3.216 \(\int \frac {(1-a^2 x^2)^2 \tanh ^{-1}(a x)^2}{x^9} \, dx\)

Optimal. Leaf size=170 \[ \frac {4}{63} a^8 \log (x)+\frac {1}{24} a^8 \tanh ^{-1}(a x)^2-\frac {a^7 \tanh ^{-1}(a x)}{12 x}+\frac {5 a^6}{504 x^2}-\frac {a^5 \tanh ^{-1}(a x)}{36 x^3}+\frac {a^4}{84 x^4}-\frac {a^4 \tanh ^{-1}(a x)^2}{4 x^4}+\frac {a^3 \tanh ^{-1}(a x)}{12 x^5}-\frac {a^2}{168 x^6}+\frac {a^2 \tanh ^{-1}(a x)^2}{3 x^6}-\frac {2}{63} a^8 \log \left (1-a^2 x^2\right )-\frac {\tanh ^{-1}(a x)^2}{8 x^8}-\frac {a \tanh ^{-1}(a x)}{28 x^7} \]

[Out]

-1/168*a^2/x^6+1/84*a^4/x^4+5/504*a^6/x^2-1/28*a*arctanh(a*x)/x^7+1/12*a^3*arctanh(a*x)/x^5-1/36*a^5*arctanh(a
*x)/x^3-1/12*a^7*arctanh(a*x)/x+1/24*a^8*arctanh(a*x)^2-1/8*arctanh(a*x)^2/x^8+1/3*a^2*arctanh(a*x)^2/x^6-1/4*
a^4*arctanh(a*x)^2/x^4+4/63*a^8*ln(x)-2/63*a^8*ln(-a^2*x^2+1)

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Rubi [A]  time = 0.84, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 56, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {6012, 5916, 5982, 266, 44, 36, 29, 31, 5948} \[ \frac {5 a^6}{504 x^2}+\frac {a^4}{84 x^4}-\frac {a^2}{168 x^6}-\frac {2}{63} a^8 \log \left (1-a^2 x^2\right )-\frac {a^5 \tanh ^{-1}(a x)}{36 x^3}-\frac {a^4 \tanh ^{-1}(a x)^2}{4 x^4}+\frac {a^3 \tanh ^{-1}(a x)}{12 x^5}+\frac {a^2 \tanh ^{-1}(a x)^2}{3 x^6}+\frac {4}{63} a^8 \log (x)+\frac {1}{24} a^8 \tanh ^{-1}(a x)^2-\frac {a^7 \tanh ^{-1}(a x)}{12 x}-\frac {a \tanh ^{-1}(a x)}{28 x^7}-\frac {\tanh ^{-1}(a x)^2}{8 x^8} \]

Antiderivative was successfully verified.

[In]

Int[((1 - a^2*x^2)^2*ArcTanh[a*x]^2)/x^9,x]

[Out]

-a^2/(168*x^6) + a^4/(84*x^4) + (5*a^6)/(504*x^2) - (a*ArcTanh[a*x])/(28*x^7) + (a^3*ArcTanh[a*x])/(12*x^5) -
(a^5*ArcTanh[a*x])/(36*x^3) - (a^7*ArcTanh[a*x])/(12*x) + (a^8*ArcTanh[a*x]^2)/24 - ArcTanh[a*x]^2/(8*x^8) + (
a^2*ArcTanh[a*x]^2)/(3*x^6) - (a^4*ArcTanh[a*x]^2)/(4*x^4) + (4*a^8*Log[x])/63 - (2*a^8*Log[1 - a^2*x^2])/63

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

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

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 266

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 5916

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcT
anh[c*x])^p)/(d*(m + 1)), x] - Dist[(b*c*p)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcTanh[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 5948

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTanh[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 5982

