3.30 \(\int \frac {\coth ^{-1}(a x)^3}{x^2} \, dx\)

Optimal. Leaf size=79 \[ -\frac {3}{2} a \text {Li}_3\left (\frac {2}{a x+1}-1\right )-3 a \text {Li}_2\left (\frac {2}{a x+1}-1\right ) \coth ^{-1}(a x)+a \coth ^{-1}(a x)^3-\frac {\coth ^{-1}(a x)^3}{x}+3 a \log \left (2-\frac {2}{a x+1}\right ) \coth ^{-1}(a x)^2 \]

[Out]

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

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Rubi [A]  time = 0.20, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5917, 5989, 5933, 5949, 6057, 6610} \[ -\frac {3}{2} a \text {PolyLog}\left (3,\frac {2}{a x+1}-1\right )-3 a \coth ^{-1}(a x) \text {PolyLog}\left (2,\frac {2}{a x+1}-1\right )+a \coth ^{-1}(a x)^3-\frac {\coth ^{-1}(a x)^3}{x}+3 a \log \left (2-\frac {2}{a x+1}\right ) \coth ^{-1}(a x)^2 \]

Antiderivative was successfully verified.

[In]

Int[ArcCoth[a*x]^3/x^2,x]

[Out]

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

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 5933

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 5949

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 5989

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]

Rule 6057

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

Rule 6610

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /
;  !FalseQ[w]] /; FreeQ[n, x]

Rubi steps

\begin {align*} \int \frac {\coth ^{-1}(a x)^3}{x^2} \, dx &=-\frac {\coth ^{-1}(a x)^3}{x}+(3 a) \int \frac {\coth ^{-1}(a x)^2}{x \left (1-a^2 x^2\right )} \, dx\\ &=a \coth ^{-1}(a x)^3-\frac {\coth ^{-1}(a x)^3}{x}+(3 a) \int \frac {\coth ^{-1}(a x)^2}{x (1+a x)} \, dx\\ &=a \coth ^{-1}(a x)^3-\frac {\coth ^{-1}(a x)^3}{x}+3 a \coth ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )-\left (6 a^2\right ) \int \frac {\coth ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=a \coth ^{-1}(a x)^3-\frac {\coth ^{-1}(a x)^3}{x}+3 a \coth ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )-3 a \coth ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+a x}\right )+\left (3 a^2\right ) \int \frac {\text {Li}_2\left (-1+\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=a \coth ^{-1}(a x)^3-\frac {\coth ^{-1}(a x)^3}{x}+3 a \coth ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )-3 a \coth ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+a x}\right )-\frac {3}{2} a \text {Li}_3\left (-1+\frac {2}{1+a x}\right )\\ \end {align*}

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Mathematica [A]  time = 0.13, size = 72, normalized size = 0.91 \[ -3 a \coth ^{-1}(a x) \text {Li}_2\left (-e^{-2 \coth ^{-1}(a x)}\right )-\frac {3}{2} a \text {Li}_3\left (-e^{-2 \coth ^{-1}(a x)}\right )+\frac {(a x-1) \coth ^{-1}(a x)^3}{x}+3 a \coth ^{-1}(a x)^2 \log \left (e^{-2 \coth ^{-1}(a x)}+1\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcCoth[a*x]^3/x^2,x]

[Out]

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

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fricas [F]  time = 0.81, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arcoth}\left (a x\right )^{3}}{x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(a*x)^3/x^2,x, algorithm="fricas")

[Out]

integral(arccoth(a*x)^3/x^2, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arcoth}\left (a x\right )^{3}}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(a*x)^3/x^2,x, algorithm="giac")

[Out]

integrate(arccoth(a*x)^3/x^2, x)

