3.2.52 \(\int -\frac {\log (1-e (\frac {a+b x}{c+d x})^n)}{(a+b x) (c+d x)} \, dx\) [152]

Optimal. Leaf size=33 \[ \frac {\text {PolyLog}\left (2,e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(b c-a d) n} \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.027, Rules used = {2598} \begin {gather*} \frac {\text {Li}_2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{n (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-(Log[1 - e*((a + b*x)/(c + d*x))^n]/((a + b*x)*(c + d*x))),x]

[Out]

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

Rule 2598

Int[Log[v_]*(u_), x_Symbol] :> With[{w = DerivativeDivides[v, u*(1 - v), x]}, Simp[w*PolyLog[2, 1 - v], x] /;
!FalseQ[w]]

Rubi steps

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

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Mathematica [F] Contains unresolved integral.
time = 1.22, size = 40, normalized size = 1.21 \begin {gather*} -\int \frac {\log \left (1-e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)} \, dx \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-(Log[1 - e*((a + b*x)/(c + d*x))^n]/((a + b*x)*(c + d*x))),x]

[Out]

-Integrate[Log[1 - e*((a + b*x)/(c + d*x))^n]/((a + b*x)*(c + d*x)), x]

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Maple [F]
time = 0.32, size = 0, normalized size = 0.00 \[\int -\frac {\ln \left (1-e \left (\frac {b x +a}{d x +c}\right )^{n}\right )}{\left (b x +a \right ) \left (d x +c \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-ln(1-e*((b*x+a)/(d*x+c))^n)/(b*x+a)/(d*x+c),x)

[Out]

int(-ln(1-e*((b*x+a)/(d*x+c))^n)/(b*x+a)/(d*x+c),x)

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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(1-e*((b*x+a)/(d*x+c))^n)/(b*x+a)/(d*x+c),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.37, size = 33, normalized size = 1.00 \begin {gather*} \frac {{\rm Li}_2\left (\left (\frac {b x + a}{d x + c}\right )^{n} e\right )}{{\left (b c - a d\right )} n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-log(1-e*((b*x+a)/(d*x+c))^n)/(b*x+a)/(d*x+c),x, algorithm="fricas")

[Out]

dilog(((b*x + a)/(d*x + c))^n*e)/((b*c - a*d)*n)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-ln(1-e*((b*x+a)/(d*x+c))**n)/(b*x+a)/(d*x+c),x)

[Out]

Timed out

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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(1-e*((b*x+a)/(d*x+c))^n)/(b*x+a)/(d*x+c),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {\ln \left (1-e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\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(1 - e*((a + b*x)/(c + d*x))^n)/((a + b*x)*(c + d*x)),x)

[Out]

int(-log(1 - e*((a + b*x)/(c + d*x))^n)/((a + b*x)*(c + d*x)), x)

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