3.1.33 \(\int \frac {\coth ^{-1}(a x)^3}{x^5} \, dx\) [33]

Optimal. Leaf size=141 \[ -\frac {a^3}{4 x}-\frac {a^2 \coth ^{-1}(a x)}{4 x^2}+a^4 \coth ^{-1}(a x)^2-\frac {a \coth ^{-1}(a x)^2}{4 x^3}-\frac {3 a^3 \coth ^{-1}(a x)^2}{4 x}+\frac {1}{4} a^4 \coth ^{-1}(a x)^3-\frac {\coth ^{-1}(a x)^3}{4 x^4}+\frac {1}{4} a^4 \tanh ^{-1}(a x)+2 a^4 \coth ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )-a^4 \text {PolyLog}\left (2,-1+\frac {2}{1+a x}\right ) \]

[Out]

-1/4*a^3/x-1/4*a^2*arccoth(a*x)/x^2+a^4*arccoth(a*x)^2-1/4*a*arccoth(a*x)^2/x^3-3/4*a^3*arccoth(a*x)^2/x+1/4*a
^4*arccoth(a*x)^3-1/4*arccoth(a*x)^3/x^4+1/4*a^4*arctanh(a*x)+2*a^4*arccoth(a*x)*ln(2-2/(a*x+1))-a^4*polylog(2
,-1+2/(a*x+1))

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Rubi [A]
time = 0.31, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 8, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {6038, 6130, 331, 212, 6136, 6080, 2497, 6096} \begin {gather*} -a^4 \text {Li}_2\left (\frac {2}{a x+1}-1\right )+\frac {1}{4} a^4 \tanh ^{-1}(a x)+\frac {1}{4} a^4 \coth ^{-1}(a x)^3+a^4 \coth ^{-1}(a x)^2+2 a^4 \log \left (2-\frac {2}{a x+1}\right ) \coth ^{-1}(a x)-\frac {a^3}{4 x}-\frac {3 a^3 \coth ^{-1}(a x)^2}{4 x}-\frac {a^2 \coth ^{-1}(a x)}{4 x^2}-\frac {\coth ^{-1}(a x)^3}{4 x^4}-\frac {a \coth ^{-1}(a x)^2}{4 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcCoth[a*x]^3/x^5,x]

[Out]

-1/4*a^3/x - (a^2*ArcCoth[a*x])/(4*x^2) + a^4*ArcCoth[a*x]^2 - (a*ArcCoth[a*x]^2)/(4*x^3) - (3*a^3*ArcCoth[a*x
]^2)/(4*x) + (a^4*ArcCoth[a*x]^3)/4 - ArcCoth[a*x]^3/(4*x^4) + (a^4*ArcTanh[a*x])/4 + 2*a^4*ArcCoth[a*x]*Log[2
- 2/(1 + a*x)] - a^4*PolyLog[2, -1 + 2/(1 + a*x)]

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 2497

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/D[u, x])]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, 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
]

Rule 6080

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

Rule 6096

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcCoth[c*x])^(p
+ 1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && NeQ[p, -1]

Rule 6130

Int[(((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/d
, Int[(f*x)^m*(a + b*ArcCoth[c*x])^p, x], x] - Dist[e/(d*f^2), Int[(f*x)^(m + 2)*((a + b*ArcCoth[c*x])^p/(d +
e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && LtQ[m, -1]

Rule 6136

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Simp[(a + b*ArcCoth[c
*x])^(p + 1)/(b*d*(p + 1)), x] + Dist[1/d, Int[(a + b*ArcCoth[c*x])^p/(x*(1 + c*x)), x], x] /; FreeQ[{a, b, c,
d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0]

