3.71 \(\int x^4 (a+b \tanh ^{-1}(c x^2))^2 \, dx\)

Optimal. Leaf size=1173 \[ \text{result too large to display} \]

[Out]

(8*b^2*x)/(15*c^2) + (2*a*b*x^3)/(15*c) - (2*a*b*x^5)/25 + (2*a*b*ArcTan[Sqrt[c]*x])/(5*c^(5/2)) - (4*b^2*ArcT
an[Sqrt[c]*x])/(15*c^(5/2)) + ((I/5)*b^2*ArcTan[Sqrt[c]*x]^2)/c^(5/2) - (4*b^2*ArcTanh[Sqrt[c]*x])/(15*c^(5/2)
) - (b^2*ArcTanh[Sqrt[c]*x]^2)/(5*c^(5/2)) + (2*b^2*ArcTanh[Sqrt[c]*x]*Log[2/(1 - Sqrt[c]*x)])/(5*c^(5/2)) - (
2*b^2*ArcTan[Sqrt[c]*x]*Log[2/(1 - I*Sqrt[c]*x)])/(5*c^(5/2)) + (b^2*ArcTan[Sqrt[c]*x]*Log[((1 + I)*(1 - Sqrt[
c]*x))/(1 - I*Sqrt[c]*x)])/(5*c^(5/2)) + (2*b^2*ArcTan[Sqrt[c]*x]*Log[2/(1 + I*Sqrt[c]*x)])/(5*c^(5/2)) - (2*b
^2*ArcTanh[Sqrt[c]*x]*Log[2/(1 + Sqrt[c]*x)])/(5*c^(5/2)) + (b^2*ArcTanh[Sqrt[c]*x]*Log[(-2*Sqrt[c]*(1 - Sqrt[
-c]*x))/((Sqrt[-c] - Sqrt[c])*(1 + Sqrt[c]*x))])/(5*c^(5/2)) + (b^2*ArcTanh[Sqrt[c]*x]*Log[(2*Sqrt[c]*(1 + Sqr
t[-c]*x))/((Sqrt[-c] + Sqrt[c])*(1 + Sqrt[c]*x))])/(5*c^(5/2)) + (b^2*ArcTan[Sqrt[c]*x]*Log[((1 - I)*(1 + Sqrt
[c]*x))/(1 - I*Sqrt[c]*x)])/(5*c^(5/2)) - (b^2*x^3*Log[1 - c*x^2])/(15*c) + (b^2*x^5*Log[1 - c*x^2])/25 - (b^2
*ArcTan[Sqrt[c]*x]*Log[1 - c*x^2])/(5*c^(5/2)) + (b*x^3*(2*a - b*Log[1 - c*x^2]))/(15*c) + (b*x^5*(2*a - b*Log
[1 - c*x^2]))/25 - (b*ArcTanh[Sqrt[c]*x]*(2*a - b*Log[1 - c*x^2]))/(5*c^(5/2)) + (x^5*(2*a - b*Log[1 - c*x^2])
^2)/20 + (2*b^2*x^3*Log[1 + c*x^2])/(15*c) + (a*b*x^5*Log[1 + c*x^2])/5 + (b^2*ArcTan[Sqrt[c]*x]*Log[1 + c*x^2
])/(5*c^(5/2)) - (b^2*ArcTanh[Sqrt[c]*x]*Log[1 + c*x^2])/(5*c^(5/2)) - (b^2*x^5*Log[1 - c*x^2]*Log[1 + c*x^2])
/10 + (b^2*x^5*Log[1 + c*x^2]^2)/20 + (b^2*PolyLog[2, 1 - 2/(1 - Sqrt[c]*x)])/(5*c^(5/2)) + ((I/5)*b^2*PolyLog
[2, 1 - 2/(1 - I*Sqrt[c]*x)])/c^(5/2) - ((I/10)*b^2*PolyLog[2, 1 - ((1 + I)*(1 - Sqrt[c]*x))/(1 - I*Sqrt[c]*x)
])/c^(5/2) + ((I/5)*b^2*PolyLog[2, 1 - 2/(1 + I*Sqrt[c]*x)])/c^(5/2) + (b^2*PolyLog[2, 1 - 2/(1 + Sqrt[c]*x)])
/(5*c^(5/2)) - (b^2*PolyLog[2, 1 + (2*Sqrt[c]*(1 - Sqrt[-c]*x))/((Sqrt[-c] - Sqrt[c])*(1 + Sqrt[c]*x))])/(10*c
^(5/2)) - (b^2*PolyLog[2, 1 - (2*Sqrt[c]*(1 + Sqrt[-c]*x))/((Sqrt[-c] + Sqrt[c])*(1 + Sqrt[c]*x))])/(10*c^(5/2
)) - ((I/10)*b^2*PolyLog[2, 1 - ((1 - I)*(1 + Sqrt[c]*x))/(1 - I*Sqrt[c]*x)])/c^(5/2)

