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

Optimal. Leaf size=61 \[ \frac {\log \left (c x^n\right )}{a \left (a x+b \log ^q\left (c x^n\right )\right )}-\frac {n (1-q) \text {Int}\left (\frac {1}{x \left (a x+b \log ^q\left (c x^n\right )\right )},x\right )}{a} \]

[Out]

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

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Rubi [A]  time = 0.15, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}

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Mathematica [A]  time = 80.03, size = 0, normalized size = 0.00 \[ \int \frac {n q-\log \left (c x^n\right )}{\left (a x+b \log ^q\left (c x^n\right )\right )^2} \, dx \]

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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fricas [A]  time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {n q - \log \left (c x^{n}\right )}{a^{2} x^{2} + 2 \, a b x \log \left (c x^{n}\right )^{q} + b^{2} \log \left (c x^{n}\right )^{2 \, q}}, x\right ) \]

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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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {n q - \log \left (c x^{n}\right )}{{\left (a x + b \log \left (c x^{n}\right )^{q}\right )}^{2}}\,{d x} \]

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)

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maple [A]  time = 3.26, size = 0, normalized size = 0.00 \[ \int \frac {n q -\ln \left (c \,x^{n}\right )}{\left (a x +b \ln \left (c \,x^{n}\right )^{q}\right )^{2}}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ n {\left (q - 1\right )} \int \frac {1}{a^{2} x^{2} + a b x {\left (\log \relax (c) + \log \left (x^{n}\right )\right )}^{q}}\,{d x} + \frac {\log \relax (c) + \log \left (x^{n}\right )}{a^{2} x + a b {\left (\log \relax (c) + \log \left (x^{n}\right )\right )}^{q}} \]

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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mupad [A]  time = 0.00, size = -1, normalized size = -0.02 \[ \int -\frac {\ln \left (c\,x^n\right )-n\,q}{{\left (b\,{\ln \left (c\,x^n\right )}^q+a\,x\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {n q - \log {\left (c x^{n} \right )}}{\left (a x + b \log {\left (c x^{n} \right )}^{q}\right )^{2}}\, dx \]

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]

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

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