3.3.15 \(\int \frac {(1-a^2 x^2)^2 \tanh ^{-1}(a x)^2}{x^8} \, dx\) [215]

Optimal. Leaf size=183 \[ -\frac {a^2}{105 x^5}+\frac {17 a^4}{630 x^3}+\frac {a^6}{210 x}-\frac {1}{210} a^7 \tanh ^{-1}(a x)-\frac {a \tanh ^{-1}(a x)}{21 x^6}+\frac {9 a^3 \tanh ^{-1}(a x)}{70 x^4}-\frac {8 a^5 \tanh ^{-1}(a x)}{105 x^2}+\frac {8}{105} a^7 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{7 x^7}+\frac {2 a^2 \tanh ^{-1}(a x)^2}{5 x^5}-\frac {a^4 \tanh ^{-1}(a x)^2}{3 x^3}+\frac {16}{105} a^7 \tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )-\frac {8}{105} a^7 \text {PolyLog}\left (2,-1+\frac {2}{1+a x}\right ) \]

[Out]

-1/105*a^2/x^5+17/630*a^4/x^3+1/210*a^6/x-1/210*a^7*arctanh(a*x)-1/21*a*arctanh(a*x)/x^6+9/70*a^3*arctanh(a*x)
/x^4-8/105*a^5*arctanh(a*x)/x^2+8/105*a^7*arctanh(a*x)^2-1/7*arctanh(a*x)^2/x^7+2/5*a^2*arctanh(a*x)^2/x^5-1/3
*a^4*arctanh(a*x)^2/x^3+16/105*a^7*arctanh(a*x)*ln(2-2/(a*x+1))-8/105*a^7*polylog(2,-1+2/(a*x+1))

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Rubi [A]
time = 0.75, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 42, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {6159, 6037, 6129, 331, 212, 6135, 6079, 2497} \begin {gather*} -\frac {8}{105} a^7 \text {Li}_2\left (\frac {2}{a x+1}-1\right )+\frac {8}{105} a^7 \tanh ^{-1}(a x)^2-\frac {1}{210} a^7 \tanh ^{-1}(a x)+\frac {16}{105} a^7 \log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x)+\frac {a^6}{210 x}-\frac {8 a^5 \tanh ^{-1}(a x)}{105 x^2}+\frac {17 a^4}{630 x^3}-\frac {a^4 \tanh ^{-1}(a x)^2}{3 x^3}+\frac {9 a^3 \tanh ^{-1}(a x)}{70 x^4}-\frac {a^2}{105 x^5}+\frac {2 a^2 \tanh ^{-1}(a x)^2}{5 x^5}-\frac {\tanh ^{-1}(a x)^2}{7 x^7}-\frac {a \tanh ^{-1}(a x)}{21 x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((1 - a^2*x^2)^2*ArcTanh[a*x]^2)/x^8,x]

[Out]

-1/105*a^2/x^5 + (17*a^4)/(630*x^3) + a^6/(210*x) - (a^7*ArcTanh[a*x])/210 - (a*ArcTanh[a*x])/(21*x^6) + (9*a^
3*ArcTanh[a*x])/(70*x^4) - (8*a^5*ArcTanh[a*x])/(105*x^2) + (8*a^7*ArcTanh[a*x]^2)/105 - ArcTanh[a*x]^2/(7*x^7
) + (2*a^2*ArcTanh[a*x]^2)/(5*x^5) - (a^4*ArcTanh[a*x]^2)/(3*x^3) + (16*a^7*ArcTanh[a*x]*Log[2 - 2/(1 + a*x)])
/105 - (8*a^7*PolyLog[2, -1 + 2/(1 + a*x)])/105

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 6037

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcTanh[c*
x^n])^p/(m + 1)), x] - Dist[b*c*n*(p/(m + 1)), Int[x^(m + n)*((a + b*ArcTanh[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 6079

