### 3.13 $$\int \frac{\log (c x^n)}{x (a x^m+b \log ^2(c x^n))} \, dx$$

Optimal. Leaf size=66 $\frac{\log \left (a x^m+b \log ^2\left (c x^n\right )\right )}{2 b n}-\frac{a m \text{CannotIntegrate}\left (\frac{x^{m-1}}{a x^m+b \log ^2\left (c x^n\right )},x\right )}{2 b n}$

[Out]

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

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Rubi [A]  time = 0.191758, 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 = {} $\int \frac{\log \left (c x^n\right )}{x \left (a x^m+b \log ^2\left (c x^n\right )\right )} \, dx$

Veriﬁcation is Not applicable to the result.

[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*} \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) \int \frac{x^{-1+m}}{a x^m+b \log ^2\left (c x^n\right )} \, dx}{2 b n}\\ \end{align*}

Mathematica [A]  time = 1.40854, size = 0, normalized size = 0. $\int \frac{\log \left (c x^n\right )}{x \left (a x^m+b \log ^2\left (c x^n\right )\right )} \, dx$

Veriﬁcation is Not applicable to the result.

[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]

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Maple [A]  time = 1.777, 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 ) }}\, 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),x)

[Out]

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

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Maxima [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 )} 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),x, algorithm="maxima")

[Out]

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

[Out]

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

[Out]

Integral(log(c*x**n)/(x*(a*x**m + b*log(c*x**n)**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 )} 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),x, algorithm="giac")

[Out]

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