Integrand size = 227, antiderivative size = 30 \[ \int \frac {-2 x^3+e^{10} \left (-3+x^2\right )+e^5 \left (-6 x+2 x^3\right )+e^{2 e^3-2 x} \left (-2 x+e^5 \left (6-2 x^2\right )\right )+e^{e^3-x} \left (-4 x^2+e^5 \left (-6+6 x+2 x^2-2 x^3\right )\right )+\left (-6 x+2 x^3+e^{2 e^3-2 x} \left (6-2 x^2\right )+e^5 \left (-6+2 x^2\right )+e^{e^3-x} \left (-6+6 x+2 x^2-2 x^3\right )\right ) \log \left (3-x^2\right )+\left (-3+x^2\right ) \log ^2\left (3-x^2\right )}{e^{10} \left (-3+x^2\right )+e^5 \left (-6+2 x^2\right ) \log \left (3-x^2\right )+\left (-3+x^2\right ) \log ^2\left (3-x^2\right )} \, dx=x+\frac {\left (e^{e^3-x}+x\right )^2}{e^5+\log \left (3-x^2\right )} \]
Time = 0.21 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.17 \[ \int \frac {-2 x^3+e^{10} \left (-3+x^2\right )+e^5 \left (-6 x+2 x^3\right )+e^{2 e^3-2 x} \left (-2 x+e^5 \left (6-2 x^2\right )\right )+e^{e^3-x} \left (-4 x^2+e^5 \left (-6+6 x+2 x^2-2 x^3\right )\right )+\left (-6 x+2 x^3+e^{2 e^3-2 x} \left (6-2 x^2\right )+e^5 \left (-6+2 x^2\right )+e^{e^3-x} \left (-6+6 x+2 x^2-2 x^3\right )\right ) \log \left (3-x^2\right )+\left (-3+x^2\right ) \log ^2\left (3-x^2\right )}{e^{10} \left (-3+x^2\right )+e^5 \left (-6+2 x^2\right ) \log \left (3-x^2\right )+\left (-3+x^2\right ) \log ^2\left (3-x^2\right )} \, dx=x+\frac {e^{-2 x} \left (e^{e^3}+e^x x\right )^2}{e^5+\log \left (3-x^2\right )} \]
Integrate[(-2*x^3 + E^10*(-3 + x^2) + E^5*(-6*x + 2*x^3) + E^(2*E^3 - 2*x) *(-2*x + E^5*(6 - 2*x^2)) + E^(E^3 - x)*(-4*x^2 + E^5*(-6 + 6*x + 2*x^2 - 2*x^3)) + (-6*x + 2*x^3 + E^(2*E^3 - 2*x)*(6 - 2*x^2) + E^5*(-6 + 2*x^2) + E^(E^3 - x)*(-6 + 6*x + 2*x^2 - 2*x^3))*Log[3 - x^2] + (-3 + x^2)*Log[3 - x^2]^2)/(E^10*(-3 + x^2) + E^5*(-6 + 2*x^2)*Log[3 - x^2] + (-3 + x^2)*Log [3 - x^2]^2),x]
Leaf count is larger than twice the leaf count of optimal. \(186\) vs. \(2(30)=60\).
Time = 4.39 (sec) , antiderivative size = 186, normalized size of antiderivative = 6.20, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.013, Rules used = {7292, 7276, 2009}
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.
