3.1.12 \(\int x^5 \coth ^{-1}(a x)^2 \, dx\) [12]

Optimal. Leaf size=105 \[ \frac {4 x^2}{45 a^4}+\frac {x^4}{60 a^2}+\frac {x \coth ^{-1}(a x)}{3 a^5}+\frac {x^3 \coth ^{-1}(a x)}{9 a^3}+\frac {x^5 \coth ^{-1}(a x)}{15 a}-\frac {\coth ^{-1}(a x)^2}{6 a^6}+\frac {1}{6} x^6 \coth ^{-1}(a x)^2+\frac {23 \log \left (1-a^2 x^2\right )}{90 a^6} \]

[Out]

4/45*x^2/a^4+1/60*x^4/a^2+1/3*x*arccoth(a*x)/a^5+1/9*x^3*arccoth(a*x)/a^3+1/15*x^5*arccoth(a*x)/a-1/6*arccoth(
a*x)^2/a^6+1/6*x^6*arccoth(a*x)^2+23/90*ln(-a^2*x^2+1)/a^6

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Rubi [A]
time = 0.17, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {6038, 6128, 272, 45, 6022, 266, 6096} \begin {gather*} -\frac {\coth ^{-1}(a x)^2}{6 a^6}+\frac {x \coth ^{-1}(a x)}{3 a^5}+\frac {4 x^2}{45 a^4}+\frac {x^3 \coth ^{-1}(a x)}{9 a^3}+\frac {x^4}{60 a^2}+\frac {23 \log \left (1-a^2 x^2\right )}{90 a^6}+\frac {1}{6} x^6 \coth ^{-1}(a x)^2+\frac {x^5 \coth ^{-1}(a x)}{15 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5*ArcCoth[a*x]^2,x]

[Out]

(4*x^2)/(45*a^4) + x^4/(60*a^2) + (x*ArcCoth[a*x])/(3*a^5) + (x^3*ArcCoth[a*x])/(9*a^3) + (x^5*ArcCoth[a*x])/(
15*a) - ArcCoth[a*x]^2/(6*a^6) + (x^6*ArcCoth[a*x]^2)/6 + (23*Log[1 - a^2*x^2])/(90*a^6)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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 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]

