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

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

[Out]

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

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Rubi [A]  time = 0.0581217, antiderivative size = 33, normalized size of antiderivative = 1., 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 = {2518} \[ \frac{\text{PolyLog}\left (2,e \left (\frac{a+b x}{c+d x}\right )^n\right )}{n (b c-a d)} \]

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 2518

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*}

Mathematica [F]  time = 1.82302, size = 40, normalized size = 1.21 \[ -\int \frac{\log \left (1-e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)} \, dx \]

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 = 0.702, size = 0, normalized size = 0. \begin{align*} \int -{\frac{1}{ \left ( bx+a \right ) \left ( dx+c \right ) }\ln \left ( 1-e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) }\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} -\frac{{\left (\log \left (b x + a\right ) - \log \left (d x + c\right )\right )} \log \left (-{\left (b x + a\right )}^{n} e +{\left (d x + c\right )}^{n}\right ) -{\left (\log \left (b x + a\right ) - \log \left (d x + c\right )\right )} \log \left ({\left (d x + c\right )}^{n}\right )}{b c - a d} + \int \frac{{\left (e n \log \left (b x + a\right ) - e n \log \left (d x + c\right )\right )}{\left (b x + a\right )}^{n}}{{\left (b d e x^{2} + a c e +{\left (b c e + a d e\right )} x\right )}{\left (b x + a\right )}^{n} -{\left (b d x^{2} + a c +{\left (b c + a d\right )} x\right )}{\left (d x + c\right )}^{n}}\,{d x} \end{align*}

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(-(b*x + a)^n*e + (d*x + c)^n) - (log(b*x + a) - log(d*x + c))*log((d*x + c
)^n))/(b*c - a*d) + integrate((e*n*log(b*x + a) - e*n*log(d*x + c))*(b*x + a)^n/((b*d*e*x^2 + a*c*e + (b*c*e +
a*d*e)*x)*(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 = 2.44507, size = 68, normalized size = 2.06 \begin{align*} \frac{{\rm Li}_2\left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right )}{{\left (b c - a d\right )} n} \end{align*}

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

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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(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., size = 0, normalized size = 0. \begin{align*} \int -\frac{\log \left (-e \left (\frac{b x + a}{d x + c}\right )^{n} + 1\right )}{{\left (b x + a\right )}{\left (d x + c\right )}}\,{d x} \end{align*}

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)