| \(\displaystyle \int \frac {-2 x^3+e^5 \left (2 x^3-6 x\right )+e^{10} \left (x^2-3\right )+e^{2 e^3-2 x} \left (e^5 \left (6-2 x^2\right )-2 x\right )+\left (x^2-3\right ) \log ^2\left (3-x^2\right )+e^{e^3-x} \left (e^5 \left (-2 x^3+2 x^2+6 x-6\right )-4 x^2\right )+\left (2 x^3+e^{2 e^3-2 x} \left (6-2 x^2\right )+e^5 \left (2 x^2-6\right )+e^{e^3-x} \left (-2 x^3+2 x^2+6 x-6\right )-6 x\right ) \log \left (3-x^2\right )}{e^{10} \left (x^2-3\right )+\left (x^2-3\right ) \log ^2\left (3-x^2\right )+e^5 \left (2 x^2-6\right ) \log \left (3-x^2\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {2 x^3-e^5 \left (2 x^3-6 x\right )-e^{10} \left (x^2-3\right )-e^{2 e^3-2 x} \left (e^5 \left (6-2 x^2\right )-2 x\right )-\left (x^2-3\right ) \log ^2\left (3-x^2\right )-e^{e^3-x} \left (e^5 \left (-2 x^3+2 x^2+6 x-6\right )-4 x^2\right )-\left (2 x^3+e^{2 e^3-2 x} \left (6-2 x^2\right )+e^5 \left (2 x^2-6\right )+e^{e^3-x} \left (-2 x^3+2 x^2+6 x-6\right )-6 x\right ) \log \left (3-x^2\right )}{\left (3-x^2\right ) \left (\log \left (3-x^2\right )+e^5\right )^2}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {\log ^2\left (3-x^2\right )}{\left (\log \left (3-x^2\right )+e^5\right )^2}-\frac {6 x \log \left (3-x^2\right )}{\left (x^2-3\right ) \left (\log \left (3-x^2\right )+e^5\right )^2}+\frac {2 e^5 x}{\left (\log \left (3-x^2\right )+e^5\right )^2}-\frac {2 e^{2 e^3-2 x} \left (e^5 x^2+x^2 \log \left (3-x^2\right )-3 \log \left (3-x^2\right )+x-3 e^5\right )}{\left (x^2-3\right ) \left (\log \left (3-x^2\right )+e^5\right )^2}+\frac {2 e^5 \log \left (3-x^2\right )}{\left (\log \left (3-x^2\right )+e^5\right )^2}+\frac {e^{10}}{\left (\log \left (3-x^2\right )+e^5\right )^2}+\frac {2 x^3 \log \left (3-x^2\right )}{\left (x^2-3\right ) \left (\log \left (3-x^2\right )+e^5\right )^2}-\frac {2 x^3}{\left (x^2-3\right ) \left (\log \left (3-x^2\right )+e^5\right )^2}+\frac {2 e^{e^3-x} \left (e^5 x^3+2 \left (1-\frac {e^5}{2}\right ) x^2-x^2 \log \left (3-x^2\right )-3 x \log \left (3-x^2\right )+3 \log \left (3-x^2\right )+x^3 \log \left (3-x^2\right )-3 e^5 x+3 e^5\right )}{\left (3-x^2\right ) \left (\log \left (3-x^2\right )+e^5\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^{2 e^3-2 x} \left (-e^5 x^2+x^2 \left (-\log \left (3-x^2\right )\right )+3 \log \left (3-x^2\right )+3 e^5\right )}{\left (3-x^2\right ) \left (\log \left (3-x^2\right )+e^5\right )^2}-\frac {3-x^2}{\log \left (3-x^2\right )+e^5}+\frac {3}{\log \left (3-x^2\right )+e^5}+\frac {2 e^{e^3-x} \left (-e^5 x^3+3 x \log \left (3-x^2\right )+x^3 \left (-\log \left (3-x^2\right )\right )+3 e^5 x\right )}{\left (3-x^2\right ) \left (\log \left (3-x^2\right )+e^5\right )^2}+x\) |