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 6012

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^9} \, dx &=\int \left (\frac {\tanh ^{-1}(a x)^2}{x^9}-\frac {2 a^2 \tanh ^{-1}(a x)^2}{x^7}+\frac {a^4 \tanh ^{-1}(a x)^2}{x^5}\right ) \, dx\\ &=-\left (\left (2 a^2\right ) \int \frac {\tanh ^{-1}(a x)^2}{x^7} \, dx\right )+a^4 \int \frac {\tanh ^{-1}(a x)^2}{x^5} \, dx+\int \frac {\tanh ^{-1}(a x)^2}{x^9} \, dx\\ &=-\frac {\tanh ^{-1}(a x)^2}{8 x^8}+\frac {a^2 \tanh ^{-1}(a x)^2}{3 x^6}-\frac {a^4 \tanh ^{-1}(a x)^2}{4 x^4}+\frac {1}{4} a \int \frac {\tanh ^{-1}(a x)}{x^8 \left (1-a^2 x^2\right )} \, dx-\frac {1}{3} \left (2 a^3\right ) \int \frac {\tanh ^{-1}(a x)}{x^6 \left (1-a^2 x^2\right )} \, dx+\frac {1}{2} a^5 \int \frac {\tanh ^{-1}(a x)}{x^4 \left (1-a^2 x^2\right )} \, dx\\ &=-\frac {\tanh ^{-1}(a x)^2}{8 x^8}+\frac {a^2 \tanh ^{-1}(a x)^2}{3 x^6}-\frac {a^4 \tanh ^{-1}(a x)^2}{4 x^4}+\frac {1}{4} a \int \frac {\tanh ^{-1}(a x)}{x^8} \, dx+\frac {1}{4} a^3 \int \frac {\tanh ^{-1}(a x)}{x^6 \left (1-a^2 x^2\right )} \, dx-\frac {1}{3} \left (2 a^3\right ) \int \frac {\tanh ^{-1}(a x)}{x^6} \, dx+\frac {1}{2} a^5 \int \frac {\tanh ^{-1}(a x)}{x^4} \, dx-\frac {1}{3} \left (2 a^5\right ) \int \frac {\tanh ^{-1}(a x)}{x^4 \left (1-a^2 x^2\right )} \, dx+\frac {1}{2} a^7 \int \frac {\tanh ^{-1}(a x)}{x^2 \left (1-a^2 x^2\right )} \, dx\\ &=-\frac {a \tanh ^{-1}(a x)}{28 x^7}+\frac {2 a^3 \tanh ^{-1}(a x)}{15 x^5}-\frac {a^5 \tanh ^{-1}(a x)}{6 x^3}-\frac {\tanh ^{-1}(a x)^2}{8 x^8}+\frac {a^2 \tanh ^{-1}(a x)^2}{3 x^6}-\frac {a^4 \tanh ^{-1}(a x)^2}{4 x^4}+\frac {1}{28} a^2 \int \frac {1}{x^7 \left (1-a^2 x^2\right )} \, dx+\frac {1}{4} a^3 \int \frac {\tanh ^{-1}(a x)}{x^6} \, dx-\frac {1}{15} \left (2 a^4\right ) \int \frac {1}{x^5 \left (1-a^2 x^2\right )} \, dx+\frac {1}{4} a^5 \int \frac {\tanh ^{-1}(a x)}{x^4 \left (1-a^2 x^2\right )} \, dx-\frac {1}{3} \left (2 a^5\right ) \int \frac {\tanh ^{-1}(a x)}{x^4} \, dx+\frac {1}{6} a^6 \int \frac {1}{x^3 \left (1-a^2 x^2\right )} \, dx+\frac {1}{2} a^7 \int \frac {\tanh ^{-1}(a x)}{x^2} \, dx-\frac {1}{3} \left (2 a^7\right ) \int \frac {\tanh ^{-1}(a x)}{x^2 \left (1-a^2 x^2\right )} \, dx+\frac {1}{2} a^9 \int \frac {\tanh ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=-\frac {a \tanh ^{-1}(a x)}{28 x^7}+\frac {a^3 \tanh ^{-1}(a x)}{12 x^5}+\frac {a^5 \tanh ^{-1}(a x)}{18 x^3}-\frac {a^7 \tanh ^{-1}(a x)}{2 x}+\frac {1}{4} a^8 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{8 x^8}+\frac {a^2 \tanh ^{-1}(a x)^2}{3 x^6}-\frac {a^4 \tanh ^{-1}(a x)^2}{4 x^4}+\frac {1}{56} a^2 \operatorname {Subst}\left (\int \frac {1}{x^4 \left (1-a^2 x\right )} \, dx,x,x^2\right )+\frac {1}{20} a^4 \int \frac {1}{x^5 \left (1-a^2 x^2\right )} \, dx-\frac {1}{15} a^4 \operatorname {Subst}\left (\int \frac {1}{x^3 \left (1-a^2 x\right )} \, dx,x,x^2\right )+\frac {1}{4} a^5 \int \frac {\tanh ^{-1}(a x)}{x^4} \, dx+\frac {1}{12} a^6 \operatorname {Subst}\left (\int \frac {1}{x^2 \left (1-a^2 x\right )} \, dx,x,x^2\right )-\frac {1}{9} \left (2 a^6\right ) \int \frac {1}{x^3 \left (1-a^2 x^2\right )} \, dx+\frac {1}{4} a^7 \int \frac {\tanh ^{-1}(a x)}{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^2} \, dx+\frac {1}{2} a^8 \int \frac {1}{x \left (1-a^2 x^2\right )} \, dx-\frac {1}{3} \left (2 a^9\right ) \int \frac {\tanh ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=-\frac {a \tanh ^{-1}(a x)}{28 x^7}+\frac {a^3 \tanh ^{-1}(a x)}{12 x^5}-\frac {a^5 \tanh ^{-1}(a x)}{36 x^3}+\frac {a^7 \tanh ^{-1}(a x)}{6 x}-\frac {1}{12} a^8 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{8 x^8}+\frac {a^2 \tanh ^{-1}(a x)^2}{3 x^6}-\frac {a^4 \tanh ^{-1}(a x)^2}{4 x^4}+\frac {1}{56} a^2 \operatorname {Subst}\left (\int \left (\frac {1}{x^4}+\frac {a^2}{x^3}+\frac {a^4}{x^2}+\frac {a^6}{x}-\frac {a^8}{-1+a^2 x}\right ) \, dx,x,x^2\right )+\frac {1}{40} a^4 \operatorname {Subst}\left (\int \frac {1}{x^3 \left (1-a^2 x\right )} \, dx,x,x^2\right )-\frac {1}{15} a^4 \operatorname {Subst}\left (\int \left (\frac {1}{x^3}+\frac {a^2}{x^2}+\frac {a^4}{x}-\frac {a^6}{-1+a^2 x}\right ) \, dx,x,x^2\right )+\frac {1}{12} a^6 \int \frac {1}{x^3 \left (1-a^2 x^2\right )} \, dx+\frac {1}{12} a^6 \operatorname {Subst}\left (\int \left (\frac {1}{x^2}+\frac {a^2}{x}-\frac {a^4}{-1+a^2 x}\right ) \, dx,x,x^2\right )-\frac {1}{9} a^6 \operatorname {Subst}\left (\int \frac {1}{x^2 \left (1-a^2 x\right )} \, dx,x,x^2\right )+\frac {1}{4} a^7 \int \frac {\tanh ^{-1}(a x)}{x^2} \, dx+\frac {1}{4} a^8 \operatorname {Subst}\left (\int \frac {1}{x \left (1-a^2 x\right )} \, dx,x,x^2\right )-\frac {1}{3} \left (2 a^8\right ) \int \frac {1}{x \left (1-a^2 x^2\right )} \, dx+\frac {1}{4} a^9 \int \frac {\tanh ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=-\frac {a^2}{168 x^6}+\frac {41 a^4}{1680 x^4}-\frac {29 a^6}{840 x^2}-\frac {a \tanh ^{-1}(a x)}{28 x^7}+\frac {a^3 \tanh ^{-1}(a x)}{12 x^5}-\frac {a^5 \tanh ^{-1}(a x)}{36 x^3}-\frac {a^7 \tanh ^{-1}(a x)}{12 x}+\frac {1}{24} a^8 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{8 x^8}+\frac {a^2 \tanh ^{-1}(a x)^2}{3 x^6}-\frac {a^4 \tanh ^{-1}(a x)^2}{4 x^4}+\frac {29}{420} a^8 \log (x)-\frac {29}{840} a^8 \log \left (1-a^2 x^2\right )+\frac {1}{40} a^4 \operatorname {Subst}\left (\int \left (\frac {1}{x^3}+\frac {a^2}{x^2}+\frac {a^4}{x}-\frac {a^6}{-1+a^2 x}\right ) \, dx,x,x^2\right )+\frac {1}{24} a^6 \operatorname {Subst}\left (\int \frac {1}{x^2 \left (1-a^2 x\right )} \, dx,x,x^2\right )-\frac {1}{9} a^6 \operatorname {Subst}\left (\int \left (\frac {1}{x^2}+\frac {a^2}{x}-\frac {a^4}{-1+a^2 x}\right ) \, dx,x,x^2\right )+\frac {1}{4} a^8 \int \frac {1}{x \left (1-a^2 x^2\right )} \, dx+\frac {1}{4} a^8 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )-\frac {1}{3} a^8 \operatorname {Subst}\left (\int \frac {1}{x \left (1-a^2 x\right )} \, dx,x,x^2\right )+\frac {1}{4} a^{10} \operatorname {Subst}\left (\int \frac {1}{1-a^2 x} \, dx,x,x^2\right )\\ &=-\frac {a^2}{168 x^6}+\frac {a^4}{84 x^4}+\frac {13 a^6}{252 x^2}-\frac {a \tanh ^{-1}(a x)}{28 x^7}+\frac {a^3 \tanh ^{-1}(a x)}{12 x^5}-\frac {a^5 \tanh ^{-1}(a x)}{36 x^3}-\frac {a^7 \tanh ^{-1}(a x)}{12 x}+\frac {1}{24} a^8 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{8 x^8}+\frac {a^2 \tanh ^{-1}(a x)^2}{3 x^6}-\frac {a^4 \tanh ^{-1}(a x)^2}{4 x^4}+\frac {25}{63} a^8 \log (x)-\frac {25}{126} a^8 \log \left (1-a^2 x^2\right )+\frac {1}{24} a^6 \operatorname {Subst}\left (\int \left (\frac {1}{x^2}+\frac {a^2}{x}-\frac {a^4}{-1+a^2 x}\right ) \, dx,x,x^2\right )+\frac {1}{8} a^8 \operatorname {Subst}\left (\int \frac {1}{x \left (1-a^2 x\right )} \, dx,x,x^2\right )-\frac {1}{3} a^8 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )-\frac {1}{3} a^{10} \operatorname {Subst}\left (\int \frac {1}{1-a^2 x} \, dx,x,x^2\right )\\ &=-\frac {a^2}{168 x^6}+\frac {a^4}{84 x^4}+\frac {5 a^6}{504 x^2}-\frac {a \tanh ^{-1}(a x)}{28 x^7}+\frac {a^3 \tanh ^{-1}(a x)}{12 x^5}-\frac {a^5 \tanh ^{-1}(a x)}{36 x^3}-\frac {a^7 \tanh ^{-1}(a x)}{12 x}+\frac {1}{24} a^8 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{8 x^8}+\frac {a^2 \tanh ^{-1}(a x)^2}{3 x^6}-\frac {a^4 \tanh ^{-1}(a x)^2}{4 x^4}-\frac {47}{252} a^8 \log (x)+\frac {47}{504} a^8 \log \left (1-a^2 x^2\right )+\frac {1}{8} a^8 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )+\frac {1}{8} a^{10} \operatorname {Subst}\left (\int \frac {1}{1-a^2 x} \, dx,x,x^2\right )\\ &=-\frac {a^2}{168 x^6}+\frac {a^4}{84 x^4}+\frac {5 a^6}{504 x^2}-\frac {a \tanh ^{-1}(a x)}{28 x^7}+\frac {a^3 \tanh ^{-1}(a x)}{12 x^5}-\frac {a^5 \tanh ^{-1}(a x)}{36 x^3}-\frac {a^7 \tanh ^{-1}(a x)}{12 x}+\frac {1}{24} a^8 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{8 x^8}+\frac {a^2 \tanh ^{-1}(a x)^2}{3 x^6}-\frac {a^4 \tanh ^{-1}(a x)^2}{4 x^4}+\frac {4}{63} a^8 \log (x)-\frac {2}{63} a^8 \log \left (1-a^2 x^2\right )\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 124, normalized size = 0.73 \[ \frac {21 \left (a^2 x^2+3\right ) \left (a^2 x^2-1\right )^3 \tanh ^{-1}(a x)^2+a^2 x^2 \left (32 a^6 x^6 \log (x)+5 a^4 x^4+6 a^2 x^2-16 a^6 x^6 \log \left (1-a^2 x^2\right )-3\right )-2 a x \left (21 a^6 x^6+7 a^4 x^4-21 a^2 x^2+9\right ) \tanh ^{-1}(a x)}{504 x^8} \]

