Integrand size = 49, antiderivative size = 204 \[ \int \frac {\log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{(a+b x) (e+f x)} \, dx=-\frac {\log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (1-\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{b e-a f}-\frac {2 \log \left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \operatorname {PolyLog}\left (2,\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{b e-a f}+\frac {2 \operatorname {PolyLog}\left (3,\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{b e-a f} \]
-ln((-a*f+b*e)*(d*x+c)/(-c*f+d*e)/(b*x+a))^2*ln(1-(-a*f+b*e)*(d*x+c)/(-c*f +d*e)/(b*x+a))/(-a*f+b*e)-2*ln((-a*f+b*e)*(d*x+c)/(-c*f+d*e)/(b*x+a))*poly log(2,(-a*f+b*e)*(d*x+c)/(-c*f+d*e)/(b*x+a))/(-a*f+b*e)+2*polylog(3,(-a*f+ b*e)*(d*x+c)/(-c*f+d*e)/(b*x+a))/(-a*f+b*e)
Leaf count is larger than twice the leaf count of optimal. \(1636\) vs. \(2(204)=408\).
Time = 0.63 (sec) , antiderivative size = 1636, normalized size of antiderivative = 8.02 \[ \int \frac {\log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{(a+b x) (e+f x)} \, dx =\text {Too large to display} \]
(-2*Log[a/b + x]^3 + 3*Log[a/b + x]^2*Log[a + b*x] - 6*Log[a/b + x]*Log[c/ d + x]*Log[a + b*x] + 3*Log[c/d + x]^2*Log[a + b*x] + 6*Log[a/b + x]*Log[c /d + x]*Log[(d*(a + b*x))/(-(b*c) + a*d)] - 3*Log[c/d + x]^2*Log[(d*(a + b *x))/(-(b*c) + a*d)] + 3*Log[a/b + x]^2*Log[(b*(c + d*x))/(b*c - a*d)] - 3 *Log[a/b + x]^2*Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))] + 6*L og[a/b + x]*Log[a + b*x]*Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x ))] - 6*Log[c/d + x]*Log[a + b*x]*Log[((b*e - a*f)*(c + d*x))/((d*e - c*f) *(a + b*x))] + 6*Log[c/d + x]*Log[(d*(a + b*x))/(-(b*c) + a*d)]*Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))] + 3*Log[(-(b*c) + a*d)/(d*(a + b*x))]*Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]^2 + 3*Log[a + b*x]*Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]^2 - 3*Log[a/b + x]^2*Log[e + f*x] + 6*Log[a/b + x]*Log[c/d + x]*Log[e + f*x] - 3*Log[c/d + x]^2*Log[e + f*x] - 6*Log[a/b + x]*Log[((b*e - a*f)*(c + d*x))/((d*e - c* f)*(a + b*x))]*Log[e + f*x] + 6*Log[c/d + x]*Log[((b*e - a*f)*(c + d*x))/( (d*e - c*f)*(a + b*x))]*Log[e + f*x] - 3*Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]^2*Log[e + f*x] + 3*Log[a/b + x]^2*Log[(b*(e + f*x))/(b *e - a*f)] - 6*Log[a/b + x]*Log[(f*(c + d*x))/(-(d*e) + c*f)]*Log[(b*(e + f*x))/(b*e - a*f)] + 3*Log[(f*(c + d*x))/(-(d*e) + c*f)]^2*Log[(b*(e + f*x ))/(b*e - a*f)] + 6*Log[a/b + x]*Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)* (a + b*x))]*Log[(b*(e + f*x))/(b*e - a*f)] - 6*Log[(f*(c + d*x))/(-(d*e...