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x_Symbol] :> Simp[(a + b*ArcTanh[c*x
])^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] - Dist[b*c*(p/d), Int[(a + b*ArcTanh[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 6129

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

Rule 6135

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Simp[(a + b*ArcTanh[c
*x])^(p + 1)/(b*d*(p + 1)), x] + Dist[1/d, Int[(a + b*ArcTanh[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 6159

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Int[E
xpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[
c^2*d + e, 0] && IGtQ[p, 0] && IGtQ[q, 1]

Rubi steps

\begin {align*} \int \frac {\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{x^8} \, dx &=\int \left (\frac {\tanh ^{-1}(a x)^2}{x^8}-\frac {2 a^2 \tanh ^{-1}(a x)^2}{x^6}+\frac {a^4 \tanh ^{-1}(a x)^2}{x^4}\right ) \, dx\\ &=-\left (\left (2 a^2\right ) \int \frac {\tanh ^{-1}(a x)^2}{x^6} \, dx\right )+a^4 \int \frac {\tanh ^{-1}(a x)^2}{x^4} \, dx+\int \frac {\tanh ^{-1}(a x)^2}{x^8} \, dx\\ &=-\frac {\tanh ^{-1}(a x)^2}{7 x^7}+\frac {2 a^2 \tanh ^{-1}(a x)^2}{5 x^5}-\frac {a^4 \tanh ^{-1}(a x)^2}{3 x^3}+\frac {1}{7} (2 a) \int \frac {\tanh ^{-1}(a x)}{x^7 \left (1-a^2 x^2\right )} \, dx-\frac {1}{5} \left (4 a^3\right ) \int \frac {\tanh ^{-1}(a x)}{x^5 \left (1-a^2 x^2\right )} \, dx+\frac {1}{3} \left (2 a^5\right ) \int \frac {\tanh ^{-1}(a x)}{x^3 \left (1-a^2 x^2\right )} \, dx\\ &=-\frac {\tanh ^{-1}(a x)^2}{7 x^7}+\frac {2 a^2 \tanh ^{-1}(a x)^2}{5 x^5}-\frac {a^4 \tanh ^{-1}(a x)^2}{3 x^3}+\frac {1}{7} (2 a) \int \frac {\tanh ^{-1}(a x)}{x^7} \, dx+\frac {1}{7} \left (2 a^3\right ) \int \frac {\tanh ^{-1}(a x)}{x^5 \left (1-a^2 x^2\right )} \, dx-\frac {1}{5} \left (4 a^3\right ) \int \frac {\tanh ^{-1}(a x)}{x^5} \, dx+\frac {1}{3} \left (2 a^5\right ) \int \frac {\tanh ^{-1}(a x)}{x^3} \, dx-\frac {1}{5} \left (4 a^5\right ) \int \frac {\tanh ^{-1}(a x)}{x^3 \left (1-a^2 x^2\right )} \, dx+\frac {1}{3} \left (2 a^7\right ) \int \frac {\tanh ^{-1}(a x)}{x \left (1-a^2 x^2\right )} \, dx\\ &=-\frac {a \tanh ^{-1}(a x)}{21 x^6}+\frac {a^3 \tanh ^{-1}(a x)}{5 x^4}-\frac {a^5 \tanh ^{-1}(a x)}{3 x^2}+\frac {1}{3} a^7 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{7 x^7}+\frac {2 a^2 \tanh ^{-1}(a x)^2}{5 x^5}-\frac {a^4 \tanh ^{-1}(a x)^2}{3 x^3}+\frac {1}{21} a^2 \int \frac {1}{x^6 \left (1-a^2 x^2\right )} \, dx+\frac {1}{7} \left (2 a^3\right ) \int \frac {\tanh ^{-1}(a x)}{x^5} \, dx-\frac {1}{5} a^4 \int \frac {1}{x^4 \left (1-a^2 x^2\right )} \, dx+\frac {1}{7} \left (2 a^5\right ) \int \frac {\tanh ^{-1}(a x)}{x^3 \left (1-a^2 x^2\right )} \, dx-\frac {1}{5} \left (4 a^5\right ) \int \frac {\tanh ^{-1}(a x)}{x^3} \, dx+\frac {1}{3} a^6 \int \frac {1}{x^2 \left (1-a^2 x^2\right )} \, dx+\frac {1}{3} \left (2 a^7\right ) \int \frac {\tanh ^{-1}(a x)}{x (1+a x)} \, dx-\frac {1}{5} \left (4 a^7\right ) \int \frac {\tanh ^{-1}(a x)}{x \left (1-a^2 x^2\right )} \, dx\\ &=-\frac {a^2}{105 x^5}+\frac {a^4}{15 x^3}-\frac {a^6}{3 x}-\frac {a \tanh ^{-1}(a x)}{21 x^6}+\frac {9 a^3 \tanh ^{-1}(a x)}{70 x^4}+\frac {a^5 \tanh ^{-1}(a x)}{15 x^2}-\frac {1}{15} a^7 