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maple [C]  time = 0.70, size = 796, normalized size = 10.08 \[ -\frac {\mathrm {arccoth}\left (a x \right )^{3}}{x}+3 a \mathrm {arccoth}\left (a x \right )^{2} \ln \left (a x \right )-\frac {3 a \mathrm {arccoth}\left (a x \right )^{2} \ln \left (a x -1\right )}{2}-\frac {3 a \mathrm {arccoth}\left (a x \right )^{2} \ln \left (a x +1\right )}{2}-\frac {3 a \mathrm {arccoth}\left (a x \right )^{2} \ln \left (\frac {a x -1}{a x +1}\right )}{2}-a \mathrm {arccoth}\left (a x \right )^{3}+3 a \,\mathrm {arccoth}\left (a x \right ) \polylog \left (2, -\frac {a x +1}{a x -1}\right )-\frac {3 a \polylog \left (3, -\frac {a x +1}{a x -1}\right )}{2}-\frac {3 i a \mathrm {arccoth}\left (a x \right )^{2} \pi \,\mathrm {csgn}\left (\frac {i}{\frac {a x +1}{a x -1}-1}\right ) \mathrm {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right ) \mathrm {csgn}\left (\frac {i \left (a x +1\right )}{\left (a x -1\right ) \left (\frac {a x +1}{a x -1}-1\right )}\right )}{4}-\frac {3 i a \mathrm {arccoth}\left (a x \right )^{2} \pi \mathrm {csgn}\left (\frac {i \left (1+\frac {a x +1}{a x -1}\right )}{\frac {a x +1}{a x -1}-1}\right )^{2} \mathrm {csgn}\left (i \left (1+\frac {a x +1}{a x -1}\right )\right )}{2}-\frac {3 i a \mathrm {arccoth}\left (a x \right )^{2} \pi \mathrm {csgn}\left (\frac {i \left (1+\frac {a x +1}{a x -1}\right )}{\frac {a x +1}{a x -1}-1}\right )^{2} \mathrm {csgn}\left (\frac {i}{\frac {a x +1}{a x -1}-1}\right )}{2}+\frac {3 i a \mathrm {arccoth}\left (a x \right )^{2} \pi \,\mathrm {csgn}\left (\frac {i}{\frac {a x +1}{a x -1}-1}\right ) \mathrm {csgn}\left (\frac {i \left (a x +1\right )}{\left (a x -1\right ) \left (\frac {a x +1}{a x -1}-1\right )}\right )^{2}}{4}-\frac {3 i a \mathrm {arccoth}\left (a x \right )^{2} \pi \mathrm {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right )^{3}}{4}+\frac {3 i a \mathrm {arccoth}\left (a x \right )^{2} \pi \mathrm {csgn}\left (\frac {i \left (1+\frac {a x +1}{a x -1}\right )}{\frac {a x +1}{a x -1}-1}\right )^{3}}{2}+\frac {3 i a \mathrm {arccoth}\left (a x \right )^{2} \pi \,\mathrm {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right ) \mathrm {csgn}\left (\frac {i \left (a x +1\right )}{\left (a x -1\right ) \left (\frac {a x +1}{a x -1}-1\right )}\right )^{2}}{4}-\frac {3 i a \mathrm {arccoth}\left (a x \right )^{2} \pi \mathrm {csgn}\left (\frac {i \left (a x +1\right )}{\left (a x -1\right ) \left (\frac {a x +1}{a x -1}-1\right )}\right )^{3}}{4}+\frac {3 i a \mathrm {arccoth}\left (a x \right )^{2} \pi \mathrm {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right )^{2} \mathrm {csgn}\left (\frac {i}{\sqrt {\frac {a x -1}{a x +1}}}\right )}{2}-\frac {3 i a \mathrm {arccoth}\left (a x \right )^{2} \pi \,\mathrm {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right ) \mathrm {csgn}\left (\frac {i}{\sqrt {\frac {a x -1}{a x +1}}}\right )^{2}}{4}+\frac {3 i a \mathrm {arccoth}\left (a x \right )^{2} \pi \,\mathrm {csgn}\left (\frac {i \left (1+\frac {a x +1}{a x -1}\right )}{\frac {a x +1}{a x -1}-1}\right ) \mathrm {csgn}\left (\frac {i}{\frac {a x +1}{a x -1}-1}\right ) \mathrm {csgn}\left (i \left (1+\frac {a x +1}{a x -1}\right )\right )}{2}+3 a \mathrm {arccoth}\left (a x \right )^{2} \ln \relax (2) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccoth(a*x)^3/x^2,x)

[Out]

-arccoth(a*x)^3/x+3*a*arccoth(a*x)^2*ln(a*x)-3/2*a*arccoth(a*x)^2*ln(a*x-1)-3/2*a*arccoth(a*x)^2*ln(a*x+1)-3/2
*a*arccoth(a*x)^2*ln((a*x-1)/(a*x+1))-a*arccoth(a*x)^3+3*a*arccoth(a*x)*polylog(2,-(a*x+1)/(a*x-1))-3/2*a*poly
log(3,-(a*x+1)/(a*x-1))-3/4*I*a*arccoth(a*x)^2*Pi*csgn(I/((a*x+1)/(a*x-1)-1))*csgn(I*(a*x+1)/(a*x-1))*csgn(I*(
a*x+1)/(a*x-1)/((a*x+1)/(a*x-1)-1))-3/2*I*a*arccoth(a*x)^2*Pi*csgn(I/((a*x+1)/(a*x-1)-1)*(1+(a*x+1)/(a*x-1)))^
2*csgn(I*(1+(a*x+1)/(a*x-1)))-3/2*I*a*arccoth(a*x)^2*Pi*csgn(I/((a*x+1)/(a*x-1)-1)*(1+(a*x+1)/(a*x-1)))^2*csgn
(I/((a*x+1)/(a*x-1)-1))+3/4*I*a*arccoth(a*x)^2*Pi*csgn(I/((a*x+1)/(a*x-1)-1))*csgn(I*(a*x+1)/(a*x-1)/((a*x+1)/
(a*x-1)-1))^2-3/4*I*a*arccoth(a*x)^2*Pi*csgn(I*(a*x+1)/(a*x-1))^3+3/2*I*a*arccoth(a*x)^2*Pi*csgn(I/((a*x+1)/(a
*x-1)-1)*(1+(a*x+1)/(a*x-1)))^3+3/4*I*a*arccoth(a*x)^2*Pi*csgn(I*(a*x+1)/(a*x-1))*csgn(I*(a*x+1)/(a*x-1)/((a*x
+1)/(a*x-1)-1))^2-3/4*I*a*arccoth(a*x)^2*Pi*csgn(I*(a*x+1)/(a*x-1)/((a*x+1)/(a*x-1)-1))^3+3/2*I*a*arccoth(a*x)
^2*Pi*csgn(I*(a*x+1)/(a*x-1))^2*csgn(I/((a*x-1)/(a*x+1))^(1/2))-3/4*I*a*arccoth(a*x)^2*Pi*csgn(I*(a*x+1)/(a*x-
1))*csgn(I/((a*x-1)/(a*x+1))^(1/2))^2+3/2*I*a*arccoth(a*x)^2*Pi*csgn(I/((a*x+1)/(a*x-1)-1)*(1+(a*x+1)/(a*x-1))
)*csgn(I/((a*x+1)/(a*x-1)-1))*csgn(I*(1+(a*x+1)/(a*x-1)))+3*a*arccoth(a*x)^2*ln(2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(a*x)^3/x^2,x, algorithm="maxima")