Int[(-2*x^3 + E^10*(-3 + x^2) + E^5*(-6*x + 2*x^3) + E^(2*E^3 - 2*x)*(-2*x + E^5*(6 - 2*x^2)) + E^(E^3 - x)*(-4*x^2 + E^5*(-6 + 6*x + 2*x^2 - 2*x^3) ) + (-6*x + 2*x^3 + E^(2*E^3 - 2*x)*(6 - 2*x^2) + E^5*(-6 + 2*x^2) + E^(E^ 3 - x)*(-6 + 6*x + 2*x^2 - 2*x^3))*Log[3 - x^2] + (-3 + x^2)*Log[3 - x^2]^ 2)/(E^10*(-3 + x^2) + E^5*(-6 + 2*x^2)*Log[3 - x^2] + (-3 + x^2)*Log[3 - x ^2]^2),x]
x + 3/(E^5 + Log[3 - x^2]) - (3 - x^2)/(E^5 + Log[3 - x^2]) + (E^(2*E^3 - 2*x)*(3*E^5 - E^5*x^2 + 3*Log[3 - x^2] - x^2*Log[3 - x^2]))/((3 - x^2)*(E^ 5 + Log[3 - x^2])^2) + (2*E^(E^3 - x)*(3*E^5*x - E^5*x^3 + 3*x*Log[3 - x^2 ] - x^3*Log[3 - x^2]))/((3 - x^2)*(E^5 + Log[3 - x^2])^2)
3.22.79.3.1 Defintions of rubi rules used
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 3.17 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.33
| method | result | size |
| risch | \(x +\frac {x^{2}+2 \,{\mathrm e}^{-x +{\mathrm e}^{3}} x +{\mathrm e}^{-2 x +2 \,{\mathrm e}^{3}}}{{\mathrm e}^{5}+\ln \left (-x^{2}+3\right )}\) | \(40\) |
| parallelrisch | \(\frac {x \,{\mathrm e}^{5}+\ln \left (-x^{2}+3\right ) x +2 \,{\mathrm e}^{-x +{\mathrm e}^{3}} x +x^{2}+{\mathrm e}^{-2 x +2 \,{\mathrm e}^{3}}}{{\mathrm e}^{5}+\ln \left (-x^{2}+3\right )}\) | \(52\) |
int(((x^2-3)*ln(-x^2+3)^2+((-2*x^2+6)*exp(-x+exp(3))^2+(-2*x^3+2*x^2+6*x-6 )*exp(-x+exp(3))+(2*x^2-6)*exp(5)+2*x^3-6*x)*ln(-x^2+3)+((-2*x^2+6)*exp(5) -2*x)*exp(-x+exp(3))^2+((-2*x^3+2*x^2+6*x-6)*exp(5)-4*x^2)*exp(-x+exp(3))+ (x^2-3)*exp(5)^2+(2*x^3-6*x)*exp(5)-2*x^3)/((x^2-3)*ln(-x^2+3)^2+(2*x^2-6) *exp(5)*ln(-x^2+3)+(x^2-3)*exp(5)^2),x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.70 \[ \int \frac {-2 x^3+e^{10} \left (-3+x^2\right )+e^5 \left (-6 x+2 x^3\right )+e^{2 e^3-2 x} \left (-2 x+e^5 \left (6-2 x^2\right )\right )+e^{e^3-x} \left (-4 x^2+e^5 \left (-6+6 x+2 x^2-2 x^3\right )\right )+\left (-6 x+2 x^3+e^{2 e^3-2 x} \left (6-2 x^2\right )+e^5 \left (-6+2 x^2\right )+e^{e^3-x} \left (-6+6 x+2 x^2-2 x^3\right )\right ) \log \left (3-x^2\right )+\left (-3+x^2\right ) \log ^2\left (3-x^2\right )}{e^{10} \left (-3+x^2\right )+e^5 \left (-6+2 x^2\right ) \log \left (3-x^2\right )+\left (-3+x^2\right ) \log ^2\left (3-x^2\right )} \, dx=\frac {x^{2} + x e^{5} + 2 \, x e^{\left (-x + e^{3}\right )} + x \log \left (-x^{2} + 3\right ) + e^{\left (-2 \, x + 2 \, e^{3}\right )}}{e^{5} + \log \left (-x^{2} + 3\right )} \]
integrate(((x^2-3)*log(-x^2+3)^2+((-2*x^2+6)*exp(-x+exp(3))^2+(-2*x^3+2*x^ 2+6*x-6)*exp(-x+exp(3))+(2*x^2-6)*exp(5)+2*x^3-6*x)*log(-x^2+3)+((-2*x^2+6 )*exp(5)-2*x)*exp(-x+exp(3))^2+((-2*x^3+2*x^2+6*x-6)*exp(5)-4*x^2)*exp(-x+ exp(3))+(x^2-3)*exp(5)^2+(2*x^3-6*x)*exp(5)-2*x^3)/((x^2-3)*log(-x^2+3)^2+ (2*x^2-6)*exp(5)*log(-x^2+3)+(x^2-3)*exp(5)^2),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (20) = 40\).