Rubi steps

\begin {align*} \int x^5 \coth ^{-1}(a x)^2 \, dx &=\frac {1}{6} x^6 \coth ^{-1}(a x)^2-\frac {1}{3} a \int \frac {x^6 \coth ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=\frac {1}{6} x^6 \coth ^{-1}(a x)^2+\frac {\int x^4 \coth ^{-1}(a x) \, dx}{3 a}-\frac {\int \frac {x^4 \coth ^{-1}(a x)}{1-a^2 x^2} \, dx}{3 a}\\ &=\frac {x^5 \coth ^{-1}(a x)}{15 a}+\frac {1}{6} x^6 \coth ^{-1}(a x)^2-\frac {1}{15} \int \frac {x^5}{1-a^2 x^2} \, dx+\frac {\int x^2 \coth ^{-1}(a x) \, dx}{3 a^3}-\frac {\int \frac {x^2 \coth ^{-1}(a x)}{1-a^2 x^2} \, dx}{3 a^3}\\ &=\frac {x^3 \coth ^{-1}(a x)}{9 a^3}+\frac {x^5 \coth ^{-1}(a x)}{15 a}+\frac {1}{6} x^6 \coth ^{-1}(a x)^2-\frac {1}{30} \text {Subst}\left (\int \frac {x^2}{1-a^2 x} \, dx,x,x^2\right )+\frac {\int \coth ^{-1}(a x) \, dx}{3 a^5}-\frac {\int \frac {\coth ^{-1}(a x)}{1-a^2 x^2} \, dx}{3 a^5}-\frac {\int \frac {x^3}{1-a^2 x^2} \, dx}{9 a^2}\\ &=\frac {x \coth ^{-1}(a x)}{3 a^5}+\frac {x^3 \coth ^{-1}(a x)}{9 a^3}+\frac {x^5 \coth ^{-1}(a x)}{15 a}-\frac {\coth ^{-1}(a x)^2}{6 a^6}+\frac {1}{6} x^6 \coth ^{-1}(a x)^2-\frac {1}{30} \text {Subst}\left (\int \left (-\frac {1}{a^4}-\frac {x}{a^2}-\frac {1}{a^4 \left (-1+a^2 x\right )}\right ) \, dx,x,x^2\right )-\frac {\int \frac {x}{1-a^2 x^2} \, dx}{3 a^4}-\frac {\text {Subst}\left (\int \frac {x}{1-a^2 x} \, dx,x,x^2\right )}{18 a^2}\\ &=\frac {x^2}{30 a^4}+\frac {x^4}{60 a^2}+\frac {x \coth ^{-1}(a x)}{3 a^5}+\frac {x^3 \coth ^{-1}(a x)}{9 a^3}+\frac {x^5 \coth ^{-1}(a x)}{15 a}-\frac {\coth ^{-1}(a x)^2}{6 a^6}+\frac {1}{6} x^6 \coth ^{-1}(a x)^2+\frac {\log \left (1-a^2 x^2\right )}{5 a^6}-\frac {\text {Subst}\left (\int \left (-\frac {1}{a^2}-\frac {1}{a^2 \left (-1+a^2 x\right )}\right ) \, dx,x,x^2\right )}{18 a^2}\\ &=\frac {4 x^2}{45 a^4}+\frac {x^4}{60 a^2}+\frac {x \coth ^{-1}(a x)}{3 a^5}+\frac {x^3 \coth ^{-1}(a x)}{9 a^3}+\frac {x^5 \coth ^{-1}(a x)}{15 a}-\frac {\coth ^{-1}(a x)^2}{6 a^6}+\frac {1}{6} x^6 \coth ^{-1}(a x)^2+\frac {23 \log \left (1-a^2 x^2\right )}{90 a^6}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 80, normalized size = 0.76 \begin {gather*} \frac {16 a^2 x^2+3 a^4 x^4+4 a x \left (15+5 a^2 x^2+3 a^4 x^4\right ) \coth ^{-1}(a x)+30 \left (-1+a^6 x^6\right ) \coth ^{-1}(a x)^2+46 \log \left (1-a^2 x^2\right )}{180 a^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^5*ArcCoth[a*x]^2,x]

[Out]

(16*a^2*x^2 + 3*a^4*x^4 + 4*a*x*(15 + 5*a^2*x^2 + 3*a^4*x^4)*ArcCoth[a*x] + 30*(-1 + a^6*x^6)*ArcCoth[a*x]^2 +
46*Log[1 - a^2*x^2])/(180*a^6)

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Maple [A]
time = 0.23, size = 168, normalized size = 1.60