Time = 0.59 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.96, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.082, Rules used = {2966, 2754, 2821, 7143}
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.
| \(\displaystyle \int \frac {\log ^2\left (\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{(a+b x) (e+f x)} \, dx\) |
| \(\Big \downarrow \) 2966 |
| \(\displaystyle \int \frac {\log ^2\left (\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{-\frac {(c+d x) (b e-a f)}{a+b x}-c f+d e}d\frac {c+d x}{a+b x}\) |
| \(\Big \downarrow \) 2754 |
| \(\displaystyle \frac {2 \int \frac {(a+b x) \log \left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (1-\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{c+d x}d\frac {c+d x}{a+b x}}{b e-a f}-\frac {\log ^2\left (\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right ) \log \left (1-\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{b e-a f}\) |
| \(\Big \downarrow \) 2821 |
| \(\displaystyle \frac {2 \left (\int \frac {(a+b x) \operatorname {PolyLog}\left (2,\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{c+d x}d\frac {c+d x}{a+b x}-\operatorname {PolyLog}\left (2,\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )\right )}{b e-a f}-\frac {\log ^2\left (\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right ) \log \left (1-\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{b e-a f}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {2 \left (\operatorname {PolyLog}\left (3,\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )-\operatorname {PolyLog}\left (2,\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )\right )}{b e-a f}-\frac {\log ^2\left (\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right ) \log \left (1-\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{b e-a f}\) |
-((Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]^2*Log[1 - ((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))])/(b*e - a*f)) + (2*(-(Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]*PolyLog[2, ((b*e - a*f)*(c + d*x ))/((d*e - c*f)*(a + b*x))]) + PolyLog[3, ((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]))/(b*e - a*f)
3.2.6.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symb ol] :> Simp[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^p/e), x] - Simp[b*n*(p/e) Int[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0]
Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b _.))^(p_.))/(x_), x_Symbol] :> Simp[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c *x^n])^p/m), x] + Simp[b*n*(p/m) Int[PolyLog[2, (-d)*f*x^m]*((a + b*Log[c *x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]
Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_ )]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.)*((h_.) + (i_.)*(x_))^(q_.), x_Sy mbol] :> Simp[(b*c - a*d)^(q + 1)*(i/d)^q Subst[Int[(b*f - a*g - (d*f - c *g)*x)^m*((A + B*Log[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, A, B, n}, x] && EqQ[n + mn , 0] && IGtQ[n, 0] && NeQ[b*c - a*d, 0] && IntegersQ[m, q] && IGtQ[p, 0] && EqQ[d*h - c*i, 0]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
Time = 5.45 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.66
| method | result | size |
| derivativedivides | \(\frac {\ln \left (-\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}+\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}\right )^{2} \ln \left (\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}-\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}+1\right )+2 \ln \left (-\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}+\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}\right ) \operatorname {Li}_{2}\left (-\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}+\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}\right )-2 \,\operatorname {Li}_{3}\left (-\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}+\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}\right )}{a f -b e}\) | \(338\) |
| default | \(\frac {\ln \left (-\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}+\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}\right )^{2} \ln \left (\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}-\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}+1\right )+2 \ln \left (-\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}+\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}\right ) \operatorname {Li}_{2}\left (-\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}+\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}\right )-2 \,\operatorname {Li}_{3}\left (-\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}+\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}\right )}{a f -b e}\) | \(338\) |