Rubi steps

\begin {align*} \int \frac {\coth ^{-1}(a x)^3}{x^5} \, dx &=-\frac {\coth ^{-1}(a x)^3}{4 x^4}+\frac {1}{4} (3 a) \int \frac {\coth ^{-1}(a x)^2}{x^4 \left (1-a^2 x^2\right )} \, dx\\ &=-\frac {\coth ^{-1}(a x)^3}{4 x^4}+\frac {1}{4} (3 a) \int \frac {\coth ^{-1}(a x)^2}{x^4} \, dx+\frac {1}{4} \left (3 a^3\right ) \int \frac {\coth ^{-1}(a x)^2}{x^2 \left (1-a^2 x^2\right )} \, dx\\ &=-\frac {a \coth ^{-1}(a x)^2}{4 x^3}-\frac {\coth ^{-1}(a x)^3}{4 x^4}+\frac {1}{2} a^2 \int \frac {\coth ^{-1}(a x)}{x^3 \left (1-a^2 x^2\right )} \, dx+\frac {1}{4} \left (3 a^3\right ) \int \frac {\coth ^{-1}(a x)^2}{x^2} \, dx+\frac {1}{4} \left (3 a^5\right ) \int \frac {\coth ^{-1}(a x)^2}{1-a^2 x^2} \, dx\\ &=-\frac {a \coth ^{-1}(a x)^2}{4 x^3}-\frac {3 a^3 \coth ^{-1}(a x)^2}{4 x}+\frac {1}{4} a^4 \coth ^{-1}(a x)^3-\frac {\coth ^{-1}(a x)^3}{4 x^4}+\frac {1}{2} a^2 \int \frac {\coth ^{-1}(a x)}{x^3} \, dx+\frac {1}{2} a^4 \int \frac {\coth ^{-1}(a x)}{x \left (1-a^2 x^2\right )} \, dx+\frac {1}{2} \left (3 a^4\right ) \int \frac {\coth ^{-1}(a x)}{x \left (1-a^2 x^2\right )} \, dx\\ &=-\frac {a^2 \coth ^{-1}(a x)}{4 x^2}+a^4 \coth ^{-1}(a x)^2-\frac {a \coth ^{-1}(a x)^2}{4 x^3}-\frac {3 a^3 \coth ^{-1}(a x)^2}{4 x}+\frac {1}{4} a^4 \coth ^{-1}(a x)^3-\frac {\coth ^{-1}(a x)^3}{4 x^4}+\frac {1}{4} a^3 \int \frac {1}{x^2 \left (1-a^2 x^2\right )} \, dx+\frac {1}{2} a^4 \int \frac {\coth ^{-1}(a x)}{x (1+a x)} \, dx+\frac {1}{2} \left (3 a^4\right ) \int \frac {\coth ^{-1}(a x)}{x (1+a x)} \, dx\\ &=-\frac {a^3}{4 x}-\frac {a^2 \coth ^{-1}(a x)}{4 x^2}+a^4 \coth ^{-1}(a x)^2-\frac {a \coth ^{-1}(a x)^2}{4 x^3}-\frac {3 a^3 \coth ^{-1}(a x)^2}{4 x}+\frac {1}{4} a^4 \coth ^{-1}(a x)^3-\frac {\coth ^{-1}(a x)^3}{4 x^4}+2 a^4 \coth ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )+\frac {1}{4} a^5 \int \frac {1}{1-a^2 x^2} \, dx-\frac {1}{2} a^5 \int \frac {\log \left (2-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx-\frac {1}{2} \left (3 a^5\right ) \int \frac {\log \left (2-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac {a^3}{4 x}-\frac {a^2 \coth ^{-1}(a x)}{4 x^2}+a^4 \coth ^{-1}(a x)^2-\frac {a \coth ^{-1}(a x)^2}{4 x^3}-\frac {3 a^3 \coth ^{-1}(a x)^2}{4 x}+\frac {1}{4} a^4 \coth ^{-1}(a x)^3-\frac {\coth ^{-1}(a x)^3}{4 x^4}+\frac {1}{4} a^4 \tanh ^{-1}(a x)+2 a^4 \coth ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )-a^4 \text {Li}_2\left (-1+\frac {2}{1+a x}\right )\\ \end {align*}