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{7 x^7}+\frac {2 a^2 \tanh ^{-1}(a x)^2}{5 x^5}-\frac {a^4 \tanh ^{-1}(a x)^2}{3 x^3}+\frac {2}{3} a^7 \tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )+\frac {1}{21} a^4 \int \frac {1}{x^4 \left (1-a^2 x^2\right )} \, dx+\frac {1}{14} a^4 \int \frac {1}{x^4 \left (1-a^2 x^2\right )} \, dx+\frac {1}{7} \left (2 a^5\right ) \int \frac {\tanh ^{-1}(a x)}{x^3} \, dx-\frac {1}{5} a^6 \int \frac {1}{x^2 \left (1-a^2 x^2\right )} \, dx-\frac {1}{5} \left (2 a^6\right ) \int \frac {1}{x^2 \left (1-a^2 x^2\right )} \, dx+\frac {1}{7} \left (2 a^7\right ) \int \frac {\tanh ^{-1}(a x)}{x \left (1-a^2 x^2\right )} \, dx-\frac {1}{5} \left (4 a^7\right ) \int \frac {\tanh ^{-1}(a x)}{x (1+a x)} \, dx+\frac {1}{3} a^8 \int \frac {1}{1-a^2 x^2} \, dx-\frac {1}{3} \left (2 a^8\right ) \int \frac {\log \left (2-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac {a^2}{105 x^5}+\frac {17 a^4}{630 x^3}+\frac {4 a^6}{15 x}+\frac {1}{3} a^7 \tanh ^{-1}(a x)-\frac {a \tanh ^{-1}(a x)}{21 x^6}+\frac {9 a^3 \tanh ^{-1}(a x)}{70 x^4}-\frac {8 a^5 \tanh ^{-1}(a x)}{105 x^2}+\frac {8}{105} a^7 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{7 x^7}+\frac {2 a^2 \tanh ^{-1}(a x)^2}{5 x^5}-\frac {a^4 \tanh ^{-1}(a x)^2}{3 x^3}-\frac {2}{15} a^7 \tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )-\frac {1}{3} a^7 \text {Li}_2\left (-1+\frac {2}{1+a x}\right )+\frac {1}{21} a^6 \int \frac {1}{x^2 \left (1-a^2 x^2\right )} \, dx+\frac {1}{14} a^6 \int \frac {1}{x^2 \left (1-a^2 x^2\right )} \, dx+\frac {1}{7} a^6 \int \frac {1}{x^2 \left (1-a^2 x^2\right )} \, dx+\frac {1}{7} \left (2 a^7\right ) \int \frac {\tanh ^{-1}(a x)}{x (1+a x)} \, dx-\frac {1}{5} a^8 \int \frac {1}{1-a^2 x^2} \, dx-\frac {1}{5} \left (2 a^8\right ) \int \frac {1}{1-a^2 x^2} \, dx+\frac {1}{5} \left (4 a^8\right ) \int \frac {\log \left (2-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac {a^2}{105 x^5}+\frac {17 a^4}{630 x^3}+\frac {a^6}{210 x}-\frac {4}{15} a^7 \tanh ^{-1}(a x)-\frac {a \tanh ^{-1}(a x)}{21 x^6}+\frac {9 a^3 \tanh ^{-1}(a x)}{70 x^4}-\frac {8 a^5 \tanh ^{-1}(a x)}{105 x^2}+\frac {8}{105} a^7 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{7 x^7}+\frac {2 a^2 \tanh ^{-1}(a x)^2}{5 x^5}-\frac {a^4 \tanh ^{-1}(a x)^2}{3 x^3}+\frac {16}{105} a^7 \tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )+\frac {1}{15} a^7 \text {Li}_2\left (-1+\frac {2}{1+a x}\right )+\frac {1}{21} a^8 \int \frac {1}{1-a^2 x^2} \, dx+\frac {1}{14} a^8 \int \frac {1}{1-a^2 x^2} \, dx+\frac {1}{7} a^8 \int \frac {1}{1-a^2 x^2} \, dx-\frac {1}{7} \left (2 a^8\right ) \int \frac {\log \left (2-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac {a^2}{105 x^5}+\frac {17 a^4}{630 x^3}+\frac {a^6}{210 x}-\frac {1}{210} a^7 \tanh ^{-1}(a x)-\frac {a \tanh ^{-1}(a x)}{21 x^6}+\frac {9 a^3 \tanh ^{-1}(a x)}{70 x^4}-\frac {8 a^5 \tanh ^{-1}(a x)}{105 x^2}+\frac {8}{105} a^7 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{7 x^7}+\frac {2 a^2 \tanh ^{-1}(a x)^2}{5 x^5}-\frac {a^4 \tanh ^{-1}(a x)^2}{3 x^3}+\frac {16}{105} a^7 \tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )-\frac {8}{105} a^7 \text {Li}_2\left (-1+\frac {2}{1+a x}\right )\\ \end {align*}

