Integrand size = 69, antiderivative size = 21 \[ \int \frac {9 x^2+e \left (-54 x+18 x^2\right )+\left (-54 x+18 x^2\right ) \log (3-x)+\left (108 x-36 x^2\right ) \log \left (x^2\right )+\left (54 x-18 x^2\right ) \log ^2\left (x^2\right )}{-3+x} \, dx=9 x^2 \left (e+\log (3-x)-\log ^2\left (x^2\right )\right ) \] Output:
9*(ln(3-x)-ln(x^2)^2+exp(1))*x^2
Time = 0.04 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.38 \[ \int \frac {9 x^2+e \left (-54 x+18 x^2\right )+\left (-54 x+18 x^2\right ) \log (3-x)+\left (108 x-36 x^2\right ) \log \left (x^2\right )+\left (54 x-18 x^2\right ) \log ^2\left (x^2\right )}{-3+x} \, dx=9 \left (e x^2+x^2 \log (3-x)-x^2 \log ^2\left (x^2\right )\right ) \] Input:
Integrate[(9*x^2 + E*(-54*x + 18*x^2) + (-54*x + 18*x^2)*Log[3 - x] + (108 *x - 36*x^2)*Log[x^2] + (54*x - 18*x^2)*Log[x^2]^2)/(-3 + x),x]
Output:
9*(E*x^2 + x^2*Log[3 - x] - x^2*Log[x^2]^2)
Leaf count is larger than twice the leaf count of optimal. \(43\) vs. \(2(21)=42\).
Time = 0.61 (sec) , antiderivative size = 43, normalized size of antiderivative = 2.05, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.058, Rules used = {7292, 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.
\(\displaystyle \int \frac {9 x^2+e \left (18 x^2-54 x\right )+\left (54 x-18 x^2\right ) \log ^2\left (x^2\right )+\left (18 x^2-54 x\right ) \log (3-x)+\left (108 x-36 x^2\right ) \log \left (x^2\right )}{x-3} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {9 x \left (2 x \log ^2\left (x^2\right )-6 \log ^2\left (x^2\right )+4 x \log \left (x^2\right )-12 \log \left (x^2\right )-(1+2 e) x-2 x \log (3-x)+6 \log (3-x)+6 e\right )}{3-x}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 9 \int \frac {x \left (2 x \log ^2\left (x^2\right )-6 \log ^2\left (x^2\right )+4 x \log \left (x^2\right )-12 \log \left (x^2\right )-(1+2 e) x-2 x \log (3-x)+6 \log (3-x)+6 e\right )}{3-x}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle 9 \int \left (-2 x \log ^2\left (x^2\right )-4 x \log \left (x^2\right )+\frac {x (-2 \log (3-x) x-(1+2 e) x+6 \log (3-x)+6 e)}{3-x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 9 \left (\frac {1}{2} (1+2 e) x^2-\frac {x^2}{2}+x^2 \left (-\log ^2\left (x^2\right )\right )+x^2 \log (3-x)\right )\) |
Input:
Int[(9*x^2 + E*(-54*x + 18*x^2) + (-54*x + 18*x^2)*Log[3 - x] + (108*x - 3 6*x^2)*Log[x^2] + (54*x - 18*x^2)*Log[x^2]^2)/(-3 + x),x]
Output:
9*(-1/2*x^2 + ((1 + 2*E)*x^2)/2 + x^2*Log[3 - x] - x^2*Log[x^2]^2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Time = 0.81 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.67
method | result | size |
