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

Optimal. Leaf size=204 \[ \frac {2 \text {Li}_3\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{b e-a f}-\frac {2 \text {Li}_2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{b e-a f}-\frac {\log \left (1-\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right ) \log ^2\left (\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{b e-a f} \]

[Out]

-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))*polylog(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)

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Rubi [A]  time = 0.25, antiderivative size = 206, normalized size of antiderivative = 1.01, number of steps used = 3, number of rules used = 3, integrand size = 49, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {2503, 2506, 6610} \[ \frac {2 \text {PolyLog}\left (3,\frac {(e+f x) (b c-a d)}{(a+b x) (d e-c f)}+1\right )}{b e-a f}-\frac {2 \log \left (\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right ) \text {PolyLog}\left (2,\frac {(e+f x) (b c-a d)}{(a+b x) (d e-c f)}+1\right )}{b e-a f}-\frac {\log \left (-\frac {(e+f x) (b c-a d)}{(a+b x) (d e-c f)}\right ) \log ^2\left (\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{b e-a f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]^2*Log[-(((b*c - a*d)*(e + f*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, 1 + ((b*c - a*d)*(e + f
*x))/((d*e - c*f)*(a + b*x))])/(b*e - a*f) + (2*PolyLog[3, 1 + ((b*c - a*d)*(e + f*x))/((d*e - c*f)*(a + b*x))
])/(b*e - a*f)

Rule 2503

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)*(u_), x_Symbol] :> Wi
th[{g = Coeff[Simplify[1/(u*(a + b*x))], x, 0], h = Coeff[Simplify[1/(u*(a + b*x))], x, 1]}, -Simp[(Log[e*(f*(
a + b*x)^p*(c + d*x)^q)^r]^s*Log[-(((b*c - a*d)*(g + h*x))/((d*g - c*h)*(a + b*x)))])/(b*g - a*h), x] + Dist[(
p*r*s*(b*c - a*d))/(b*g - a*h), Int[(Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1)*Log[-(((b*c - a*d)*(g + h*x)
)/((d*g - c*h)*(a + b*x)))])/((a + b*x)*(c + d*x)), x], x] /; NeQ[b*g - a*h, 0] && NeQ[d*g - c*h, 0]] /; FreeQ
[{a, b, c, d, e, f, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && IGtQ[s, 0] && EqQ[p + q, 0] && LinearQ[Simplify[1/
(u*(a + b*x))], x]

Rule 2506

Int[Log[v_]*Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)*(u_), x_Symbo
l] :> With[{g = Simplify[((v - 1)*(c + d*x))/(a + b*x)], h = Simplify[u*(a + b*x)*(c + d*x)]}, -Simp[(h*PolyLo
g[2, 1 - v]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/(b*c - a*d), x] + Dist[h*p*r*s, Int[(PolyLog[2, 1 - v]*Log
[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1))/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{g, h}, x]] /; FreeQ[{a, b,
c, d, e, f, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && IGtQ[s, 0] && EqQ[p + q, 0]

Rule 6610

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /
;  !FalseQ[w]] /; FreeQ[n, x]

Rubi steps

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

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Mathematica [B]  time = 0.52, size = 1636, normalized size = 8.02 \[ \text {result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-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*Log[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]*Lo
g[(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*L
og[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) + c*f)]*Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]*
Log[(b*(e + f*x))/(b*e - a*f)] + 3*Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]^2*Log[(b*(e + f*x))/(b
*e - a*f)] - 6*Log[a/b + x]*Log[c/d + x]*Log[(d*(e + f*x))/(d*e - c*f)] + 3*Log[c/d + x]^2*Log[(d*(e + f*x))/(
d*e - c*f)] + 6*Log[a/b + x]*Log[(f*(c + d*x))/(-(d*e) + c*f)]*Log[(d*(e + f*x))/(d*e - c*f)] - 3*Log[(f*(c +
d*x))/(-(d*e) + c*f)]^2*Log[(d*(e + f*x))/(d*e - c*f)] - 6*Log[c/d + x]*Log[((b*e - a*f)*(c + d*x))/((d*e - c*
f)*(a + b*x))]*Log[(d*(e + f*x))/(d*e - c*f)] + 6*Log[(f*(c + d*x))/(-(d*e) + c*f)]*Log[((b*e - a*f)*(c + d*x)
)/((d*e - c*f)*(a + b*x))]*Log[(d*(e + f*x))/(d*e - c*f)] - 3*Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*
x))]^2*Log[((-(b*c) + a*d)*(e + f*x))/((d*e - c*f)*(a + b*x))] + 6*Log[a/b + x]*PolyLog[2, (d*(a + b*x))/(-(b*
c) + a*d)] + 6*(Log[a/b + x] + Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))])*PolyLog[2, (b*(c + d*x))/
(b*c - a*d)] + 6*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))]
- 6*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))] - 6*PolyLog[3, (d*(a + b*x))/(-(b*c) + a*d)] - 6*PolyLog[3, (b*(c + d*x))/(b*c - a*d)] - 6*PolyLog[3, (
b*(c + d*x))/(d*(a + b*x))] + 6*PolyLog[3, ((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))])/(3*b*e - 3*a*f)

