Integrand size = 80, antiderivative size = 31 \[ \int \frac {36-42 x+16 x^2-16 x^3+6 x^4-x^5+e^x \left (-9 x^3+6 x^4-x^5\right )+\left (-36+33 x-10 x^2+x^3\right ) \log \left (x^2\right )}{9 x^3-6 x^4+x^5} \, dx=-2-e^x-\frac {5}{3-x}-x-\frac {(-2+x) \log \left (x^2\right )}{x^2} \] Output:
-x-exp(x)-2-ln(x^2)/x^2*(-2+x)-5/(3-x)
Time = 0.16 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.90 \[ \int \frac {36-42 x+16 x^2-16 x^3+6 x^4-x^5+e^x \left (-9 x^3+6 x^4-x^5\right )+\left (-36+33 x-10 x^2+x^3\right ) \log \left (x^2\right )}{9 x^3-6 x^4+x^5} \, dx=-e^x+\frac {5}{-3+x}-x-\frac {(-2+x) \log \left (x^2\right )}{x^2} \] Input:
Integrate[(36 - 42*x + 16*x^2 - 16*x^3 + 6*x^4 - x^5 + E^x*(-9*x^3 + 6*x^4 - x^5) + (-36 + 33*x - 10*x^2 + x^3)*Log[x^2])/(9*x^3 - 6*x^4 + x^5),x]
Output:
-E^x + 5/(-3 + x) - x - ((-2 + x)*Log[x^2])/x^2
Time = 1.26 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.35, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {2026, 7277, 27, 7293, 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.
\(\displaystyle \int \frac {-x^5+6 x^4-16 x^3+16 x^2+\left (x^3-10 x^2+33 x-36\right ) \log \left (x^2\right )+e^x \left (-x^5+6 x^4-9 x^3\right )-42 x+36}{x^5-6 x^4+9 x^3} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {-x^5+6 x^4-16 x^3+16 x^2+\left (x^3-10 x^2+33 x-36\right ) \log \left (x^2\right )+e^x \left (-x^5+6 x^4-9 x^3\right )-42 x+36}{x^3 \left (x^2-6 x+9\right )}dx\) |
\(\Big \downarrow \) 7277 |
\(\displaystyle 4 \int \frac {-x^5+6 x^4-16 x^3+16 x^2-42 x-e^x \left (x^5-6 x^4+9 x^3\right )-\left (-x^3+10 x^2-33 x+36\right ) \log \left (x^2\right )+36}{4 (3-x)^2 x^3}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {-x^5+6 x^4-16 x^3+16 x^2-\left (-x^3+10 x^2-33 x+36\right ) \log \left (x^2\right )-e^x \left (x^5-6 x^4+9 x^3\right )-42 x+36}{(3-x)^2 x^3}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {36}{(x-3)^2 x^3}-\frac {x^2}{(x-3)^2}-\frac {42}{(x-3)^2 x^2}+\frac {(x-4) \log \left (x^2\right )}{x^3}+\frac {6 x}{(x-3)^2}-e^x-\frac {16}{(x-3)^2}+\frac {16}{(x-3)^2 x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {(4-x)^2 \log \left (x^2\right )}{8 x^2}-e^x-x-\frac {5}{3-x}-\frac {\log (x)}{4}\) |
Input:
Int[(36 - 42*x + 16*x^2 - 16*x^3 + 6*x^4 - x^5 + E^x*(-9*x^3 + 6*x^4 - x^5 ) + (-36 + 33*x - 10*x^2 + x^3)*Log[x^2])/(9*x^3 - 6*x^4 + x^5),x]
Output:
-E^x - 5/(3 - x) - x - Log[x]/4 + ((4 - x)^2*Log[x^2])/(8*x^2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Int[(u_)*((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_.), x_Symbol] :> Simp[1/(4^p*c^p) Int[u*(b + 2*c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n} , x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p] && !AlgebraicFu nctionQ[u, x]
Time = 1.29 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.10
method | result | size |
default | \(-\frac {\ln \left (x^{2}\right )}{x}+\frac {2 \ln \left (x^{2}\right )}{x^{2}}-x +\frac {5}{-3+x}-{\mathrm e}^{x}\) | \(34\) |
parts | \(-\frac {\ln \left (x^{2}\right )}{x}+\frac {2 \ln \left (x^{2}\right )}{x^{2}}-x +\frac {5}{-3+x}-{\mathrm e}^{x}\) | \(34\) |
