3.1.23 \(\int x^5 \coth ^{-1}(a x)^3 \, dx\) [23]

Optimal. Leaf size=186 \[ \frac {19 x}{60 a^5}+\frac {x^3}{60 a^3}+\frac {4 x^2 \coth ^{-1}(a x)}{15 a^4}+\frac {x^4 \coth ^{-1}(a x)}{20 a^2}+\frac {23 \coth ^{-1}(a x)^2}{30 a^6}+\frac {x \coth ^{-1}(a x)^2}{2 a^5}+\frac {x^3 \coth ^{-1}(a x)^2}{6 a^3}+\frac {x^5 \coth ^{-1}(a x)^2}{10 a}-\frac {\coth ^{-1}(a x)^3}{6 a^6}+\frac {1}{6} x^6 \coth ^{-1}(a x)^3-\frac {19 \tanh ^{-1}(a x)}{60 a^6}-\frac {23 \coth ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{15 a^6}-\frac {23 \text {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{30 a^6} \]

[Out]

19/60*x/a^5+1/60*x^3/a^3+4/15*x^2*arccoth(a*x)/a^4+1/20*x^4*arccoth(a*x)/a^2+23/30*arccoth(a*x)^2/a^6+1/2*x*ar
ccoth(a*x)^2/a^5+1/6*x^3*arccoth(a*x)^2/a^3+1/10*x^5*arccoth(a*x)^2/a-1/6*arccoth(a*x)^3/a^6+1/6*x^6*arccoth(a
*x)^3-19/60*arctanh(a*x)/a^6-23/15*arccoth(a*x)*ln(2/(-a*x+1))/a^6-23/30*polylog(2,1-2/(-a*x+1))/a^6

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Rubi [A]
time = 0.49, antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 33, number of rules used = 11, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.100, Rules used = {6038, 6128, 308, 212, 327, 6132, 6056, 2449, 2352, 6022, 6096} \begin {gather*} -\frac {23 \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{30 a^6}-\frac {19 \tanh ^{-1}(a x)}{60 a^6}-\frac {\coth ^{-1}(a x)^3}{6 a^6}+\frac {23 \coth ^{-1}(a x)^2}{30 a^6}-\frac {23 \log \left (\frac {2}{1-a x}\right ) \coth ^{-1}(a x)}{15 a^6}+\frac {19 x}{60 a^5}+\frac {x \coth ^{-1}(a x)^2}{2 a^5}+\frac {4 x^2 \coth ^{-1}(a x)}{15 a^4}+\frac {x^3}{60 a^3}+\frac {x^3 \coth ^{-1}(a x)^2}{6 a^3}+\frac {x^4 \coth ^{-1}(a x)}{20 a^2}+\frac {1}{6} x^6 \coth ^{-1}(a x)^3+\frac {x^5 \coth ^{-1}(a x)^2}{10 a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(19*x)/(60*a^5) + x^3/(60*a^3) + (4*x^2*ArcCoth[a*x])/(15*a^4) + (x^4*ArcCoth[a*x])/(20*a^2) + (23*ArcCoth[a*x
]^2)/(30*a^6) + (x*ArcCoth[a*x]^2)/(2*a^5) + (x^3*ArcCoth[a*x]^2)/(6*a^3) + (x^5*ArcCoth[a*x]^2)/(10*a) - ArcC
oth[a*x]^3/(6*a^6) + (x^6*ArcCoth[a*x]^3)/6 - (19*ArcTanh[a*x])/(60*a^6) - (23*ArcCoth[a*x]*Log[2/(1 - a*x)])/
(15*a^6) - (23*PolyLog[2, 1 - 2/(1 - a*x)])/(30*a^6)

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 308

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 327

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

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2449

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Dist[-e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 6022

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

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 6056

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcCoth[c*x])^p)
*(Log[2/(1 + e*(x/d))]/e), x] + Dist[b*c*(p/e), Int[(a + b*ArcCoth[c*x])^(p - 1)*(Log[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 6128

