\(\int \frac {\log (1+e (f^{c (a+b x)})^n)}{x} \, dx\) [122]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \frac {\log \left (1+e \left (f^{c (a+b x)}\right )^n\right )}{x} \, dx=\text {Int}\left (\frac {\log \left (1+e \left (f^{c (a+b x)}\right )^n\right )}{x},x\right ) \]

[Out]

CannotIntegrate(ln(1+e*(f^(c*(b*x+a)))^n)/x,x)

Rubi [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 20, 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 = {} \[ \int \frac {\log \left (1+e \left (f^{c (a+b x)}\right )^n\right )}{x} \, dx=\int \frac {\log \left (1+e \left (f^{c (a+b x)}\right )^n\right )}{x} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.17 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\log \left (1+e \left (f^{c (a+b x)}\right )^n\right )}{x} \, dx=\int \frac {\log \left (1+e \left (f^{c (a+b x)}\right )^n\right )}{x} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15 \[ \int \frac {\log \left (1+e \left (f^{c (a+b x)}\right )^n\right )}{x} \, dx=\int { \frac {\log \left (e {\left (f^{{\left (b x + a\right )} c}\right )}^{n} + 1\right )}{x} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 1.70 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {\log \left (1+e \left (f^{c (a+b x)}\right )^n\right )}{x} \, dx=\int \frac {\log {\left (e \left (f^{a c + b c x}\right )^{n} + 1 \right )}}{x}\, dx \]

[In]

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

[Out]

Integral(log(e*(f**(a*c + b*c*x))**n + 1)/x, x)

Maxima [N/A]

Not integrable

Time = 0.36 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.05 \[ \int \frac {\log \left (1+e \left (f^{c (a+b x)}\right )^n\right )}{x} \, dx=\int { \frac {\log \left (e {\left (f^{{\left (b x + a\right )} c}\right )}^{n} + 1\right )}{x} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.46 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\log \left (1+e \left (f^{c (a+b x)}\right )^n\right )}{x} \, dx=\int { \frac {\log \left (e {\left (f^{{\left (b x + a\right )} c}\right )}^{n} + 1\right )}{x} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.51 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\log \left (1+e \left (f^{c (a+b x)}\right )^n\right )}{x} \, dx=\int \frac {\ln \left (e\,{\left (f^{c\,\left (a+b\,x\right )}\right )}^n+1\right )}{x} \,d x \]

[In]

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

[Out]

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