\(\int \frac {1}{(a+b \log (c (d+\frac {e}{f+g x})^p))^2} \, dx\) [641]

Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int \frac {1}{\left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^2} \, dx=\text {Int}\left (\frac {1}{\left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^2},x\right ) \]

[Out]

Unintegrable(1/(a+b*ln(c*(d+e/(g*x+f))^p))^2,x)

Rubi [N/A]

Not integrable

Time = 0.01 (sec) , antiderivative size = 22, 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 = {} \[ \int \frac {1}{\left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^2} \, dx=\int \frac {1}{\left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^2} \, dx \]

[In]

Int[(a + b*Log[c*(d + e/(f + g*x))^p])^(-2),x]

[Out]

Defer[Int][(a + b*Log[c*(d + e/(f + g*x))^p])^(-2), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.39 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{\left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^2} \, dx=\int \frac {1}{\left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^2} \, dx \]

[In]

Integrate[(a + b*Log[c*(d + e/(f + g*x))^p])^(-2),x]

[Out]

Integrate[(a + b*Log[c*(d + e/(f + g*x))^p])^(-2), x]

Maple [N/A]

Not integrable

Time = 0.08 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00

\[\int \frac {1}{{\left (a +b \ln \left (c \left (d +\frac {e}{g x +f}\right )^{p}\right )\right )}^{2}}d x\]

[In]

int(1/(a+b*ln(c*(d+e/(g*x+f))^p))^2,x)

[Out]

int(1/(a+b*ln(c*(d+e/(g*x+f))^p))^2,x)

Fricas [N/A]

Not integrable

Time = 0.36 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.82 \[ \int \frac {1}{\left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^2} \, dx=\int { \frac {1}{{\left (b \log \left (c {\left (d + \frac {e}{g x + f}\right )}^{p}\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(1/(a+b*log(c*(d+e/(g*x+f))^p))^2,x, algorithm="fricas")

[Out]

integral(1/(b^2*log(c*((d*g*x + d*f + e)/(g*x + f))^p)^2 + 2*a*b*log(c*((d*g*x + d*f + e)/(g*x + f))^p) + a^2)
, x)

Sympy [N/A]

Not integrable

Time = 3.46 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^2} \, dx=\int \frac {1}{\left (a + b \log {\left (c \left (d + \frac {e}{f + g x}\right )^{p} \right )}\right )^{2}}\, dx \]

[In]

integrate(1/(a+b*ln(c*(d+e/(g*x+f))**p))**2,x)

[Out]

Integral((a + b*log(c*(d + e/(f + g*x))**p))**(-2), x)

Maxima [N/A]

Not integrable

Time = 0.40 (sec) , antiderivative size = 148, normalized size of antiderivative = 6.73 \[ \int \frac {1}{\left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^2} \, dx=\int { \frac {1}{{\left (b \log \left (c {\left (d + \frac {e}{g x + f}\right )}^{p}\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(1/(a+b*log(c*(d+e/(g*x+f))^p))^2,x, algorithm="maxima")

[Out]

(d*g^2*x^2 + d*f^2 + e*f + (2*d*f*g + e*g)*x)/(b^2*e*g*p*log((d*g*x + d*f + e)^p) - b^2*e*g*p*log((g*x + f)^p)
+ b^2*e*g*p*log(c) + a*b*e*g*p) - integrate((2*d*g*x + 2*d*f + e)/(b^2*e*p*log((d*g*x + d*f + e)^p) - b^2*e*p
*log((g*x + f)^p) + b^2*e*p*log(c) + a*b*e*p), x)

Giac [N/A]

Not integrable

Time = 0.48 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{\left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^2} \, dx=\int { \frac {1}{{\left (b \log \left (c {\left (d + \frac {e}{g x + f}\right )}^{p}\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(1/(a+b*log(c*(d+e/(g*x+f))^p))^2,x, algorithm="giac")

[Out]

integrate((b*log(c*(d + e/(g*x + f))^p) + a)^(-2), x)

Mupad [N/A]

Not integrable

Time = 1.60 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{\left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^2} \, dx=\int \frac {1}{{\left (a+b\,\ln \left (c\,{\left (d+\frac {e}{f+g\,x}\right )}^p\right )\right )}^2} \,d x \]

[In]

int(1/(a + b*log(c*(d + e/(f + g*x))^p))^2,x)

[Out]

int(1/(a + b*log(c*(d + e/(f + g*x))^p))^2, x)