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

Rule 6132

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

Rubi steps

\begin {align*} \int x^5 \coth ^{-1}(a x)^3 \, dx &=\frac {1}{6} x^6 \coth ^{-1}(a x)^3-\frac {1}{2} a \int \frac {x^6 \coth ^{-1}(a x)^2}{1-a^2 x^2} \, dx\\ &=\frac {1}{6} x^6 \coth ^{-1}(a x)^3+\frac {\int x^4 \coth ^{-1}(a x)^2 \, dx}{2 a}-\frac {\int \frac {x^4 \coth ^{-1}(a x)^2}{1-a^2 x^2} \, dx}{2 a}\\ &=\frac {x^5 \coth ^{-1}(a x)^2}{10 a}+\frac {1}{6} x^6 \coth ^{-1}(a x)^3-\frac {1}{5} \int \frac {x^5 \coth ^{-1}(a x)}{1-a^2 x^2} \, dx+\frac {\int x^2 \coth ^{-1}(a x)^2 \, dx}{2 a^3}-\frac {\int \frac {x^2 \coth ^{-1}(a x)^2}{1-a^2 x^2} \, dx}{2 a^3}\\ &=\frac {x^3 \coth ^{-1}(a x)^2}{6 a^3}+\frac {x^5 \coth ^{-1}(a x)^2}{10 a}+\frac {1}{6} x^6 \coth ^{-1}(a x)^3+\frac {\int \coth ^{-1}(a x)^2 \, dx}{2 a^5}-\frac {\int \frac {\coth ^{-1}(a x)^2}{1-a^2 x^2} \, dx}{2 a^5}+\frac {\int x^3 \coth ^{-1}(a x) \, dx}{5 a^2}-\frac {\int \frac {x^3 \coth ^{-1}(a x)}{1-a^2 x^2} \, dx}{5 a^2}-\frac {\int \frac {x^3 \coth ^{-1}(a x)}{1-a^2 x^2} \, dx}{3 a^2}\\ &=\frac {x^4 \coth ^{-1}(a x)}{20 a^2}+\frac {x \coth ^{-1}(a x)^2}{2 a^5}+\frac {x^3 \coth ^{-1}(a x)^2}{6 a^3}+\frac {x^5 \coth ^{-1}(a x)^2}{10 a}-\frac {\coth ^{-1}(a x)^3}{6 a^6}+\frac {1}{6} x^6 \coth ^{-1}(a x)^3+\frac {\int x \coth ^{-1}(a x) \, dx}{5 a^4}-\frac {\int \frac {x \coth ^{-1}(a x)}{1-a^2 x^2} \, dx}{5 a^4}+\frac {\int x \coth ^{-1}(a x) \, dx}{3 a^4}-\frac {\int \frac {x \coth ^{-1}(a x)}{1-a^2 x^2} \, dx}{3 a^4}-\frac {\int \frac {x \coth ^{-1}(a x)}{1-a^2 x^2} \, dx}{a^4}-\frac {\int \frac {x^4}{1-a^2 x^2} \, dx}{20 a}\\ &=\frac {4 x^2 \coth ^{-1}(a x)}{15 a^4}+\frac {x^4 \coth ^{-1}(a x)}{20 a^2}+\frac {23 \coth ^{-1}(a x)^2}{30 a^6}+\frac {x \coth ^{-1}(a x)^2}{2 a^5}+\frac {x^3 \coth ^{-1}(a x)^2}{6 a^3}+\frac {x^5 \coth ^{-1}(a x)^2}{10 a}-\frac {\coth ^{-1}(a x)^3}{6 a^6}+\frac {1}{6} x^6 \coth ^{-1}(a x)^3-\frac {\int \frac {\coth ^{-1}(a x)}{1-a x} \, dx}{5 a^5}-\frac {\int \frac {\coth ^{-1}(a x)}{1-a x} \, dx}{3 a^5}-\frac {\int \frac {\coth ^{-1}(a x)}{1-a x} \, dx}{a^5}-\frac {\int \frac {x^2}{1-a^2 x^2} \, dx}{10 a^3}-\frac {\int \frac {x^2}{1-a^2 