3.10 \(\int \frac{\coth ^{-1}(a x)}{x^4} \, dx\)

Optimal. Leaf size=47 \[ -\frac{1}{6} a^3 \log \left (1-a^2 x^2\right )+\frac{1}{3} a^3 \log (x)-\frac{a}{6 x^2}-\frac{\coth ^{-1}(a x)}{3 x^3} \]

[Out]

-a/(6*x^2) - ArcCoth[a*x]/(3*x^3) + (a^3*Log[x])/3 - (a^3*Log[1 - a^2*x^2])/6

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Rubi [A]  time = 0.0302444, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {5917, 266, 44} \[ -\frac{1}{6} a^3 \log \left (1-a^2 x^2\right )+\frac{1}{3} a^3 \log (x)-\frac{a}{6 x^2}-\frac{\coth ^{-1}(a x)}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[ArcCoth[a*x]/x^4,x]

[Out]

-a/(6*x^2) - ArcCoth[a*x]/(3*x^3) + (a^3*Log[x])/3 - (a^3*Log[1 - a^2*x^2])/6

Rule 5917

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcC
oth[c*x])^p)/(d*(m + 1)), x] - Dist[(b*c*p)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcCoth[c*x])^(p - 1))/(1 -
c^2*x^2), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && (EqQ[p, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{\coth ^{-1}(a x)}{x^4} \, dx &=-\frac{\coth ^{-1}(a x)}{3 x^3}+\frac{1}{3} a \int \frac{1}{x^3 \left (1-a^2 x^2\right )} \, dx\\ &=-\frac{\coth ^{-1}(a x)}{3 x^3}+\frac{1}{6} a \operatorname{Subst}\left (\int \frac{1}{x^2 \left (1-a^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac{\coth ^{-1}(a x)}{3 x^3}+\frac{1}{6} a \operatorname{Subst}\left (\int \left (\frac{1}{x^2}+\frac{a^2}{x}-\frac{a^4}{-1+a^2 x}\right ) \, dx,x,x^2\right )\\ &=-\frac{a}{6 x^2}-\frac{\coth ^{-1}(a x)}{3 x^3}+\frac{1}{3} a^3 \log (x)-\frac{1}{6} a^3 \log \left (1-a^2 x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0095672, size = 47, normalized size = 1. \[ -\frac{1}{6} a^3 \log \left (1-a^2 x^2\right )+\frac{1}{3} a^3 \log (x)-\frac{a}{6 x^2}-\frac{\coth ^{-1}(a x)}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCoth[a*x]/x^4,x]

[Out]

-a/(6*x^2) - ArcCoth[a*x]/(3*x^3) + (a^3*Log[x])/3 - (a^3*Log[1 - a^2*x^2])/6

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Maple [A]  time = 0.039, size = 48, normalized size = 1. \begin{align*} -{\frac{{\rm arccoth} \left (ax\right )}{3\,{x}^{3}}}-{\frac{{a}^{3}\ln \left ( ax-1 \right ) }{6}}-{\frac{a}{6\,{x}^{2}}}+{\frac{{a}^{3}\ln \left ( ax \right ) }{3}}-{\frac{{a}^{3}\ln \left ( ax+1 \right ) }{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccoth(a*x)/x^4,x)

[Out]

-1/3*arccoth(a*x)/x^3-1/6*a^3*ln(a*x-1)-1/6*a/x^2+1/3*a^3*ln(a*x)-1/6*a^3*ln(a*x+1)

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Maxima [A]  time = 0.945418, size = 54, normalized size = 1.15 \begin{align*} -\frac{1}{6} \,{\left (a^{2} \log \left (a^{2} x^{2} - 1\right ) - a^{2} \log \left (x^{2}\right ) + \frac{1}{x^{2}}\right )} a - \frac{\operatorname{arcoth}\left (a x\right )}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(a*x)/x^4,x, algorithm="maxima")

[Out]

-1/6*(a^2*log(a^2*x^2 - 1) - a^2*log(x^2) + 1/x^2)*a - 1/3*arccoth(a*x)/x^3

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Fricas [A]  time = 1.5701, size = 120, normalized size = 2.55 \begin{align*} -\frac{a^{3} x^{3} \log \left (a^{2} x^{2} - 1\right ) - 2 \, a^{3} x^{3} \log \left (x\right ) + a x + \log \left (\frac{a x + 1}{a x - 1}\right )}{6 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(a*x)/x^4,x, algorithm="fricas")

[Out]

-1/6*(a^3*x^3*log(a^2*x^2 - 1) - 2*a^3*x^3*log(x) + a*x + log((a*x + 1)/(a*x - 1)))/x^3

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Sympy [A]  time = 4.12088, size = 46, normalized size = 0.98 \begin{align*} \frac{a^{3} \log{\left (x \right )}}{3} - \frac{a^{3} \log{\left (a x + 1 \right )}}{3} + \frac{a^{3} \operatorname{acoth}{\left (a x \right )}}{3} - \frac{a}{6 x^{2}} - \frac{\operatorname{acoth}{\left (a x \right )}}{3 x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acoth(a*x)/x**4,x)

[Out]

a**3*log(x)/3 - a**3*log(a*x + 1)/3 + a**3*acoth(a*x)/3 - a/(6*x**2) - acoth(a*x)/(3*x**3)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arcoth}\left (a x\right )}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(a*x)/x^4,x, algorithm="giac")

[Out]

integrate(arccoth(a*x)/x^4, x)