3.1.76 \(\int \frac {(a+b \tanh ^{-1}(c x^2))^2}{x^6} \, dx\) [76]

Optimal. Leaf size=1176 \[ -\frac {2 a b c}{15 x^3}+\frac {2 a b c^2}{5 x}-\frac {8 b^2 c^2}{15 x}+\frac {2}{5} a b c^{5/2} \text {ArcTan}\left (\sqrt {c} x\right )-\frac {4}{15} b^2 c^{5/2} \text {ArcTan}\left (\sqrt {c} x\right )+\frac {1}{5} i b^2 c^{5/2} \text {ArcTan}\left (\sqrt {c} x\right )^2+\frac {4}{15} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right )+\frac {1}{5} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right )^2-\frac {2}{5} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt {c} x}\right )-\frac {2}{5} b^2 c^{5/2} \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-i \sqrt {c} x}\right )+\frac {1}{5} b^2 c^{5/2} \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )+\frac {2}{5} b^2 c^{5/2} \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+i \sqrt {c} x}\right )+\frac {2}{5} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+\sqrt {c} x}\right )-\frac {1}{5} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (-\frac {2 \sqrt {c} \left (1-\sqrt {-c} x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (1+\sqrt {c} x\right )}\right )-\frac {1}{5} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2 \sqrt {c} \left (1+\sqrt {-c} x\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (1+\sqrt {c} x\right )}\right )+\frac {1}{5} b^2 c^{5/2} \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (\frac {(1-i) \left (1+\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )+\frac {b^2 c \log \left (1-c x^2\right )}{15 x^3}-\frac {b^2 c^2 \log \left (1-c x^2\right )}{5 x}-\frac {1}{5} b^2 c^{5/2} \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )-\frac {b c \left (2 a-b \log \left (1-c x^2\right )\right )}{15 x^3}-\frac {b c^2 \left (2 a-b \log \left (1-c x^2\right )\right )}{5 x}+\frac {1}{5} b c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )-\frac {\left (2 a-b \log \left (1-c x^2\right )\right )^2}{20 x^5}-\frac {a b \log \left (1+c x^2\right )}{5 x^5}-\frac {2 b^2 c \log \left (1+c x^2\right )}{15 x^3}+\frac {1}{5} b^2 c^{5/2} \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )+\frac {1}{5} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )+\frac {b^2 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )}{10 x^5}-\frac {b^2 \log ^2\left (1+c x^2\right )}{20 x^5}-\frac {1}{5} b^2 c^{5/2} \text {PolyLog}\left (2,1-\frac {2}{1-\sqrt {c} x}\right )+\frac {1}{5} i b^2 c^{5/2} \text {PolyLog}\left (2,1-\frac {2}{1-i \sqrt {c} x}\right )-\frac {1}{10} i b^2 c^{5/2} \text {PolyLog}\left (2,1-\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )+\frac {1}{5} i b^2 c^{5/2} \text {PolyLog}\left (2,1-\frac {2}{1+i \sqrt {c} x}\right )-\frac {1}{5} b^2 c^{5/2} \text {PolyLog}\left (2,1-\frac {2}{1+\sqrt {c} x}\right )+\frac {1}{10} b^2 c^{5/2} \text {PolyLog}\left (2,1+\frac {2 \sqrt {c} \left (1-\sqrt {-c} x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (1+\sqrt {c} x\right )}\right )+\frac {1}{10} b^2 c^{5/2} \text {PolyLog}\left (2,1-\frac {2 \sqrt {c} \left (1+\sqrt {-c} x\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (1+\sqrt {c} x\right )}\right )-\frac {1}{10} i b^2 c^{5/2} \text {PolyLog}\left (2,1-\frac {(1-i) \left (1+\sqrt {c} x\right )}{1-i \sqrt {c} x}\right ) \]

[Out]

-4/15*b^2*c^(5/2)*arctan(x*c^(1/2))+4/15*b^2*c^(5/2)*arctanh(x*c^(1/2))+1/5*b^2*c^(5/2)*arctanh(x*c^(1/2))^2-1
/20*b^2*ln(c*x^2+1)^2/x^5-1/5*b^2*c^(5/2)*polylog(2,1-2/(1-x*c^(1/2)))-1/5*b^2*c^(5/2)*polylog(2,1-2/(1+x*c^(1
/2)))+1/10*b^2*c^(5/2)*polylog(2,1+2*(1-x*(-c)^(1/2))*c^(1/2)/((-c)^(1/2)-c^(1/2))/(1+x*c^(1/2)))+1/10*b^2*c^(
5/2)*polylog(2,1-2*(1+x*(-c)^(1/2))*c^(1/2)/((-c)^(1/2)+c^(1/2))/(1+x*c^(1/2)))-8/15*b^2*c^2/x+2/5*a*b*c^(5/2)
*arctan(x*c^(1/2))+1/15*b^2*c*ln(-c*x^2+1)/x^3-1/5*b^2*c^2*ln(-c*x^2+1)/x-1/5*b^2*c^(5/2)*arctan(x*c^(1/2))*ln
(-c*x^2+1)-1/15*b*c*(2*a-b*ln(-c*x^2+1))/x^3-1/5*b*c^2*(2*a-b*ln(-c*x^2+1))/x+1/5*b*c^(5/2)*arctanh(x*c^(1/2))
*(2*a-b*ln(-c*x^2+1))-1/5*a*b*ln(c*x^2+1)/x^5-2/15*b^2*c*ln(c*x^2+1)/x^3+1/5*b^2*c^(5/2)*arctan(x*c^(1/2))*ln(
c*x^2+1)+1/5*b^2*c^(5/2)*arctanh(x*c^(1/2))*ln(c*x^2+1)+1/10*b^2*ln(-c*x^2+1)*ln(c*x^2+1)/x^5-2/5*b^2*c^(5/2)*
arctanh(x*c^(1/2))*ln(2/(1-x*c^(1/2)))-2/5*b^2*c^(5/2)*arctan(x*c^(1/2))*ln(2/(1-I*x*c^(1/2)))+1/5*b^2*c^(5/2)
*arctan(x*c^(1/2))*ln((1+I)*(1-x*c^(1/2))/(1-I*x*c^(1/2)))+2/5*b^2*c^(5/2)*arctan(x*c^(1/2))*ln(2/(1+I*x*c^(1/
2)))+2/5*b^2*c^(5/2)*arctanh(x*c^(1/2))*ln(2/(1+x*c^(1/2)))-1/5*b^2*c^(5/2)*arctanh(x*c^(1/2))*ln(-2*(1-x*(-c)
^(1/2))*c^(1/2)/((-c)^(1/2)-c^(1/2))/(1+x*c^(1/2)))-1/5*b^2*c^(5/2)*arctanh(x*c^(1/2))*ln(2*(1+x*(-c)^(1/2))*c
^(1/2)/((-c)^(1/2)+c^(1/2))/(1+x*c^(1/2)))+1/5*b^2*c^(5/2)*arctan(x*c^(1/2))*ln((1-I)*(1+x*c^(1/2))/(1-I*x*c^(
1/2)))-1/10*I*b^2*c^(5/2)*polylog(2,1-(1+I)*(1-x*c^(1/2))/(1-I*x*c^(1/2)))-1/10*I*b^2*c^(5/2)*polylog(2,1+(-1+
I)*(1+x*c^(1/2))/(1-I*x*c^(1/2)))+1/5*I*b^2*c^(5/2)*polylog(2,1-2/(1-I*x*c^(1/2)))+1/5*I*b^2*c^(5/2)*polylog(2
,1-2/(1+I*x*c^(1/2)))+1/5*I*b^2*c^(5/2)*arctan(x*c^(1/2))^2-1/20*(2*a-b*ln(-c*x^2+1))^2/x^5-2/15*a*b*c/x^3+2/5
*a*b*c^2/x