Antiderivative was successfully verified.

[In]

Integrate[((1 - a^2*x^2)^2*ArcTanh[a*x]^2)/x^9,x]

[Out]

(-2*a*x*(9 - 21*a^2*x^2 + 7*a^4*x^4 + 21*a^6*x^6)*ArcTanh[a*x] + 21*(-1 + a^2*x^2)^3*(3 + a^2*x^2)*ArcTanh[a*x
]^2 + a^2*x^2*(-3 + 6*a^2*x^2 + 5*a^4*x^4 + 32*a^6*x^6*Log[x] - 16*a^6*x^6*Log[1 - a^2*x^2]))/(504*x^8)

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fricas [A]  time = 0.61, size = 148, normalized size = 0.87 \[ -\frac {64 \, a^{8} x^{8} \log \left (a^{2} x^{2} - 1\right ) - 128 \, a^{8} x^{8} \log \relax (x) - 20 \, a^{6} x^{6} - 24 \, a^{4} x^{4} + 12 \, a^{2} x^{2} - 21 \, {\left (a^{8} x^{8} - 6 \, a^{4} x^{4} + 8 \, a^{2} x^{2} - 3\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} + 4 \, {\left (21 \, a^{7} x^{7} + 7 \, a^{5} x^{5} - 21 \, a^{3} x^{3} + 9 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{2016 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*x^2+1)^2*arctanh(a*x)^2/x^9,x, algorithm="fricas")