________________________________________________________________________________________

Rubi [A]  time = 2.33394, antiderivative size = 1173, normalized size of antiderivative = 1., number of steps used = 102, number of rules used = 26, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.625, Rules used = {6099, 2457, 2476, 2448, 321, 206, 2455, 302, 2470, 12, 5984, 5918, 2402, 2315, 6742, 203, 30, 2557, 207, 5992, 5920, 2447, 4928, 4856, 4920, 4854} \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

Int[x^4*(a + b*ArcTanh[c*x^2])^2,x]

[Out]

(8*b^2*x)/(15*c^2) + (2*a*b*x^3)/(15*c) - (2*a*b*x^5)/25 + (2*a*b*ArcTan[Sqrt[c]*x])/(5*c^(5/2)) - (4*b^2*ArcT
an[Sqrt[c]*x])/(15*c^(5/2)) + ((I/5)*b^2*ArcTan[Sqrt[c]*x]^2)/c^(5/2) - (4*b^2*ArcTanh[Sqrt[c]*x])/(15*c^(5/2)
) - (b^2*ArcTanh[Sqrt[c]*x]^2)/(5*c^(5/2)) + (2*b^2*ArcTanh[Sqrt[c]*x]*Log[2/(1 - Sqrt[c]*x)])/(5*c^(5/2)) - (
2*b^2*ArcTan[Sqrt[c]*x]*Log[2/(1 - I*Sqrt[c]*x)])/(5*c^(5/2)) + (b^2*ArcTan[Sqrt[c]*x]*Log[((1 + I)*(1 - Sqrt[
c]*x))/(1 - I*Sqrt[c]*x)])/(5*c^(5/2)) + (2*b^2*ArcTan[Sqrt[c]*x]*Log[2/(1 + I*Sqrt[c]*x)])/(5*c^(5/2)) - (2*b
^2*ArcTanh[Sqrt[c]*x]*Log[2/(1 + Sqrt[c]*x)])/(5*c^(5/2)) + (b^2*ArcTanh[Sqrt[c]*x]*Log[(-2*Sqrt[c]*(1 - Sqrt[
-c]*x))/((Sqrt[-c] - Sqrt[c])*(1 + Sqrt[c]*x))])/(5*c^(5/2)) + (b^2*ArcTanh[Sqrt[c]*x]*Log[(2*Sqrt[c]*(1 + Sqr
t[-c]*x))/((Sqrt[-c] + Sqrt[c])*(1 + Sqrt[c]*x))])/(5*c^(5/2)) + (b^2*ArcTan[Sqrt[c]*x]*Log[((1 - I)*(1 + Sqrt
[c]*x))/(1 - I*Sqrt[c]*x)])/(5*c^(5/2)) - (b^2*x^3*Log[1 - c*x^2])/(15*c) + (b^2*x^5*Log[1 - c*x^2])/25 - (b^2
*ArcTan[Sqrt[c]*x]*Log[1 - c*x^2])/(5*c^(5/2)) + (b*x^3*(2*a - b*Log[1 - c*x^2]))/(15*c) + (b*x^5*(2*a - b*Log
[1 - c*x^2]))/25 - (b*ArcTanh[Sqrt[c]*x]*(2*a - b*Log[1 - c*x^2]))/(5*c^(5/2)) + (x^5*(2*a - b*Log[1 - c*x^2])
^2)/20 + (2*b^2*x^3*Log[1 + c*x^2])/(15*c) + (a*b*x^5*Log[1 + c*x^2])/5 + (b^2*ArcTan[Sqrt[c]*x]*Log[1 + c*x^2
])/(5*c^(5/2)) - (b^2*ArcTanh[Sqrt[c]*x]*Log[1 + c*x^2])/(5*c^(5/2)) - (b^2*x^5*Log[1 - c*x^2]*Log[1 + c*x^2])
/10 + (b^2*x^5*Log[1 + c*x^2]^2)/20 + (b^2*PolyLog[2, 1 - 2/(1 - Sqrt[c]*x)])/(5*c^(5/2)) + ((I/5)*b^2*PolyLog
[2, 1 - 2/(1 - I*Sqrt[c]*x)])/c^(5/2) - ((I/10)*b^2*PolyLog[2, 1 - ((1 + I)*(1 - Sqrt[c]*x))/(1 - I*Sqrt[c]*x)
])/c^(5/2) + ((I/5)*b^2*PolyLog[2, 1 - 2/(1 + I*Sqrt[c]*x)])/c^(5/2) + (b^2*PolyLog[2, 1 - 2/(1 + Sqrt[c]*x)])
/(5*c^(5/2)) - (b^2*PolyLog[2, 1 + (2*Sqrt[c]*(1 - Sqrt[-c]*x))/((Sqrt[-c] - Sqrt[c])*(1 + Sqrt[c]*x))])/(10*c
^(5/2)) - (b^2*PolyLog[2, 1 - (2*Sqrt[c]*(1 + Sqrt[-c]*x))/((Sqrt[-c] + Sqrt[c])*(1 + Sqrt[c]*x))])/(10*c^(5/2
)) - ((I/10)*b^2*PolyLog[2, 1 - ((1 - I)*(1 + Sqrt[c]*x))/(1 - I*Sqrt[c]*x)])/c^(5/2)