________________________________________________________________________________________

Rubi [A]
time = 1.39, antiderivative size = 1176, normalized size of antiderivative = 1.00, number of steps used = 77, number of rules used = 24, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.500, Rules used = {6041, 2507, 2526, 2505, 331, 212, 213, 2520, 12, 6131, 6055, 2449, 2352, 6874, 209, 30, 2637, 6139, 6057, 2497, 5048, 4966, 5040, 4964} \begin {gather*} \frac {1}{5} i b^2 \text {ArcTan}\left (\sqrt {c} x\right )^2 c^{5/2}+\frac {1}{5} b^2 \tanh ^{-1}\left (\sqrt {c} x\right )^2 c^{5/2}-\frac {4}{15} b^2 \text {ArcTan}\left (\sqrt {c} x\right ) c^{5/2}+\frac {2}{5} a b \text {ArcTan}\left (\sqrt {c} x\right ) c^{5/2}+\frac {4}{15} b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) c^{5/2}-\frac {2}{5} b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt {c} x}\right ) c^{5/2}-\frac {2}{5} b^2 \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-i \sqrt {c} x}\right ) c^{5/2}+\frac {1}{5} b^2 \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right ) c^{5/2}+\frac {2}{5} b^2 \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (\frac {2}{i \sqrt {c} x+1}\right ) c^{5/2}+\frac {2}{5} b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{\sqrt {c} x+1}\right ) c^{5/2}-\frac {1}{5} b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (-\frac {2 \sqrt {c} \left (1-\sqrt {-c} x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (\sqrt {c} x+1\right )}\right ) c^{5/2}-\frac {1}{5} b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2 \sqrt {c} \left (\sqrt {-c} x+1\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (\sqrt {c} x+1\right )}\right ) c^{5/2}+\frac {1}{5} b^2 \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (\frac {(1-i) \left (\sqrt {c} x+1\right )}{1-i \sqrt {c} x}\right ) c^{5/2}-\frac {1}{5} b^2 \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right ) c^{5/2}+\frac {1}{5} b \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right ) c^{5/2}+\frac {1}{5} b^2 \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (c x^2+1\right ) c^{5/2}+\frac {1}{5} b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (c x^2+1\right ) c^{5/2}-\frac {1}{5} b^2 \text {Li}_2\left (1-\frac {2}{1-\sqrt {c} x}\right ) c^{5/2}+\frac {1}{5} i b^2 \text {Li}_2\left (1-\frac {2}{1-i \sqrt {c} x}\right ) c^{5/2}-\frac {1}{10} i b^2 \text {Li}_2\left (1-\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right ) c^{5/2}+\frac {1}{5} i b^2 \text {Li}_2\left (1-\frac {2}{i \sqrt {c} x+1}\right ) c^{5/2}-\frac {1}{5} b^2 \text {Li}_2\left (1-\frac {2}{\sqrt {c} x+1}\right ) c^{5/2}+\frac {1}{10} b^2 \text {Li}_2\left (\frac {2 \sqrt {c} \left (1-\sqrt {-c} x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (\sqrt {c} x+1\right )}+1\right ) c^{5/2}+\frac {1}{10} b^2 \text {Li}_2\left (1-\frac {2 \sqrt {c} \left (\sqrt {-c} x+1\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (\sqrt {c} x+1\right )}\right ) c^{5/2}-\frac {1}{10} i b^2 \text {Li}_2\left (1-\frac {(1-i) \left (\sqrt {c} x+1\right )}{1-i \sqrt {c} x}\right ) c^{5/2}-\frac {b^2 \log \left (1-c x^2\right ) c^2}{5 x}-\frac {b \left (2 a-b \log \left (1-c x^2\right )\right ) c^2}{5 x}-\frac {8 b^2 c^2}{15 x}+\frac {2 a b c^2}{5 x}+\frac {b^2 \log \left (1-c x^2\right ) c}{15 x^3}-\frac {b \left (2 a-b \log \left (1-c x^2\right )\right ) c}{15 x^3}-\frac {2 b^2 \log \left (c x^2+1\right ) c}{15 x^3}-\frac {2 a b c}{15 x^3}-\frac {\left (2 a-b \log \left (1-c x^2\right )\right )^2}{20 x^5}-\frac {b^2 \log ^2\left (c x^2+1\right )}{20 x^5}+\frac {b^2 \log \left (1-c x^2\right ) \log \left (c x^2+1\right )}{10 x^5}-\frac {a b \log \left (c x^2+1\right )}{5 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTanh[c*x^2])^2/x^6,x]

[Out]