[Out]

-1/2016*(64*a^8*x^8*log(a^2*x^2 - 1) - 128*a^8*x^8*log(x) - 20*a^6*x^6 - 24*a^4*x^4 + 12*a^2*x^2 - 21*(a^8*x^8
- 6*a^4*x^4 + 8*a^2*x^2 - 3)*log(-(a*x + 1)/(a*x - 1))^2 + 4*(21*a^7*x^7 + 7*a^5*x^5 - 21*a^3*x^3 + 9*a*x)*lo
g(-(a*x + 1)/(a*x - 1)))/x^8

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giac [B]  time = 0.24, size = 651, normalized size = 3.83 \[ \frac {2}{63} \, {\left (2 \, a^{7} \log \left (-\frac {a x + 1}{a x - 1} - 1\right ) - 2 \, a^{7} \log \left (-\frac {a x + 1}{a x - 1}\right ) + \frac {84 \, {\left (\frac {{\left (a x + 1\right )}^{5} a^{7}}{{\left (a x - 1\right )}^{5}} - \frac {{\left (a x + 1\right )}^{4} a^{7}}{{\left (a x - 1\right )}^{4}} + \frac {{\left (a x + 1\right )}^{3} a^{7}}{{\left (a x - 1\right )}^{3}}\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2}}{\frac {{\left (a x + 1\right )}^{8}}{{\left (a x - 1\right )}^{8}} + \frac {8 \, {\left (a x + 1\right )}^{7}}{{\left (a x - 1\right )}^{7}} + \frac {28 \, {\left (a x + 1\right )}^{6}}{{\left (a x - 1\right )}^{6}} + \frac {56 \, {\left (a x + 1\right )}^{5}}{{\left (a x - 1\right )}^{5}} + \frac {70 \, {\left (a x + 1\right )}^{4}}{{\left (a x - 1\right )}^{4}} + \frac {56 \, {\left (a x + 1\right )}^{3}}{{\left (a x - 1\right )}^{3}} + \frac {28 \, {\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}} + \frac {8 \, {\left (a x + 1\right )}}{a x - 1} + 1} + \frac {2 \, {\left (\frac {28 \, {\left (a x + 1\right )}^{4} a^{7}}{{\left (a x - 1\right )}^{4}} + \frac {7 \, {\left (a x + 1\right )}^{3} a^{7}}{{\left (a x - 1\right )}^{3}} + \frac {21 \, {\left (a x + 1\right )}^{2} a^{7}}{{\left (a x - 1\right )}^{2}} + \frac {7 \, {\left (a x + 1\right )} a^{7}}{a x - 1} + a^{7}\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{\frac {{\left (a x + 1\right )}^{7}}{{\left (a x - 1\right )}^{7}} + \frac {7 \, {\left (a x + 1\right )}^{6}}{{\left (a x - 1\right )}^{6}} + \frac {21 \, {\left (a x + 1\right )}^{5}}{{\left (a x - 1\right )}^{5}} + \frac {35 \, {\left (a x + 1\right )}^{4}}{{\left (a x - 1\right )}^{4}} + \frac {35 \, {\left (a x + 1\right )}^{3}}{{\left (a x - 1\right )}^{3}} + \frac {21 \, {\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}} + \frac {7 \, {\left (a x + 1\right )}}{a x - 1} + 1} - \frac {\frac {2 \, {\left (a x + 1\right )}^{5} a^{7}}{{\left (a x - 1\right )}^{5}} + \frac {11 \, {\left (a x + 1\right )}^{4} a^{7}}{{\left (a x - 1\right )}^{4}} + \frac {6 \, {\left (a x + 1\right )}^{3} a^{7}}{{\left (a x - 1\right )}^{3}} + \frac {11 \, {\left (a x + 1\right )}^{2} a^{7}}{{\left (a x - 1\right )}^{2}} + \frac {2 \, {\left (a x + 1\right )} a^{7}}{a x - 1}}{\frac {{\left (a x + 1\right )}^{6}}{{\left (a x - 1\right )}^{6}} + \frac {6 \, {\left (a x + 1\right )}^{5}}{{\left (a x - 1\right )}^{5}} + \frac {15 \, {\left (a x + 1\right )}^{4}}{{\left (a x - 1\right )}^{4}} + \frac {20 \, {\left (a x + 1\right )}^{3}}{{\left (a x - 1\right )}^{3}} + \frac {15 \, {\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}} + \frac {6 \, {\left (a x + 1\right )}}{a x - 1} + 1}\right )} a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*x^2+1)^2*arctanh(a*x)^2/x^9,x, algorithm="giac")