method	result	size
derivativedivides	\(\frac {\frac {a^{6} x^{6} \mathrm {arccoth}\left (a x \right )^{2}}{6}+\frac {a^{5} x^{5} \mathrm {arccoth}\left (a x \right )}{15}+\frac {a^{3} x^{3} \mathrm {arccoth}\left (a x \right )}{9}+\frac {a x \,\mathrm {arccoth}\left (a x \right )}{3}+\frac {\mathrm {arccoth}\left (a x \right ) \ln \left (a x -1\right )}{6}-\frac {\mathrm {arccoth}\left (a x \right ) \ln \left (a x +1\right )}{6}+\frac {\ln \left (a x -1\right )^{2}}{24}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{12}-\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{12}+\frac {\ln \left (a x +1\right )^{2}}{24}+\frac {a^{4} x^{4}}{60}+\frac {4 a^{2} x^{2}}{45}+\frac {23 \ln \left (a x -1\right )}{90}+\frac {23 \ln \left (a x +1\right )}{90}}{a^{6}}\)	\(168\)
default	\(\frac {\frac {a^{6} x^{6} \mathrm {arccoth}\left (a x \right )^{2}}{6}+\frac {a^{5} x^{5} \mathrm {arccoth}\left (a x \right )}{15}+\frac {a^{3} x^{3} \mathrm {arccoth}\left (a x \right )}{9}+\frac {a x \,\mathrm {arccoth}\left (a x \right )}{3}+\frac {\mathrm {arccoth}\left (a x \right ) \ln \left (a x -1\right )}{6}-\frac {\mathrm {arccoth}\left (a x \right ) \ln \left (a x +1\right )}{6}+\frac {\ln \left (a x -1\right )^{2}}{24}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{12}-\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{12}+\frac {\ln \left (a x +1\right )^{2}}{24}+\frac {a^{4} x^{4}}{60}+\frac {4 a^{2} x^{2}}{45}+\frac {23 \ln \left (a x -1\right )}{90}+\frac {23 \ln \left (a x +1\right )}{90}}{a^{6}}\)	\(168\)
risch	\(\frac {\left (a^{6} x^{6}-1\right ) \ln \left (a x +1\right )^{2}}{24 a^{6}}-\frac {\left (15 x^{6} \ln \left (a x -1\right ) a^{6}-6 a^{5} x^{5}-10 a^{3} x^{3}-30 a x -15 \ln \left (a x -1\right )\right ) \ln \left (a x +1\right )}{180 a^{6}}+\frac {x^{6} \ln \left (a x -1\right )^{2}}{24}-\frac {x^{5} \ln \left (a x -1\right )}{30 a}+\frac {x^{4}}{60 a^{2}}-\frac {x^{3} \ln \left (a x -1\right )}{18 a^{3}}+\frac {4 x^{2}}{45 a^{4}}-\frac {x \ln \left (a x -1\right )}{6 a^{5}}-\frac {\ln \left (a x -1\right )^{2}}{24 a^{6}}+\frac {23 \ln \left (a^{2} x^{2}-1\right )}{90 a^{6}}+\frac {16}{135 a^{6}}\)	\(180\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*arccoth(a*x)^2,x,method=_RETURNVERBOSE)

[Out]

1/a^6*(1/6*a^6*x^6*arccoth(a*x)^2+1/15*a^5*x^5*arccoth(a*x)+1/9*a^3*x^3*arccoth(a*x)+1/3*a*x*arccoth(a*x)+1/6*
arccoth(a*x)*ln(a*x-1)-1/6*arccoth(a*x)*ln(a*x+1)+1/24*ln(a*x-1)^2-1/12*ln(a*x-1)*ln(1/2*a*x+1/2)-1/12*(ln(a*x
+1)-ln(1/2*a*x+1/2))*ln(-1/2*a*x+1/2)+1/24*ln(a*x+1)^2+1/60*a^4*x^4+4/45*a^2*x^2+23/90*ln(a*x-1)+23/90*ln(a*x+
1))

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Maxima [A]
time = 0.26, size = 135, normalized size = 1.29 \begin {gather*} \frac {1}{6} \, x^{6} \operatorname {arcoth}\left (a x\right )^{2} + \frac {1}{90} \, 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 ) + \frac {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 )}{360 \, a^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*arccoth(a*x)^2,x, algorithm="maxima")

[Out]

1/6*x^6*arccoth(a*x)^2 + 1/90*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) + 1/360*(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))/a^6

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Fricas [A]
time = 0.37, size = 98, normalized size = 0.93 \begin {gather*} \frac {6 \, a^{4} x^{4} + 32 \, a^{2} x^{2} + 15 \, {\left (a^{6} x^{6} - 1\right )} \log \left (\frac {a x + 1}{a x - 1}\right )^{2} + 4 \, {\left (3 \, a^{5} x^{5} + 5 \, a^{3} x^{3} + 15 \, a x\right )} \log \left (\frac {a x + 1}{a x - 1}\right ) + 92 \, \log \left (a^{2} x^{2} - 1\right )}{360 \, a^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*arccoth(a*x)^2,x, algorithm="fricas")

[Out]

1/360*(6*a^4*x^4 + 32*a^2*x^2 + 15*(a^6*x^6 - 1)*log((a*x + 1)/(a*x - 1))^2 + 4*(3*a^5*x^5 + 5*a^3*x^3 + 15*a*
x)*log((a*x + 1)/(a*x - 1)) + 92*log(a^2*x^2 - 1))/a^6

