Integrand size = 59, antiderivative size = 24 \[ \int \frac {\left (-48 x+2 x^2+2 x^3\right ) \log (x)+\left (-48 x+3 x^2+4 x^3\right ) \log ^2(x)}{-64+\left (-24 x^2+x^3+x^4\right ) \log ^2(x)} \, dx=\log \left (16+\frac {1}{2} x^2 \left (12-\frac {1}{2} x (1+x)\right ) \log ^2(x)\right ) \] Output:
ln(16+1/2*x^2*ln(x)^2*(12-1/2*x*(1+x)))
\[ \int \frac {\left (-48 x+2 x^2+2 x^3\right ) \log (x)+\left (-48 x+3 x^2+4 x^3\right ) \log ^2(x)}{-64+\left (-24 x^2+x^3+x^4\right ) \log ^2(x)} \, dx=\int \frac {\left (-48 x+2 x^2+2 x^3\right ) \log (x)+\left (-48 x+3 x^2+4 x^3\right ) \log ^2(x)}{-64+\left (-24 x^2+x^3+x^4\right ) \log ^2(x)} \, dx \] Input:
Integrate[((-48*x + 2*x^2 + 2*x^3)*Log[x] + (-48*x + 3*x^2 + 4*x^3)*Log[x] ^2)/(-64 + (-24*x^2 + x^3 + x^4)*Log[x]^2),x]
Output:
Integrate[((-48*x + 2*x^2 + 2*x^3)*Log[x] + (-48*x + 3*x^2 + 4*x^3)*Log[x] ^2)/(-64 + (-24*x^2 + x^3 + x^4)*Log[x]^2), x]
Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {7259, 16}
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.
\(\displaystyle \int \frac {\left (4 x^3+3 x^2-48 x\right ) \log ^2(x)+\left (2 x^3+2 x^2-48 x\right ) \log (x)}{\left (x^4+x^3-24 x^2\right ) \log ^2(x)-64} \, dx\) |
\(\Big \downarrow \) 7259 |
\(\displaystyle \int \frac {1}{-\left (\left (-x^4-x^3+24 x^2\right ) \log ^2(x)\right )-64}d\left (-\left (\left (-x^4-x^3+24 x^2\right ) \log ^2(x)\right )\right )\) |
\(\Big \downarrow \) 16 |
\(\displaystyle \log \left (\left (-x^4-x^3+24 x^2\right ) \log ^2(x)+64\right )\) |
Input:
Int[((-48*x + 2*x^2 + 2*x^3)*Log[x] + (-48*x + 3*x^2 + 4*x^3)*Log[x]^2)/(- 64 + (-24*x^2 + x^3 + x^4)*Log[x]^2),x]
Output:
Log[64 + (24*x^2 - x^3 - x^4)*Log[x]^2]
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(u_)*((a_) + (b_.)*(v_)^(p_.)*(w_)^(p_.))^(m_.), x_Symbol] :> With[{c = Simplify[u/(w*D[v, x] + v*D[w, x])]}, Simp[c Subst[Int[(a + b*x^p)^m, x] , x, v*w], x] /; FreeQ[c, x]] /; FreeQ[{a, b, m, p}, x] && IntegerQ[p]
Time = 0.30 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.21
method | result | size |
default | \(\ln \left (x^{4} \ln \left (x \right )^{2}+x^{3} \ln \left (x \right )^{2}-24 x^{2} \ln \left (x \right )^{2}-64\right )\) | \(29\) |
norman | \(\ln \left (x^{4} \ln \left (x \right )^{2}+x^{3} \ln \left (x \right )^{2}-24 x^{2} \ln \left (x \right )^{2}-64\right )\) | \(29\) |
parallelrisch | \(\ln \left (x^{4} \ln \left (x \right )^{2}+x^{3} \ln \left (x \right )^{2}-24 x^{2} \ln \left (x \right )^{2}-64\right )\) | \(29\) |
risch | \(2 \ln \left (x \right )+\ln \left (x^{2}+x -24\right )+\ln \left (\ln \left (x \right )^{2}-\frac {64}{x^{2} \left (x^{2}+x -24\right )}\right )\) | \(32\) |
Input:
int(((4*x^3+3*x^2-48*x)*ln(x)^2+(2*x^3+2*x^2-48*x)*ln(x))/((x^4+x^3-24*x^2 )*ln(x)^2-64),x,method=_RETURNVERBOSE)
Output:
ln(x^4*ln(x)^2+x^3*ln(x)^2-24*x^2*ln(x)^2-64)
Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (19) = 38\).
