3.2.5 \(\int \frac {\log (\frac {e (c+d x)}{a+b x}) \log (\frac {(-b c+a d) (e+f x)}{(d e-c f) (a+b x)})}{(a+b x) (c+d x)} \, dx\) [105]

Optimal. Leaf size=109 \[ \frac {\log \left (\frac {e (c+d x)}{a+b x}\right ) \text {Li}_2\left (1+\frac {(b c-a d) (e+f x)}{(d e-c f) (a+b x)}\right )}{b c-a d}-\frac {\text {Li}_3\left (1+\frac {(b c-a d) (e+f x)}{(d e-c f) (a+b x)}\right )}{b c-a d} \]

[Out]

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

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Rubi [A]
time = 0.10, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 62, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {2588, 6745} \begin {gather*} \frac {\log \left (\frac {e (c+d x)}{a+b x}\right ) \text {PolyLog}\left (2,\frac {(e+f x) (b c-a d)}{(a+b x) (d e-c f)}+1\right )}{b c-a d}-\frac {\text {PolyLog}\left (3,\frac {(e+f x) (b c-a d)}{(a+b x) (d e-c f)}+1\right )}{b c-a d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2588

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)*PolyL
og[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]*(L
og[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 6745

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 \left (\frac {e (c+d x)}{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 &=\frac {\log \left (\frac {e (c+d x)}{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 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\\ &=\frac {\log \left (\frac {e (c+d x)}{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 c-a d}-\frac {\text {Li}_3\left (1+\frac {(b c-a d) (e+f x)}{(d e-c f) (a+b x)}\right )}{b c-a d}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 96, normalized size = 0.88 \begin {gather*} \frac {\log \left (\frac {e (c+d x)}{a+b x}\right ) \text {Li}_2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )-\text {Li}_3\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{b c-a d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Log[(e*(c + d*x))/(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*c - a*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(523\) vs. \(2(109)=218\).
time = 8.53, size = 524, normalized size = 4.81

method	result	size
default	\(\frac {\ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )^{2} \ln \left (-\frac {\frac {e \left (d x +c \right ) a f}{b x +a}-\frac {e^{2} \left (d x +c \right ) b}{b x +a}-c e f +d \,e^{2}}{e \left (c f -e d \right )}\right )}{2 a d -2 c b}-\frac {a f \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )^{2} \ln \left (1-\frac {\left (a f -b e \right ) e \left (d x +c \right )}{\left (b x +a \right ) \left (c e f -d \,e^{2}\right )}\right )}{2 \left (a d -c b \right ) \left (a f -b e \right )}-\frac {a f \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) \polylog \left (2, \frac {\left (a f -b e \right ) e \left (d x +c \right )}{\left (b x +a \right ) \left (c e f -d \,e^{2}\right )}\right )}{\left (a d -c b \right ) \left (a f -b e \right )}+\frac {a f \polylog \left (3, \frac {\left (a f -b e \right ) e \left (d x +c \right )}{\left (b x +a \right ) \left (c e f -d \,e^{2}\right )}\right )}{\left (a d -c b \right ) \left (a f -b e \right )}+\frac {b e \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )^{2} \ln \left (1-\frac {\left (a f -b e \right ) e \left (d x +c \right )}{\left (b x +a \right ) \left (c e f -d \,e^{2}\right )}\right )}{2 \left (a d -c b \right ) \left (a f -b e \right )}+\frac {b e \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) \polylog \left (2, \frac {\left (a f -b e \right ) e \left (d x +c \right )}{\left (b x +a \right ) \left (c e f -d \,e^{2}\right )}\right )}{\left (a d -c b \right ) \left (a f -b e \right )}-\frac {b e \polylog \left (3, \frac {\left (a f -b e \right ) e \left (d x +c \right )}{\left (b x +a \right ) \left (c e f -d \,e^{2}\right )}\right )}{\left (a d -c b \right ) \left (a f -b e \right )}\)	\(524\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(log(b*x + a)^2 - 2*(log(b*x + a) - 1)*log(d*x + c) + log(d*x + c)^2 - 2*log(b*x + a))*log(f*x + e)/(b*c
- a*d) + integrate(1/2*((b*d*f*x^2 + a*c*f + (3*b*c*f - a*d*f)*x + 2*(b*c - a*d)*e)*log(b*x + a)^2 + 2*((f*log
(-b*c + a*d) - f*log(-c*f + d*e))*b*c - (d*f*log(-b*c + a*d) - d*f*log(-c*f + d*e))*a)*x + 2*(b*c*(log(-b*c +
a*d) - log(-c*f + d*e)) - (d*log(-b*c + a*d) - d*log(-c*f + d*e))*a)*e - 2*(b*d*f*x^2 + a*c*f + ((f*log(-b*c +
a*d) - f*log(-c*f + d*e) + 2*f)*b*c - (d*f*log(-b*c + a*d) - d*f*log(-c*f + d*e))*a)*x + (b*c*(log(-b*c + a*d
) - log(-c*f + d*e) + 1) + (d*(log(-c*f + d*e) - 1) - d*log(-b*c + a*d))*a)*e)*log(b*x + a) + 2*(b*d*f*x^2 + a
*c*f + ((f*log(-b*c + a*d) - f*log(-c*f + d*e) + f)*b*c - (d*f*log(-b*c + a*d) - (f*log(-c*f + d*e) + f)*d)*a)
*x + (b*c*(log(-b*c + a*d) - log(-c*f + d*e)) - (d*log(-b*c + a*d) - d*log(-c*f + d*e))*a)*e - (b*d*f*x^2 + 2*
b*c*f*x + a*c*f + (b*c - a*d)*e)*log(b*x + a))*log(d*x + c))/((b^2*c*d*f - a*b*d^2*f)*x^3 + (b^2*c^2*f - a^2*d
^2*f + (b^2*c*d - a*b*d^2)*e)*x^2 + (a*b*c^2*f - a^2*c*d*f + (b^2*c^2 - a^2*d^2)*e)*x + (a*b*c^2 - a^2*c*d)*e)
, x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(e*(d*x+c)/(b*x+a))*log((a*d-b*c)*(f*x+e)/(-c*f+d*e)/(b*x+a))/(b*x+a)/(d*x+c),x, algorithm="frica
s")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\ln \left (\frac {e\,\left (c+d\,x\right )}{a+b\,x}\right )\,\ln \left (-\frac {\left (e+f\,x\right )\,\left (a\,d-b\,c\right )}{\left (c\,f-d\,e\right )\,\left (a+b\,x\right )}\right )}{\left (a+b\,x\right )\,\left (c+d\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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