[Out]

2/63*(2*a^7*log(-(a*x + 1)/(a*x - 1) - 1) - 2*a^7*log(-(a*x + 1)/(a*x - 1)) + 84*((a*x + 1)^5*a^7/(a*x - 1)^5
- (a*x + 1)^4*a^7/(a*x - 1)^4 + (a*x + 1)^3*a^7/(a*x - 1)^3)*log(-(a*x + 1)/(a*x - 1))^2/((a*x + 1)^8/(a*x - 1
)^8 + 8*(a*x + 1)^7/(a*x - 1)^7 + 28*(a*x + 1)^6/(a*x - 1)^6 + 56*(a*x + 1)^5/(a*x - 1)^5 + 70*(a*x + 1)^4/(a*
x - 1)^4 + 56*(a*x + 1)^3/(a*x - 1)^3 + 28*(a*x + 1)^2/(a*x - 1)^2 + 8*(a*x + 1)/(a*x - 1) + 1) + 2*(28*(a*x +
1)^4*a^7/(a*x - 1)^4 + 7*(a*x + 1)^3*a^7/(a*x - 1)^3 + 21*(a*x + 1)^2*a^7/(a*x - 1)^2 + 7*(a*x + 1)*a^7/(a*x
- 1) + a^7)*log(-(a*x + 1)/(a*x - 1))/((a*x + 1)^7/(a*x - 1)^7 + 7*(a*x + 1)^6/(a*x - 1)^6 + 21*(a*x + 1)^5/(a
*x - 1)^5 + 35*(a*x + 1)^4/(a*x - 1)^4 + 35*(a*x + 1)^3/(a*x - 1)^3 + 21*(a*x + 1)^2/(a*x - 1)^2 + 7*(a*x + 1)
/(a*x - 1) + 1) - (2*(a*x + 1)^5*a^7/(a*x - 1)^5 + 11*(a*x + 1)^4*a^7/(a*x - 1)^4 + 6*(a*x + 1)^3*a^7/(a*x - 1
)^3 + 11*(a*x + 1)^2*a^7/(a*x - 1)^2 + 2*(a*x + 1)*a^7/(a*x - 1))/((a*x + 1)^6/(a*x - 1)^6 + 6*(a*x + 1)^5/(a*
x - 1)^5 + 15*(a*x + 1)^4/(a*x - 1)^4 + 20*(a*x + 1)^3/(a*x - 1)^3 + 15*(a*x + 1)^2/(a*x - 1)^2 + 6*(a*x + 1)/
(a*x - 1) + 1))*a

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maple [A]  time = 0.07, size = 253, normalized size = 1.49 \[ -\frac {\arctanh \left (a x \right )^{2}}{8 x^{8}}-\frac {a^{4} \arctanh \left (a x \right )^{2}}{4 x^{4}}+\frac {a^{2} \arctanh \left (a x \right )^{2}}{3 x^{6}}-\frac {a \arctanh \left (a x \right )}{28 x^{7}}+\frac {a^{3} \arctanh \left (a x \right )}{12 x^{5}}-\frac {a^{5} \arctanh \left (a x \right )}{36 x^{3}}-\frac {a^{7} \arctanh \left (a x \right )}{12 x}-\frac {a^{8} \arctanh \left (a x \right ) \ln \left (a x -1\right )}{24}+\frac {a^{8} \arctanh \left (a x \right ) \ln \left (a x +1\right )}{24}-\frac {a^{8} \ln \left (a x -1\right )^{2}}{96}+\frac {a^{8} \ln \left (a x -1\right ) \ln \left (\frac {1}{2}+\frac {a x}{2}\right )}{48}-\frac {a^{8} \ln \left (a x +1\right )^{2}}{96}-\frac {a^{8} \ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {a x}{2}\right )}{48}+\frac {a^{8} \ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (a x +1\right )}{48}-\frac {a^{2}}{168 x^{6}}+\frac {a^{4}}{84 x^{4}}+\frac {5 a^{6}}{504 x^{2}}+\frac {4 a^{8} \ln \left (a x \right )}{63}-\frac {2 a^{8} \ln \left (a x -1\right )}{63}-\frac {2 a^{8} \ln \left (a x +1\right )}{63} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a^2*x^2+1)^2*arctanh(a*x)^2/x^9,x)

[Out]