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Mathematica [A]
time = 1.01, size = 140, normalized size = 0.77 \begin {gather*} \frac {a^2 x^2 \left (-6+17 a^2 x^2+3 a^4 x^4\right )+6 \left (-15+42 a^2 x^2-35 a^4 x^4+8 a^7 x^7\right ) \tanh ^{-1}(a x)^2+3 a x \tanh ^{-1}(a x) \left (-10+27 a^2 x^2-16 a^4 x^4-a^6 x^6+32 a^6 x^6 \log \left (1-e^{-2 \tanh ^{-1}(a x)}\right )\right )-48 a^7 x^7 \text {PolyLog}\left (2,e^{-2 \tanh ^{-1}(a x)}\right )}{630 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 - a^2*x^2)^2*ArcTanh[a*x]^2)/x^8,x]

[Out]

(a^2*x^2*(-6 + 17*a^2*x^2 + 3*a^4*x^4) + 6*(-15 + 42*a^2*x^2 - 35*a^4*x^4 + 8*a^7*x^7)*ArcTanh[a*x]^2 + 3*a*x*
ArcTanh[a*x]*(-10 + 27*a^2*x^2 - 16*a^4*x^4 - a^6*x^6 + 32*a^6*x^6*Log[1 - E^(-2*ArcTanh[a*x])]) - 48*a^7*x^7*
PolyLog[2, E^(-2*ArcTanh[a*x])])/(630*x^7)

