Integrand size = 86, antiderivative size = 22 \[ \int \frac {33 x+20 x^3+3 x^5+\left (-726+33 x-352 x^2+16 x^3-66 x^4+3 x^5\right ) \log (22-x)}{\left (-726 x+33 x^2-440 x^3+20 x^4-66 x^5+3 x^6\right ) \log (22-x)} \, dx=\log \left (\left (x+\frac {2 x}{3 \left (3+x^2\right )}\right ) \log (22-x)\right ) \] Output:
ln((x+2*x/(3*x^2+9))*ln(22-x))
Time = 0.03 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int \frac {33 x+20 x^3+3 x^5+\left (-726+33 x-352 x^2+16 x^3-66 x^4+3 x^5\right ) \log (22-x)}{\left (-726 x+33 x^2-440 x^3+20 x^4-66 x^5+3 x^6\right ) \log (22-x)} \, dx=\log (x)-\log \left (3+x^2\right )+\log \left (11+3 x^2\right )+\log (\log (22-x)) \] Input:
Integrate[(33*x + 20*x^3 + 3*x^5 + (-726 + 33*x - 352*x^2 + 16*x^3 - 66*x^ 4 + 3*x^5)*Log[22 - x])/((-726*x + 33*x^2 - 440*x^3 + 20*x^4 - 66*x^5 + 3* x^6)*Log[22 - x]),x]
Output:
Log[x] - Log[3 + x^2] + Log[11 + 3*x^2] + Log[Log[22 - x]]
Time = 2.24 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.035, Rules used = {2026, 2463, 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.
\(\displaystyle \int \frac {3 x^5+20 x^3+\left (3 x^5-66 x^4+16 x^3-352 x^2+33 x-726\right ) \log (22-x)+33 x}{\left (3 x^6-66 x^5+20 x^4-440 x^3+33 x^2-726 x\right ) \log (22-x)} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {3 x^5+20 x^3+\left (3 x^5-66 x^4+16 x^3-352 x^2+33 x-726\right ) \log (22-x)+33 x}{x \left (3 x^5-66 x^4+20 x^3-440 x^2+33 x-726\right ) \log (22-x)}dx\) |
\(\Big \downarrow \) 2463 |
\(\displaystyle \int \left (\frac {3 x^5+20 x^3+\left (3 x^5-66 x^4+16 x^3-352 x^2+33 x-726\right ) \log (22-x)+33 x}{712481 (x-22) x \log (22-x)}+\frac {(-x-22) \left (3 x^5+20 x^3+\left (3 x^5-66 x^4+16 x^3-352 x^2+33 x-726\right ) \log (22-x)+33 x\right )}{974 x \left (x^2+3\right ) \log (22-x)}+\frac {9 (x+22) \left (3 x^5+20 x^3+\left (3 x^5-66 x^4+16 x^3-352 x^2+33 x-726\right ) \log (22-x)+33 x\right )}{2926 x \left (3 x^2+11\right ) \log (22-x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\log \left (x^2+3\right )+\log \left (3 x^2+11\right )+\log (x)+\log (\log (22-x))\) |
Input:
Int[(33*x + 20*x^3 + 3*x^5 + (-726 + 33*x - 352*x^2 + 16*x^3 - 66*x^4 + 3* x^5)*Log[22 - x])/((-726*x + 33*x^2 - 440*x^3 + 20*x^4 - 66*x^5 + 3*x^6)*L og[22 - x]),x]
Output:
Log[x] - Log[3 + x^2] + Log[11 + 3*x^2] + Log[Log[22 - 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_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u, Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Time = 0.20 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14
method | result | size |
parallelrisch | \(\ln \left (x \right )+\ln \left (\ln \left (22-x \right )\right )+\ln \left (x^{2}+\frac {11}{3}\right )-\ln \left (x^{2}+3\right )\) | \(25\) |
norman | \(-\ln \left (x^{2}+3\right )+\ln \left (x \right )+\ln \left (\ln \left (22-x \right )\right )+\ln \left (3 x^{2}+11\right )\) | \(27\) |
risch | \(-\ln \left (x^{2}+3\right )+\ln \left (3 x^{3}+11 x \right )+\ln \left (\ln \left (22-x \right )\right )\) | \(27\) |
parts | \(-\ln \left (x^{2}+3\right )+\ln \left (x \right )+\ln \left (\ln \left (22-x \right )\right )+\ln \left (3 x^{2}+11\right )\) | \(27\) |
derivativedivides | \(\ln \left (\ln \left (22-x \right )\right )+\ln \left (3 \left (22-x \right )^{2}-1441+132 x \right )+\ln \left (-x \right )-\ln \left (\left (22-x \right )^{2}-481+44 x \right )\) | \(43\) |
default | \(\ln \left (\ln \left (22-x \right )\right )+\ln \left (3 \left (22-x \right )^{2}-1441+132 x \right )+\ln \left (-x \right )-\ln \left (\left (22-x \right )^{2}-481+44 x \right )\) | \(43\) |
Input:
int(((3*x^5-66*x^4+16*x^3-352*x^2+33*x-726)*ln(22-x)+3*x^5+20*x^3+33*x)/(3 *x^6-66*x^5+20*x^4-440*x^3+33*x^2-726*x)/ln(22-x),x,method=_RETURNVERBOSE)
Output:
ln(x)+ln(ln(22-x))+ln(x^2+11/3)-ln(x^2+3)
Time = 0.10 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int \frac {33 x+20 x^3+3 x^5+\left (-726+33 x-352 x^2+16 x^3-66 x^4+3 x^5\right ) \log (22-x)}{\left (-726 x+33 x^2-440 x^3+20 x^4-66 x^5+3 x^6\right ) \log (22-x)} \, dx=\log \left (3 \, x^{3} + 11 \, x\right ) - \log \left (x^{2} + 3\right ) + \log \left (\log \left (-x + 22\right )\right ) \] Input:
integrate(((3*x^5-66*x^4+16*x^3-352*x^2+33*x-726)*log(22-x)+3*x^5+20*x^3+3 3*x)/(3*x^6-66*x^5+20*x^4-440*x^3+33*x^2-726*x)/log(22-x),x, algorithm="fr icas")
Output:
log(3*x^3 + 11*x) - log(x^2 + 3) + log(log(-x + 22))
Time = 0.10 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {33 x+20 x^3+3 x^5+\left (-726+33 x-352 x^2+16 x^3-66 x^4+3 x^5\right ) \log (22-x)}{\left (-726 x+33 x^2-440 x^3+20 x^4-66 x^5+3 x^6\right ) \log (22-x)} \, dx=- \log {\left (x^{2} + 3 \right )} + \log {\left (3 x^{3} + 11 x \right )} + \log {\left (\log {\left (22 - x \right )} \right )} \] Input:
integrate(((3*x**5-66*x**4+16*x**3-352*x**2+33*x-726)*ln(22-x)+3*x**5+20*x **3+33*x)/(3*x**6-66*x**5+20*x**4-440*x**3+33*x**2-726*x)/ln(22-x),x)
Output:
-log(x**2 + 3) + log(3*x**3 + 11*x) + log(log(22 - x))
Time = 0.19 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int \frac {33 x+20 x^3+3 x^5+\left (-726+33 x-352 x^2+16 x^3-66 x^4+3 x^5\right ) \log (22-x)}{\left (-726 x+33 x^2-440 x^3+20 x^4-66 x^5+3 x^6\right ) \log (22-x)} \, dx=\log \left (3 \, x^{2} + 11\right ) - \log \left (x^{2} + 3\right ) + \log \left (x\right ) + \log \left (\log \left (-x + 22\right )\right ) \] Input:
integrate(((3*x^5-66*x^4+16*x^3-352*x^2+33*x-726)*log(22-x)+3*x^5+20*x^3+3 3*x)/(3*x^6-66*x^5+20*x^4-440*x^3+33*x^2-726*x)/log(22-x),x, algorithm="ma xima")
Output:
log(3*x^2 + 11) - log(x^2 + 3) + log(x) + log(log(-x + 22))
Time = 0.12 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.86 \[ \int \frac {33 x+20 x^3+3 x^5+\left (-726+33 x-352 x^2+16 x^3-66 x^4+3 x^5\right ) \log (22-x)}{\left (-726 x+33 x^2-440 x^3+20 x^4-66 x^5+3 x^6\right ) \log (22-x)} \, dx=\log \left (-3 \, {\left (x - 22\right )}^{3} - 198 \, {\left (x - 22\right )}^{2} - 4367 \, x + 63888\right ) - \log \left ({\left (x - 22\right )}^{2} + 44 \, x - 481\right ) + \log \left (\log \left (-x + 22\right )\right ) \] Input:
integrate(((3*x^5-66*x^4+16*x^3-352*x^2+33*x-726)*log(22-x)+3*x^5+20*x^3+3 3*x)/(3*x^6-66*x^5+20*x^4-440*x^3+33*x^2-726*x)/log(22-x),x, algorithm="gi ac")
Output:
log(-3*(x - 22)^3 - 198*(x - 22)^2 - 4367*x + 63888) - log((x - 22)^2 + 44 *x - 481) + log(log(-x + 22))
Time = 3.99 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {33 x+20 x^3+3 x^5+\left (-726+33 x-352 x^2+16 x^3-66 x^4+3 x^5\right ) \log (22-x)}{\left (-726 x+33 x^2-440 x^3+20 x^4-66 x^5+3 x^6\right ) \log (22-x)} \, dx=\ln \left (\ln \left (22-x\right )\right )+\ln \left (x^3+\frac {11\,x}{3}\right )-\ln \left (x^2+3\right ) \] Input:
int(-(33*x + log(22 - x)*(33*x - 352*x^2 + 16*x^3 - 66*x^4 + 3*x^5 - 726) + 20*x^3 + 3*x^5)/(log(22 - x)*(726*x - 33*x^2 + 440*x^3 - 20*x^4 + 66*x^5 - 3*x^6)),x)
Output:
log(log(22 - x)) + log((11*x)/3 + x^3) - log(x^2 + 3)
Time = 0.21 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int \frac {33 x+20 x^3+3 x^5+\left (-726+33 x-352 x^2+16 x^3-66 x^4+3 x^5\right ) \log (22-x)}{\left (-726 x+33 x^2-440 x^3+20 x^4-66 x^5+3 x^6\right ) \log (22-x)} \, dx=\mathrm {log}\left (\mathrm {log}\left (-x +22\right )\right )-\mathrm {log}\left (x^{2}+3\right )+\mathrm {log}\left (3 x^{2}+11\right )+\mathrm {log}\left (x \right ) \] Input:
int(((3*x^5-66*x^4+16*x^3-352*x^2+33*x-726)*log(22-x)+3*x^5+20*x^3+33*x)/( 3*x^6-66*x^5+20*x^4-440*x^3+33*x^2-726*x)/log(22-x),x)
Output:
log(log( - x + 22)) - log(x**2 + 3) + log(3*x**2 + 11) + log(x)