3.3.45 \(\int \frac {1}{(a+b x) (c+d x) \log (e (\frac {a+b x}{c+d x})^n)} \, dx\) [245]

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

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {2561, 2339, 29} \begin {gather*} \frac {\log \left (\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{n (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 2339

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 2561

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m
_.)*((h_.) + (i_.)*(x_))^(q_.), x_Symbol] :> Dist[(b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q, Subst[Int[x^m*((A +
B*Log[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, h, i,
A, B, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c*i, 0] && IntegersQ[m, q]

Rubi steps

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

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Mathematica [A]
time = 0.03, size = 34, normalized size = 1.03 \begin {gather*} -\frac {\log \left (\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(-b c+a d) n} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.20, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (b x +a \right ) \left (d x +c \right ) \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.44, size = 34, normalized size = 1.03 \begin {gather*} \frac {\log \left (-\log \left ({\left (b x + a\right )}^{n}\right ) + \log \left ({\left (d x + c\right )}^{n}\right ) - 1\right )}{b c n - a d n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.41, size = 33, normalized size = 1.00 \begin {gather*} \frac {\log \left (n \log \left (\frac {b x + a}{d x + c}\right ) + 1\right )}{{\left (b c - a d\right )} n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (34) = 68\).
time = 4.15, size = 82, normalized size = 2.48 \begin {gather*} \frac {{\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )} \log \left (\frac {1}{4} \, \pi ^{2} {\left (\mathrm {sgn}\left (b x + a\right ) \mathrm {sgn}\left (d x + c\right ) - 1\right )}^{2} n^{2} + {\left (n \log \left (\frac {{\left | b x + a \right |}}{{\left | d x + c \right |}}\right ) + 1\right )}^{2}\right )}{2 \, n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b*c/(b*c - a*d)^2 - a*d/(b*c - a*d)^2)*log(1/4*pi^2*(sgn(b*x + a)*sgn(d*x + c) - 1)^2*n^2 + (n*log(abs(b*
x + a)/abs(d*x + c)) + 1)^2)/n

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Mupad [B]
time = 4.48, size = 33, normalized size = 1.00 \begin {gather*} -\frac {\ln \left (\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}{a\,d\,n-b\,c\,n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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