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fricas [F]  time = 0.71, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left (\frac {b c e - a c f + {\left (b d e - a d f\right )} x}{a d e - a c f + {\left (b d e - b c f\right )} x}\right )^{2}}{b f x^{2} + a e + {\left (b e + a f\right )} x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [A]  time = 0.05, size = 357, normalized size = 1.75 \[ \frac {\ln \left (\frac {\left (a f -b e \right ) d}{\left (c f -d e \right ) b}-\frac {\left (a f -b e \right ) \left (a d -b c \right )}{\left (c f -d e \right ) \left (b x +a \right ) b}\right )^{2} \ln \left (-\frac {\left (a f -b e \right ) d}{\left (c f -d e \right ) b}+\frac {\left (a f -b e \right ) \left (a d -b c \right )}{\left (c f -d e \right ) \left (b x +a \right ) b}+1\right )}{a f -b e}+\frac {2 \polylog \left (2, \frac {\left (a f -b e \right ) d}{\left (c f -d e \right ) b}-\frac {\left (a f -b e \right ) \left (a d -b c \right )}{\left (c f -d e \right ) \left (b x +a \right ) b}\right ) \ln \left (\frac {\left (a f -b e \right ) d}{\left (c f -d e \right ) b}-\frac {\left (a f -b e \right ) \left (a d -b c \right )}{\left (c f -d e \right ) \left (b x +a \right ) b}\right )}{a f -b e}-\frac {2 \polylog \left (3, \frac {\left (a f -b e \right ) d}{\left (c f -d e \right ) b}-\frac {\left (a f -b e \right ) \left (a d -b c \right )}{\left (c f -d e \right ) \left (b x +a \right ) b}\right )}{a f -b e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(a*f-b*e)*ln(1/(c*f-d*e)*(a*f-b*e)/b*d-(a*f-b*e)*(a*d-b*c)/(c*f-d*e)/(b*x+a)/b)^2*ln(1+(a*f-b*e)*(a*d-b*c)/(
c*f-d*e)/(b*x+a)/b-1/(c*f-d*e)*(a*f-b*e)/b*d)+2/(a*f-b*e)*ln(1/(c*f-d*e)*(a*f-b*e)/b*d-(a*f-b*e)*(a*d-b*c)/(c*
f-d*e)/(b*x+a)/b)*polylog(2,1/(c*f-d*e)*(a*f-b*e)/b*d-(a*f-b*e)*(a*d-b*c)/(c*f-d*e)/(b*x+a)/b)-2/(a*f-b*e)*pol
ylog(3,1/(c*f-d*e)*(a*f-b*e)/b*d-(a*f-b*e)*(a*d-b*c)/(c*f-d*e)/(b*x+a)/b)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: Memory limit reached. Please jump to an outer pointer, quit progra
m and enlarge thememory limits before executing the program again.

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\ln \left (\frac {\left (a\,f-b\,e\right )\,\left (c+d\,x\right )}{\left (c\,f-d\,e\right )\,\left (a+b\,x\right )}\right )}^2}{\left (e+f\,x\right )\,\left (a+b\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log {\left (- \frac {a c f}{- a c f + a d e - b c f x + b d e x} - \frac {a d f x}{- a c f + a d e - b c f x + b d e x} + \frac {b c e}{- a c f + a d e - b c f x + b d e x} + \frac {b d e x}{- a c f + a d e - b c f x + b d e x} \right )}^{2}}{\left (a + b x\right ) \left (e + f x\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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