3.78 \(\int \frac{1}{x^2 (a+b \log (c x^n))^2} \, dx\)

Optimal. Leaf size=73 \[ -\frac{e^{\frac{a}{b n}} \left (c x^n\right )^{\frac{1}{n}} \text{Ei}\left (-\frac{a+b \log \left (c x^n\right )}{b n}\right )}{b^2 n^2 x}-\frac{1}{b n x \left (a+b \log \left (c x^n\right )\right )} \]

[Out]

-((E^(a/(b*n))*(c*x^n)^n^(-1)*ExpIntegralEi[-((a + b*Log[c*x^n])/(b*n))])/(b^2*n^2*x)) - 1/(b*n*x*(a + b*Log[c
*x^n]))

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Rubi [A]  time = 0.0745142, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {2306, 2310, 2178} \[ -\frac{e^{\frac{a}{b n}} \left (c x^n\right )^{\frac{1}{n}} \text{Ei}\left (-\frac{a+b \log \left (c x^n\right )}{b n}\right )}{b^2 n^2 x}-\frac{1}{b n x \left (a+b \log \left (c x^n\right )\right )} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(a + b*Log[c*x^n])^2),x]

[Out]

-((E^(a/(b*n))*(c*x^n)^n^(-1)*ExpIntegralEi[-((a + b*Log[c*x^n])/(b*n))])/(b^2*n^2*x)) - 1/(b*n*x*(a + b*Log[c
*x^n]))

Rule 2306

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log
[c*x^n])^(p + 1))/(b*d*n*(p + 1)), x] - Dist[(m + 1)/(b*n*(p + 1)), Int[(d*x)^m*(a + b*Log[c*x^n])^(p + 1), x]
, x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1] && LtQ[p, -1]

Rule 2310

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol] :> Dist[(d*x)^(m + 1)/(d*n*(c*x^n
)^((m + 1)/n)), Subst[Int[E^(((m + 1)*x)/n)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}
, x]

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rubi steps

\begin{align*} \int \frac{1}{x^2 \left (a+b \log \left (c x^n\right )\right )^2} \, dx &=-\frac{1}{b n x \left (a+b \log \left (c x^n\right )\right )}-\frac{\int \frac{1}{x^2 \left (a+b \log \left (c x^n\right )\right )} \, dx}{b n}\\ &=-\frac{1}{b n x \left (a+b \log \left (c x^n\right )\right )}-\frac{\left (c x^n\right )^{\frac{1}{n}} \operatorname{Subst}\left (\int \frac{e^{-\frac{x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{b n^2 x}\\ &=-\frac{e^{\frac{a}{b n}} \left (c x^n\right )^{\frac{1}{n}} \text{Ei}\left (-\frac{a+b \log \left (c x^n\right )}{b n}\right )}{b^2 n^2 x}-\frac{1}{b n x \left (a+b \log \left (c x^n\right )\right )}\\ \end{align*}

Mathematica [A]  time = 0.0889523, size = 76, normalized size = 1.04 \[ -\frac{e^{\frac{a}{b n}} \left (c x^n\right )^{\frac{1}{n}} \left (a+b \log \left (c x^n\right )\right ) \text{Ei}\left (-\frac{a+b \log \left (c x^n\right )}{b n}\right )+b n}{b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*n + E^(a/(b*n))*(c*x^n)^n^(-1)*ExpIntegralEi[-((a + b*Log[c*x^n])/(b*n))]*(a + b*Log[c*x^n]))/(b^2*n^2*x*
(a + b*Log[c*x^n])))

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Maple [F]  time = 0.659, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2} \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(a+b*ln(c*x^n))^2,x)

[Out]

int(1/x^2/(a+b*ln(c*x^n))^2,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(a+b*log(c*x^n))^2,x, algorithm="maxima")

[Out]

-1/(b^2*n*x*log(x^n) + (b^2*n*log(c) + a*b*n)*x) - integrate(1/(b^2*n*x^2*log(x^n) + (b^2*n*log(c) + a*b*n)*x^
2), x)

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Fricas [A]  time = 0.772961, size = 220, normalized size = 3.01 \begin{align*} -\frac{{\left (b n x \log \left (x\right ) + b x \log \left (c\right ) + a x\right )} e^{\left (\frac{b \log \left (c\right ) + a}{b n}\right )} \logintegral \left (\frac{e^{\left (-\frac{b \log \left (c\right ) + a}{b n}\right )}}{x}\right ) + b n}{b^{3} n^{3} x \log \left (x\right ) + b^{3} n^{2} x \log \left (c\right ) + a b^{2} n^{2} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(a+b*log(c*x^n))^2,x, algorithm="fricas")

[Out]

-((b*n*x*log(x) + b*x*log(c) + a*x)*e^((b*log(c) + a)/(b*n))*log_integral(e^(-(b*log(c) + a)/(b*n))/x) + b*n)/
(b^3*n^3*x*log(x) + b^3*n^2*x*log(c) + a*b^2*n^2*x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/(a+b*ln(c*x**n))**2,x)

[Out]

Integral(1/(x**2*(a + b*log(c*x**n))**2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(a+b*log(c*x^n))^2,x, algorithm="giac")

[Out]

integrate(1/((b*log(c*x^n) + a)^2*x^2), x)