.3.1 Defintions of rubi rules used
Time = 12.86 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.81
method | result | size |
risch | \(\frac {x}{{\mathrm e}^{\frac {1}{x^{3} \ln \left (x \right ) \left (3+x \right )}}-4}\) | \(21\) |
parallelrisch | \(\frac {x}{{\mathrm e}^{\frac {1}{x^{3} \ln \left (x \right ) \left (3+x \right )}}-4}\) | \(21\) |
int((((x^5+6*x^4+9*x^3)*ln(x)^2+(4*x+9)*ln(x)+3+x)*exp(1/(x^4+3*x^3)/ln(x) )+(-4*x^5-24*x^4-36*x^3)*ln(x)^2)/((x^5+6*x^4+9*x^3)*ln(x)^2*exp(1/(x^4+3* x^3)/ln(x))^2+(-8*x^5-48*x^4-72*x^3)*ln(x)^2*exp(1/(x^4+3*x^3)/ln(x))+(16* x^5+96*x^4+144*x^3)*ln(x)^2),x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88 \[ \int \frac {\left (-36 x^3-24 x^4-4 x^5\right ) \log ^2(x)+e^{\frac {1}{\left (3 x^3+x^4\right ) \log (x)}} \left (3+x+(9+4 x) \log (x)+\left (9 x^3+6 x^4+x^5\right ) \log ^2(x)\right )}{e^{\frac {1}{\left (3 x^3+x^4\right ) \log (x)}} \left (-72 x^3-48 x^4-8 x^5\right ) \log ^2(x)+e^{\frac {2}{\left (3 x^3+x^4\right ) \log (x)}} \left (9 x^3+6 x^4+x^5\right ) \log ^2(x)+\left (144 x^3+96 x^4+16 x^5\right ) \log ^2(x)} \, dx=\frac {x}{e^{\left (\frac {1}{{\left (x^{4} + 3 \, x^{3}\right )} \log \left (x\right )}\right )} - 4} \]
integrate((((x^5+6*x^4+9*x^3)*log(x)^2+(4*x+9)*log(x)+3+x)*exp(1/(x^4+3*x^ 3)/log(x))+(-4*x^5-24*x^4-36*x^3)*log(x)^2)/((x^5+6*x^4+9*x^3)*log(x)^2*ex p(1/(x^4+3*x^3)/log(x))^2+(-8*x^5-48*x^4-72*x^3)*log(x)^2*exp(1/(x^4+3*x^3 )/log(x))+(16*x^5+96*x^4+144*x^3)*log(x)^2),x, algorithm=\
Time = 0.17 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.65 \[ \int \frac {\left (-36 x^3-24 x^4-4 x^5\right ) \log ^2(x)+e^{\frac {1}{\left (3 x^3+x^4\right ) \log (x)}} \left (3+x+(9+4 x) \log (x)+\left (9 x^3+6 x^4+x^5\right ) \log ^2(x)\right )}{e^{\frac {1}{\left (3 x^3+x^4\right ) \log (x)}} \left (-72 x^3-48 x^4-8 x^5\right ) \log ^2(x)+e^{\frac {2}{\left (3 x^3+x^4\right ) \log (x)}} \left (9 x^3+6 x^4+x^5\right ) \log ^2(x)+\left (144 x^3+96 x^4+16 x^5\right ) \log ^2(x)} \, dx=\frac {x}{e^{\frac {1}{\left (x^{4} + 3 x^{3}\right ) \log {\left (x \right )}}} - 4} \]
integrate((((x**5+6*x**4+9*x**3)*ln(x)**2+(4*x+9)*ln(x)+3+x)*exp(1/(x**4+3 *x**3)/ln(x))+(-4*x**5-24*x**4-36*x**3)*ln(x)**2)/((x**5+6*x**4+9*x**3)*ln (x)**2*exp(1/(x**4+3*x**3)/ln(x))**2+(-8*x**5-48*x**4-72*x**3)*ln(x)**2*ex p(1/(x**4+3*x**3)/ln(x))+(16*x**5+96*x**4+144*x**3)*ln(x)**2),x)
Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (22) = 44\).