-1/8*arctanh(a*x)^2/x^8-1/4*a^4*arctanh(a*x)^2/x^4+1/3*a^2*arctanh(a*x)^2/x^6-1/28*a*arctanh(a*x)/x^7+1/12*a^3
*arctanh(a*x)/x^5-1/36*a^5*arctanh(a*x)/x^3-1/12*a^7*arctanh(a*x)/x-1/24*a^8*arctanh(a*x)*ln(a*x-1)+1/24*a^8*a
rctanh(a*x)*ln(a*x+1)-1/96*a^8*ln(a*x-1)^2+1/48*a^8*ln(a*x-1)*ln(1/2+1/2*a*x)-1/96*a^8*ln(a*x+1)^2-1/48*a^8*ln
(-1/2*a*x+1/2)*ln(1/2+1/2*a*x)+1/48*a^8*ln(-1/2*a*x+1/2)*ln(a*x+1)-1/168*a^2/x^6+1/84*a^4/x^4+5/504*a^6/x^2+4/
63*a^8*ln(a*x)-2/63*a^8*ln(a*x-1)-2/63*a^8*ln(a*x+1)

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maxima [A]  time = 0.32, size = 204, normalized size = 1.20 \[ \frac {1}{2016} \, {\left (128 \, a^{6} \log \relax (x) - \frac {21 \, a^{6} x^{6} \log \left (a x + 1\right )^{2} + 21 \, a^{6} x^{6} \log \left (a x - 1\right )^{2} + 64 \, a^{6} x^{6} \log \left (a x - 1\right ) - 20 \, a^{4} x^{4} - 24 \, a^{2} x^{2} - 2 \, {\left (21 \, a^{6} x^{6} \log \left (a x - 1\right ) - 32 \, a^{6} x^{6}\right )} \log \left (a x + 1\right ) + 12}{x^{6}}\right )} a^{2} + \frac {1}{504} \, {\left (21 \, a^{7} \log \left (a x + 1\right ) - 21 \, a^{7} \log \left (a x - 1\right ) - \frac {2 \, {\left (21 \, a^{6} x^{6} + 7 \, a^{4} x^{4} - 21 \, a^{2} x^{2} + 9\right )}}{x^{7}}\right )} a \operatorname {artanh}\left (a x\right ) - \frac {{\left (6 \, a^{4} x^{4} - 8 \, a^{2} x^{2} + 3\right )} \operatorname {artanh}\left (a x\right )^{2}}{24 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*x^2+1)^2*arctanh(a*x)^2/x^9,x, algorithm="maxima")

[Out]

1/2016*(128*a^6*log(x) - (21*a^6*x^6*log(a*x + 1)^2 + 21*a^6*x^6*log(a*x - 1)^2 + 64*a^6*x^6*log(a*x - 1) - 20
*a^4*x^4 - 24*a^2*x^2 - 2*(21*a^6*x^6*log(a*x - 1) - 32*a^6*x^6)*log(a*x + 1) + 12)/x^6)*a^2 + 1/504*(21*a^7*l
og(a*x + 1) - 21*a^7*log(a*x - 1) - 2*(21*a^6*x^6 + 7*a^4*x^4 - 21*a^2*x^2 + 9)/x^7)*a*arctanh(a*x) - 1/24*(6*
a^4*x^4 - 8*a^2*x^2 + 3)*arctanh(a*x)^2/x^8

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mupad [B]  time = 2.72, size = 357, normalized size = 2.10 \[ \frac {4\,a^8\,\ln \relax (x)}{63}+\frac {a^8\,{\ln \left (a\,x+1\right )}^2}{96}+\frac {a^8\,{\ln \left (1-a\,x\right )}^2}{96}-\frac {{\ln \left (a\,x+1\right )}^2}{32\,x^8}-\frac {{\ln \left (1-a\,x\right )}^2}{32\,x^8}-\frac {2\,a^8\,\ln \left (a^2\,x^2-1\right )}{63}-\frac {a^2}{168\,x^6}+\frac {a^4}{84\,x^4}+\frac {5\,a^6}{504\,x^2}-\frac {a^8\,\ln \left (a\,x+1\right )\,\ln \left (1-a\,x\right )}{48}+\frac {\ln \left (a\,x+1\right )\,\ln \left (1-a\,x\right )}{16\,x^8}+\frac {a^2\,{\ln \left (a\,x+1\right )}^2}{12\,x^6}-\frac {a^4\,{\ln \left (a\,x+1\right )}^2}{16\,x^4}+\frac {a^2\,{\ln \left (1-a\,x\right )}^2}{12\,x^6}-\frac {a^4\,{\ln \left (1-a\,x\right )}^2}{16\,x^4}-\frac {a\,\ln \left (a\,x+1\right )}{56\,x^7}+\frac {a\,\ln \left (1-a\,x\right )}{56\,x^7}+\frac {a^3\,\ln \left (a\,x+1\right )}{24\,x^5}-\frac {a^5\,\ln \left (a\,x+1\right )}{72\,x^3}-\frac {a^7\,\ln \left (a\,x+1\right )}{24\,x}-\frac {a^3\,\ln \left (1-a\,x\right )}{24\,x^5}+\frac {a^5\,\ln \left (1-a\,x\right )}{72\,x^3}+\frac {a^7\,\ln \left (1-a\,x\right )}{24\,x}-\frac {a^2\,\ln \left (a\,x+1\right )\,\ln \left (1-a\,x\right )}{6\,x^6}+\frac {a^4\,\ln \left (a\,x+1\right )\,\ln \left (1-a\,x\right )}{8\,x^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((atanh(a*x)^2*(a^2*x^2 - 1)^2)/x^9,x)

