\(\int \frac {1}{A+B \log (\frac {e (a+b x)}{c+d x})} \, dx\) [252]

Optimal result

Integrand size = 21, antiderivative size = 21 \[ \int \frac {1}{A+B \log \left (\frac {e (a+b x)}{c+d x}\right )} \, dx=\text {Int}\left (\frac {1}{A+B \log \left (\frac {e (a+b x)}{c+d x}\right )},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 21, 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 = {} \[ \int \frac {1}{A+B \log \left (\frac {e (a+b x)}{c+d x}\right )} \, dx=\int \frac {1}{A+B \log \left (\frac {e (a+b x)}{c+d x}\right )} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {1}{A+B \log \left (\frac {e (a+b x)}{c+d x}\right )} \, dx=\int \frac {1}{A+B \log \left (\frac {e (a+b x)}{c+d x}\right )} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 1.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \frac {1}{A+B \log \left (\frac {e (a+b x)}{c+d x}\right )} \, dx=\int { \frac {1}{B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.85 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {1}{A+B \log \left (\frac {e (a+b x)}{c+d x}\right )} \, dx=\int \frac {1}{A + B \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.21 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {1}{A+B \log \left (\frac {e (a+b x)}{c+d x}\right )} \, dx=\int { \frac {1}{B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 11.53 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {1}{A+B \log \left (\frac {e (a+b x)}{c+d x}\right )} \, dx=\int { \frac {1}{B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.99 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {1}{A+B \log \left (\frac {e (a+b x)}{c+d x}\right )} \, dx=\int \frac {1}{A+B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )} \,d x \]

[In]

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

[Out]

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