### 3.39 $$\int \frac{n q-\log (c x^n)}{(a x+b \log ^q(c x^n))^2} \, dx$$

Optimal. Leaf size=60 $\frac{\log \left (c x^n\right )}{a \left (a x+b \log ^q\left (c x^n\right )\right )}-\frac{n (1-q) \text{CannotIntegrate}\left (\frac{1}{x \left (a x+b \log ^q\left (c x^n\right )\right )},x\right )}{a}$

[Out]

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

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Rubi [A]  time = 0.152569, 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 = {} $\int \frac{n q-\log \left (c x^n\right )}{\left (a x+b \log ^q\left (c x^n\right )\right )^2} \, dx$

Veriﬁcation is Not applicable to the result.

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

Mathematica [A]  time = 76.6005, size = 0, normalized size = 0. $\int \frac{n q-\log \left (c x^n\right )}{\left (a x+b \log ^q\left (c x^n\right )\right )^2} \, dx$

Veriﬁcation is Not applicable to the result.

[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]

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Maple [A]  time = 0.368, size = 0, normalized size = 0. \begin{align*} \int{\frac{nq-\ln \left ( c{x}^{n} \right ) }{ \left ( ax+b \left ( \ln \left ( c{x}^{n} \right ) \right ) ^{q} \right ) ^{2}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} n{\left (q - 1\right )} \int \frac{1}{a^{2} x^{2} + a b x{\left (\log \left (c\right ) + \log \left (x^{n}\right )\right )}^{q}}\,{d x} + \frac{\log \left (c\right ) + \log \left (x^{n}\right )}{a^{2} x + a b{\left (\log \left (c\right ) + \log \left (x^{n}\right )\right )}^{q}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)