(-2*a*b*c)/(15*x^3) + (2*a*b*c^2)/(5*x) - (8*b^2*c^2)/(15*x) + (2*a*b*c^(5/2)*ArcTan[Sqrt[c]*x])/5 - (4*b^2*c^
(5/2)*ArcTan[Sqrt[c]*x])/15 + (I/5)*b^2*c^(5/2)*ArcTan[Sqrt[c]*x]^2 + (4*b^2*c^(5/2)*ArcTanh[Sqrt[c]*x])/15 +
(b^2*c^(5/2)*ArcTanh[Sqrt[c]*x]^2)/5 - (2*b^2*c^(5/2)*ArcTanh[Sqrt[c]*x]*Log[2/(1 - Sqrt[c]*x)])/5 - (2*b^2*c^
(5/2)*ArcTan[Sqrt[c]*x]*Log[2/(1 - I*Sqrt[c]*x)])/5 + (b^2*c^(5/2)*ArcTan[Sqrt[c]*x]*Log[((1 + I)*(1 - Sqrt[c]
*x))/(1 - I*Sqrt[c]*x)])/5 + (2*b^2*c^(5/2)*ArcTan[Sqrt[c]*x]*Log[2/(1 + I*Sqrt[c]*x)])/5 + (2*b^2*c^(5/2)*Arc
Tanh[Sqrt[c]*x]*Log[2/(1 + Sqrt[c]*x)])/5 - (b^2*c^(5/2)*ArcTanh[Sqrt[c]*x]*Log[(-2*Sqrt[c]*(1 - Sqrt[-c]*x))/
((Sqrt[-c] - Sqrt[c])*(1 + Sqrt[c]*x))])/5 - (b^2*c^(5/2)*ArcTanh[Sqrt[c]*x]*Log[(2*Sqrt[c]*(1 + Sqrt[-c]*x))/
((Sqrt[-c] + Sqrt[c])*(1 + Sqrt[c]*x))])/5 + (b^2*c^(5/2)*ArcTan[Sqrt[c]*x]*Log[((1 - I)*(1 + Sqrt[c]*x))/(1 -
I*Sqrt[c]*x)])/5 + (b^2*c*Log[1 - c*x^2])/(15*x^3) - (b^2*c^2*Log[1 - c*x^2])/(5*x) - (b^2*c^(5/2)*ArcTan[Sqr
t[c]*x]*Log[1 - c*x^2])/5 - (b*c*(2*a - b*Log[1 - c*x^2]))/(15*x^3) - (b*c^2*(2*a - b*Log[1 - c*x^2]))/(5*x) +
(b*c^(5/2)*ArcTanh[Sqrt[c]*x]*(2*a - b*Log[1 - c*x^2]))/5 - (2*a - b*Log[1 - c*x^2])^2/(20*x^5) - (a*b*Log[1
+ c*x^2])/(5*x^5) - (2*b^2*c*Log[1 + c*x^2])/(15*x^3) + (b^2*c^(5/2)*ArcTan[Sqrt[c]*x]*Log[1 + c*x^2])/5 + (b^
2*c^(5/2)*ArcTanh[Sqrt[c]*x]*Log[1 + c*x^2])/5 + (b^2*Log[1 - c*x^2]*Log[1 + c*x^2])/(10*x^5) - (b^2*Log[1 + c
*x^2]^2)/(20*x^5) - (b^2*c^(5/2)*PolyLog[2, 1 - 2/(1 - Sqrt[c]*x)])/5 + (I/5)*b^2*c^(5/2)*PolyLog[2, 1 - 2/(1
- I*Sqrt[c]*x)] - (I/10)*b^2*c^(5/2)*PolyLog[2, 1 - ((1 + I)*(1 - Sqrt[c]*x))/(1 - I*Sqrt[c]*x)] + (I/5)*b^2*c
^(5/2)*PolyLog[2, 1 - 2/(1 + I*Sqrt[c]*x)] - (b^2*c^(5/2)*PolyLog[2, 1 - 2/(1 + Sqrt[c]*x)])/5 + (b^2*c^(5/2)*
PolyLog[2, 1 + (2*Sqrt[c]*(1 - Sqrt[-c]*x))/((Sqrt[-c] - Sqrt[c])*(1 + Sqrt[c]*x))])/10 + (b^2*c^(5/2)*PolyLog
[2, 1 - (2*Sqrt[c]*(1 + Sqrt[-c]*x))/((Sqrt[-c] + Sqrt[c])*(1 + Sqrt[c]*x))])/10 - (I/10)*b^2*c^(5/2)*PolyLog[
2, 1 - ((1 - I)*(1 + Sqrt[c]*x))/(1 - I*Sqrt[c]*x)]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

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 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[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 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 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 2505

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

Rule 2507

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_)*((f_.)*(x_))^(m_.), x_Symbol] :> Simp[(f*x)
^(m + 1)*((a + b*Log[c*(d + e*x^n)^p])^q/(f*(m + 1))), x] - Dist[b*e*n*p*(q/(f^n*(m + 1))), Int[(f*x)^(m + n)*
((a + b*Log[c*(d + e*x^n)^p])^(q - 1)/(d + e*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && IGtQ[q, 1]
&& IntegerQ[n] && NeQ[m, -1]

Rule 2520

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))/((f_) + (g_.)*(x_)^2), x_Symbol] :> With[{u = In
tHide[1/(f + g*x^2), x]}, Simp[u*(a + b*Log[c*(d + e*x^n)^p]), x] - Dist[b*e*n*p, Int[u*(x^(n - 1)/(d + e*x^n)
), x], x]] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && IntegerQ[n]

Rule 2526

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.)*((f_) + (g_.)*(x_)^(s_))^(r_.),
x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x^n)^p])^q, x^m*(f + g*x^s)^r, x], x] /; FreeQ[{a, b, c,
d, e, f, g, m, n, p, q, r, s}, x] && IGtQ[q, 0] && IntegerQ[m] && IntegerQ[r] && IntegerQ[s]

Rule 2637

Int[Log[v_]*Log[w_]*(u_), x_Symbol] :> With[{z = IntHide[u, x]}, Dist[Log[v]*Log[w], z, x] + (-Int[SimplifyInt
egrand[z*Log[w]*(D[v, x]/v), x], x] - Int[SimplifyIntegrand[z*Log[v]*(D[w, x]/w), x], x]) /; InverseFunctionFr
eeQ[z, x]] /; InverseFunctionFreeQ[v, x] && InverseFunctionFreeQ[w, x]

Rule 4964

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTan[c*x])^p)*(
Log[2/(1 + e*(x/d))]/e), x] + Dist[b*c*(p/e), Int[(a + b*ArcTan[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 4966

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

Rule 5040

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

Rule 5048

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*(x_)^(m_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[a
+ b*ArcTan[c*x], x^m/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[m] &&  !(EqQ[m, 1] && NeQ[a,
0])

Rule 6041

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_)]*(b_.))^(p_)*(x_)^(m_.), x_Symbol] :> Int[ExpandIntegrand[x^m*(a + b*(Log
[1 + c*x^n]/2) - b*(Log[1 - c*x^n]/2))^p, x], x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 1] && IGtQ[n, 0] && Integer
Q[m]

Rule 6055

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTanh[c*x])^p)
*(Log[2/(1 + e*(x/d))]/e), x] + Dist[b*c*(p/e), Int[(a + b*ArcTanh[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 6057