Time = 0.10 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.96 \[ \int \frac {\left (-48 x+2 x^2+2 x^3\right ) \log (x)+\left (-48 x+3 x^2+4 x^3\right ) \log ^2(x)}{-64+\left (-24 x^2+x^3+x^4\right ) \log ^2(x)} \, dx=\log \left (x^{2} + x - 24\right ) + 2 \, \log \left (x\right ) + \log \left (\frac {{\left (x^{4} + x^{3} - 24 \, x^{2}\right )} \log \left (x\right )^{2} - 64}{x^{4} + x^{3} - 24 \, x^{2}}\right ) \] Input:
integrate(((4*x^3+3*x^2-48*x)*log(x)^2+(2*x^3+2*x^2-48*x)*log(x))/((x^4+x^ 3-24*x^2)*log(x)^2-64),x, algorithm="fricas")
Output:
log(x^2 + x - 24) + 2*log(x) + log(((x^4 + x^3 - 24*x^2)*log(x)^2 - 64)/(x ^4 + x^3 - 24*x^2))
Time = 0.36 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.33 \[ \int \frac {\left (-48 x+2 x^2+2 x^3\right ) \log (x)+\left (-48 x+3 x^2+4 x^3\right ) \log ^2(x)}{-64+\left (-24 x^2+x^3+x^4\right ) \log ^2(x)} \, dx=2 \log {\left (x \right )} + \log {\left (\log {\left (x \right )}^{2} - \frac {64}{x^{4} + x^{3} - 24 x^{2}} \right )} + \log {\left (x^{2} + x - 24 \right )} \] Input:
integrate(((4*x**3+3*x**2-48*x)*ln(x)**2+(2*x**3+2*x**2-48*x)*ln(x))/((x** 4+x**3-24*x**2)*ln(x)**2-64),x)
Output:
2*log(x) + log(log(x)**2 - 64/(x**4 + x**3 - 24*x**2)) + log(x**2 + x - 24 )
Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (19) = 38\).
Time = 0.06 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.96 \[ \int \frac {\left (-48 x+2 x^2+2 x^3\right ) \log (x)+\left (-48 x+3 x^2+4 x^3\right ) \log ^2(x)}{-64+\left (-24 x^2+x^3+x^4\right ) \log ^2(x)} \, dx=\log \left (x^{2} + x - 24\right ) + 2 \, \log \left (x\right ) + \log \left (\frac {{\left (x^{4} + x^{3} - 24 \, x^{2}\right )} \log \left (x\right )^{2} - 64}{x^{4} + x^{3} - 24 \, x^{2}}\right ) \] Input:
integrate(((4*x^3+3*x^2-48*x)*log(x)^2+(2*x^3+2*x^2-48*x)*log(x))/((x^4+x^ 3-24*x^2)*log(x)^2-64),x, algorithm="maxima")
Output:
log(x^2 + x - 24) + 2*log(x) + log(((x^4 + x^3 - 24*x^2)*log(x)^2 - 64)/(x ^4 + x^3 - 24*x^2))
Time = 0.14 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.17 \[ \int \frac {\left (-48 x+2 x^2+2 x^3\right ) \log (x)+\left (-48 x+3 x^2+4 x^3\right ) \log ^2(x)}{-64+\left (-24 x^2+x^3+x^4\right ) \log ^2(x)} \, dx=\log \left (x^{4} \log \left (x\right )^{2} + x^{3} \log \left (x\right )^{2} - 24 \, x^{2} \log \left (x\right )^{2} - 64\right ) \] Input:
integrate(((4*x^3+3*x^2-48*x)*log(x)^2+(2*x^3+2*x^2-48*x)*log(x))/((x^4+x^ 3-24*x^2)*log(x)^2-64),x, algorithm="giac")
Output:
log(x^4*log(x)^2 + x^3*log(x)^2 - 24*x^2*log(x)^2 - 64)
