\(\int \frac {\log (c x^n)}{x (a x^m+b \log ^2(c x^n))} \, dx\) [13]

Optimal result

Integrand size = 28, antiderivative size = 28 \[ \int \frac {\log \left (c x^n\right )}{x \left (a x^m+b \log ^2\left (c x^n\right )\right )} \, dx=\frac {\log \left (a x^m+b \log ^2\left (c x^n\right )\right )}{2 b n}-\frac {a m \text {Int}\left (\frac {x^{-1+m}}{a x^m+b \log ^2\left (c x^n\right )},x\right )}{2 b n} \]

[Out]

-1/2*a*m*CannotIntegrate(x^(-1+m)/(a*x^m+b*ln(c*x^n)^2),x)/b/n+1/2*ln(a*x^m+b*ln(c*x^n)^2)/b/n

Rubi [N/A]

Not integrable

Time = 0.13 (sec) , antiderivative size = 28, 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 = {} \[ \int \frac {\log \left (c x^n\right )}{x \left (a x^m+b \log ^2\left (c x^n\right )\right )} \, dx=\int \frac {\log \left (c x^n\right )}{x \left (a x^m+b \log ^2\left (c x^n\right )\right )} \, dx \]

[In]

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

[Out]

Log[a*x^m + b*Log[c*x^n]^2]/(2*b*n) - (a*m*Defer[Int][x^(-1 + m)/(a*x^m + b*Log[c*x^n]^2), x])/(2*b*n)

Rubi steps \begin{align*} \text {integral}& = \frac {\log \left (a x^m+b \log ^2\left (c x^n\right )\right )}{2 b n}-\frac {(a m) \int \frac {x^{-1+m}}{a x^m+b \log ^2\left (c x^n\right )} \, dx}{2 b n} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 1.30 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\log \left (c x^n\right )}{x \left (a x^m+b \log ^2\left (c x^n\right )\right )} \, dx=\int \frac {\log \left (c x^n\right )}{x \left (a x^m+b \log ^2\left (c x^n\right )\right )} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.38 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {\log \left (c x^n\right )}{x \left (a x^m+b \log ^2\left (c x^n\right )\right )} \, dx=\int { \frac {\log \left (c x^{n}\right )}{{\left (b \log \left (c x^{n}\right )^{2} + a x^{m}\right )} x} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 5.88 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {\log \left (c x^n\right )}{x \left (a x^m+b \log ^2\left (c x^n\right )\right )} \, dx=\int \frac {\log {\left (c x^{n} \right )}}{x \left (a x^{m} + b \log {\left (c x^{n} \right )}^{2}\right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\log \left (c x^n\right )}{x \left (a x^m+b \log ^2\left (c x^n\right )\right )} \, dx=\int { \frac {\log \left (c x^{n}\right )}{{\left (b \log \left (c x^{n}\right )^{2} + a x^{m}\right )} x} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\log \left (c x^n\right )}{x \left (a x^m+b \log ^2\left (c x^n\right )\right )} \, dx=\int { \frac {\log \left (c x^{n}\right )}{{\left (b \log \left (c x^{n}\right )^{2} + a x^{m}\right )} x} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.43 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\log \left (c x^n\right )}{x \left (a x^m+b \log ^2\left (c x^n\right )\right )} \, dx=\int \frac {\ln \left (c\,x^n\right )}{x\,\left (a\,x^m+b\,{\ln \left (c\,x^n\right )}^2\right )} \,d x \]

[In]

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

[Out]

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