3.14 $$\int \frac{\log (c x^n)}{x (a x^m+b \log ^2(c x^n))^2} \, dx$$

Optimal. Leaf size=67 $-\frac{a m \text{CannotIntegrate}\left (\frac{x^{m-1}}{\left (a x^m+b \log ^2\left (c x^n\right )\right )^2},x\right )}{2 b n}-\frac{1}{2 b n \left (a x^m+b \log ^2\left (c x^n\right )\right )}$

[Out]

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

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Rubi [A]  time = 0.196432, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0., Rules used = {} $\int \frac{\log \left (c x^n\right )}{x \left (a x^m+b \log ^2\left (c x^n\right )\right )^2} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

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

[Out]

-1/(2*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*} \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) \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 [A]  time = 1.56721, size = 0, normalized size = 0. $\int \frac{\log \left (c x^n\right )}{x \left (a x^m+b \log ^2\left (c x^n\right )\right )^2} \, dx$

Veriﬁcation is Not applicable to the result.

[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]

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Maple [A]  time = 51.749, size = 0, normalized size = 0. \begin{align*} \int{\frac{\ln \left ( c{x}^{n} \right ) }{x \left ( a{x}^{m}+b \left ( \ln \left ( c{x}^{n} \right ) \right ) ^{2} \right ) ^{2}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{m \log \left (c\right ) + m \log \left (x^{n}\right ) + 2 \, n}{4 \, b^{2} n^{2} \log \left (c\right )^{2} + a^{2} m^{2} x^{2 \, m} +{\left (m^{2} \log \left (c\right )^{2} + 4 \, n^{2}\right )} a b x^{m} +{\left (a b m^{2} x^{m} + 4 \, b^{2} n^{2}\right )} \log \left (x^{n}\right )^{2} + 2 \,{\left (a b m^{2} x^{m} \log \left (c\right ) + 4 \, b^{2} n^{2} \log \left (c\right )\right )} \log \left (x^{n}\right )} - \int \frac{a m^{4} x^{m} \log \left (x^{n}\right ) + 4 \, b m n^{3} +{\left (m^{4} \log \left (c\right ) + 3 \, m^{3} n\right )} a x^{m}}{16 \, b^{3} n^{4} x \log \left (c\right )^{2} + a^{3} m^{4} x x^{3 \, m} +{\left (m^{4} \log \left (c\right )^{2} + 8 \, m^{2} n^{2}\right )} a^{2} b x x^{2 \, m} + 8 \,{\left (m^{2} n^{2} \log \left (c\right )^{2} + 2 \, n^{4}\right )} a b^{2} x x^{m} +{\left (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\right )} \log \left (x^{n}\right )^{2} + 2 \,{\left (a^{2} b m^{4} x x^{2 \, m} \log \left (c\right ) + 8 \, a b^{2} m^{2} n^{2} x x^{m} \log \left (c\right ) + 16 \, b^{3} n^{4} x \log \left (c\right )\right )} \log \left (x^{n}\right )}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left (c x^{n}\right )}{b^{2} x \log \left (c x^{n}\right )^{4} + 2 \, a b x x^{m} \log \left (c x^{n}\right )^{2} + a^{2} x x^{2 \, m}}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log{\left (c x^{n} \right )}}{x \left (a x^{m} + b \log{\left (c x^{n} \right )}^{2}\right )^{2}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)