Time = 0.22 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.73 \[ \int \frac {-2 x^3+e^{10} \left (-3+x^2\right )+e^5 \left (-6 x+2 x^3\right )+e^{2 e^3-2 x} \left (-2 x+e^5 \left (6-2 x^2\right )\right )+e^{e^3-x} \left (-4 x^2+e^5 \left (-6+6 x+2 x^2-2 x^3\right )\right )+\left (-6 x+2 x^3+e^{2 e^3-2 x} \left (6-2 x^2\right )+e^5 \left (-6+2 x^2\right )+e^{e^3-x} \left (-6+6 x+2 x^2-2 x^3\right )\right ) \log \left (3-x^2\right )+\left (-3+x^2\right ) \log ^2\left (3-x^2\right )}{e^{10} \left (-3+x^2\right )+e^5 \left (-6+2 x^2\right ) \log \left (3-x^2\right )+\left (-3+x^2\right ) \log ^2\left (3-x^2\right )} \, dx=\frac {x^{2}}{\log {\left (3 - x^{2} \right )} + e^{5}} + x + \frac {\left (2 x \log {\left (3 - x^{2} \right )} + 2 x e^{5}\right ) e^{- x + e^{3}} + \left (\log {\left (3 - x^{2} \right )} + e^{5}\right ) e^{- 2 x + 2 e^{3}}}{\log {\left (3 - x^{2} \right )}^{2} + 2 e^{5} \log {\left (3 - x^{2} \right )} + e^{10}} \]
integrate(((x**2-3)*ln(-x**2+3)**2+((-2*x**2+6)*exp(-x+exp(3))**2+(-2*x**3 +2*x**2+6*x-6)*exp(-x+exp(3))+(2*x**2-6)*exp(5)+2*x**3-6*x)*ln(-x**2+3)+(( -2*x**2+6)*exp(5)-2*x)*exp(-x+exp(3))**2+((-2*x**3+2*x**2+6*x-6)*exp(5)-4* x**2)*exp(-x+exp(3))+(x**2-3)*exp(5)**2+(2*x**3-6*x)*exp(5)-2*x**3)/((x**2 -3)*ln(-x**2+3)**2+(2*x**2-6)*exp(5)*ln(-x**2+3)+(x**2-3)*exp(5)**2),x)
x**2/(log(3 - x**2) + exp(5)) + x + ((2*x*log(3 - x**2) + 2*x*exp(5))*exp( -x + exp(3)) + (log(3 - x**2) + exp(5))*exp(-2*x + 2*exp(3)))/(log(3 - x** 2)**2 + 2*exp(5)*log(3 - x**2) + exp(10))
Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (27) = 54\).
Time = 0.25 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.13 \[ \int \frac {-2 x^3+e^{10} \left (-3+x^2\right )+e^5 \left (-6 x+2 x^3\right )+e^{2 e^3-2 x} \left (-2 x+e^5 \left (6-2 x^2\right )\right )+e^{e^3-x} \left (-4 x^2+e^5 \left (-6+6 x+2 x^2-2 x^3\right )\right )+\left (-6 x+2 x^3+e^{2 e^3-2 x} \left (6-2 x^2\right )+e^5 \left (-6+2 x^2\right )+e^{e^3-x} \left (-6+6 x+2 x^2-2 x^3\right )\right ) \log \left (3-x^2\right )+\left (-3+x^2\right ) \log ^2\left (3-x^2\right )}{e^{10} \left (-3+x^2\right )+e^5 \left (-6+2 x^2\right ) \log \left (3-x^2\right )+\left (-3+x^2\right ) \log ^2\left (3-x^2\right )} \, dx=\frac {x e^{\left (2 \, x\right )} \log \left (-x^{2} + 3\right ) + {\left (x^{2} + x e^{5}\right )} e^{\left (2 \, x\right )} + 2 \, x e^{\left (x + e^{3}\right )} + e^{\left (2 \, e^{3}\right )}}{e^{\left (2 \, x\right )} \log \left (-x^{2} + 3\right ) + e^{\left (2 \, x + 5\right )}} \]
integrate(((x^2-3)*log(-x^2+3)^2+((-2*x^2+6)*exp(-x+exp(3))^2+(-2*x^3+2*x^ 2+6*x-6)*exp(-x+exp(3))+(2*x^2-6)*exp(5)+2*x^3-6*x)*log(-x^2+3)+((-2*x^2+6 )*exp(5)-2*x)*exp(-x+exp(3))^2+((-2*x^3+2*x^2+6*x-6)*exp(5)-4*x^2)*exp(-x+ exp(3))+(x^2-3)*exp(5)^2+(2*x^3-6*x)*exp(5)-2*x^3)/((x^2-3)*log(-x^2+3)^2+ (2*x^2-6)*exp(5)*log(-x^2+3)+(x^2-3)*exp(5)^2),x, algorithm=\
(x*e^(2*x)*log(-x^2 + 3) + (x^2 + x*e^5)*e^(2*x) + 2*x*e^(x + e^3) + e^(2* e^3))/(e^(2*x)*log(-x^2 + 3) + e^(2*x + 5))
Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (27) = 54\).