x^2} \, dx}{6 a^3}-\frac {\int \left (-\frac {1}{a^4}-\frac {x^2}{a^2}+\frac {1}{a^4 \left (1-a^2 x^2\right )}\right ) \, dx}{20 a}\\ &=\frac {19 x}{60 a^5}+\frac {x^3}{60 a^3}+\frac {4 x^2 \coth ^{-1}(a x)}{15 a^4}+\frac {x^4 \coth ^{-1}(a x)}{20 a^2}+\frac {23 \coth ^{-1}(a x)^2}{30 a^6}+\frac {x \coth ^{-1}(a x)^2}{2 a^5}+\frac {x^3 \coth ^{-1}(a x)^2}{6 a^3}+\frac {x^5 \coth ^{-1}(a x)^2}{10 a}-\frac {\coth ^{-1}(a x)^3}{6 a^6}+\frac {1}{6} x^6 \coth ^{-1}(a x)^3-\frac {23 \coth ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{15 a^6}-\frac {\int \frac {1}{1-a^2 x^2} \, dx}{20 a^5}-\frac {\int \frac {1}{1-a^2 x^2} \, dx}{10 a^5}-\frac {\int \frac {1}{1-a^2 x^2} \, dx}{6 a^5}+\frac {\int \frac {\log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{5 a^5}+\frac {\int \frac {\log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{3 a^5}+\frac {\int \frac {\log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{a^5}\\ &=\frac {19 x}{60 a^5}+\frac {x^3}{60 a^3}+\frac {4 x^2 \coth ^{-1}(a x)}{15 a^4}+\frac {x^4 \coth ^{-1}(a x)}{20 a^2}+\frac {23 \coth ^{-1}(a x)^2}{30 a^6}+\frac {x \coth ^{-1}(a x)^2}{2 a^5}+\frac {x^3 \coth ^{-1}(a x)^2}{6 a^3}+\frac {x^5 \coth ^{-1}(a x)^2}{10 a}-\frac {\coth ^{-1}(a x)^3}{6 a^6}+\frac {1}{6} x^6 \coth ^{-1}(a x)^3-\frac {19 \tanh ^{-1}(a x)}{60 a^6}-\frac {23 \coth ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{15 a^6}-\frac {\text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-a x}\right )}{5 a^6}-\frac {\text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-a x}\right )}{3 a^6}-\frac {\text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-a x}\right )}{a^6}\\ &=\frac {19 x}{60 a^5}+\frac {x^3}{60 a^3}+\frac {4 x^2 \coth ^{-1}(a x)}{15 a^4}+\frac {x^4 \coth ^{-1}(a x)}{20 a^2}+\frac {23 \coth ^{-1}(a x)^2}{30 a^6}+\frac {x \coth ^{-1}(a x)^2}{2 a^5}+\frac {x^3 \coth ^{-1}(a x)^2}{6 a^3}+\frac {x^5 \coth ^{-1}(a x)^2}{10 a}-\frac {\coth ^{-1}(a x)^3}{6 a^6}+\frac {1}{6} x^6 \coth ^{-1}(a x)^3-\frac {19 \tanh ^{-1}(a x)}{60 a^6}-\frac {23 \coth ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{15 a^6}-\frac {23 \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{30 a^6}\\ \end {align*}