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Mathematica [A]
time = 0.16, size = 118, normalized size = 0.84 \begin {gather*} \frac {-a^3 x^3+a x \left (-1-3 a^2 x^2+4 a^3 x^3\right ) \coth ^{-1}(a x)^2+\left (-1+a^4 x^4\right ) \coth ^{-1}(a x)^3+a^2 x^2 \coth ^{-1}(a x) \left (-1+a^2 x^2+8 a^2 x^2 \log \left (1+e^{-2 \coth ^{-1}(a x)}\right )\right )-4 a^4 x^4 \text {PolyLog}\left (2,-e^{-2 \coth ^{-1}(a x)}\right )}{4 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[ArcCoth[a*x]^3/x^5,x]

[Out]

(-(a^3*x^3) + a*x*(-1 - 3*a^2*x^2 + 4*a^3*x^3)*ArcCoth[a*x]^2 + (-1 + a^4*x^4)*ArcCoth[a*x]^3 + a^2*x^2*ArcCot
h[a*x]*(-1 + a^2*x^2 + 8*a^2*x^2*Log[1 + E^(-2*ArcCoth[a*x])]) - 4*a^4*x^4*PolyLog[2, -E^(-2*ArcCoth[a*x])])/(
4*x^4)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 3.61, size = 659, normalized size = 4.67 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccoth(a*x)^3/x^5,x,method=_RETURNVERBOSE)

[Out]

a^4*(-1/4/a^4/x^4*arccoth(a*x)^3+3/8*arccoth(a*x)^2*ln(a*x+1)-1/4/a^3/x^3*arccoth(a*x)^2-3/4/a/x*arccoth(a*x)^
2-3/8*arccoth(a*x)^2*ln(a*x-1)+2*dilog(1+I/((a*x-1)/(a*x+1))^(1/2))+2*dilog(1-I/((a*x-1)/(a*x+1))^(1/2))+1/4*(
a*x-1)/a/x-1/2*arccoth(a*x)*(a*x+1)/a/x-arccoth(a*x)^2+1/4*arccoth(a*x)^3-3/8*I*arccoth(a*x)^2*Pi*csgn(I/((a*x
-1)/(a*x+1))^(1/2))*csgn(I/(a*x-1)*(a*x+1))^2+2*arccoth(a*x)*ln(1+I/((a*x-1)/(a*x+1))^(1/2))+2*arccoth(a*x)*ln
(1-I/((a*x-1)/(a*x+1))^(1/2))+3/16*I*arccoth(a*x)^2*Pi*csgn(I/(a*x-1)*(a*x+1))^3+3/16*I*arccoth(a*x)^2*Pi*csgn
(I/((a*x-1)/(a*x+1))^(1/2))^2*csgn(I/(a*x-1)*(a*x+1))+1/4*arccoth(a*x)*(a*x+1)^2/a^2/x^2+1/2*arccoth(a*x)*(a*x
-1)*(a*x+1)/a^2/x^2-3/16*I*arccoth(a*x)^2*Pi*csgn(I/(1/(a*x-1)*(a*x+1)-1))*csgn(I/(a*x-1)*(a*x+1)/(1/(a*x-1)*(
a*x+1)-1))^2-3/16*I*arccoth(a*x)^2*Pi*csgn(I/(a*x-1)*(a*x+1))*csgn(I/(a*x-1)*(a*x+1)/(1/(a*x-1)*(a*x+1)-1))^2+
3/16*I*arccoth(a*x)^2*Pi*csgn(I/(1/(a*x-1)*(a*x+1)-1))*csgn(I/(a*x-1)*(a*x+1))*csgn(I/(a*x-1)*(a*x+1)/(1/(a*x-
1)*(a*x+1)-1))+3/16*I*arccoth(a*x)^2*Pi*csgn(I/(a*x-1)*(a*x+1)/(1/(a*x-1)*(a*x+1)-1))^3+3/8*arccoth(a*x)^2*ln(
(a*x-1)/(a*x+1)))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 342 vs. \(2 (126) = 252\).
time = 0.28, size = 342, normalized size = 2.43 \begin {gather*} \frac {1}{8} \, {\left (3 \, a^{3} \log \left (a x + 1\right ) - 3 \, a^{3} \log \left (a x - 1\right ) - \frac {2 \, {\left (3 \, a^{2} x^{2} + 1\right )}}{x^{3}}\right )} a \operatorname {arcoth}\left (a x\right )^{2} + \frac {1}{32} \, {\left ({\left (32 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )} a - 32 \, {\left (\log \left (a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-a x\right )\right )} a + 32 \, {\left (\log \left (-a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (a x\right )\right )} a + 4 \, a \log \left (a x + 1\right ) - 4 \, a \log \left (a x - 1\right ) + \frac {a x \log \left (a x + 1\right )^{3} - a x \log \left (a x - 1\right )^{3} - 8 \, a x \log \left (a x - 1\right )^{2} - {\left (3 \, a x \log \left (a x - 1\right ) - 8 \, a x\right )} \log \left (a x + 1\right )^{2} + {\left (3 \, a x \log \left (a x - 1\right )^{2} - 16 \, a x \log \left (a x - 1\right )\right )} \log \left (a x + 1\right ) - 8}{x}\right )} a^{2} + 2 \, {\left (32 \, a^{2} \log \left (x\right ) - \frac {3 \, a^{2} x^{2} \log \left (a x + 1\right )^{2} + 3 \, a^{2} x^{2} \log \left (a x - 1\right )^{2} + 16 \, a^{2} x^{2} \log \left (a x - 1\right ) - 2 \, {\left (3 \, a^{2} x^{2} \log \left (a x - 1\right ) - 8 \, a^{2} x^{2}\right )} \log \left (a x + 1\right ) + 4}{x^{2}}\right )} a \operatorname {arcoth}\left (a x\right )\right )} a - \frac {\operatorname {arcoth}\left (a x\right )^{3}}{4 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(a*x)^3/x^5,x, algorithm="maxima")

