\(\int \frac {\log ^2(\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)})}{a+b x} \, dx\) [104]

Optimal result

Integrand size = 42, antiderivative size = 140 \[ \int \frac {\log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{a+b x} \, dx=-\frac {\log \left (\frac {-b c+a d}{d (a+b x)}\right ) \log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{b}-\frac {2 \log \left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{b}+\frac {2 \operatorname {PolyLog}\left (3,\frac {b (c+d x)}{d (a+b x)}\right )}{b} \]

[Out]

-ln((a*d-b*c)/d/(b*x+a))*ln((-a*f+b*e)*(d*x+c)/(-c*f+d*e)/(b*x+a))^2/b-2*ln((-a*f+b*e)*(d*x+c)/(-c*f+d*e)/(b*x
+a))*polylog(2,b*(d*x+c)/d/(b*x+a))/b+2*polylog(3,b*(d*x+c)/d/(b*x+a))/b

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2552, 2354, 2421, 6724} \[ \int \frac {\log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{a+b x} \, dx=-\frac {2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right ) \log \left (\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{b}-\frac {\log \left (\frac {a d-b c}{d (a+b x)}\right ) \log ^2\left (\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{b}+\frac {2 \operatorname {PolyLog}\left (3,\frac {b (c+d x)}{d (a+b x)}\right )}{b} \]

[In]

Int[Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]^2/(a + b*x),x]

[Out]

-((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)/b) - (2*Log[((b*e
- a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/b + (2*PolyLog[3, (b*(c +
d*x))/(d*(a + b*x))])/b

Rule 2354

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[Log[1 + e*(x/d)]*((a +
b*Log[c*x^n])^p/e), x] - Dist[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]

Rule 2421

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] + Dist[b*n*(p/m), Int[PolyLog[2, (-d)*f*x^m]*((a + b*L
og[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]

Rule 2552

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_)
)^(m_.), x_Symbol] :> Dist[(b*c - a*d)^(m + 1)*(g/d)^m, Subst[Int[(A + B*Log[e*x^n])^p/(b - d*x)^(m + 2), x],
x, (a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, A, B, n}, x] && EqQ[n + mn, 0] && IGtQ[n, 0] && NeQ
[b*c - a*d, 0] && IntegersQ[m, p] && EqQ[d*f - c*g, 0] && (GtQ[p, 0] || LtQ[m, -1])

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> 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]

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {\log ^2\left (\frac {(b e-a f) x}{d e-c f}\right )}{d-b x} \, dx,x,\frac {c+d x}{a+b x}\right ) \\ & = -\frac {\log \left (\frac {-b c+a d}{d (a+b x)}\right ) \log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{b}+\frac {2 \text {Subst}\left (\int \frac {\log \left (\frac {(b e-a f) x}{d e-c f}\right ) \log \left (1-\frac {b x}{d}\right )}{x} \, dx,x,\frac {c+d x}{a+b x}\right )}{b} \\ & = -\frac {\log \left (\frac {-b c+a d}{d (a+b x)}\right ) \log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{b}-\frac {2 \log \left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b}+\frac {2 \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {b x}{d}\right )}{x} \, dx,x,\frac {c+d x}{a+b x}\right )}{b} \\ & = -\frac {\log \left (\frac {-b c+a d}{d (a+b x)}\right ) \log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{b}-\frac {2 \log \left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b}+\frac {2 \text {Li}_3\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.96 \[ \int \frac {\log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{a+b x} \, dx=\frac {-\log \left (\frac {-b c+a d}{d (a+b x)}\right ) \log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )-2 \log \left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )+2 \operatorname {PolyLog}\left (3,\frac {b (c+d x)}{d (a+b x)}\right )}{b} \]

[In]

Integrate[Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]^2/(a + b*x),x]

[Out]

(-(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) - 2*Log[((b*e - a
*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))] + 2*PolyLog[3, (b*(c + d*x))/(
d*(a + b*x))])/b

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(418\) vs. \(2(140)=280\).

Time = 3.77 (sec) , antiderivative size = 419, normalized size of antiderivative = 2.99

method	result	size
derivativedivides	\(\frac {\left (c f -d e \right ) \left (\ln \left (-\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}+\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}\right )^{2} \ln \left (1-\frac {\left (-b c f +b d e \right ) \left (-\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}+\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}\right )}{-a d f +b d e}\right )+2 \ln \left (-\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}+\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}\right ) \operatorname {Li}_{2}\left (\frac {\left (-b c f +b d e \right ) \left (-\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}+\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}\right )}{-a d f +b d e}\right )-2 \,\operatorname {Li}_{3}\left (\frac {\left (-b c f +b d e \right ) \left (-\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}+\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}\right )}{-a d f +b d e}\right )\right )}{-b c f +b d e}\)	\(419\)
default	\(\frac {\left (c f -d e \right ) \left (\ln \left (-\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}+\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}\right )^{2} \ln \left (1-\frac {\left (-b c f +b d e \right ) \left (-\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}+\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}\right )}{-a d f +b d e}\right )+2 \ln \left (-\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}+\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}\right ) \operatorname {Li}_{2}\left (\frac {\left (-b c f +b d e \right ) \left (-\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}+\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}\right )}{-a d f +b d e}\right )-2 \,\operatorname {Li}_{3}\left (\frac {\left (-b c f +b d e \right ) \left (-\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}+\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}\right )}{-a d f +b d e}\right )\right )}{-b c f +b d e}\)	\(419\)
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 (1-\frac {\left (-b c f +b d e \right ) \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 d f +b d e}\right ) c f}{-b c f +b d e}-\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 (1-\frac {\left (-b c f +b d e \right ) \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 d f +b d e}\right ) d e}{-b c f +b d 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 (-b c f +b d e \right ) \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 d f +b d e}\right ) c f}{-b c f +b d 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 (-b c f +b d e \right ) \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 d f +b d e}\right ) d e}{-b c f +b d e}-\frac {2 \,\operatorname {Li}_{3}\left (\frac {\left (-b c f +b d e \right ) \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 d f +b d e}\right ) c f}{-b c f +b d e}+\frac {2 \,\operatorname {Li}_{3}\left (\frac {\left (-b c f +b d e \right ) \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 d f +b d e}\right ) d e}{-b c f +b d e}\)	\(879\)