Time = 4.07 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.17 \[ \int \frac {\left (-48 x+2 x^2+2 x^3\right ) \log (x)+\left (-48 x+3 x^2+4 x^3\right ) \log ^2(x)}{-64+\left (-24 x^2+x^3+x^4\right ) \log ^2(x)} \, dx=\ln \left (x^4\,{\ln \left (x\right )}^2+x^3\,{\ln \left (x\right )}^2-24\,x^2\,{\ln \left (x\right )}^2-64\right ) \] Input:
int((log(x)^2*(3*x^2 - 48*x + 4*x^3) + log(x)*(2*x^2 - 48*x + 2*x^3))/(log (x)^2*(x^3 - 24*x^2 + x^4) - 64),x)
Output:
log(x^3*log(x)^2 - 24*x^2*log(x)^2 + x^4*log(x)^2 - 64)
\[ \int \frac {\left (-48 x+2 x^2+2 x^3\right ) \log (x)+\left (-48 x+3 x^2+4 x^3\right ) \log ^2(x)}{-64+\left (-24 x^2+x^3+x^4\right ) \log ^2(x)} \, dx=4 \left (\int \frac {\mathrm {log}\left (x \right )^{2} x^{3}}{\mathrm {log}\left (x \right )^{2} x^{4}+\mathrm {log}\left (x \right )^{2} x^{3}-24 \mathrm {log}\left (x \right )^{2} x^{2}-64}d x \right )+3 \left (\int \frac {\mathrm {log}\left (x \right )^{2} x^{2}}{\mathrm {log}\left (x \right )^{2} x^{4}+\mathrm {log}\left (x \right )^{2} x^{3}-24 \mathrm {log}\left (x \right )^{2} x^{2}-64}d x \right )-48 \left (\int \frac {\mathrm {log}\left (x \right )^{2} x}{\mathrm {log}\left (x \right )^{2} x^{4}+\mathrm {log}\left (x \right )^{2} x^{3}-24 \mathrm {log}\left (x \right )^{2} x^{2}-64}d x \right )+2 \left (\int \frac {\mathrm {log}\left (x \right ) x^{3}}{\mathrm {log}\left (x \right )^{2} x^{4}+\mathrm {log}\left (x \right )^{2} x^{3}-24 \mathrm {log}\left (x \right )^{2} x^{2}-64}d x \right )+2 \left (\int \frac {\mathrm {log}\left (x \right ) x^{2}}{\mathrm {log}\left (x \right )^{2} x^{4}+\mathrm {log}\left (x \right )^{2} x^{3}-24 \mathrm {log}\left (x \right )^{2} x^{2}-64}d x \right )-48 \left (\int \frac {\mathrm {log}\left (x \right ) x}{\mathrm {log}\left (x \right )^{2} x^{4}+\mathrm {log}\left (x \right )^{2} x^{3}-24 \mathrm {log}\left (x \right )^{2} x^{2}-64}d x \right ) \] Input:
int(((4*x^3+3*x^2-48*x)*log(x)^2+(2*x^3+2*x^2-48*x)*log(x))/((x^4+x^3-24*x ^2)*log(x)^2-64),x)
Output:
4*int((log(x)**2*x**3)/(log(x)**2*x**4 + log(x)**2*x**3 - 24*log(x)**2*x** 2 - 64),x) + 3*int((log(x)**2*x**2)/(log(x)**2*x**4 + log(x)**2*x**3 - 24* log(x)**2*x**2 - 64),x) - 48*int((log(x)**2*x)/(log(x)**2*x**4 + log(x)**2 *x**3 - 24*log(x)**2*x**2 - 64),x) + 2*int((log(x)*x**3)/(log(x)**2*x**4 + log(x)**2*x**3 - 24*log(x)**2*x**2 - 64),x) + 2*int((log(x)*x**2)/(log(x) **2*x**4 + log(x)**2*x**3 - 24*log(x)**2*x**2 - 64),x) - 48*int((log(x)*x) /(log(x)**2*x**4 + log(x)**2*x**3 - 24*log(x)**2*x**2 - 64),x)