[Out]

1/8*(3*a^3*log(a*x + 1) - 3*a^3*log(a*x - 1) - 2*(3*a^2*x^2 + 1)/x^3)*a*arccoth(a*x)^2 + 1/32*((32*(log(a*x -
1)*log(1/2*a*x + 1/2) + dilog(-1/2*a*x + 1/2))*a - 32*(log(a*x + 1)*log(x) + dilog(-a*x))*a + 32*(log(-a*x + 1
)*log(x) + dilog(a*x))*a + 4*a*log(a*x + 1) - 4*a*log(a*x - 1) + (a*x*log(a*x + 1)^3 - a*x*log(a*x - 1)^3 - 8*
a*x*log(a*x - 1)^2 - (3*a*x*log(a*x - 1) - 8*a*x)*log(a*x + 1)^2 + (3*a*x*log(a*x - 1)^2 - 16*a*x*log(a*x - 1)
)*log(a*x + 1) - 8)/x)*a^2 + 2*(32*a^2*log(x) - (3*a^2*x^2*log(a*x + 1)^2 + 3*a^2*x^2*log(a*x - 1)^2 + 16*a^2*
x^2*log(a*x - 1) - 2*(3*a^2*x^2*log(a*x - 1) - 8*a^2*x^2)*log(a*x + 1) + 4)/x^2)*a*arccoth(a*x))*a - 1/4*arcco
th(a*x)^3/x^4

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(a*x)^3/x^5,x, algorithm="fricas")

[Out]

integral(arccoth(a*x)^3/x^5, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {acoth}^{3}{\left (a x \right )}}{x^{5}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acoth(a*x)**3/x**5,x)

[Out]

Integral(acoth(a*x)**3/x**5, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(a*x)^3/x^5,x, algorithm="giac")

[Out]

integrate(arccoth(a*x)^3/x^5, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {acoth}\left (a\,x\right )}^3}{x^5} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acoth(a*x)^3/x^5,x)

[Out]

int(acoth(a*x)^3/x^5, x)

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