[In]

int(ln((-a*f+b*e)*(d*x+c)/(-c*f+d*e)/(b*x+a))^2/(b*x+a),x,method=_RETURNVERBOSE)

[Out]

(c*f-d*e)/(-b*c*f+b*d*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(1-(-b*c*f+
b*d*e)/(-a*d*f+b*d*e)*(-(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)*polylog(2,(-b*c*f+b*d*e)/(-a*d*f+b*d*e)*(-(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,(-b*c*f+b*d*e)/(-a*d*f+b*d*e)*(-(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)))

Fricas [F]

\[ \int \frac {\log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{a+b x} \, dx=\int { \frac {\log \left (\frac {{\left (b e - a f\right )} {\left (d x + c\right )}}{{\left (d e - c f\right )} {\left (b x + a\right )}}\right )^{2}}{b x + a} \,d x } \]

[In]

integrate(log((-a*f+b*e)*(d*x+c)/(-c*f+d*e)/(b*x+a))^2/(b*x+a),x, algorithm="fricas")

[Out]

integral(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/(b*x + a), x)

Sympy [F]

\[ \int \frac {\log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{a+b x} \, dx=\int \frac {\log {\left (- \frac {a c f}{- a c f + a d e - b c f x + b d e x} - \frac {a d f x}{- a c f + a d e - b c f x + b d e x} + \frac {b c e}{- a c f + a d e - b c f x + b d e x} + \frac {b d e x}{- a c f + a d e - b c f x + b d e x} \right )}^{2}}{a + b x}\, dx \]

[In]

integrate(ln((-a*f+b*e)*(d*x+c)/(-c*f+d*e)/(b*x+a))**2/(b*x+a),x)

[Out]

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), x)

Maxima [F]

\[ \int \frac {\log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{a+b x} \, dx=\int { \frac {\log \left (\frac {{\left (b e - a f\right )} {\left (d x + c\right )}}{{\left (d e - c f\right )} {\left (b x + a\right )}}\right )^{2}}{b x + a} \,d x } \]

[In]

integrate(log((-a*f+b*e)*(d*x+c)/(-c*f+d*e)/(b*x+a))^2/(b*x+a),x, algorithm="maxima")

[Out]

log(d*x + c)^3/a - integrate(-((log(-b*e + a*f)^2 - 2*log(-b*e + a*f)*log(-d*e + c*f) + log(-d*e + c*f)^2)*b*d
*x + (log(-b*e + a*f)^2 - 2*log(-b*e + a*f)*log(-d*e + c*f) + log(-d*e + c*f)^2)*b*c + (b*d*x + b*c)*log(b*x +
a)^2 - 2*(b*d*x*(log(-b*e + a*f) - log(-d*e + c*f)) + b*c*(log(-b*e + a*f) - log(-d*e + c*f)))*log(b*x + a) +
2*(b*d*x*(log(-b*e + a*f) - log(-d*e + c*f)) + b*c*(log(-b*e + a*f) - log(-d*e + c*f)) - (2*b*d*x + b*c + a*d
)*log(b*x + a))*log(d*x + c))/(b^2*d*x^2 + a*b*c + (b^2*c + a*b*d)*x), x)

Giac [F]

\[ \int \frac {\log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{a+b x} \, dx=\int { \frac {\log \left (\frac {{\left (b e - a f\right )} {\left (d x + c\right )}}{{\left (d e - c f\right )} {\left (b x + a\right )}}\right )^{2}}{b x + a} \,d x } \]

[In]

integrate(log((-a*f+b*e)*(d*x+c)/(-c*f+d*e)/(b*x+a))^2/(b*x+a),x, algorithm="giac")

[Out]

integrate(log((b*e - a*f)*(d*x + c)/((d*e - c*f)*(b*x + a)))^2/(b*x + a), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{a+b x} \, dx=\int \frac {{\ln \left (\frac {\left (a\,f-b\,e\right )\,\left (c+d\,x\right )}{\left (c\,f-d\,e\right )\,\left (a+b\,x\right )}\right )}^2}{a+b\,x} \,d x \]

[In]

int(log(((a*f - b*e)*(c + d*x))/((c*f - d*e)*(a + b*x)))^2/(a + b*x),x)

[Out]

int(log(((a*f - b*e)*(c + d*x))/((c*f - d*e)*(a + b*x)))^2/(a + b*x), x)