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Maple [A]
time = 0.28, size = 253, normalized size = 1.38

method	result	size
derivativedivides	\(a^{7} \left (-\frac {\arctanh \left (a x \right )^{2}}{3 a^{3} x^{3}}-\frac {\arctanh \left (a x \right )^{2}}{7 a^{7} x^{7}}+\frac {2 \arctanh \left (a x \right )^{2}}{5 a^{5} x^{5}}-\frac {\arctanh \left (a x \right )}{21 a^{6} x^{6}}+\frac {9 \arctanh \left (a x \right )}{70 a^{4} x^{4}}-\frac {8 \arctanh \left (a x \right )}{105 a^{2} x^{2}}+\frac {16 \arctanh \left (a x \right ) \ln \left (a x \right )}{105}-\frac {8 \arctanh \left (a x \right ) \ln \left (a x -1\right )}{105}-\frac {8 \arctanh \left (a x \right ) \ln \left (a x +1\right )}{105}-\frac {8 \dilog \left (a x +1\right )}{105}-\frac {8 \ln \left (a x \right ) \ln \left (a x +1\right )}{105}-\frac {8 \dilog \left (a x \right )}{105}-\frac {2 \ln \left (a x -1\right )^{2}}{105}+\frac {8 \dilog \left (\frac {a x}{2}+\frac {1}{2}\right )}{105}+\frac {4 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{105}+\frac {2 \ln \left (a x +1\right )^{2}}{105}-\frac {4 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{105}+\frac {1}{210 a x}-\frac {1}{105 a^{5} x^{5}}+\frac {17}{630 a^{3} x^{3}}+\frac {\ln \left (a x -1\right )}{420}-\frac {\ln \left (a x +1\right )}{420}\right )\)	\(253\)
default	\(a^{7} \left (-\frac {\arctanh \left (a x \right )^{2}}{3 a^{3} x^{3}}-\frac {\arctanh \left (a x \right )^{2}}{7 a^{7} x^{7}}+\frac {2 \arctanh \left (a x \right )^{2}}{5 a^{5} x^{5}}-\frac {\arctanh \left (a x \right )}{21 a^{6} x^{6}}+\frac {9 \arctanh \left (a x \right )}{70 a^{4} x^{4}}-\frac {8 \arctanh \left (a x \right )}{105 a^{2} x^{2}}+\frac {16 \arctanh \left (a x \right ) \ln \left (a x \right )}{105}-\frac {8 \arctanh \left (a x \right ) \ln \left (a x -1\right )}{105}-\frac {8 \arctanh \left (a x \right ) \ln \left (a x +1\right )}{105}-\frac {8 \dilog \left (a x +1\right )}{105}-\frac {8 \ln \left (a x \right ) \ln \left (a x +1\right )}{105}-\frac {8 \dilog \left (a x \right )}{105}-\frac {2 \ln \left (a x -1\right )^{2}}{105}+\frac {8 \dilog \left (\frac {a x}{2}+\frac {1}{2}\right )}{105}+\frac {4 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{105}+\frac {2 \ln \left (a x +1\right )^{2}}{105}-\frac {4 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{105}+\frac {1}{210 a x}-\frac {1}{105 a^{5} x^{5}}+\frac {17}{630 a^{3} x^{3}}+\frac {\ln \left (a x -1\right )}{420}-\frac {\ln \left (a x +1\right )}{420}\right )\)	\(253\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a^2*x^2+1)^2*arctanh(a*x)^2/x^8,x,method=_RETURNVERBOSE)

[Out]

a^7*(-1/3*arctanh(a*x)^2/a^3/x^3-1/7*arctanh(a*x)^2/a^7/x^7+2/5*arctanh(a*x)^2/a^5/x^5-1/21*arctanh(a*x)/a^6/x
^6+9/70*arctanh(a*x)/a^4/x^4-8/105*arctanh(a*x)/a^2/x^2+16/105*arctanh(a*x)*ln(a*x)-8/105*arctanh(a*x)*ln(a*x-
1)-8/105*arctanh(a*x)*ln(a*x+1)-8/105*dilog(a*x+1)-8/105*ln(a*x)*ln(a*x+1)-8/105*dilog(a*x)-2/105*ln(a*x-1)^2+
8/105*dilog(1/2*a*x+1/2)+4/105*ln(a*x-1)*ln(1/2*a*x+1/2)+2/105*ln(a*x+1)^2-4/105*(ln(a*x+1)-ln(1/2*a*x+1/2))*l
n(-1/2*a*x+1/2)+1/210/a/x-1/105/a^5/x^5+17/630/a^3/x^3+1/420*ln(a*x-1)-1/420*ln(a*x+1))

