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

Optimal. Leaf size=51 \[ \frac{e^{\frac{2 a}{b n}} \left (c x^n\right )^{2/n} \text{Ei}\left (-\frac{2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b n x^2} \]

[Out]

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

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Rubi [A]  time = 0.0513079, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2310, 2178} \[ \frac{e^{\frac{2 a}{b n}} \left (c x^n\right )^{2/n} \text{Ei}\left (-\frac{2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b n x^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(a + b*Log[c*x^n])),x]

[Out]

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

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

Mathematica [A]  time = 0.0500862, size = 51, normalized size = 1. \[ \frac{e^{\frac{2 a}{b n}} \left (c x^n\right )^{2/n} \text{Ei}\left (-\frac{2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b n x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.165, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3} \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Fricas [A]  time = 0.785426, size = 108, normalized size = 2.12 \begin{align*} \frac{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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**3/(a+b*ln(c*x**n)),x)

[Out]

Integral(1/(x**3*(a + b*log(c*x**n))), 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 )} 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)),x, algorithm="giac")

[Out]

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