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Sympy [A]
time = 0.48, size = 114, normalized size = 1.09 \begin {gather*} \begin {cases} \frac {x^{6} \operatorname {acoth}^{2}{\left (a x \right )}}{6} + \frac {x^{5} \operatorname {acoth}{\left (a x \right )}}{15 a} + \frac {x^{4}}{60 a^{2}} + \frac {x^{3} \operatorname {acoth}{\left (a x \right )}}{9 a^{3}} + \frac {4 x^{2}}{45 a^{4}} + \frac {x \operatorname {acoth}{\left (a x \right )}}{3 a^{5}} + \frac {23 \log {\left (a x + 1 \right )}}{45 a^{6}} - \frac {\operatorname {acoth}^{2}{\left (a x \right )}}{6 a^{6}} - \frac {23 \operatorname {acoth}{\left (a x \right )}}{45 a^{6}} & \text {for}\: a \neq 0 \\- \frac {\pi ^{2} x^{6}}{24} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*acoth(a*x)**2,x)

[Out]

Piecewise((x**6*acoth(a*x)**2/6 + x**5*acoth(a*x)/(15*a) + x**4/(60*a**2) + x**3*acoth(a*x)/(9*a**3) + 4*x**2/
(45*a**4) + x*acoth(a*x)/(3*a**5) + 23*log(a*x + 1)/(45*a**6) - acoth(a*x)**2/(6*a**6) - 23*acoth(a*x)/(45*a**
6), Ne(a, 0)), (-pi**2*x**6/24, True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 534 vs. \(2 (89) = 178\).
time = 0.39, size = 534, normalized size = 5.09 \begin {gather*} \frac {1}{90} \, {\left (\frac {15 \, {\left (\frac {3 \, {\left (a x + 1\right )}^{5}}{{\left (a x - 1\right )}^{5}} + \frac {10 \, {\left (a x + 1\right )}^{3}}{{\left (a x - 1\right )}^{3}} + \frac {3 \, {\left (a x + 1\right )}}{a x - 1}\right )} \log \left (\frac {a x + 1}{a x - 1}\right )^{2}}{\frac {{\left (a x + 1\right )}^{6} a^{7}}{{\left (a x - 1\right )}^{6}} - \frac {6 \, {\left (a x + 1\right )}^{5} a^{7}}{{\left (a x - 1\right )}^{5}} + \frac {15 \, {\left (a x + 1\right )}^{4} a^{7}}{{\left (a x - 1\right )}^{4}} - \frac {20 \, {\left (a x + 1\right )}^{3} a^{7}}{{\left (a x - 1\right )}^{3}} + \frac {15 \, {\left (a x + 1\right )}^{2} a^{7}}{{\left (a x - 1\right )}^{2}} - \frac {6 \, {\left (a x + 1\right )} a^{7}}{a x - 1} + a^{7}} + \frac {2 \, {\left (\frac {45 \, {\left (a x + 1\right )}^{4}}{{\left (a x - 1\right )}^{4}} - \frac {90 \, {\left (a x + 1\right )}^{3}}{{\left (a x - 1\right )}^{3}} + \frac {140 \, {\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}} - \frac {70 \, {\left (a x + 1\right )}}{a x - 1} + 23\right )} \log \left (\frac {a x + 1}{a x - 1}\right )}{\frac {{\left (a x + 1\right )}^{5} a^{7}}{{\left (a x - 1\right )}^{5}} - \frac {5 \, {\left (a x + 1\right )}^{4} a^{7}}{{\left (a x - 1\right )}^{4}} + \frac {10 \, {\left (a x + 1\right )}^{3} a^{7}}{{\left (a x - 1\right )}^{3}} - \frac {10 \, {\left (a x + 1\right )}^{2} a^{7}}{{\left (a x - 1\right )}^{2}} + \frac {5 \, {\left (a x + 1\right )} a^{7}}{a x - 1} - a^{7}} + \frac {4 \, {\left (\frac {11 \, {\left (a x + 1\right )}^{3}}{{\left (a x - 1\right )}^{3}} - \frac {16 \, {\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}} + \frac {11 \, {\left (a x + 1\right )}}{a x - 1}\right )}}{\frac {{\left (a x + 1\right )}^{4} a^{7}}{{\left (a x - 1\right )}^{4}} - \frac {4 \, {\left (a x + 1\right )}^{3} a^{7}}{{\left (a x - 1\right )}^{3}} + \frac {6 \, {\left (a x + 1\right )}^{2} a^{7}}{{\left (a x - 1\right )}^{2}} - \frac {4 \, {\left (a x + 1\right )} a^{7}}{a x - 1} + a^{7}} - \frac {46 \, \log \left (\frac {a x + 1}{a x - 1} - 1\right )}{a^{7}} + \frac {46 \, \log \left (\frac {a x + 1}{a x - 1}\right )}{a^{7}}\right )} a \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*arccoth(a*x)^2,x, algorithm="giac")