| risch | \(\frac {\ln \left (-\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}+\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}\right )^{2} \ln \left (\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}-\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}+1\right )}{a f -b e}+\frac {2 \ln \left (-\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}+\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}\right ) \operatorname {Li}_{2}\left (-\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}+\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}\right )}{a f -b e}-\frac {2 \,\operatorname {Li}_{3}\left (-\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}+\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}\right )}{a f -b e}\) | \(357\) |
1/(a*f-b*e)*(ln(-(a*f-b*e)*(a*d-b*c)/b/(c*f-d*e)/(b*x+a)+d/(c*f-d*e)*(a*f- b*e)/b)^2*ln((a*f-b*e)*(a*d-b*c)/b/(c*f-d*e)/(b*x+a)-d/(c*f-d*e)*(a*f-b*e) /b+1)+2*ln(-(a*f-b*e)*(a*d-b*c)/b/(c*f-d*e)/(b*x+a)+d/(c*f-d*e)*(a*f-b*e)/ b)*polylog(2,-(a*f-b*e)*(a*d-b*c)/b/(c*f-d*e)/(b*x+a)+d/(c*f-d*e)*(a*f-b*e )/b)-2*polylog(3,-(a*f-b*e)*(a*d-b*c)/b/(c*f-d*e)/(b*x+a)+d/(c*f-d*e)*(a*f -b*e)/b))
Time = 0.29 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.29 \[ \int \frac {\log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{(a+b x) (e+f x)} \, dx=-\frac {\log \left (\frac {b c e - a c f + {\left (b d e - a d f\right )} x}{a d e - a c f + {\left (b d e - b c f\right )} x}\right )^{2} \log \left (-\frac {{\left (b c - a d\right )} f x + {\left (b c - a d\right )} e}{a d e - a c f + {\left (b d e - b c f\right )} x}\right ) + 2 \, {\rm Li}_2\left (\frac {{\left (b c - a d\right )} f x + {\left (b c - a d\right )} e}{a d e - a c f + {\left (b d e - b c f\right )} x} + 1\right ) \log \left (\frac {b c e - a c f + {\left (b d e - a d f\right )} x}{a d e - a c f + {\left (b d e - b c f\right )} x}\right ) - 2 \, {\rm polylog}\left (3, \frac {b c e - a c f + {\left (b d e - a d f\right )} x}{a d e - a c f + {\left (b d e - b c f\right )} x}\right )}{b e - a f} \]
-(log((b*c*e - a*c*f + (b*d*e - a*d*f)*x)/(a*d*e - a*c*f + (b*d*e - b*c*f) *x))^2*log(-((b*c - a*d)*f*x + (b*c - a*d)*e)/(a*d*e - a*c*f + (b*d*e - b* c*f)*x)) + 2*dilog(((b*c - a*d)*f*x + (b*c - a*d)*e)/(a*d*e - a*c*f + (b*d *e - b*c*f)*x) + 1)*log((b*c*e - a*c*f + (b*d*e - a*d*f)*x)/(a*d*e - a*c*f + (b*d*e - b*c*f)*x)) - 2*polylog(3, (b*c*e - a*c*f + (b*d*e - a*d*f)*x)/ (a*d*e - a*c*f + (b*d*e - b*c*f)*x)))/(b*e - a*f)
\[ \int \frac {\log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{(a+b x) (e+f x)} \, dx=\int \frac {\log {\left (- \frac {a c f}{- a c f + a d e - b c f x + b d e x} - \frac {a d f x}{- a c f + a d e - b c f x + b d e x} + \frac {b c e}{- a c f + a d e - b c f x + b d e x} + \frac {b d e x}{- a c f + a d e - b c f x + b d e x} \right )}^{2}}{\left (a + b x\right ) \left (e + f x\right )}\, dx \]
Integral(log(-a*c*f/(-a*c*f + a*d*e - b*c*f*x + b*d*e*x) - a*d*f*x/(-a*c*f + a*d*e - b*c*f*x + b*d*e*x) + b*c*e/(-a*c*f + a*d*e - b*c*f*x + b*d*e*x) + b*d*e*x/(-a*c*f + a*d*e - b*c*f*x + b*d*e*x))**2/((a + b*x)*(e + f*x)), x)
Exception generated. \[ \int \frac {\log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{(a+b x) (e+f x)} \, dx=\text {Exception raised: RuntimeError} \]
Exception raised: RuntimeError >> ECL says: Memory limit reached. Please j ump to an outer pointer, quit program and enlarge thememory limits before executing the program again.
\[ \int \frac {\log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{(a+b x) (e+f x)} \, dx=\int { \frac {\log \left (\frac {{\left (b e - a f\right )} {\left (d x + c\right )}}{{\left (d e - c f\right )} {\left (b x + a\right )}}\right )^{2}}{{\left (b x + a\right )} {\left (f x + e\right )}} \,d x } \]
Timed out. \[ \int \frac {\log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{(a+b x) (e+f x)} \, dx=\int \frac {{\ln \left (\frac {\left (a\,f-b\,e\right )\,\left (c+d\,x\right )}{\left (c\,f-d\,e\right )\,\left (a+b\,x\right )}\right )}^2}{\left (e+f\,x\right )\,\left (a+b\,x\right )} \,d x \]