\(\int \frac {\log (c x^n)}{x (a x^m+b \log ^2(c x^n))^2} \, dx\) [14]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.12 (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 = {} \[ \int \frac {\log \left (c x^n\right )}{x \left (a x^m+b \log ^2\left (c x^n\right )\right )^2} \, dx=\int \frac {\log \left (c x^n\right )}{x \left (a x^m+b \log ^2\left (c x^n\right )\right )^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 1.13 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\log \left (c x^n\right )}{x \left (a x^m+b \log ^2\left (c x^n\right )\right )^2} \, dx=\int \frac {\log \left (c x^n\right )}{x \left (a x^m+b \log ^2\left (c x^n\right )\right )^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.06 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.82 \[ \int \frac {\log \left (c x^n\right )}{x \left (a x^m+b \log ^2\left (c x^n\right )\right )^2} \, dx=\int { \frac {\log \left (c x^{n}\right )}{{\left (b \log \left (c x^{n}\right )^{2} + a x^{m}\right )}^{2} x} \,d x } \]

[In]

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

[Out]

integral(log(c*x^n)/(b^2*x*log(c*x^n)^4 + 2*a*b*x*x^m*log(c*x^n)^2 + a^2*x*x^(2*m)), x)

Sympy [N/A]

Not integrable

Time = 10.04 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {\log \left (c x^n\right )}{x \left (a x^m+b \log ^2\left (c x^n\right )\right )^2} \, dx=\int \frac {\log {\left (c x^{n} \right )}}{x \left (a x^{m} + b \log {\left (c x^{n} \right )}^{2}\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 343, normalized size of antiderivative = 12.25 \[ \int \frac {\log \left (c x^n\right )}{x \left (a x^m+b \log ^2\left (c x^n\right )\right )^2} \, dx=\int { \frac {\log \left (c x^{n}\right )}{{\left (b \log \left (c x^{n}\right )^{2} + a x^{m}\right )}^{2} x} \,d x } \]

[In]

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

[Out]

-(m*log(c) + m*log(x^n) + 2*n)/(4*b^2*n^2*log(c)^2 + a^2*m^2*x^(2*m) + (m^2*log(c)^2 + 4*n^2)*a*b*x^m + (a*b*m
^2*x^m + 4*b^2*n^2)*log(x^n)^2 + 2*(a*b*m^2*x^m*log(c) + 4*b^2*n^2*log(c))*log(x^n)) - integrate((a*m^4*x^m*lo
g(x^n) + 4*b*m*n^3 + (m^4*log(c) + 3*m^3*n)*a*x^m)/(16*b^3*n^4*x*log(c)^2 + a^3*m^4*x*x^(3*m) + (m^4*log(c)^2
+ 8*m^2*n^2)*a^2*b*x*x^(2*m) + 8*(m^2*n^2*log(c)^2 + 2*n^4)*a*b^2*x*x^m + (a^2*b*m^4*x*x^(2*m) + 8*a*b^2*m^2*n
^2*x*x^m + 16*b^3*n^4*x)*log(x^n)^2 + 2*(a^2*b*m^4*x*x^(2*m)*log(c) + 8*a*b^2*m^2*n^2*x*x^m*log(c) + 16*b^3*n^
4*x*log(c))*log(x^n)), x)

Giac [N/A]

Not integrable

Time = 0.37 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\log \left (c x^n\right )}{x \left (a x^m+b \log ^2\left (c x^n\right )\right )^2} \, dx=\int { \frac {\log \left (c x^{n}\right )}{{\left (b \log \left (c x^{n}\right )^{2} + a x^{m}\right )}^{2} x} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.53 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\log \left (c x^n\right )}{x \left (a x^m+b \log ^2\left (c x^n\right )\right )^2} \, dx=\int \frac {\ln \left (c\,x^n\right )}{x\,{\left (a\,x^m+b\,{\ln \left (c\,x^n\right )}^2\right )}^2} \,d x \]

[In]

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

[Out]

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