3.1.9 \(\int \frac {\coth ^{-1}(a x)}{x^3} \, dx\) [9]

Optimal. Leaf size=31 \[ -\frac {a}{2 x}-\frac {\coth ^{-1}(a x)}{2 x^2}+\frac {1}{2} a^2 \tanh ^{-1}(a x) \]

[Out]

-1/2*a/x-1/2*arccoth(a*x)/x^2+1/2*a^2*arctanh(a*x)

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Rubi [A]
time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6038, 331, 212} \begin {gather*} \frac {1}{2} a^2 \tanh ^{-1}(a x)-\frac {\coth ^{-1}(a x)}{2 x^2}-\frac {a}{2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcCoth[a*x]/x^3,x]

[Out]

-1/2*a/x - ArcCoth[a*x]/(2*x^2) + (a^2*ArcTanh[a*x])/2

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 6038

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

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 47, normalized size = 1.52 \begin {gather*} -\frac {a}{2 x}-\frac {\coth ^{-1}(a x)}{2 x^2}-\frac {1}{4} a^2 \log (1-a x)+\frac {1}{4} a^2 \log (1+a x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[ArcCoth[a*x]/x^3,x]

[Out]

-1/2*a/x - ArcCoth[a*x]/(2*x^2) - (a^2*Log[1 - a*x])/4 + (a^2*Log[1 + a*x])/4

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Maple [A]
time = 0.04, size = 42, normalized size = 1.35

method	result	size
derivativedivides	\(a^{2} \left (-\frac {\mathrm {arccoth}\left (a x \right )}{2 a^{2} x^{2}}-\frac {\ln \left (a x -1\right )}{4}+\frac {\ln \left (a x +1\right )}{4}-\frac {1}{2 a x}\right )\)	\(42\)
default	\(a^{2} \left (-\frac {\mathrm {arccoth}\left (a x \right )}{2 a^{2} x^{2}}-\frac {\ln \left (a x -1\right )}{4}+\frac {\ln \left (a x +1\right )}{4}-\frac {1}{2 a x}\right )\)	\(42\)
risch	\(-\frac {\ln \left (a x +1\right )}{4 x^{2}}+\frac {\ln \left (-a x -1\right ) a^{2} x^{2}-\ln \left (-a x +1\right ) a^{2} x^{2}-2 a x +\ln \left (a x -1\right )}{4 x^{2}}\)	\(58\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccoth(a*x)/x^3,x,method=_RETURNVERBOSE)

[Out]

a^2*(-1/2/a^2/x^2*arccoth(a*x)-1/4*ln(a*x-1)+1/4*ln(a*x+1)-1/2/a/x)

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Maxima [A]
time = 0.25, size = 36, normalized size = 1.16 \begin {gather*} \frac {1}{4} \, {\left (a \log \left (a x + 1\right ) - a \log \left (a x - 1\right ) - \frac {2}{x}\right )} a - \frac {\operatorname {arcoth}\left (a x\right )}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(a*x)/x^3,x, algorithm="maxima")

[Out]

1/4*(a*log(a*x + 1) - a*log(a*x - 1) - 2/x)*a - 1/2*arccoth(a*x)/x^2

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Fricas [A]
time = 0.35, size = 35, normalized size = 1.13 \begin {gather*} -\frac {2 \, a x - {\left (a^{2} x^{2} - 1\right )} \log \left (\frac {a x + 1}{a x - 1}\right )}{4 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(a*x)/x^3,x, algorithm="fricas")

[Out]

-1/4*(2*a*x - (a^2*x^2 - 1)*log((a*x + 1)/(a*x - 1)))/x^2

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Sympy [A]
time = 0.24, size = 24, normalized size = 0.77 \begin {gather*} \frac {a^{2} \operatorname {acoth}{\left (a x \right )}}{2} - \frac {a}{2 x} - \frac {\operatorname {acoth}{\left (a x \right )}}{2 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acoth(a*x)/x**3,x)

[Out]

a**2*acoth(a*x)/2 - a/(2*x) - acoth(a*x)/(2*x**2)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 140 vs. \(2 (25) = 50\).
time = 0.41, size = 140, normalized size = 4.52 \begin {gather*} a {\left (\frac {a}{\frac {a x + 1}{a x - 1} + 1} + \frac {{\left (a x + 1\right )} a \log \left (-\frac {\frac {\frac {{\left (a x + 1\right )} a}{a x - 1} - a}{a {\left (\frac {a x + 1}{a x - 1} + 1\right )}} + 1}{\frac {\frac {{\left (a x + 1\right )} a}{a x - 1} - a}{a {\left (\frac {a x + 1}{a x - 1} + 1\right )}} - 1}\right )}{{\left (a x - 1\right )} {\left (\frac {a x + 1}{a x - 1} + 1\right )}^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(a*x)/x^3,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 1.19, size = 40, normalized size = 1.29 \begin {gather*} \frac {a\,\mathrm {atan}\left (\frac {a^2\,x}{\sqrt {-a^2}}\right )\,\sqrt {-a^2}}{2}-\frac {\frac {\mathrm {acoth}\left (a\,x\right )}{2}+\frac {a\,x}{2}}{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acoth(a*x)/x^3,x)

[Out]

(a*atan((a^2*x)/(-a^2)^(1/2))*(-a^2)^(1/2))/2 - (acoth(a*x)/2 + (a*x)/2)/x^2

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