parallelrisch | \(\frac {2 x^{3} \ln \left (x \right )-3 x^{4}-3 \,{\mathrm e}^{x} x^{3}-x^{3} \ln \left (x^{2}\right )-6 x^{2} \ln \left (x \right )+9 \,{\mathrm e}^{x} x^{2}+42 x^{2}+15 x \ln \left (x^{2}\right )-18 \ln \left (x^{2}\right )}{3 x^{2} \left (-3+x \right )}\) | \(72\) |
risch | \(-\frac {2 \left (-2+x \right ) \ln \left (x \right )}{x^{2}}-\frac {-i \pi \,x^{2} \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )+2 i \pi \,x^{2} \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}-i \pi \,x^{2} \operatorname {csgn}\left (i x^{2}\right )^{3}+5 i \pi x \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-10 i \pi x \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+5 i \pi x \operatorname {csgn}\left (i x^{2}\right )^{3}-6 i \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )+12 i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}-6 i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+2 x^{4}+2 \,{\mathrm e}^{x} x^{3}-6 x^{3}-6 \,{\mathrm e}^{x} x^{2}-10 x^{2}}{2 x^{2} \left (-3+x \right )}\) | \(211\) |
orering | \(\text {Expression too large to display}\) | \(1044\) |
Input:
int(((x^3-10*x^2+33*x-36)*ln(x^2)+(-x^5+6*x^4-9*x^3)*exp(x)-x^5+6*x^4-16*x ^3+16*x^2-42*x+36)/(x^5-6*x^4+9*x^3),x,method=_RETURNVERBOSE)
Output:
-ln(x^2)/x+2/x^2*ln(x^2)-x+5/(-3+x)-exp(x)
Time = 0.08 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.68 \[ \int \frac {36-42 x+16 x^2-16 x^3+6 x^4-x^5+e^x \left (-9 x^3+6 x^4-x^5\right )+\left (-36+33 x-10 x^2+x^3\right ) \log \left (x^2\right )}{9 x^3-6 x^4+x^5} \, dx=-\frac {x^{4} - 3 \, x^{3} - 5 \, x^{2} + {\left (x^{3} - 3 \, x^{2}\right )} e^{x} + {\left (x^{2} - 5 \, x + 6\right )} \log \left (x^{2}\right )}{x^{3} - 3 \, x^{2}} \] Input:
integrate(((x^3-10*x^2+33*x-36)*log(x^2)+(-x^5+6*x^4-9*x^3)*exp(x)-x^5+6*x ^4-16*x^3+16*x^2-42*x+36)/(x^5-6*x^4+9*x^3),x, algorithm="fricas")
Output:
-(x^4 - 3*x^3 - 5*x^2 + (x^3 - 3*x^2)*e^x + (x^2 - 5*x + 6)*log(x^2))/(x^3 - 3*x^2)
Time = 0.12 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.65 \[ \int \frac {36-42 x+16 x^2-16 x^3+6 x^4-x^5+e^x \left (-9 x^3+6 x^4-x^5\right )+\left (-36+33 x-10 x^2+x^3\right ) \log \left (x^2\right )}{9 x^3-6 x^4+x^5} \, dx=- x - e^{x} + \frac {5}{x - 3} + \frac {\left (2 - x\right ) \log {\left (x^{2} \right )}}{x^{2}} \] Input:
integrate(((x**3-10*x**2+33*x-36)*ln(x**2)+(-x**5+6*x**4-9*x**3)*exp(x)-x* *5+6*x**4-16*x**3+16*x**2-42*x+36)/(x**5-6*x**4+9*x**3),x)
Output:
-x - exp(x) + 5/(x - 3) + (2 - x)*log(x**2)/x**2
\[ \int \frac {36-42 x+16 x^2-16 x^3+6 x^4-x^5+e^x \left (-9 x^3+6 x^4-x^5\right )+\left (-36+33 x-10 x^2+x^3\right ) \log \left (x^2\right )}{9 x^3-6 x^4+x^5} \, dx=\int { -\frac {x^{5} - 6 \, x^{4} + 16 \, x^{3} - 16 \, x^{2} + {\left (x^{5} - 6 \, x^{4} + 9 \, x^{3}\right )} e^{x} - {\left (x^{3} - 10 \, x^{2} + 33 \, x - 36\right )} \log \left (x^{2}\right ) + 42 \, x - 36}{x^{5} - 6 \, x^{4} + 9 \, x^{3}} \,d x } \] Input:
integrate(((x^3-10*x^2+33*x-36)*log(x^2)+(-x^5+6*x^4-9*x^3)*exp(x)-x^5+6*x ^4-16*x^3+16*x^2-42*x+36)/(x^5-6*x^4+9*x^3),x, algorithm="maxima")
Output:
-x + 9*e^3*exp_integral_e(2, -x + 3)/(x - 3) - 2*(2*x^2 - 3*x - 3)/(x^3 - 3*x^2) + 14/3*(2*x - 3)/(x^2 - 3*x) + 5/3/(x - 3) - 2*((x - 2)*log(x) + x - 1)/x^2 - integrate((x^2 - 6*x)*e^x/(x^2 - 6*x + 9), x)
Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (28) = 56\).