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

Rule 6131

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

Rule 6139

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*(x_)^(m_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[a
+ b*ArcTanh[c*x], x^m/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[m] &&  !(EqQ[m, 1] && NeQ[
a, 0])

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int \frac {\left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2}{x^6} \, dx &=\int \left (\frac {\left (2 a-b \log \left (1-c x^2\right )\right )^2}{4 x^6}-\frac {b \left (-2 a+b \log \left (1-c x^2\right )\right ) \log \left (1+c x^2\right )}{2 x^6}+\frac {b^2 \log ^2\left (1+c x^2\right )}{4 x^6}\right ) \, dx\\ &=\frac {1}{4} \int \frac {\left (2 a-b \log \left (1-c x^2\right )\right )^2}{x^6} \, dx-\frac {1}{2} b \int \frac {\left (-2 a+b \log \left (1-c x^2\right )\right ) \log \left (1+c x^2\right )}{x^6} \, dx+\frac {1}{4} b^2 \int \frac {\log ^2\left (1+c x^2\right )}{x^6} \, dx\\ &=-\frac {\left (2 a-b \log \left (1-c x^2\right )\right )^2}{20 x^5}-\frac {b^2 \log ^2\left (1+c x^2\right )}{20 x^5}-\frac {1}{2} b \int \left (-\frac {2 a \log \left (1+c x^2\right )}{x^6}+\frac {b \log \left (1-c x^2\right ) \log \left (1+c x^2\right )}{x^6}\right ) \, dx+\frac {1}{5} (b c) \int \frac {2 a-b \log \left (1-c x^2\right )}{x^4 \left (1-c x^2\right )} \, dx+\frac {1}{5} \left (b^2 c\right ) \int \frac {\log \left (1+c x^2\right )}{x^4 \left (1+c x^2\right )} \, dx\\ &=-\frac {\left (2 a-b \log \left (1-c x^2\right )\right )^2}{20 x^5}-\frac {b^2 \log ^2\left (1+c x^2\right )}{20 x^5}+(a b) \int \frac {\log \left (1+c x^2\right )}{x^6} \, dx-\frac {1}{2} b^2 \int \frac {\log \left (1-c x^2\right ) \log \left (1+c x^2\right )}{x^6} \, dx+\frac {1}{5} (b c) \int \left (\frac {2 a-b \log \left (1-c x^2\right )}{x^4}+\frac {c \left (2 a-b \log \left (1-c x^2\right )\right )}{x^2}-\frac {c^2 \left (2 a-b \log \left (1-c x^2\right )\right )}{-1+c x^2}\right ) \, dx+\frac {1}{5} \left (b^2 c\right ) \int \left (\frac {\log \left (1+c x^2\right )}{x^4}-\frac {c \log \left (1+c x^2\right )}{x^2}+\frac {c^2 \log \left (1+c x^2\right )}{1+c x^2}\right ) \, dx\\ &=-\frac {\left (2 a-b \log \left (1-c x^2\right )\right )^2}{20 x^5}-\frac {a b \log \left (1+c x^2\right )}{5 x^5}+\frac {b^2 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )}{10 x^5}-\frac {b^2 \log ^2\left (1+c x^2\right )}{20 x^5}+\frac {1}{2} b^2 \int -\frac {2 c \log \left (1-c x^2\right )}{5 x^4 \left (1+c x^2\right )} \, dx+\frac {1}{2} b^2 \int \frac {2 c \log \left (1+c x^2\right )}{5 x^4 \left (1-c x^2\right )} \, dx+\frac {1}{5} (b c) \int \frac {2 a-b \log \left (1-c x^2\right )}{x^4} \, dx+\frac {1}{5} (2 a b c) \int \frac {1}{x^4 \left (1+c x^2\right )} \, dx+\frac {1}{5} \left (b^2 c\right ) \int \frac {\log \left (1+c x^2\right )}{x^4} \, dx+\frac {1}{5} \left (b c^2\right ) \int \frac {2 a-b \log \left (1-c x^2\right )}{x^2} \, dx-\frac {1}{5} \left (b^2 c^2\right ) \int \frac {\log \left (1+c x^2\right )}{x^2} \, dx-\frac {1}{5} \left (b c^3\right ) \int \frac {2 a-b \log \left (1-c x^2\right )}{-1+c x^2} \, dx+\frac {1}{5} \left (b^2 c^3\right ) \int \frac {\log \left (1+c x^2\right )}{1+c x^2} \, dx\\ &=-\frac {2 a b c}{15 x^3}-\frac {b c \left (2 a-b \log \left (1-c x^2\right )\right )}{15 x^3}-\frac {b c^2 \left (2 a-b \log \left (1-c x^2\right )\right )}{5 x}+\frac {1}{5} b c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )-\frac {\left (2 a-b \log \left (1-c x^2\right )\right )^2}{20 x^5}-\frac {a b \log \left (1+c x^2\right )}{5 x^5}-\frac {b^2 c \log \left (1+c x^2\right )}{15 x^3}+\frac {b^2 c^2 \log \left (1+c x^2\right )}{5 x}+\frac {1}{5} b^2 c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )+\frac {b^2 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )}{10 x^5}-\frac {b^2 \log ^2\left (1+c x^2\right )}{20 x^5}-\frac {1}{5} \left (b^2 c\right ) \int \frac {\log \left (1-c x^2\right )}{x^4 \left (1+c x^2\right )} \, dx+\frac {1}{5} \left (b^2 c\right ) \int \frac {\log \left (1+c x^2\right )}{x^4 \left (1-c x^2\right )} \, dx-\frac {1}{5} \left (2 a b c^2\right ) \int \frac {1}{x^2 \left (1+c x^2\right )} \, dx+\frac {1}{15} \left (2 b^2 c^2\right ) \int \frac {1}{x^2 \left (1-c x^2\right )} \, dx+\frac {1}{15} \left (2 b^2 c^2\right ) \int \frac {1}{x^2 \left (1+c x^2\right )} \, dx+\frac {1}{5} \left (2 b^2 c^3\right ) \int \frac {1}{1-c x^2} \, dx-\frac {1}{5} \left (2 b^2 c^3\right ) \int \frac {1}{1+c x^2} \, dx-\frac {1}{5} \left (2 b^2 c^4\right ) \int \frac {x \tan ^{-1}\left (\sqrt {c} x\right )}{\sqrt {c} \left (1+c x^2\right )} \, dx-\frac {1}{5} \left (2 b^2 c^4\right ) \int \frac {x \tanh ^{-1}\left (\sqrt {c} x\right )}{\sqrt {c} \left (1-c x^2\right )} \, dx\\ &=-\frac {2 a b c}{15 x^3}+\frac {2 a b c^2}{5 x}-\frac {4 b^2 c^2}{15 x}-\frac {2}{5} b^2 c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right )+\frac {2}{5} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right )-\frac {b c \left (2 a-b \log \left (1-c x^2\right )\right )}{15 x^3}-\frac {b c^2 \left (2 a-b \log \left (1-c x^2\right )\right )}{5 x}+\frac {1}{5} b c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )-\frac {\left (2 a-b \log \left (1-c x^2\right )\right )^2}{20 x^5}-\frac {a b \log \left (1+c x^2\right )}{5 x^5}-\frac {b^2 c \log \left (1+c x^2\right )}{15 x^3}+\frac {b^2 c^2 \log \left (1+c x^2\right )}{5 x}+\frac {1}{5} b^2 c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )+\frac {b^2 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )}{10 x^5}-\frac {b^2 \log ^2\left (1+c x^2\right )}{20 x^5}-\frac {1}{5} \left (b^2 c\right ) \int \left (\frac {\log \left (1-c x^2\right )}{x^4}-\frac {c \log \left (1-c x^2\right )}{x^2}+\frac {c^2 \log \left (1-c x^2\right )}{1+c x^2}\right ) \, dx+\frac {1}{5} \left (b^2 c\right ) \int \left (\frac {\log \left (1+c x^2\right )}{x^4}+\frac {c \log \left (1+c x^2\right )}{x^2}-\frac {c^2 \log \left (1+c x^2\right )}{-1+c x^2}\right ) \, dx+\frac {1}{5} \left (2 a b c^3\right ) \int \frac {1}{1+c x^2} \, dx+\frac {1}{15} \left (2 b^2 c^3\right ) \int \frac {1}{1-c x^2} \, dx-\frac {1}{15} \left (2 b^2 c^3\right ) \int \frac {1}{1+c x^2} \, dx-\frac {1}{5} \left (2 b^2 c^{7/2}\right ) \int \frac {x \tan ^{-1}\left (\sqrt {c} x\right )}{1+c x^2} \, dx-\frac {1}{5} \left (2 b^2 c^{7/2}\right ) \int \frac {x \tanh ^{-1}\left (\sqrt {c} x\right )}{1-c x^2} \, dx\\ &=-\frac {2 a b c}{15 x^3}+\frac {2 a b c^2}{5 x}-\frac {4 b^2 c^2}{15 x}+\frac {2}{5} a b c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right )-\frac {8}{15} b^2 