Time = 0.31 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.73 \[ \int \frac {-2 x^3+e^{10} \left (-3+x^2\right )+e^5 \left (-6 x+2 x^3\right )+e^{2 e^3-2 x} \left (-2 x+e^5 \left (6-2 x^2\right )\right )+e^{e^3-x} \left (-4 x^2+e^5 \left (-6+6 x+2 x^2-2 x^3\right )\right )+\left (-6 x+2 x^3+e^{2 e^3-2 x} \left (6-2 x^2\right )+e^5 \left (-6+2 x^2\right )+e^{e^3-x} \left (-6+6 x+2 x^2-2 x^3\right )\right ) \log \left (3-x^2\right )+\left (-3+x^2\right ) \log ^2\left (3-x^2\right )}{e^{10} \left (-3+x^2\right )+e^5 \left (-6+2 x^2\right ) \log \left (3-x^2\right )+\left (-3+x^2\right ) \log ^2\left (3-x^2\right )} \, dx=\frac {x^{3} e^{\left (3 \, x\right )} + x^{2} e^{\left (3 \, x\right )} \log \left (-x^{2} + 3\right ) + x^{2} e^{\left (3 \, x + 5\right )} + 2 \, x^{2} e^{\left (2 \, x + e^{3}\right )} + x e^{\left (x + 2 \, e^{3}\right )}}{x e^{\left (3 \, x\right )} \log \left (-x^{2} + 3\right ) + x e^{\left (3 \, x + 5\right )}} \]
integrate(((x^2-3)*log(-x^2+3)^2+((-2*x^2+6)*exp(-x+exp(3))^2+(-2*x^3+2*x^ 2+6*x-6)*exp(-x+exp(3))+(2*x^2-6)*exp(5)+2*x^3-6*x)*log(-x^2+3)+((-2*x^2+6 )*exp(5)-2*x)*exp(-x+exp(3))^2+((-2*x^3+2*x^2+6*x-6)*exp(5)-4*x^2)*exp(-x+ exp(3))+(x^2-3)*exp(5)^2+(2*x^3-6*x)*exp(5)-2*x^3)/((x^2-3)*log(-x^2+3)^2+ (2*x^2-6)*exp(5)*log(-x^2+3)+(x^2-3)*exp(5)^2),x, algorithm=\
(x^3*e^(3*x) + x^2*e^(3*x)*log(-x^2 + 3) + x^2*e^(3*x + 5) + 2*x^2*e^(2*x + e^3) + x*e^(x + 2*e^3))/(x*e^(3*x)*log(-x^2 + 3) + x*e^(3*x + 5))
Time = 11.96 (sec) , antiderivative size = 95, normalized size of antiderivative = 3.17 \[ \int \frac {-2 x^3+e^{10} \left (-3+x^2\right )+e^5 \left (-6 x+2 x^3\right )+e^{2 e^3-2 x} \left (-2 x+e^5 \left (6-2 x^2\right )\right )+e^{e^3-x} \left (-4 x^2+e^5 \left (-6+6 x+2 x^2-2 x^3\right )\right )+\left (-6 x+2 x^3+e^{2 e^3-2 x} \left (6-2 x^2\right )+e^5 \left (-6+2 x^2\right )+e^{e^3-x} \left (-6+6 x+2 x^2-2 x^3\right )\right ) \log \left (3-x^2\right )+\left (-3+x^2\right ) \log ^2\left (3-x^2\right )}{e^{10} \left (-3+x^2\right )+e^5 \left (-6+2 x^2\right ) \log \left (3-x^2\right )+\left (-3+x^2\right ) \log ^2\left (3-x^2\right )} \, dx=x+\frac {3\,{\mathrm {e}}^5-\ln \left (3-x^2\right )\,\left (x^2-3\right )-x^2\,{\mathrm {e}}^5+x^2}{\ln \left (3-x^2\right )+{\mathrm {e}}^5}+x^2+\frac {{\mathrm {e}}^{2\,{\mathrm {e}}^3-2\,x}}{\ln \left (3-x^2\right )+{\mathrm {e}}^5}+\frac {2\,x\,{\mathrm {e}}^{{\mathrm {e}}^3-x}}{\ln \left (3-x^2\right )+{\mathrm {e}}^5} \]
int((exp(exp(3) - x)*(exp(5)*(6*x + 2*x^2 - 2*x^3 - 6) - 4*x^2) - exp(2*ex p(3) - 2*x)*(2*x + exp(5)*(2*x^2 - 6)) - exp(5)*(6*x - 2*x^3) + log(3 - x^ 2)*(exp(exp(3) - x)*(6*x + 2*x^2 - 2*x^3 - 6) - 6*x - exp(2*exp(3) - 2*x)* (2*x^2 - 6) + exp(5)*(2*x^2 - 6) + 2*x^3) - 2*x^3 + log(3 - x^2)^2*(x^2 - 3) + exp(10)*(x^2 - 3))/(log(3 - x^2)^2*(x^2 - 3) + exp(10)*(x^2 - 3) + lo g(3 - x^2)*exp(5)*(2*x^2 - 6)),x)