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Mathematica [A]
time = 0.37, size = 117, normalized size = 0.63 \begin {gather*} \frac {a x \left (19+a^2 x^2\right )+2 \left (-23+15 a x+5 a^3 x^3+3 a^5 x^5\right ) \coth ^{-1}(a x)^2+10 \left (-1+a^6 x^6\right ) \coth ^{-1}(a x)^3+\coth ^{-1}(a x) \left (-19+16 a^2 x^2+3 a^4 x^4-92 \log \left (1-e^{-2 \coth ^{-1}(a x)}\right )\right )+46 \text {PolyLog}\left (2,e^{-2 \coth ^{-1}(a x)}\right )}{60 a^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a*x*(19 + a^2*x^2) + 2*(-23 + 15*a*x + 5*a^3*x^3 + 3*a^5*x^5)*ArcCoth[a*x]^2 + 10*(-1 + a^6*x^6)*ArcCoth[a*x]
^3 + ArcCoth[a*x]*(-19 + 16*a^2*x^2 + 3*a^4*x^4 - 92*Log[1 - E^(-2*ArcCoth[a*x])]) + 46*PolyLog[2, E^(-2*ArcCo
th[a*x])])/(60*a^6)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 5.72, size = 2135, normalized size = 11.48

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(2135\)
default	\(\text {Expression too large to display}\)	\(2135\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a^6*(1/6*a^3*x^3*arccoth(a*x)^2+1/10*a^5*x^5*arccoth(a*x)^2-23/15*arccoth(a*x)*ln(1+1/((a*x-1)/(a*x+1))^(1/2
))-1/6*arccoth(a*x)^3+23/30*arccoth(a*x)^2-1/20*(2*((a*x-1)/(a*x+1))^(1/2)*a^2*x^2-2*a^2*x^2-((a*x-1)/(a*x+1))
^(1/2)+2*a*x)*arccoth(a*x)*(a*x+1)^2+1/80*(2*((a*x-1)/(a*x+1))^(1/2)*a^2*x^2+2*((a*x-1)/(a*x+1))^(1/2)*a*x-2*a
^2*x^2+1)*arccoth(a*x)*(a*x+1)^2-1/8*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))+1/8*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))^2-23/
15*dilog(1+1/((a*x-1)/(a*x+1))^(1/2))+23/15*dilog(1/((a*x-1)/(a*x+1))^(1/2))-47/240*(((a*x-1)/(a*x+1))^(1/2)*a
*x+((a*x-1)/(a*x+1))^(1/2)+a*x)*arccoth(a*x)*(a*x+1)-1/120*(a*x-1)*((a*x-1)/(a*x+1))^(1/2)/(2*((a*x-1)/(a*x+1)
)^(1/2)*a*x+((a*x-1)/(a*x+1))^(1/2)-2*a*x+1)-3/40*(2*((a*x-1)/(a*x+1))^(1/2)*a^2*x^2+((a*x-1)/(a*x+1))^(1/2)*a
*x+2*a^2*x^2-((a*x-1)/(a*x+1))^(1/2)-a*x-1)*arccoth(a*x)*(a*x+1)+47/240*(((a*x-1)/(a*x+1))^(1/2)*a*x+((a*x-1)/
(a*x+1))^(1/2)-a*x)*arccoth(a*x)*(a*x+1)+1/4*arccoth(a*x)^2*ln(a*x-1)-1/4*arccoth(a*x)^2*ln(a*x+1)+1/80*(a*x-1
)/(((a*x-1)/(a*x+1))^(1/2)*a*x+((a*x-1)/(a*x+1))^(1/2)+a*x)-1/80*(a*x-1)/(((a*x-1)/(a*x+1))^(1/2)*a*x+((a*x-1)