c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right )+\frac {1}{5} i b^2 c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right )^2+\frac {8}{15} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right )+\frac {1}{5} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right )^2-\frac {b c \left (2 a-b \log \left (1-c x^2\right )\right )}{15 x^3}-\frac {b c^2 \left (2 a-b \log \left (1-c x^2\right )\right )}{5 x}+\frac {1}{5} b c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )-\frac {\left (2 a-b \log \left (1-c x^2\right )\right )^2}{20 x^5}-\frac {a b \log \left (1+c x^2\right )}{5 x^5}-\frac {b^2 c \log \left (1+c x^2\right )}{15 x^3}+\frac {b^2 c^2 \log \left (1+c x^2\right )}{5 x}+\frac {1}{5} b^2 c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )+\frac {b^2 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )}{10 x^5}-\frac {b^2 \log ^2\left (1+c x^2\right )}{20 x^5}-\frac {1}{5} \left (b^2 c\right ) \int \frac {\log \left (1-c x^2\right )}{x^4} \, dx+\frac {1}{5} \left (b^2 c\right ) \int \frac {\log \left (1+c x^2\right )}{x^4} \, dx+\frac {1}{5} \left (b^2 c^2\right ) \int \frac {\log \left (1-c x^2\right )}{x^2} \, dx+\frac {1}{5} \left (b^2 c^2\right ) \int \frac {\log \left (1+c x^2\right )}{x^2} \, dx-\frac {1}{5} \left (b^2 c^3\right ) \int \frac {\log \left (1-c x^2\right )}{1+c x^2} \, dx-\frac {1}{5} \left (b^2 c^3\right ) \int \frac {\log \left (1+c x^2\right )}{-1+c x^2} \, dx+\frac {1}{5} \left (2 b^2 c^3\right ) \int \frac {\tan ^{-1}\left (\sqrt {c} x\right )}{i-\sqrt {c} x} \, dx-\frac {1}{5} \left (2 b^2 c^3\right ) \int \frac {\tanh ^{-1}\left (\sqrt {c} x\right )}{1-\sqrt {c} x} \, dx\\ &=-\frac {2 a b c}{15 x^3}+\frac {2 a b c^2}{5 x}-\frac {4 b^2 c^2}{15 x}+\frac {2}{5} a b c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right )-\frac {8}{15} b^2 c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right )+\frac {1}{5} i b^2 c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right )^2+\frac {8}{15} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right )+\frac {1}{5} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right )^2-\frac {2}{5} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt {c} x}\right )+\frac {2}{5} b^2 c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+i \sqrt {c} x}\right )+\frac {b^2 c \log \left (1-c x^2\right )}{15 x^3}-\frac {b^2 c^2 \log \left (1-c x^2\right )}{5 x}-\frac {1}{5} b^2 c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )-\frac {b c \left (2 a-b \log \left (1-c x^2\right )\right )}{15 x^3}-\frac {b c^2 \left (2 a-b \log \left (1-c x^2\right )\right )}{5 x}+\frac {1}{5} b c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )-\frac {\left (2 a-b \log \left (1-c x^2\right )\right )^2}{20 x^5}-\frac {a b \log \left (1+c x^2\right )}{5 x^5}-\frac {2 b^2 c \log \left (1+c x^2\right )}{15 x^3}+\frac {1}{5} b^2 c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )+\frac {1}{5} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )+\frac {b^2 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )}{10 x^5}-\frac {b^2 \log ^2\left (1+c x^2\right )}{20 x^5}+\frac {1}{15} \left (2 b^2 c^2\right ) \int \frac {1}{x^2 \left (1-c x^2\right )} \, dx+\frac {1}{15} \left (2 b^2 c^2\right ) \int \frac {1}{x^2 \left (1+c x^2\right )} \, dx-\frac {1}{5} \left (2 b^2 c^3\right ) \int \frac {1}{1-c x^2} \, dx+\frac {1}{5} \left (2 b^2 c^3\right ) \int \frac {1}{1+c x^2} \, dx+\frac {1}{5} \left (2 b^2 c^3\right ) \int \frac {\log \left (\frac {2}{1-\sqrt {c} x}\right )}{1-c x^2} \, dx-\frac {1}{5} \left (2 b^2 c^3\right ) \int \frac {\log \left (\frac {2}{1+i \sqrt {c} x}\right )}{1+c x^2} \, dx-\frac {1}{5} \left (2 b^2 c^4\right ) \int \frac {x \tan ^{-1}\left (\sqrt {c} x\right )}{\sqrt {c} \left (1-c x^2\right )} \, dx-\frac {1}{5} \left (2 b^2 c^4\right ) \int \frac {x \tanh ^{-1}\left (\sqrt {c} x\right )}{\sqrt {c} \left (1+c x^2\right )} \, dx\\ &=-\frac {2 a b c}{15 x^3}+\frac {2 a b c^2}{5 x}-\frac {8 b^2 c^2}{15 x}+\frac {2}{5} a b c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right )-\frac {2}{15} b^2 c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right )+\frac {1}{5} i b^2 c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right )^2+\frac {2}{15} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right )+\frac {1}{5} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right )^2-\frac {2}{5} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt {c} x}\right )+\frac {2}{5} b^2 c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+i \sqrt {c} x}\right )+\frac {b^2 c \log \left (1-c x^2\right )}{15 x^3}-\frac {b^2 c^2 \log \left (1-c x^2\right )}{5 x}-\frac {1}{5} b^2 c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )-\frac {b c \left (2 a-b \log \left (1-c x^2\right )\right )}{15 x^3}-\frac {b c^2 \left (2 a-b \log \left (1-c x^2\right )\right )}{5 x}+\frac {1}{5} b c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )-\frac {\left (2 a-b \log \left (1-c x^2\right )\right )^2}{20 x^5}-\frac {a b \log \left (1+c x^2\right )}{5 x^5}-\frac {2 b^2 c \log \left (1+c x^2\right )}{15 x^3}+\frac {1}{5} b^2 c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )+\frac {1}{5} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )+\frac {b^2 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )}{10 x^5}-\frac {b^2 \log ^2\left (1+c x^2\right )}{20 x^5}+\frac {1}{5} \left (2 i b^2 c^{5/2}\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i \sqrt {c} x}\right )-\frac {1}{5} \left (2 b^2 c^{5/2}\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-\sqrt {c} x}\right )+\frac {1}{15} \left (2 b^2 c^3\right ) \int \frac {1}{1-c x^2} \, dx-\frac {1}{15} \left (2 b^2 c^3\right ) \int \frac {1}{1+c x^2} \, dx-\frac {1}{5} \left (2 b^2 c^{7/2}\right ) \int \frac {x \tan ^{-1}\left (\sqrt {c} x\right )}{1-c x^2} \, dx-\frac {1}{5} \left (2 b^2 c^{7/2}\right ) \int \frac {x \tanh ^{-1}\left (\sqrt {c} x\right )}{1+c x^2} \, dx\\ &=-\frac {2 a b c}{15 x^3}+\frac {2 a b c^2}{5 x}-\frac {8 b^2 c^2}{15 x}+\frac {2}{5} a b c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right )-\frac {4}{15} b^2 c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right )+\frac {1}{5} i b^2 c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right )^2+\frac {4}{15} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right )+\frac {1}{5} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right )^2-\frac {2}{5} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt {c} x}\right )+\frac {2}{5} b^2 c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+i \sqrt {c} x}\right )+\frac {b^2 c \log \left (1-c x^2\right )}{15 x^3}-\frac {b^2 c^2 \log \left (1-c x^2\right )}{5 x}-\frac {1}{5} b^2 c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )-\frac {b c \left (2 a-b \log \left (1-c x^2\right )\right )}{15 x^3}-\frac {b c^2 \left (2 a-b \log \left (1-c x^2\right )\right )}{5 x}+\frac {1}{5} b c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )-\frac {\left (2 a-b \log \left (1-c x^2\right )\right )^2}{20 x^5}-\frac {a b \log \left (1+c x^2\right )}{5 x^5}-\frac {2 b^2 c \log \left (1+c x^2\right )}{15 x^3}+\frac {1}{5} b^2 c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )+\frac {1}{5} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )+\frac {b^2 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )}{10 x^5}-\frac {b^2 \log ^2\left (1+c x^2\right )}{20 x^5}-\frac {1}{5} b^2 c^{5/2} \text {Li}_2\left (1-\frac {2}{1-\sqrt {c} x}\right )+\frac {1}{5} i b^2 c^{5/2} \text {Li}_2\left (1-\frac {2}{1+i \sqrt {c} x}\right )-\frac {1}{5} \left (2 b^2 c^{7/2}\right ) \int \left (\frac {\tan ^{-1}\left (\sqrt {c} x\right )}{2 \sqrt {c} \left (1-\sqrt {c} x\right )}-\frac {\tan ^{-1}\left (\sqrt {c} x\right )}{2 \sqrt {c} \left (1+\sqrt {c} x\right )}\right ) \, dx-\frac {1}{5} \left (2 b^2 c^{7/2}\right ) \int \left (-\frac {\sqrt {-c} \tanh ^{-1}\left (\sqrt {c} x\right )}{2 c \left (1-\sqrt {-c} x\right )}+\frac {\sqrt {-c} \tanh ^{-1}\left (\sqrt {c} x\right )}{2 c \left (1+\sqrt {-c} x\right )}\right ) \, dx\\ &=-\frac {2 a b c}{15 x^3}+\frac {2 a b c^2}{5 x}-\frac {8 b^2 c^2}{15 x}+\frac {2}{5} a b c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right )-\frac {4}{15} b^2 c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right )+\frac {1}{5} i b^2 c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right )^2+\frac {4}{15} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right )+\frac {1}{5} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right )^2-\frac {2}{5} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt {c} x}\right )+\frac {2}{5} b^2 c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+i \sqrt {c} x}\right )+\frac {b^2 c \log \left (1-c x^2\right )}{15 x^3}-\frac {b^2 c^2 \log \left (1-c x^2\right )}{5 x}-\frac {1}{5} b^2 c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )-\frac {b c \left (2 a-b \log \left (1-c x^2\right )\right )}{15 x^3}-\frac {b c^2 \left (2 a-b \log \left (1-c x^2\right )\right )}{5 x}+\frac {1}{5} b c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )-\frac {\left (2 a-b \log \left (1-c x^2\right )\right )^2}{20 x^5}-\frac {a b \log \left (1+c x^2\right )}{5 x^5}-\frac {2 b^2 c \log \left (1+c x^2\right )}{15 x^3}+\frac {1}{5} b^2 c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )+\frac {1}{5} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )+\frac {b^2 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )}{10 x^5}-\frac {b^2 \log ^2\left (1+c x^2\right )}{20 x^5}-\frac {1}{5} b^2 c^{5/2} \text {Li}_2\left (1-\frac {2}{1-\sqrt {c} x}\right )+\frac {1}{5} i b^2 c^{5/2} \text {Li}_2\left (1-\frac {2}{1+i \sqrt {c} x}\right )-\frac {1}{5} \left (b^2 c^3\right ) \int \frac {\tan ^{-1}\left (\sqrt {c} x\right )}{1-\sqrt {c} x} \, dx+\frac {1}{5} \left (b^2 c^3\right ) \int \frac {\tan ^{-1}\left (\sqrt {c} x\right )}{1+\sqrt {c} x} \, dx-\frac {\left (b^2 c^{7/2}\right ) \int \frac {\tanh ^{-1}\left (\sqrt {c} x\right )}{1-\sqrt {-c} x} \, dx}{5 \sqrt {-c}}+\frac {\left (b^2 c^{7/2}\right ) \int \frac {\tanh ^{-1}\left (\sqrt {c} x\right )}{1+\sqrt {-c} x} \, dx}{5 \sqrt {-c}}\\ &=-\frac {2 a b c}{15 x^3}+\frac {2 a b c^2}{5 x}-\frac {8 b^2 c^2}{15 x}+\frac {2}{5} a b c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right )-\frac {4}{15} b^2 c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right )+\frac {1}{5} i b^2 c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right )^2+\frac {4}{15} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right )+\frac {1}{5} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right )^2-\frac {2}{5} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt {c} x}\right )-\frac {2}{5} b^2 c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-i \sqrt {c} x}\right )+\frac {1}{5} b^2 c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )+\frac {2}{5} b^2 c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+i \sqrt {c} x}\right )+\frac {2}{5} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+\sqrt {c} x}\right )-\frac {1}{5} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (-\frac {2 \sqrt {c} \left (1-\sqrt {-c} x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (1+\sqrt {c} x\right )}\right )-\frac {1}{5} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2 \sqrt {c} \left (1+\sqrt {-c} x\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (1+\sqrt {c} x\right )}\right )+\frac {1}{5} b^2 c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {(1-i) \left (1+\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )+\frac {b^2 c \log \left (1-c x^2\right )}{15 x^3}-\frac {b^2 c^2 \log \left (1-c x^2\right )}{5 x}-\frac {1}{5} b^2 c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )-\frac {b c \left (2 a-b \log \left (1-c x^2\right )\right )}{15 x^3}-\frac {b c^2 \left (2 a-b \log \left (1-c x^2\right )\right )}{5 x}+\frac {1}{5} b c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )-\frac {\left (2 a-b \log \left (1-c x^2\right )\right )^2}{20 x^5}-\frac {a b \log \left (1+c x^2\right )}{5 x^5}-\frac {2 b^2 c \log \left (1+c x^2\right )}{15 x^3}+\frac {1}{5} b^2 c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )+\frac {1}{5} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )+\frac {b^2 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )}{10 x^5}-\frac {b^2 \log ^2\left (1+c x^2\right )}{20 x^5}-\frac {1}{5} b^2 c^{5/2} \text {Li}_2\left (1-\frac {2}{1-\sqrt {c} x}\right )+\frac {1}{5} i b^2 c^{5/2} \text {Li}_2\left (1-\frac {2}{1+i \sqrt {c} x}\right )+2 \left (\frac {1}{5} \left (b^2 c^3\right ) \int \frac {\log \left (\frac {2}{1-i \sqrt {c} x}\right )}{1+c x^2} \, dx\right )-\frac {1}{5} \left (b^2 c^3\right ) \int \frac {\log \left (\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )}{1+c x^2} \, dx-2 \left (\frac {1}{5} \left (b^2 c^3\right ) \int \frac {\log \left (\frac {2}{1+\sqrt {c} x}\right )}{1-c x^2} \, dx\right )+\frac {1}{5} \left (b^2 c^3\right ) \int \frac {\log \left (\frac {2 \sqrt {c} \left (1-\sqrt {-c} x\right )}{\left (-\sqrt {-c}+\sqrt {c}\right ) \left (1+\sqrt {c} x\right )}\right )}{1-c x^2} \, dx+\frac {1}{5} \left (b^2 c^3\right ) \int \frac {\log \left (\frac {2 \sqrt {c} \left (1+\sqrt {-c} x\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (1+\sqrt {c} x\right )}\right )}{1-c x^2} \, dx-\frac {1}{5} \left (b^2 c^3\right ) \int \frac {\log \left (\frac {(1-i) \left (1+\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )}{1+c x^2} \, dx\\ &=-\frac {2 a b c}{15 x^3}+\frac {2 a b c^2}{5 x}-\frac {8 b^2 c^2}{15 x}+\frac {2}{5} a b c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right )-\frac {4}{15} b^2 c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right )+\frac {1}{5} i b^2 c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right )^2+\frac {4}{15} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right )+\frac {1}{5} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right )^2-\frac {2}{5} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt {c} x}\right )-\frac {2}{5} b^2 c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-i \sqrt {c} x}\right )+\frac {1}{5} b^2 c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )+\frac {2}{5} b^2 c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+i \sqrt {c} x}\right )+\frac {2}{5} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+\sqrt {c} x}\right )-\frac {1}{5} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (-\frac {2 \sqrt {c} \left (1-\sqrt {-c} x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (1+\sqrt {c} x\right )}\right )-\frac {1}{5} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2 \sqrt {c} \left (1+\sqrt {-c} x\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (1+\sqrt {c} x\right )}\right )+\frac {1}{5} b^2 c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {(1-i) \left (1+\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )+\frac {b^2 c \log \left (1-c x^2\right )}{15 x^3}-\frac {b^2 c^2 \log \left (1-c x^2\right )}{5 x}-\frac {1}{5} b^2 c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )-\frac {b c \left (2 a-b \log \left (1-c x^2\right )\right )}{15 x^3}-\frac {b c^2 \left (2 a-b \log \left (1-c x^2\right )\right )}{5 x}+\frac {1}{5} b c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )-\frac {\left (2 a-b \log \left (1-c x^2\right )\right )^2}{20 x^5}-\frac {a b \log \left (1+c x^2\right )}{5 x^5}-\frac {2 b^2 c \log \left (1+c x^2\right )}{15 x^3}+\frac {1}{5} b^2 c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )+\frac {1}{5} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )+\frac {b^2 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )}{10 x^5}-\frac {b^2 \log ^2\left (1+c x^2\right )}{20 x^5}-\frac {1}{5} b^2 c^{5/2} \text {Li}_2\left (1-\frac {2}{1-\sqrt {c} x}\right )-\frac {1}{10} i b^2 c^{5/2} \text {Li}_2\left (1-\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )+\frac {1}{5} i b^2 c^{5/2} \text {Li}_2\left (1-\frac {2}{1+i \sqrt {c} x}\right )+\frac {1}{10} b^2 c^{5/2} \text {Li}_2\left (1+\frac {2 \sqrt {c} \left (1-\sqrt {-c} x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (1+\sqrt {c} x\right )}\right )+\frac {1}{10} b^2 c^{5/2} \text {Li}_2\left (1-\frac {2 \sqrt {c} \left (1+\sqrt {-c} x\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (1+\sqrt {c} x\right )}\right )-\frac {1}{10} i b^2 c^{5/2} \text {Li}_2\left (1-\frac {(1-i) \left (1+\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )+2 \left (\frac {1}{5} \left (i b^2 c^{5/2}\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-i \sqrt {c} x}\right )\right )-2 \left (\frac {1}{5} \left (b^2 c^{5/2}\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+\sqrt {c} x}\right )\right )\\ &=-\frac {2 a b c}{15 x^3}+\frac {2 a b c^2}{5 x}-\frac {8 b^2 c^2}{15 x}+\frac {2}{5} a b c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right )-\frac {4}{15} b^2 c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right )+\frac {1}{5} i b^2 c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right )^2+\frac {4}{15} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right )+\frac {1}{5} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right )^2-\frac {2}{5} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt {c} x}\right )-\frac {2}{5} b^2 c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-i \sqrt {c} x}\right )+\frac {1}{5} b^2 c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )+\frac {2}{5} b^2 c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+i \sqrt {c} x}\right )+\frac {2}{5} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+\sqrt {c} x}\right )-\frac {1}{5} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (-\frac {2 \sqrt {c} \left (1-\sqrt {-c} x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (1+\sqrt {c} x\right )}\right )-\frac {1}{5} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2 \sqrt {c} \left (1+\sqrt {-c} x\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (1+\sqrt {c} x\right )}\right )+\frac {1}{5} b^2 c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {(1-i) \left (1+\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )+\frac {b^2 c \log \left (1-c x^2\right )}{15 x^3}-\frac {b^2 c^2 \log \left (1-c x^2\right )}{5 x}-\frac {1}{5} b^2 c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )-\frac {b c \left (2 a-b \log \left (1-c x^2\right )\right )}{15 x^3}-\frac {b c^2 \left (2 a-b \log \left (1-c x^2\right )\right )}{5 x}+\frac {1}{5} b c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )-\frac {\left (2 a-b \log \left (1-c x^2\right )\right )^2}{20 x^5}-\frac {a b \log \left (1+c x^2\right )}{5 x^5}-\frac {2 b^2 c \log \left (1+c x^2\right )}{15 x^3}+\frac {1}{5} b^2 c^{5/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )+\frac {1}{5} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )+\frac {b^2 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )}{10 x^5}-\frac {b^2 \log ^2\left (1+c x^2\right )}{20 x^5}-\frac {1}{5} b^2 c^{5/2} \text {Li}_2\left (1-\frac {2}{1-\sqrt {c} x}\right )+\frac {1}{5} i b^2 c^{5/2} \text {Li}_2\left (1-\frac {2}{1-i \sqrt {c} x}\right )-\frac {1}{10} i b^2 c^{5/2} \text {Li}_2\left (1-\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )+\frac {1}{5} i b^2 c^{5/2} \text {Li}_2\left (1-\frac {2}{1+i \sqrt {c} x}\right )-\frac {1}{5} b^2 c^{5/2} \text {Li}_2\left (1-\frac {2}{1+\sqrt {c} x}\right )+\frac {1}{10} b^2 c^{5/2} \text {Li}_2\left (1+\frac {2 \sqrt {c} \left (1-\sqrt {-c} x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (1+\sqrt {c} x\right )}\right )+\frac {1}{10} b^2 c^{5/2} \text {Li}_2\left (1-\frac {2 \sqrt {c} \left (1+\sqrt {-c} x\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (1+\sqrt {c} x\right )}\right )-\frac {1}{10} i b^2 c^{5/2} \text {Li}_2\left (1-\frac {(1-i) \left (1+\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )\\ \end {align*}