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Maxima [A]
time = 0.26, size = 254, normalized size = 1.39 \begin {gather*} \frac {1}{1260} \, {\left (96 \, {\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^{5} - 96 \, {\left (\log \left (a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-a x\right )\right )} a^{5} + 96 \, {\left (\log \left (-a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (a x\right )\right )} a^{5} - 3 \, a^{5} \log \left (a x + 1\right ) + 3 \, a^{5} \log \left (a x - 1\right ) + \frac {2 \, {\left (12 \, a^{5} x^{5} \log \left (a x + 1\right )^{2} - 24 \, a^{5} x^{5} \log \left (a x + 1\right ) \log \left (a x - 1\right ) - 12 \, a^{5} x^{5} \log \left (a x - 1\right )^{2} + 3 \, a^{4} x^{4} + 17 \, a^{2} x^{2} - 6\right )}}{x^{5}}\right )} a^{2} - \frac {1}{210} \, {\left (16 \, a^{6} \log \left (a^{2} x^{2} - 1\right ) - 16 \, a^{6} \log \left (x^{2}\right ) + \frac {16 \, a^{4} x^{4} - 27 \, a^{2} x^{2} + 10}{x^{6}}\right )} a \operatorname {artanh}\left (a x\right ) - \frac {{\left (35 \, a^{4} x^{4} - 42 \, a^{2} x^{2} + 15\right )} \operatorname {artanh}\left (a x\right )^{2}}{105 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*x^2+1)^2*arctanh(a*x)^2/x^8,x, algorithm="maxima")

[Out]

1/1260*(96*(log(a*x - 1)*log(1/2*a*x + 1/2) + dilog(-1/2*a*x + 1/2))*a^5 - 96*(log(a*x + 1)*log(x) + dilog(-a*
x))*a^5 + 96*(log(-a*x + 1)*log(x) + dilog(a*x))*a^5 - 3*a^5*log(a*x + 1) + 3*a^5*log(a*x - 1) + 2*(12*a^5*x^5
*log(a*x + 1)^2 - 24*a^5*x^5*log(a*x + 1)*log(a*x - 1) - 12*a^5*x^5*log(a*x - 1)^2 + 3*a^4*x^4 + 17*a^2*x^2 -
6)/x^5)*a^2 - 1/210*(16*a^6*log(a^2*x^2 - 1) - 16*a^6*log(x^2) + (16*a^4*x^4 - 27*a^2*x^2 + 10)/x^6)*a*arctanh
(a*x) - 1/105*(35*a^4*x^4 - 42*a^2*x^2 + 15)*arctanh(a*x)^2/x^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((-a^2*x^2+1)^2*arctanh(a*x)^2/x^8,x, algorithm="fricas")

[Out]

integral((a^4*x^4 - 2*a^2*x^2 + 1)*arctanh(a*x)^2/x^8, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a x - 1\right )^{2} \left (a x + 1\right )^{2} \operatorname {atanh}^{2}{\left (a x \right )}}{x^{8}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a**2*x**2+1)**2*atanh(a*x)**2/x**8,x)

[Out]

Integral((a*x - 1)**2*(a*x + 1)**2*atanh(a*x)**2/x**8, 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((-a^2*x^2+1)^2*arctanh(a*x)^2/x^8,x, algorithm="giac")

[Out]

integrate((a^2*x^2 - 1)^2*arctanh(a*x)^2/x^8, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {atanh}\left (a\,x\right )}^2\,{\left (a^2\,x^2-1\right )}^2}{x^8} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((atanh(a*x)^2*(a^2*x^2 - 1)^2)/x^8,x)

[Out]

int((atanh(a*x)^2*(a^2*x^2 - 1)^2)/x^8, x)

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