parallelrisch | \(-9 x^{2} \ln \left (x^{2}\right )^{2}+9 x^{2} {\mathrm e}+9 \ln \left (-x +3\right ) x^{2}-81 \,{\mathrm e}\) | \(35\) |
default | \(-9 x^{2} \ln \left (x^{2}\right )^{2}-54 \ln \left (-x +3\right ) \left (-x +3\right )+\frac {243}{2}+9 \ln \left (-x +3\right ) \left (-x +3\right )^{2}+9 x^{2} {\mathrm e}+81 \ln \left (-3+x \right )\) | \(55\) |
parts | \(-9 x^{2} \ln \left (x^{2}\right )^{2}-54 \ln \left (-x +3\right ) \left (-x +3\right )+\frac {243}{2}+9 \ln \left (-x +3\right ) \left (-x +3\right )^{2}+9 x^{2} {\mathrm e}+81 \ln \left (-3+x \right )\) | \(55\) |
risch | \(9 \ln \left (-x +3\right ) x^{2}-36 x^{2} \ln \left (x \right )^{2}+18 i \pi \,x^{2} \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right ) \ln \left (x \right )-36 i \pi \,x^{2} \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2} \ln \left (x \right )+18 i \pi \,x^{2} \operatorname {csgn}\left (i x^{2}\right )^{3} \ln \left (x \right )+\frac {9 \pi ^{2} x^{2} \operatorname {csgn}\left (i x \right )^{4} \operatorname {csgn}\left (i x^{2}\right )^{2}}{4}-9 \pi ^{2} x^{2} \operatorname {csgn}\left (i x \right )^{3} \operatorname {csgn}\left (i x^{2}\right )^{3}+\frac {27 \pi ^{2} x^{2} \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )^{4}}{2}-9 \pi ^{2} x^{2} \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{5}+\frac {9 \pi ^{2} x^{2} \operatorname {csgn}\left (i x^{2}\right )^{6}}{4}+9 x^{2} {\mathrm e}\) | \(204\) |
orering | \(\frac {\left (28 x^{5}-246 x^{4}+675 x^{3}-432 x^{2}-972 x +972\right ) \left (\left (-18 x^{2}+54 x \right ) \ln \left (x^{2}\right )^{2}+\left (-36 x^{2}+108 x \right ) \ln \left (x^{2}\right )+\left (18 x^{2}-54 x \right ) \ln \left (-x +3\right )+\left (18 x^{2}-54 x \right ) {\mathrm e}+9 x^{2}\right )}{2 x \left (16 x^{3}-141 x^{2}+405 x -432\right ) \left (-3+x \right )}-\frac {3 \left (4 x^{5}-35 x^{4}+90 x^{3}-324 x +324\right ) \left (\frac {\left (-36 x +54\right ) \ln \left (x^{2}\right )^{2}+\frac {4 \left (-18 x^{2}+54 x \right ) \ln \left (x^{2}\right )}{x}+\left (-72 x +108\right ) \ln \left (x^{2}\right )+\frac {-72 x^{2}+216 x}{x}+\left (36 x -54\right ) \ln \left (-x +3\right )-\frac {18 x^{2}-54 x}{-x +3}+\left (36 x -54\right ) {\mathrm e}+18 x}{-3+x}-\frac {\left (-18 x^{2}+54 x \right ) \ln \left (x^{2}\right )^{2}+\left (-36 x^{2}+108 x \right ) \ln \left (x^{2}\right )+\left (18 x^{2}-54 x \right ) \ln \left (-x +3\right )+\left (18 x^{2}-54 x \right ) {\mathrm e}+9 x^{2}}{\left (-3+x \right )^{2}}\right )}{2 \left (16 x^{3}-141 x^{2}+405 x -432\right )}+\frac {\left (8 x^{4}-45 x^{3}+432 x -648\right ) x \left (-3+x \right ) \left (\frac {-36 \ln \left (x^{2}\right )^{2}+\frac {8 \left (-36 x +54\right ) \ln \left (x^{2}\right )}{x}+\frac {-144 x^{2}+432 x}{x^{2}}-\frac {4 \left (-18 x^{2}+54 x \right ) \ln \left (x^{2}\right )}{x^{2}}-72 \ln \left (x^{2}\right )+\frac {-288 x +432}{x}-\frac {2 \left (-36 x^{2}+108 x \right )}{x^{2}}+36 \ln \left (-x +3\right )-\frac {2 \left (36 x -54\right )}{-x +3}-\frac {18 x^{2}-54 x}{\left (-x +3\right )^{2}}+36 \,{\mathrm e}+18}{-3+x}-\frac {2 \left (\left (-36 x +54\right ) \ln \left (x^{2}\right )^{2}+\frac {4 \left (-18 x^{2}+54 x \right ) \ln \left (x^{2}\right )}{x}+\left (-72 x +108\right ) \ln \left (x^{2}\right )+\frac {-72 x^{2}+216 x}{x}+\left (36 x -54\right ) \ln \left (-x +3\right )-\frac {18 x^{2}-54 x}{-x +3}+\left (36 x -54\right ) {\mathrm e}+18 x \right )}{\left (-3+x \right )^{2}}+\frac {2 \left (-18 x^{2}+54 x \right ) \ln \left (x^{2}\right )^{2}+2 \left (-36 x^{2}+108 x \right ) \ln \left (x^{2}\right )+2 \left (18 x^{2}-54 x \right ) \ln \left (-x +3\right )+2 \left (18 x^{2}-54 x \right ) {\mathrm e}+18 x^{2}}{\left (-3+x \right )^{3}}\right )}{64 x^{3}-564 x^{2}+1620 x -1728}\) | \(680\) |
Input:
int(((-18*x^2+54*x)*ln(x^2)^2+(-36*x^2+108*x)*ln(x^2)+(18*x^2-54*x)*ln(-x+ 3)+(18*x^2-54*x)*exp(1)+9*x^2)/(-3+x),x,method=_RETURNVERBOSE)
Output:
-9*x^2*ln(x^2)^2+9*x^2*exp(1)+9*ln(-x+3)*x^2-81*exp(1)
Time = 0.09 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.43 \[ \int \frac {9 x^2+e \left (-54 x+18 x^2\right )+\left (-54 x+18 x^2\right ) \log (3-x)+\left (108 x-36 x^2\right ) \log \left (x^2\right )+\left (54 x-18 x^2\right ) \log ^2\left (x^2\right )}{-3+x} \, dx=-9 \, x^{2} \log \left (x^{2}\right )^{2} + 9 \, x^{2} e + 9 \, x^{2} \log \left (-x + 3\right ) \] Input:
integrate(((-18*x^2+54*x)*log(x^2)^2+(-36*x^2+108*x)*log(x^2)+(18*x^2-54*x )*log(3-x)+(18*x^2-54*x)*exp(1)+9*x^2)/(-3+x),x, algorithm="fricas")
Output:
-9*x^2*log(x^2)^2 + 9*x^2*e + 9*x^2*log(-x + 3)
Time = 0.26 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.76 \[ \int \frac {9 x^2+e \left (-54 x+18 x^2\right )+\left (-54 x+18 x^2\right ) \log (3-x)+\left (108 x-36 x^2\right ) \log \left (x^2\right )+\left (54 x-18 x^2\right ) \log ^2\left (x^2\right )}{-3+x} \, dx=- 9 x^{2} \log {\left (x^{2} \right )}^{2} + 9 e x^{2} + \left (9 x^{2} - 27\right ) \log {\left (3 - x \right )} + 27 \log {\left (x - 3 \right )} \] Input:
integrate(((-18*x**2+54*x)*ln(x**2)**2+(-36*x**2+108*x)*ln(x**2)+(18*x**2- 54*x)*ln(3-x)+(18*x**2-54*x)*exp(1)+9*x**2)/(-3+x),x)
Output:
-9*x**2*log(x**2)**2 + 9*E*x**2 + (9*x**2 - 27)*log(3 - x) + 27*log(x - 3)
Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (24) = 48\).
