3.79 \(\int \frac{1}{x^3 (a+b \log (c x^n))^2} \, dx\)

Optimal. Leaf size=76 \[ -\frac{2 e^{\frac{2 a}{b n}} \left (c x^n\right )^{2/n} \text{Ei}\left (-\frac{2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b^2 n^2 x^2}-\frac{1}{b n x^2 \left (a+b \log \left (c x^n\right )\right )} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0777883, antiderivative size = 76, 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} \[ -\frac{2 e^{\frac{2 a}{b n}} \left (c x^n\right )^{2/n} \text{Ei}\left (-\frac{2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b^2 n^2 x^2}-\frac{1}{b n x^2 \left (a+b \log \left (c x^n\right )\right )} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(a + b*Log[c*x^n])^2),x]

[Out]

(-2*E^((2*a)/(b*n))*(c*x^n)^(2/n)*ExpIntegralEi[(-2*(a + b*Log[c*x^n]))/(b*n)])/(b^2*n^2*x^2) - 1/(b*n*x^2*(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^3 \left (a+b \log \left (c x^n\right )\right )^2} \, dx &=-\frac{1}{b n x^2 \left (a+b \log \left (c x^n\right )\right )}-\frac{2 \int \frac{1}{x^3 \left (a+b \log \left (c x^n\right )\right )} \, dx}{b n}\\ &=-\frac{1}{b n x^2 \left (a+b \log \left (c x^n\right )\right )}-\frac{\left (2 \left (c x^n\right )^{2/n}\right ) \operatorname{Subst}\left (\int \frac{e^{-\frac{2 x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{b n^2 x^2}\\ &=-\frac{2 e^{\frac{2 a}{b n}} \left (c x^n\right )^{2/n} \text{Ei}\left (-\frac{2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b^2 n^2 x^2}-\frac{1}{b n x^2 \left (a+b \log \left (c x^n\right )\right )}\\ \end{align*}

Mathematica [A]  time = 0.0910069, size = 80, normalized size = 1.05 \[ -\frac{2 e^{\frac{2 a}{b n}} \left (c x^n\right )^{2/n} \left (a+b \log \left (c x^n\right )\right ) \text{Ei}\left (-\frac{2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )+b n}{b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*(a + b*Log[c*x^n])^2),x]

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.725, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3} \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^3/(a+b*ln(c*x^n))^2,x)

[Out]

int(1/x^3/(a+b*ln(c*x^n))^2,x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\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}} - 2 \, \int \frac{1}{b^{2} n x^{3} \log \left (x^{n}\right ) +{\left (b^{2} n \log \left (c\right ) + a b n\right )} x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(a+b*log(c*x^n))^2,x, algorithm="maxima")

[Out]

-1/(b^2*n*x^2*log(x^n) + (b^2*n*log(c) + a*b*n)*x^2) - 2*integrate(1/(b^2*n*x^3*log(x^n) + (b^2*n*log(c) + a*b
*n)*x^3), x)

________________________________________________________________________________________

Fricas [A]  time = 0.952235, size = 247, normalized size = 3.25 \begin{align*} -\frac{2 \,{\left (b n x^{2} \log \left (x\right ) + b x^{2} \log \left (c\right ) + a x^{2}\right )} e^{\left (\frac{2 \,{\left (b \log \left (c\right ) + a\right )}}{b n}\right )} \logintegral \left (\frac{e^{\left (-\frac{2 \,{\left (b \log \left (c\right ) + a\right )}}{b n}\right )}}{x^{2}}\right ) + b n}{b^{3} n^{3} x^{2} \log \left (x\right ) + b^{3} n^{2} x^{2} \log \left (c\right ) + a b^{2} n^{2} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(a+b*log(c*x^n))^2,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{3} \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**3/(a+b*ln(c*x**n))**2,x)

[Out]

Integral(1/(x**3*(a + b*log(c*x**n))**2), x)

________________________________________________________________________________________

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^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(a+b*log(c*x^n))^2,x, algorithm="giac")

[Out]

integrate(1/((b*log(c*x^n) + a)^2*x^3), x)