3.1.30 \(\int \frac {\coth ^{-1}(a x)^3}{x^2} \, dx\) [30]

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

[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 = 0.14, 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 = {6038, 6136, 6080, 6096, 6204, 6745} \begin {gather*} -\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 \end {gather*}

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 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 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]

Rule 6204

Int[(Log[u_]*((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcCot
h[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 6745

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 = 0.10, size = 72, normalized size = 0.91 \begin {gather*} \frac {(-1+a x) \coth ^{-1}(a x)^3}{x}+3 a \coth ^{-1}(a x)^2 \log \left (1+e^{-2 \coth ^{-1}(a x)}\right )-3 a \coth ^{-1}(a x) \text {PolyLog}\left (2,-e^{-2 \coth ^{-1}(a x)}\right )-\frac {3}{2} a \text {PolyLog}\left (3,-e^{-2 \coth ^{-1}(a x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[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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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 3.21, size = 719, normalized size = 9.10

method	result	size
derivativedivides	\(a \left (-\frac {\mathrm {arccoth}\left (a x \right )^{3}}{a x}-\frac {3 \mathrm {arccoth}\left (a x \right )^{2} \ln \left (a x -1\right )}{2}+3 \mathrm {arccoth}\left (a x \right )^{2} \ln \left (a x \right )-\frac {3 \mathrm {arccoth}\left (a x \right )^{2} \ln \left (a x +1\right )}{2}-\frac {3 \mathrm {arccoth}\left (a x \right )^{2} \ln \left (\frac {a x -1}{a x +1}\right )}{2}-\mathrm {arccoth}\left (a x \right )^{3}+\frac {3 \left (2 i \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 i \pi \,\mathrm {csgn}\left (\frac {i}{\frac {a x +1}{a x -1}-1}\right ) \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}-2 i \pi \,\mathrm {csgn}\left (i \left (1+\frac {a x +1}{a x -1}\right )\right ) \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}+2 i \pi \,\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 ) \mathrm {csgn}\left (\frac {i \left (1+\frac {a x +1}{a x -1}\right )}{\frac {a x +1}{a x -1}-1}\right )-i \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 )+i \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}-i \pi \mathrm {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right )^{3}+2 i \pi \,\mathrm {csgn}\left (\frac {i}{\sqrt {\frac {a x -1}{a x +1}}}\right ) \mathrm {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right )^{2}+i \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}-i \pi \mathrm {csgn}\left (\frac {i}{\sqrt {\frac {a x -1}{a x +1}}}\right )^{2} \mathrm {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right )-i \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 \ln \left (2\right )\right ) \mathrm {arccoth}\left (a x \right )^{2}}{4}+3 \,\mathrm {arccoth}\left (a x \right ) \polylog \left (2, -\frac {a x +1}{a x -1}\right )-\frac {3 \polylog \left (3, -\frac {a x +1}{a x -1}\right )}{2}\right )\)	\(719\)
default	\(a \left (-\frac {\mathrm {arccoth}\left (a x \right )^{3}}{a x}-\frac {3 \mathrm {arccoth}\left (a x \right )^{2} \ln \left (a x -1\right )}{2}+3 \mathrm {arccoth}\left (a x \right )^{2} \ln \left (a x \right )-\frac {3 \mathrm {arccoth}\left (a x \right )^{2} \ln \left (a x +1\right )}{2}-\frac {3 \mathrm {arccoth}\left (a x \right )^{2} \ln \left (\frac {a x -1}{a x +1}\right )}{2}-\mathrm {arccoth}\left (a x \right )^{3}+\frac {3 \left (2 i \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 i \pi \,\mathrm {csgn}\left (\frac {i}{\frac {a x +1}{a x -1}-1}\right ) \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}-2 i \pi \,\mathrm {csgn}\left (i \left (1+\frac {a x +1}{a x -1}\right )\right ) \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}+2 i \pi \,\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 ) \mathrm {csgn}\left (\frac {i \left (1+\frac {a x +1}{a x -1}\right )}{\frac {a x +1}{a x -1}-1}\right )-i \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 )+i \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}-i \pi \mathrm {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right )^{3}+2 i \pi \,\mathrm {csgn}\left (\frac {i}{\sqrt {\frac {a x -1}{a x +1}}}\right ) \mathrm {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right )^{2}+i \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}-i \pi \mathrm {csgn}\left (\frac {i}{\sqrt {\frac {a x -1}{a x +1}}}\right )^{2} \mathrm {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right )-i \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 \ln \left (2\right )\right ) \mathrm {arccoth}\left (a x \right )^{2}}{4}+3 \,\mathrm {arccoth}\left (a x \right ) \polylog \left (2, -\frac {a x +1}{a x -1}\right )-\frac {3 \polylog \left (3, -\frac {a x +1}{a x -1}\right )}{2}\right )\)	\(719\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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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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^2,x, algorithm="fricas")

[Out]

integral(arccoth(a*x)^3/x^2, 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^{2}}\, dx \end {gather*}

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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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^2,x, algorithm="giac")

[Out]

integrate(arccoth(a*x)^3/x^2, 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^2} \,d x \end {gather*}

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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