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

Optimal. Leaf size=48 \[ \frac {e^{\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \text {Ei}\left (-\frac {a+b \log \left (c x^n\right )}{b n}\right )}{b n x} \]

[Out]

exp(a/b/n)*(c*x^n)^(1/n)*Ei((-a-b*ln(c*x^n))/b/n)/b/n/x

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Rubi [A]  time = 0.05, antiderivative size = 48, normalized size of antiderivative = 1.00, 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} \[ \frac {e^{\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \text {Ei}\left (-\frac {a+b \log \left (c x^n\right )}{b n}\right )}{b n x} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(a + b*Log[c*x^n])),x]

[Out]

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

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]

Rubi steps

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

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Mathematica [A]  time = 0.05, size = 48, normalized size = 1.00 \[ \frac {e^{\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \text {Ei}\left (-\frac {a+b \log \left (c x^n\right )}{b n}\right )}{b n x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.42, size = 41, normalized size = 0.85 \[ \frac {e^{\left (\frac {b \log \relax (c) + a}{b n}\right )} \operatorname {log\_integral}\left (\frac {e^{\left (-\frac {b \log \relax (c) + a}{b n}\right )}}{x}\right )}{b n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.28, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \ln \left (c \,x^{n}\right )+a \right ) x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(a + b*log(c*x^n))),x)

[Out]

int(1/(x^2*(a + b*log(c*x^n))), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{2} \left (a + b \log {\left (c x^{n} \right )}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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