[Out]

1/8*a*(log(a*x + 1) - log(x))*log(a)^3 + 3/8*a*integrate(x*log(a*x - 1)/(a*x^3 + x^2), x)*log(a)^2 - 3/8*a*int
egrate(x*log(x)/(a*x^3 + x^2), x)*log(a)^2 - 1/8*(a*log(a*x + 1) - a*log(x) - 1/x)*log(a)^3 + 3/4*a^2*integrat
e(x^2*log(a*x + 1)*log(a*x - 1)/(a*x^3 + x^2), x) - 3/2*a^2*integrate(x^2*log(a*x + 1)*log(x)/(a*x^3 + x^2), x
) + 3/4*a*integrate(x*log(a*x - 1)*log(x)/(a*x^3 + x^2), x)*log(a) - 3/8*a*integrate(x*log(x)^2/(a*x^3 + x^2),
x)*log(a) + 3/8*integrate(log(a*x - 1)/(a*x^3 + x^2), x)*log(a)^2 - 3/8*integrate(log(x)/(a*x^3 + x^2), x)*lo
g(a)^2 + 3/8*a*integrate(x*log(a*x + 1)*log(a*x - 1)^2/(a*x^3 + x^2), x) - 3/8*a*integrate(x*log(a*x - 1)^2*lo
g(x)/(a*x^3 + x^2), x) + 3/8*a*integrate(x*log(a*x - 1)*log(x)^2/(a*x^3 + x^2), x) - 1/8*a*integrate(x*log(x)^
3/(a*x^3 + x^2), x) - 3/4*a*integrate(x*log(a*x + 1)*log(a*x - 1)/(a*x^3 + x^2), x) - 3/8*integrate(a*x*log(a*
x - 1)^2/(a*x^3 + x^2), x)*log(a) - 3/8*integrate(log(a*x - 1)^2/(a*x^3 + x^2), x)*log(a) + 3/4*integrate(log(
a*x - 1)*log(x)/(a*x^3 + x^2), x)*log(a) - 3/8*integrate(log(x)^2/(a*x^3 + x^2), x)*log(a) - 3/8*(a^2*log(a*x
- 1) - a^2*log(x) + a/x)*log(-1/(a*x) + 1)^2/a + 1/8*log(-1/(a*x) + 1)^3/x - 1/8*((a*x + 1)*log(a*x + 1)^3 - 3
*(2*a*x*log(x) - (a*x - 1)*log(a*x - 1))*log(a*x + 1)^2)/x + 1/8*(3*(a^3*x*log(a*x - 1)^2 + a^3*x*log(x)^2 - 2
*a^3*x*log(x) + 2*a^2 - 2*(a^3*x*log(x) - a^3*x)*log(a*x - 1))*log(-1/(a*x) + 1)/(a*x) - (a^4*x*log(a*x - 1)^3
- a^4*x*log(x)^3 + 3*a^4*x*log(x)^2 - 6*a^4*x*log(x) + 6*a^3 - 3*(a^4*x*log(x) - a^4*x)*log(a*x - 1)^2 + 3*(a
^4*x*log(x)^2 - 2*a^4*x*log(x) + 2*a^4*x)*log(a*x - 1))/(a^2*x))/a + 3/8*integrate(log(a*x + 1)*log(a*x - 1)^2
/(a*x^3 + x^2), x) - 3/8*integrate(log(a*x - 1)^2*log(x)/(a*x^3 + x^2), x) + 3/8*integrate(log(a*x - 1)*log(x)
^2/(a*x^3 + x^2), x) - 1/8*integrate(log(x)^3/(a*x^3 + x^2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {acoth}\left (a\,x\right )}^3}{x^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acoth(a*x)^3/x^2,x)

[Out]

int(acoth(a*x)^3/x^2, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {acoth}^{3}{\left (a x \right )}}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acoth(a*x)**3/x**2,x)

[Out]

Integral(acoth(a*x)**3/x**2, x)

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