/(a*x+1))^(1/2)-a*x)+1/8*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+1/4*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+3/40*(2*((a*x-1)/(a*x
+1))^(1/2)*a^2*x^2+2*((a*x-1)/(a*x+1))^(1/2)*a*x-2*a^2*x^2+1)*arccoth(a*x)*(a*x-1)*(a*x+1)+1/20*(2*((a*x-1)/(a
*x+1))^(1/2)*a^2*x^2+2*a^2*x^2-((a*x-1)/(a*x+1))^(1/2)-2*a*x)*(a*x-1)*arccoth(a*x)*(a*x+1)-1/8*I*arccoth(a*x)^
2*Pi*csgn(I/(a*x-1)*(a*x+1)/(1/(a*x-1)*(a*x+1)-1))^3-41/240*(((a*x-1)/(a*x+1))^(1/2)*a*x+((a*x-1)/(a*x+1))^(1/
2)-a*x-1)*arccoth(a*x)-1/4*arccoth(a*x)^2*ln((a*x-1)/(a*x+1))-41/120/(((a*x-1)/(a*x+1))^(1/2)+1)*((a*x-1)/(a*x
+1))^(1/2)-3/40*(2*((a*x-1)/(a*x+1))^(1/2)*a^2*x^2+2*((a*x-1)/(a*x+1))^(1/2)*a*x+2*a^2*x^2-1)*arccoth(a*x)*(a*
x-1)*(a*x+1)-1/8*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))-1/8*I*arccoth(a*x)^2*Pi*csgn(I/(a*x-1)*(a*x+1))^3-1/20*(2*((a*x-1)/(a*x+1))^(1/2)*a
^2*x^2-2*a^2*x^2-((a*x-1)/(a*x+1))^(1/2)+2*a*x)*(a*x-1)*arccoth(a*x)*(a*x+1)+3/40*(2*((a*x-1)/(a*x+1))^(1/2)*a
^2*x^2+((a*x-1)/(a*x+1))^(1/2)*a*x-2*a^2*x^2-((a*x-1)/(a*x+1))^(1/2)+a*x+1)*arccoth(a*x)*(a*x+1)+1/2*arccoth(a
*x)^2*a*x+1/6*a^6*x^6*arccoth(a*x)^3-1/40*(2*((a*x-1)/(a*x+1))^(1/2)*a^2*x^2+3*((a*x-1)/(a*x+1))^(1/2)*a*x-2*a
^2*x^2+((a*x-1)/(a*x+1))^(1/2)-a*x+1)*arccoth(a*x)*(a*x+1)-47/120*(((a*x-1)/(a*x+1))^(1/2)*a*x-a*x+1)*arccoth(
a*x)*(a*x+1)+1/20*(2*((a*x-1)/(a*x+1))^(1/2)*a^2*x^2+2*a^2*x^2-((a*x-1)/(a*x+1))^(1/2)-2*a*x)*arccoth(a*x)*(a*
x+1)^2-1/80*(2*((a*x-1)/(a*x+1))^(1/2)*a^2*x^2+2*((a*x-1)/(a*x+1))^(1/2)*a*x+2*a^2*x^2-1)*arccoth(a*x)*(a*x+1)
^2-3/40*(2*((a*x-1)/(a*x+1))^(1/2)*a^2*x^2+3*((a*x-1)/(a*x+1))^(1/2)*a*x-2*a^2*x^2+((a*x-1)/(a*x+1))^(1/2)-a*x
+1)*arccoth(a*x)*(a*x-1)+3/40*(2*((a*x-1)/(a*x+1))^(1/2)*a^2*x^2+3*((a*x-1)/(a*x+1))^(1/2)*a*x+2*a^2*x^2+((a*x
-1)/(a*x+1))^(1/2)+a*x-1)*arccoth(a*x)*(a*x-1)-41/120/(((a*x-1)/(a*x+1))^(1/2)-1)*((a*x-1)/(a*x+1))^(1/2)+41/2
40*(((a*x-1)/(a*x+1))^(1/2)*a*x+((a*x-1)/(a*x+1))^(1/2)+a*x+1)*arccoth(a*x)+1/40*(2*((a*x-1)/(a*x+1))^(1/2)*a^
2*x^2+3*((a*x-1)/(a*x+1))^(1/2)*a*x+2*a^2*x^2+((a*x-1)/(a*x+1))^(1/2)+a*x-1)*arccoth(a*x)*(a*x+1)-1/120*(a*x-1
)*((a*x-1)/(a*x+1))^(1/2)/(2*((a*x-1)/(a*x+1))^(1/2)*a*x+((a*x-1)/(a*x+1))^(1/2)+2*a*x-1)+47/120*(((a*x-1)/(a*
x+1))^(1/2)*a*x+a*x-1)*arccoth(a*x)*(a*x+1))