[Out]

1/90*(15*(3*(a*x + 1)^5/(a*x - 1)^5 + 10*(a*x + 1)^3/(a*x - 1)^3 + 3*(a*x + 1)/(a*x - 1))*log((a*x + 1)/(a*x -
1))^2/((a*x + 1)^6*a^7/(a*x - 1)^6 - 6*(a*x + 1)^5*a^7/(a*x - 1)^5 + 15*(a*x + 1)^4*a^7/(a*x - 1)^4 - 20*(a*x
+ 1)^3*a^7/(a*x - 1)^3 + 15*(a*x + 1)^2*a^7/(a*x - 1)^2 - 6*(a*x + 1)*a^7/(a*x - 1) + a^7) + 2*(45*(a*x + 1)^
4/(a*x - 1)^4 - 90*(a*x + 1)^3/(a*x - 1)^3 + 140*(a*x + 1)^2/(a*x - 1)^2 - 70*(a*x + 1)/(a*x - 1) + 23)*log((a
*x + 1)/(a*x - 1))/((a*x + 1)^5*a^7/(a*x - 1)^5 - 5*(a*x + 1)^4*a^7/(a*x - 1)^4 + 10*(a*x + 1)^3*a^7/(a*x - 1)
^3 - 10*(a*x + 1)^2*a^7/(a*x - 1)^2 + 5*(a*x + 1)*a^7/(a*x - 1) - a^7) + 4*(11*(a*x + 1)^3/(a*x - 1)^3 - 16*(a
*x + 1)^2/(a*x - 1)^2 + 11*(a*x + 1)/(a*x - 1))/((a*x + 1)^4*a^7/(a*x - 1)^4 - 4*(a*x + 1)^3*a^7/(a*x - 1)^3 +
6*(a*x + 1)^2*a^7/(a*x - 1)^2 - 4*(a*x + 1)*a^7/(a*x - 1) + a^7) - 46*log((a*x + 1)/(a*x - 1) - 1)/a^7 + 46*l
og((a*x + 1)/(a*x - 1))/a^7)*a

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Mupad [B]
time = 1.36, size = 85, normalized size = 0.81 \begin {gather*} \frac {x^6\,{\mathrm {acoth}\left (a\,x\right )}^2}{6}+\frac {\frac {23\,\ln \left (a^2\,x^2-1\right )}{90}+\frac {4\,a^2\,x^2}{45}+\frac {a^4\,x^4}{60}-\frac {{\mathrm {acoth}\left (a\,x\right )}^2}{6}+\frac {a^3\,x^3\,\mathrm {acoth}\left (a\,x\right )}{9}+\frac {a^5\,x^5\,\mathrm {acoth}\left (a\,x\right )}{15}+\frac {a\,x\,\mathrm {acoth}\left (a\,x\right )}{3}}{a^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*acoth(a*x)^2,x)

[Out]

(x^6*acoth(a*x)^2)/6 + ((23*log(a^2*x^2 - 1))/90 + (4*a^2*x^2)/45 + (a^4*x^4)/60 - acoth(a*x)^2/6 + (a^3*x^3*a
coth(a*x))/9 + (a^5*x^5*acoth(a*x))/15 + (a*x*acoth(a*x))/3)/a^6

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