Time = 0.12 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.97 \[ \int \frac {36-42 x+16 x^2-16 x^3+6 x^4-x^5+e^x \left (-9 x^3+6 x^4-x^5\right )+\left (-36+33 x-10 x^2+x^3\right ) \log \left (x^2\right )}{9 x^3-6 x^4+x^5} \, dx=-\frac {x^{4} + x^{3} e^{x} - 3 \, x^{3} - 3 \, x^{2} e^{x} + x^{2} \log \left (x^{2}\right ) - 5 \, x^{2} - 5 \, x \log \left (x^{2}\right ) + 6 \, \log \left (x^{2}\right )}{x^{3} - 3 \, x^{2}} \] Input:
integrate(((x^3-10*x^2+33*x-36)*log(x^2)+(-x^5+6*x^4-9*x^3)*exp(x)-x^5+6*x ^4-16*x^3+16*x^2-42*x+36)/(x^5-6*x^4+9*x^3),x, algorithm="giac")
Output:
-(x^4 + x^3*e^x - 3*x^3 - 3*x^2*e^x + x^2*log(x^2) - 5*x^2 - 5*x*log(x^2) + 6*log(x^2))/(x^3 - 3*x^2)
Time = 4.24 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {36-42 x+16 x^2-16 x^3+6 x^4-x^5+e^x \left (-9 x^3+6 x^4-x^5\right )+\left (-36+33 x-10 x^2+x^3\right ) \log \left (x^2\right )}{9 x^3-6 x^4+x^5} \, dx=\frac {5}{x-3}-{\mathrm {e}}^x-x-\frac {\ln \left (x^2\right )\,\left (x-2\right )}{x^2} \] Input:
int(-(42*x - 16*x^2 + 16*x^3 - 6*x^4 + x^5 + exp(x)*(9*x^3 - 6*x^4 + x^5) - log(x^2)*(33*x - 10*x^2 + x^3 - 36) - 36)/(9*x^3 - 6*x^4 + x^5),x)
Output:
5/(x - 3) - exp(x) - x - (log(x^2)*(x - 2))/x^2
Time = 0.18 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.06 \[ \int \frac {36-42 x+16 x^2-16 x^3+6 x^4-x^5+e^x \left (-9 x^3+6 x^4-x^5\right )+\left (-36+33 x-10 x^2+x^3\right ) \log \left (x^2\right )}{9 x^3-6 x^4+x^5} \, dx=\frac {-3 e^{x} x^{3}+9 e^{x} x^{2}-3 \,\mathrm {log}\left (x^{2}\right ) x^{2}+15 \,\mathrm {log}\left (x^{2}\right ) x -18 \,\mathrm {log}\left (x^{2}\right )-3 x^{4}+16 x^{3}-6 x^{2}}{3 x^{2} \left (x -3\right )} \] Input:
int(((x^3-10*x^2+33*x-36)*log(x^2)+(-x^5+6*x^4-9*x^3)*exp(x)-x^5+6*x^4-16* x^3+16*x^2-42*x+36)/(x^5-6*x^4+9*x^3),x)
Output:
( - 3*e**x*x**3 + 9*e**x*x**2 - 3*log(x**2)*x**2 + 15*log(x**2)*x - 18*log (x**2) - 3*x**4 + 16*x**3 - 6*x**2)/(3*x**2*(x - 3))