\(\int \frac {1}{A+B \log (e (\frac {a+b x}{c+d x})^n)} \, dx\) [78]

Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int \frac {1}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\text {Int}\left (\frac {1}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\int \frac {1}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\int \frac {1}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.06 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00

\[\int \frac {1}{A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\int { \frac {1}{B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 4.04 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {1}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\int \frac {1}{A + B \log {\left (e \left (\frac {a + b x}{c + d x}\right )^{n} \right )}}\, dx \]

[In]

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

[Out]

Integral(1/(A + B*log(e*((a + b*x)/(c + d*x))**n)), x)

Maxima [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\int { \frac {1}{B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 11.36 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\int { \frac {1}{B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.59 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\int \frac {1}{A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )} \,d x \]

[In]

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

[Out]

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