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 \[ \frac{2 \text{PolyLog}\left (3,\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{b e-a f}-\frac{2 \log \left (\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right ) \text{PolyLog}\left (2,\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{b e-a f}-\frac{\log \left (1-\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right ) \log ^2\left (\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{b e-a f} \]

[Out]

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

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Rubi [A]  time = 0.253539, 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*}

Mathematica [B]  time = 0.469379, 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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Maple [A]  time = 0.063, size = 357, normalized size = 1.8 \begin{align*}{\frac{1}{af-be} \left ( \ln \left ( -{\frac{ \left ( af-be \right ) \left ( ad-bc \right ) }{b \left ( cf-de \right ) \left ( bx+a \right ) }}+{\frac{ \left ( af-be \right ) d}{b \left ( cf-de \right ) }} \right ) \right ) ^{2}\ln \left ( 1+{\frac{ \left ( af-be \right ) \left ( ad-bc \right ) }{b \left ( cf-de \right ) \left ( bx+a \right ) }}-{\frac{ \left ( af-be \right ) d}{b \left ( cf-de \right ) }} \right ) }+2\,{\frac{1}{af-be}\ln \left ( -{\frac{ \left ( af-be \right ) \left ( ad-bc \right ) }{b \left ( cf-de \right ) \left ( bx+a \right ) }}+{\frac{ \left ( af-be \right ) d}{b \left ( cf-de \right ) }} \right ){\it polylog} \left ( 2,-{\frac{ \left ( af-be \right ) \left ( ad-bc \right ) }{b \left ( cf-de \right ) \left ( bx+a \right ) }}+{\frac{ \left ( af-be \right ) d}{b \left ( cf-de \right ) }} \right ) }-2\,{\frac{1}{af-be}{\it polylog} \left ( 3,-{\frac{ \left ( af-be \right ) \left ( ad-bc \right ) }{b \left ( cf-de \right ) \left ( bx+a \right ) }}+{\frac{ \left ( af-be \right ) d}{b \left ( cf-de \right ) }} \right ) } \end{align*}

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(-(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+(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/(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)*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/(a*f-b*e)*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)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

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]

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