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Mathematica [F]
time = 2.32, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2}{x^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(a + b*ArcTanh[c*x^2])^2/x^6,x]

[Out]

Integrate[(a + b*ArcTanh[c*x^2])^2/x^6, x]

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \arctanh \left (c \,x^{2}\right )\right )^{2}}{x^{6}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arctanh(c*x^2))^2/x^6,x)

[Out]

int((a+b*arctanh(c*x^2))^2/x^6,x)

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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((a+b*arctanh(c*x^2))^2/x^6,x, algorithm="maxima")

[Out]

1/15*((6*c^(3/2)*arctan(sqrt(c)*x) - 3*c^(3/2)*log((c*x - sqrt(c))/(c*x + sqrt(c))) - 4/x^3)*c - 6*arctanh(c*x
^2)/x^5)*a*b - 1/20*b^2*(log(-c*x^2 + 1)^2/x^5 + 5*integrate(-1/5*(5*(c*x^2 - 1)*log(c*x^2 + 1)^2 + 2*(2*c*x^2
- 5*(c*x^2 - 1)*log(c*x^2 + 1))*log(-c*x^2 + 1))/(c*x^8 - x^6), x)) - 1/5*a^2/x^5

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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+b*arctanh(c*x^2))^2/x^6,x, algorithm="fricas")

[Out]

integral((b^2*arctanh(c*x^2)^2 + 2*a*b*arctanh(c*x^2) + a^2)/x^6, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \operatorname {atanh}{\left (c x^{2} \right )}\right )^{2}}{x^{6}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*atanh(c*x**2))**2/x**6,x)

[Out]

Integral((a + b*atanh(c*x**2))**2/x**6, 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+b*arctanh(c*x^2))^2/x^6,x, algorithm="giac")

[Out]

integrate((b*arctanh(c*x^2) + a)^2/x^6, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x^2\right )\right )}^2}{x^6} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atanh(c*x^2))^2/x^6,x)

[Out]

int((a + b*atanh(c*x^2))^2/x^6, x)

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