[Out]

(4*a^8*log(x))/63 + (a^8*log(a*x + 1)^2)/96 + (a^8*log(1 - a*x)^2)/96 - log(a*x + 1)^2/(32*x^8) - log(1 - a*x)
^2/(32*x^8) - (2*a^8*log(a^2*x^2 - 1))/63 - a^2/(168*x^6) + a^4/(84*x^4) + (5*a^6)/(504*x^2) - (a^8*log(a*x +
1)*log(1 - a*x))/48 + (log(a*x + 1)*log(1 - a*x))/(16*x^8) + (a^2*log(a*x + 1)^2)/(12*x^6) - (a^4*log(a*x + 1)
^2)/(16*x^4) + (a^2*log(1 - a*x)^2)/(12*x^6) - (a^4*log(1 - a*x)^2)/(16*x^4) - (a*log(a*x + 1))/(56*x^7) + (a*
log(1 - a*x))/(56*x^7) + (a^3*log(a*x + 1))/(24*x^5) - (a^5*log(a*x + 1))/(72*x^3) - (a^7*log(a*x + 1))/(24*x)
- (a^3*log(1 - a*x))/(24*x^5) + (a^5*log(1 - a*x))/(72*x^3) + (a^7*log(1 - a*x))/(24*x) - (a^2*log(a*x + 1)*l
og(1 - a*x))/(6*x^6) + (a^4*log(a*x + 1)*log(1 - a*x))/(8*x^4)

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sympy [A]  time = 3.99, size = 168, normalized size = 0.99 \[ \begin {cases} \frac {4 a^{8} \log {\relax (x )}}{63} - \frac {4 a^{8} \log {\left (x - \frac {1}{a} \right )}}{63} + \frac {a^{8} \operatorname {atanh}^{2}{\left (a x \right )}}{24} - \frac {4 a^{8} \operatorname {atanh}{\left (a x \right )}}{63} - \frac {a^{7} \operatorname {atanh}{\left (a x \right )}}{12 x} + \frac {5 a^{6}}{504 x^{2}} - \frac {a^{5} \operatorname {atanh}{\left (a x \right )}}{36 x^{3}} - \frac {a^{4} \operatorname {atanh}^{2}{\left (a x \right )}}{4 x^{4}} + \frac {a^{4}}{84 x^{4}} + \frac {a^{3} \operatorname {atanh}{\left (a x \right )}}{12 x^{5}} + \frac {a^{2} \operatorname {atanh}^{2}{\left (a x \right )}}{3 x^{6}} - \frac {a^{2}}{168 x^{6}} - \frac {a \operatorname {atanh}{\left (a x \right )}}{28 x^{7}} - \frac {\operatorname {atanh}^{2}{\left (a x \right )}}{8 x^{8}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a**2*x**2+1)**2*atanh(a*x)**2/x**9,x)

[Out]

Piecewise((4*a**8*log(x)/63 - 4*a**8*log(x - 1/a)/63 + a**8*atanh(a*x)**2/24 - 4*a**8*atanh(a*x)/63 - a**7*ata
nh(a*x)/(12*x) + 5*a**6/(504*x**2) - a**5*atanh(a*x)/(36*x**3) - a**4*atanh(a*x)**2/(4*x**4) + a**4/(84*x**4)
+ a**3*atanh(a*x)/(12*x**5) + a**2*atanh(a*x)**2/(3*x**6) - a**2/(168*x**6) - a*atanh(a*x)/(28*x**7) - atanh(a
*x)**2/(8*x**8), Ne(a, 0)), (0, True))

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