\(\int \frac {n q-\log (c x^n)}{(a x+b \log ^q(c x^n))^2} \, dx\) [39]

Optimal result

Integrand size = 29, antiderivative size = 29 \[ \int \frac {n q-\log \left (c x^n\right )}{\left (a x+b \log ^q\left (c x^n\right )\right )^2} \, dx=\frac {\log \left (c x^n\right )}{a \left (a x+b \log ^q\left (c x^n\right )\right )}-\frac {n (1-q) \text {Int}\left (\frac {1}{x \left (a x+b \log ^q\left (c x^n\right )\right )},x\right )}{a} \]

[Out]

-n*(1-q)*CannotIntegrate(1/x/(a*x+b*ln(c*x^n)^q),x)/a+ln(c*x^n)/a/(a*x+b*ln(c*x^n)^q)

Rubi [N/A]

Not integrable

Time = 0.10 (sec) , antiderivative size = 29, 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 = {} \[ \int \frac {n q-\log \left (c x^n\right )}{\left (a x+b \log ^q\left (c x^n\right )\right )^2} \, dx=\int \frac {n q-\log \left (c x^n\right )}{\left (a x+b \log ^q\left (c x^n\right )\right )^2} \, dx \]

[In]

Int[(n*q - Log[c*x^n])/(a*x + b*Log[c*x^n]^q)^2,x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 79.50 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {n q-\log \left (c x^n\right )}{\left (a x+b \log ^q\left (c x^n\right )\right )^2} \, dx=\int \frac {n q-\log \left (c x^n\right )}{\left (a x+b \log ^q\left (c x^n\right )\right )^2} \, dx \]

[In]

Integrate[(n*q - Log[c*x^n])/(a*x + b*Log[c*x^n]^q)^2,x]

[Out]

Integrate[(n*q - Log[c*x^n])/(a*x + b*Log[c*x^n]^q)^2, x]

Maple [N/A]

Not integrable

Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00

\[\int \frac {n q -\ln \left (c \,x^{n}\right )}{{\left (a x +b \ln \left (c \,x^{n}\right )^{q}\right )}^{2}}d x\]

[In]

int((n*q-ln(c*x^n))/(a*x+b*ln(c*x^n)^q)^2,x)

[Out]

int((n*q-ln(c*x^n))/(a*x+b*ln(c*x^n)^q)^2,x)

Fricas [N/A]

Not integrable

Time = 0.37 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.79 \[ \int \frac {n q-\log \left (c x^n\right )}{\left (a x+b \log ^q\left (c x^n\right )\right )^2} \, dx=\int { \frac {n q - \log \left (c x^{n}\right )}{{\left (a x + b \log \left (c x^{n}\right )^{q}\right )}^{2}} \,d x } \]

[In]

integrate((n*q-log(c*x^n))/(a*x+b*log(c*x^n)^q)^2,x, algorithm="fricas")

[Out]

integral((n*q - log(c*x^n))/(a^2*x^2 + 2*a*b*x*log(c*x^n)^q + b^2*log(c*x^n)^(2*q)), x)

Sympy [N/A]

Not integrable

Time = 15.64 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int \frac {n q-\log \left (c x^n\right )}{\left (a x+b \log ^q\left (c x^n\right )\right )^2} \, dx=\int \frac {n q - \log {\left (c x^{n} \right )}}{\left (a x + b \log {\left (c x^{n} \right )}^{q}\right )^{2}}\, dx \]

[In]

integrate((n*q-ln(c*x**n))/(a*x+b*ln(c*x**n)**q)**2,x)

[Out]

Integral((n*q - log(c*x**n))/(a*x + b*log(c*x**n)**q)**2, x)

Maxima [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.03 \[ \int \frac {n q-\log \left (c x^n\right )}{\left (a x+b \log ^q\left (c x^n\right )\right )^2} \, dx=\int { \frac {n q - \log \left (c x^{n}\right )}{{\left (a x + b \log \left (c x^{n}\right )^{q}\right )}^{2}} \,d x } \]

[In]

integrate((n*q-log(c*x^n))/(a*x+b*log(c*x^n)^q)^2,x, algorithm="maxima")

[Out]

n*(q - 1)*integrate(1/(a^2*x^2 + a*b*x*(log(c) + log(x^n))^q), x) + (log(c) + log(x^n))/(a^2*x + a*b*(log(c) +
log(x^n))^q)

Giac [N/A]

Not integrable

Time = 0.36 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {n q-\log \left (c x^n\right )}{\left (a x+b \log ^q\left (c x^n\right )\right )^2} \, dx=\int { \frac {n q - \log \left (c x^{n}\right )}{{\left (a x + b \log \left (c x^{n}\right )^{q}\right )}^{2}} \,d x } \]

[In]

integrate((n*q-log(c*x^n))/(a*x+b*log(c*x^n)^q)^2,x, algorithm="giac")

[Out]

integrate((n*q - log(c*x^n))/(a*x + b*log(c*x^n)^q)^2, x)

Mupad [N/A]

Not integrable

Time = 1.62 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {n q-\log \left (c x^n\right )}{\left (a x+b \log ^q\left (c x^n\right )\right )^2} \, dx=\int -\frac {\ln \left (c\,x^n\right )-n\,q}{{\left (b\,{\ln \left (c\,x^n\right )}^q+a\,x\right )}^2} \,d x \]

[In]

int(-(log(c*x^n) - n*q)/(b*log(c*x^n)^q + a*x)^2,x)

[Out]

int(-(log(c*x^n) - n*q)/(b*log(c*x^n)^q + a*x)^2, x)