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Maxima [A]
time = 0.27, size = 289, normalized size = 1.55 \begin {gather*} \frac {1}{6} \, x^{6} \operatorname {arcoth}\left (a x\right )^{3} + \frac {1}{60} \, a {\left (\frac {2 \, {\left (3 \, a^{4} x^{5} + 5 \, a^{2} x^{3} + 15 \, x\right )}}{a^{6}} - \frac {15 \, \log \left (a x + 1\right )}{a^{7}} + \frac {15 \, \log \left (a x - 1\right )}{a^{7}}\right )} \operatorname {arcoth}\left (a x\right )^{2} + \frac {1}{240} \, a {\left (\frac {\frac {4 \, a^{3} x^{3} + {\left (15 \, \log \left (a x - 1\right ) - 46\right )} \log \left (a x + 1\right )^{2} - 5 \, \log \left (a x + 1\right )^{3} + 5 \, \log \left (a x - 1\right )^{3} + 76 \, a x - {\left (15 \, \log \left (a x - 1\right )^{2} - 92 \, \log \left (a x - 1\right )\right )} \log \left (a x + 1\right ) + 46 \, \log \left (a x - 1\right )^{2} + 38 \, \log \left (a x - 1\right )}{a} - \frac {184 \, {\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} - \frac {38 \, \log \left (a x + 1\right )}{a}}{a^{6}} + \frac {2 \, {\left (6 \, a^{4} x^{4} + 32 \, a^{2} x^{2} - 2 \, {\left (15 \, \log \left (a x - 1\right ) - 46\right )} \log \left (a x + 1\right ) + 15 \, \log \left (a x + 1\right )^{2} + 15 \, \log \left (a x - 1\right )^{2} + 92 \, \log \left (a x - 1\right )\right )} \operatorname {arcoth}\left (a x\right )}{a^{7}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*x^6*arccoth(a*x)^3 + 1/60*a*(2*(3*a^4*x^5 + 5*a^2*x^3 + 15*x)/a^6 - 15*log(a*x + 1)/a^7 + 15*log(a*x - 1)/
a^7)*arccoth(a*x)^2 + 1/240*a*(((4*a^3*x^3 + (15*log(a*x - 1) - 46)*log(a*x + 1)^2 - 5*log(a*x + 1)^3 + 5*log(
a*x - 1)^3 + 76*a*x - (15*log(a*x - 1)^2 - 92*log(a*x - 1))*log(a*x + 1) + 46*log(a*x - 1)^2 + 38*log(a*x - 1)
)/a - 184*(log(a*x - 1)*log(1/2*a*x + 1/2) + dilog(-1/2*a*x + 1/2))/a - 38*log(a*x + 1)/a)/a^6 + 2*(6*a^4*x^4
+ 32*a^2*x^2 - 2*(15*log(a*x - 1) - 46)*log(a*x + 1) + 15*log(a*x + 1)^2 + 15*log(a*x - 1)^2 + 92*log(a*x - 1)
)*arccoth(a*x)/a^7)

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

[Out]

integral(x^5*arccoth(a*x)^3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**5*acoth(a*x)**3, 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(x^5*arccoth(a*x)^3,x, algorithm="giac")

[Out]

integrate(x^5*arccoth(a*x)^3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^5*acoth(a*x)^3, x)

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