Time = 0.28 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.85 \[ \int \frac {\left (-36 x^3-24 x^4-4 x^5\right ) \log ^2(x)+e^{\frac {1}{\left (3 x^3+x^4\right ) \log (x)}} \left (3+x+(9+4 x) \log (x)+\left (9 x^3+6 x^4+x^5\right ) \log ^2(x)\right )}{e^{\frac {1}{\left (3 x^3+x^4\right ) \log (x)}} \left (-72 x^3-48 x^4-8 x^5\right ) \log ^2(x)+e^{\frac {2}{\left (3 x^3+x^4\right ) \log (x)}} \left (9 x^3+6 x^4+x^5\right ) \log ^2(x)+\left (144 x^3+96 x^4+16 x^5\right ) \log ^2(x)} \, dx=-\frac {x e^{\left (\frac {1}{27 \, {\left (x + 3\right )} \log \left (x\right )} + \frac {1}{9 \, x^{2} \log \left (x\right )}\right )}}{4 \, e^{\left (\frac {1}{27 \, {\left (x + 3\right )} \log \left (x\right )} + \frac {1}{9 \, x^{2} \log \left (x\right )}\right )} - e^{\left (\frac {1}{27 \, x \log \left (x\right )} + \frac {1}{3 \, x^{3} \log \left (x\right )}\right )}} \]
integrate((((x^5+6*x^4+9*x^3)*log(x)^2+(4*x+9)*log(x)+3+x)*exp(1/(x^4+3*x^ 3)/log(x))+(-4*x^5-24*x^4-36*x^3)*log(x)^2)/((x^5+6*x^4+9*x^3)*log(x)^2*ex p(1/(x^4+3*x^3)/log(x))^2+(-8*x^5-48*x^4-72*x^3)*log(x)^2*exp(1/(x^4+3*x^3 )/log(x))+(16*x^5+96*x^4+144*x^3)*log(x)^2),x, algorithm=\
-x*e^(1/27/((x + 3)*log(x)) + 1/9/(x^2*log(x)))/(4*e^(1/27/((x + 3)*log(x) ) + 1/9/(x^2*log(x))) - e^(1/27/(x*log(x)) + 1/3/(x^3*log(x))))
Time = 0.33 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88 \[ \int \frac {\left (-36 x^3-24 x^4-4 x^5\right ) \log ^2(x)+e^{\frac {1}{\left (3 x^3+x^4\right ) \log (x)}} \left (3+x+(9+4 x) \log (x)+\left (9 x^3+6 x^4+x^5\right ) \log ^2(x)\right )}{e^{\frac {1}{\left (3 x^3+x^4\right ) \log (x)}} \left (-72 x^3-48 x^4-8 x^5\right ) \log ^2(x)+e^{\frac {2}{\left (3 x^3+x^4\right ) \log (x)}} \left (9 x^3+6 x^4+x^5\right ) \log ^2(x)+\left (144 x^3+96 x^4+16 x^5\right ) \log ^2(x)} \, dx=\frac {x}{e^{\left (\frac {1}{x^{4} \log \left (x\right ) + 3 \, x^{3} \log \left (x\right )}\right )} - 4} \]
integrate((((x^5+6*x^4+9*x^3)*log(x)^2+(4*x+9)*log(x)+3+x)*exp(1/(x^4+3*x^ 3)/log(x))+(-4*x^5-24*x^4-36*x^3)*log(x)^2)/((x^5+6*x^4+9*x^3)*log(x)^2*ex p(1/(x^4+3*x^3)/log(x))^2+(-8*x^5-48*x^4-72*x^3)*log(x)^2*exp(1/(x^4+3*x^3 )/log(x))+(16*x^5+96*x^4+144*x^3)*log(x)^2),x, algorithm=\
Time = 8.43 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88 \[ \int \frac {\left (-36 x^3-24 x^4-4 x^5\right ) \log ^2(x)+e^{\frac {1}{\left (3 x^3+x^4\right ) \log (x)}} \left (3+x+(9+4 x) \log (x)+\left (9 x^3+6 x^4+x^5\right ) \log ^2(x)\right )}{e^{\frac {1}{\left (3 x^3+x^4\right ) \log (x)}} \left (-72 x^3-48 x^4-8 x^5\right ) \log ^2(x)+e^{\frac {2}{\left (3 x^3+x^4\right ) \log (x)}} \left (9 x^3+6 x^4+x^5\right ) \log ^2(x)+\left (144 x^3+96 x^4+16 x^5\right ) \log ^2(x)} \, dx=\frac {x}{{\mathrm {e}}^{\frac {1}{3\,x^3\,\ln \left (x\right )+x^4\,\ln \left (x\right )}}-4} \]
int(-(log(x)^2*(36*x^3 + 24*x^4 + 4*x^5) - exp(1/(log(x)*(3*x^3 + x^4)))*( x + log(x)^2*(9*x^3 + 6*x^4 + x^5) + log(x)*(4*x + 9) + 3))/(log(x)^2*(144 *x^3 + 96*x^4 + 16*x^5) + exp(2/(log(x)*(3*x^3 + x^4)))*log(x)^2*(9*x^3 + 6*x^4 + x^5) - exp(1/(log(x)*(3*x^3 + x^4)))*log(x)^2*(72*x^3 + 48*x^4 + 8 *x^5)),x)