Time = 0.08 (sec) , antiderivative size = 76, normalized size of antiderivative = 3.62 \[ \int \frac {9 x^2+e \left (-54 x+18 x^2\right )+\left (-54 x+18 x^2\right ) \log (3-x)+\left (108 x-36 x^2\right ) \log \left (x^2\right )+\left (54 x-18 x^2\right ) \log ^2\left (x^2\right )}{-3+x} \, dx=-36 \, x^{2} \log \left (x\right )^{2} + 9 \, {\left (x^{2} + 6 \, x + 18 \, \log \left (x - 3\right )\right )} e - 54 \, {\left (x + 3 \, \log \left (x - 3\right )\right )} e + 9 \, {\left (x^{2} + 6 \, x + 18 \, \log \left (x - 3\right )\right )} \log \left (-x + 3\right ) - 54 \, {\left (x + 3 \, \log \left (x - 3\right )\right )} \log \left (-x + 3\right ) \] Input:
integrate(((-18*x^2+54*x)*log(x^2)^2+(-36*x^2+108*x)*log(x^2)+(18*x^2-54*x )*log(3-x)+(18*x^2-54*x)*exp(1)+9*x^2)/(-3+x),x, algorithm="maxima")
Output:
-36*x^2*log(x)^2 + 9*(x^2 + 6*x + 18*log(x - 3))*e - 54*(x + 3*log(x - 3)) *e + 9*(x^2 + 6*x + 18*log(x - 3))*log(-x + 3) - 54*(x + 3*log(x - 3))*log (-x + 3)
Time = 0.12 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.43 \[ \int \frac {9 x^2+e \left (-54 x+18 x^2\right )+\left (-54 x+18 x^2\right ) \log (3-x)+\left (108 x-36 x^2\right ) \log \left (x^2\right )+\left (54 x-18 x^2\right ) \log ^2\left (x^2\right )}{-3+x} \, dx=-9 \, x^{2} \log \left (x^{2}\right )^{2} + 9 \, x^{2} e + 9 \, x^{2} \log \left (-x + 3\right ) \] Input:
integrate(((-18*x^2+54*x)*log(x^2)^2+(-36*x^2+108*x)*log(x^2)+(18*x^2-54*x )*log(3-x)+(18*x^2-54*x)*exp(1)+9*x^2)/(-3+x),x, algorithm="giac")
Output:
-9*x^2*log(x^2)^2 + 9*x^2*e + 9*x^2*log(-x + 3)
Time = 2.48 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.05 \[ \int \frac {9 x^2+e \left (-54 x+18 x^2\right )+\left (-54 x+18 x^2\right ) \log (3-x)+\left (108 x-36 x^2\right ) \log \left (x^2\right )+\left (54 x-18 x^2\right ) \log ^2\left (x^2\right )}{-3+x} \, dx=9\,x^2\,\left (-{\ln \left (x^2\right )}^2+\mathrm {e}+\ln \left (3-x\right )\right ) \] Input:
int((log(x^2)*(108*x - 36*x^2) - exp(1)*(54*x - 18*x^2) - log(3 - x)*(54*x - 18*x^2) + log(x^2)^2*(54*x - 18*x^2) + 9*x^2)/(x - 3),x)
Output:
9*x^2*(exp(1) + log(3 - x) - log(x^2)^2)
Time = 0.16 (sec) , antiderivative size = 43, normalized size of antiderivative = 2.05 \[ \int \frac {9 x^2+e \left (-54 x+18 x^2\right )+\left (-54 x+18 x^2\right ) \log (3-x)+\left (108 x-36 x^2\right ) \log \left (x^2\right )+\left (54 x-18 x^2\right ) \log ^2\left (x^2\right )}{-3+x} \, dx=-9 \mathrm {log}\left (x^{2}\right )^{2} x^{2}+9 \,\mathrm {log}\left (-x +3\right ) x^{2}-81 \,\mathrm {log}\left (-x +3\right )+81 \,\mathrm {log}\left (x -3\right )+9 e \,x^{2} \] Input:
int(((-18*x^2+54*x)*log(x^2)^2+(-36*x^2+108*x)*log(x^2)+(18*x^2-54*x)*log( 3-x)+(18*x^2-54*x)*exp(1)+9*x^2)/(-3+x),x)
Output:
9*( - log(x**2)**2*x**2 + log( - x + 3)*x**2 - 9*log( - x + 3) + 9*log(x - 3) + e*x**2)