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

Optimal. Leaf size=70 \[ \frac{x e^{-\frac{a}{b n}} \left (c x^n\right )^{-1/n} \text{Ei}\left (\frac{a+b \log \left (c x^n\right )}{b n}\right )}{b^2 n^2}-\frac{x}{b n \left (a+b \log \left (c x^n\right )\right )} \]

[Out]

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

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Rubi [A]  time = 0.039918, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2297, 2300, 2178} \[ \frac{x e^{-\frac{a}{b n}} \left (c x^n\right )^{-1/n} \text{Ei}\left (\frac{a+b \log \left (c x^n\right )}{b n}\right )}{b^2 n^2}-\frac{x}{b n \left (a+b \log \left (c x^n\right )\right )} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*x^n])^(-2),x]

[Out]

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

Rule 2297

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Simp[(x*(a + b*Log[c*x^n])^(p + 1))/(b*n*(p + 1))
, x] - Dist[1/(b*n*(p + 1)), Int[(a + b*Log[c*x^n])^(p + 1), x], x] /; FreeQ[{a, b, c, n}, x] && LtQ[p, -1] &&
IntegerQ[2*p]

Rule 2300

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[E^(x/n)*(a +
b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, 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}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx &=-\frac{x}{b n \left (a+b \log \left (c x^n\right )\right )}+\frac{\int \frac{1}{a+b \log \left (c x^n\right )} \, dx}{b n}\\ &=-\frac{x}{b n \left (a+b \log \left (c x^n\right )\right )}+\frac{\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{b n^2}\\ &=\frac{e^{-\frac{a}{b n}} x \left (c x^n\right )^{-1/n} \text{Ei}\left (\frac{a+b \log \left (c x^n\right )}{b n}\right )}{b^2 n^2}-\frac{x}{b n \left (a+b \log \left (c x^n\right )\right )}\\ \end{align*}

Mathematica [A]  time = 0.0975662, size = 66, normalized size = 0.94 \[ \frac{x \left (e^{-\frac{a}{b n}} \left (c x^n\right )^{-1/n} \text{Ei}\left (\frac{a+b \log \left (c x^n\right )}{b n}\right )-\frac{b n}{a+b \log \left (c x^n\right )}\right )}{b^2 n^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*x^n])^(-2),x]

[Out]

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

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Maple [C]  time = 0.265, size = 351, normalized size = 5. \begin{align*} -2\,{\frac{x}{bn \left ( 2\,a+2\,b\ln \left ( c \right ) +2\,b\ln \left ({x}^{n} \right ) +ib\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-ib\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -ib\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+ib\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) \right ) }}-{\frac{1}{{b}^{2}{n}^{2}}{\it Ei} \left ( 1,-\ln \left ( x \right ) -{\frac{ib\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-ib\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -ib\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+ib\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +2\,b\ln \left ( c \right ) +2\,b \left ( \ln \left ({x}^{n} \right ) -n\ln \left ( x \right ) \right ) +2\,a}{2\,bn}} \right ){{\rm e}^{{\frac{-ib\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+ib\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) +ib\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}-ib\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +2\,\ln \left ( x \right ) bn-2\,b\ln \left ({x}^{n} \right ) -2\,b\ln \left ( c \right ) -2\,a}{2\,bn}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/b/n*x/(2*a+2*b*ln(c)+2*b*ln(x^n)+I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I
*c)-I*b*Pi*csgn(I*c*x^n)^3+I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c))-1/b^2/n^2*Ei(1,-ln(x)-1/2*(I*b*Pi*csgn(I*x^n)*csg
n(I*c*x^n)^2-I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-I*b*Pi*csgn(I*c*x^n)^3+I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c
)+2*b*ln(c)+2*b*(ln(x^n)-n*ln(x))+2*a)/b/n)*exp(1/2*(-I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2+I*b*Pi*csgn(I*x^n)*cs
gn(I*c*x^n)*csgn(I*c)+I*b*Pi*csgn(I*c*x^n)^3-I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)+2*ln(x)*b*n-2*b*ln(x^n)-2*b*ln(c
)-2*a)/b/n)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0.951567, size = 240, normalized size = 3.43 \begin{align*} -\frac{{\left (b n x e^{\left (\frac{b \log \left (c\right ) + a}{b n}\right )} -{\left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )} \logintegral \left (x e^{\left (\frac{b \log \left (c\right ) + a}{b n}\right )}\right )\right )} e^{\left (-\frac{b \log \left (c\right ) + a}{b n}\right )}}{b^{3} n^{3} \log \left (x\right ) + b^{3} n^{2} \log \left (c\right ) + a b^{2} n^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Giac [B]  time = 1.22164, size = 321, normalized size = 4.59 \begin{align*} \frac{b n{\rm Ei}\left (\frac{\log \left (c\right )}{n} + \frac{a}{b n} + \log \left (x\right )\right ) e^{\left (-\frac{a}{b n}\right )} \log \left (x\right )}{{\left (b^{3} n^{3} \log \left (x\right ) + b^{3} n^{2} \log \left (c\right ) + a b^{2} n^{2}\right )} c^{\left (\frac{1}{n}\right )}} - \frac{b n x}{b^{3} n^{3} \log \left (x\right ) + b^{3} n^{2} \log \left (c\right ) + a b^{2} n^{2}} + \frac{b{\rm Ei}\left (\frac{\log \left (c\right )}{n} + \frac{a}{b n} + \log \left (x\right )\right ) e^{\left (-\frac{a}{b n}\right )} \log \left (c\right )}{{\left (b^{3} n^{3} \log \left (x\right ) + b^{3} n^{2} \log \left (c\right ) + a b^{2} n^{2}\right )} c^{\left (\frac{1}{n}\right )}} + \frac{a{\rm Ei}\left (\frac{\log \left (c\right )}{n} + \frac{a}{b n} + \log \left (x\right )\right ) e^{\left (-\frac{a}{b n}\right )}}{{\left (b^{3} n^{3} \log \left (x\right ) + b^{3} n^{2} \log \left (c\right ) + a b^{2} n^{2}\right )} c^{\left (\frac{1}{n}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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