Rule 6099

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

Rule 2457

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 2476

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 2448

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)], x_Symbol] :> Simp[x*Log[c*(d + e*x^n)^p], x] - Dist[e*n*p, Int[
x^n/(d + e*x^n), x], x] /; FreeQ[{c, d, e, n, p}, x]

Rule 321

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

Rule 206

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

Rule 2455

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 302

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

Rule 2470

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 12

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

Rule 5984

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 5918

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 2402

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 2315

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

Rule 6742

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

Rule 203

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

Rule 30

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

Rule 2557

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 207

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

Rule 5992

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 5920

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 2447

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 4928

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 4856

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 4920

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Simp[(I*(a + b*ArcTan
[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 4854

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcTan[c*x])^p*Lo
g[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]

Rubi steps

\begin{align*} \int x^4 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2 \, dx &=\int \left (\frac{1}{4} x^4 \left (2 a-b \log \left (1-c x^2\right )\right )^2-\frac{1}{2} b x^4 \left (-2 a+b \log \left (1-c x^2\right )\right ) \log \left (1+c x^2\right )+\frac{1}{4} b^2 x^4 \log ^2\left (1+c x^2\right )\right ) \, dx\\ &=\frac{1}{4} \int x^4 \left (2 a-b \log \left (1-c x^2\right )\right )^2 \, dx-\frac{1}{2} b \int x^4 \left (-2 a+b \log \left (1-c x^2\right )\right ) \log \left (1+c x^2\right ) \, dx+\frac{1}{4} b^2 \int x^4 \log ^2\left (1+c x^2\right ) \, dx\\ &=\frac{1}{20} x^5 \left (2 a-b \log \left (1-c x^2\right )\right )^2+\frac{1}{20} b^2 x^5 \log ^2\left (1+c x^2\right )-\frac{1}{2} b \int \left (-2 a x^4 \log \left (1+c x^2\right )+b x^4 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )\right ) \, dx-\frac{1}{5} (b c) \int \frac{x^6 \left (2 a-b \log \left (1-c x^2\right )\right )}{1-c x^2} \, dx-\frac{1}{5} \left (b^2 c\right ) \int \frac{x^6 \log \left (1+c x^2\right )}{1+c x^2} \, dx\\ &=\frac{1}{20} x^5 \left (2 a-b \log \left (1-c x^2\right )\right )^2+\frac{1}{20} b^2 x^5 \log ^2\left (1+c x^2\right )+(a b) \int x^4 \log \left (1+c x^2\right ) \, dx-\frac{1}{2} b^2 \int x^4 \log \left (1-c x^2\right ) \log \left (1+c x^2\right ) \, dx-\frac{1}{5} (b c) \int \left (-\frac{2 a-b \log \left (1-c x^2\right )}{c^3}-\frac{x^2 \left (2 a-b \log \left (1-c x^2\right )\right )}{c^2}-\frac{x^4 \left (2 a-b \log \left (1-c x^2\right )\right )}{c}+\frac{2 a-b \log \left (1-c x^2\right )}{c^3 \left (1-c x^2\right )}\right ) \, dx-\frac{1}{5} \left (b^2 c\right ) \int \left (\frac{\log \left (1+c x^2\right )}{c^3}-\frac{x^2 \log \left (1+c x^2\right )}{c^2}+\frac{x^4 \log \left (1+c x^2\right )}{c}-\frac{\log \left (1+c x^2\right )}{c^3 \left (1+c x^2\right )}\right ) \, dx\\ &=\frac{1}{20} x^5 \left (2 a-b \log \left (1-c x^2\right )\right )^2+\frac{1}{5} a b x^5 \log \left (1+c x^2\right )-\frac{1}{10} b^2 x^5 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac{1}{20} b^2 x^5 \log ^2\left (1+c x^2\right )+\frac{1}{5} b \int x^4 \left (2 a-b \log \left (1-c x^2\right )\right ) \, dx-\frac{1}{5} b^2 \int x^4 \log \left (1+c x^2\right ) \, dx+\frac{1}{2} b^2 \int \frac{2 c x^6 \log \left (1-c x^2\right )}{5+5 c x^2} \, dx+\frac{1}{2} b^2 \int \frac{2 c x^6 \log \left (1+c x^2\right )}{-5+5 c x^2} \, dx+\frac{b \int \left (2 a-b \log \left (1-c x^2\right )\right ) \, dx}{5 c^2}-\frac{b \int \frac{2 a-b \log \left (1-c x^2\right )}{1-c x^2} \, dx}{5 c^2}-\frac{b^2 \int \log \left (1+c x^2\right ) \, dx}{5 c^2}+\frac{b^2 \int \frac{\log \left (1+c x^2\right )}{1+c x^2} \, dx}{5 c^2}+\frac{b \int x^2 \left (2 a-b \log \left (1-c x^2\right )\right ) \, dx}{5 c}+\frac{b^2 \int x^2 \log \left (1+c x^2\right ) \, dx}{5 c}-\frac{1}{5} (2 a b c) \int \frac{x^6}{1+c x^2} \, dx\\ &=\frac{2 a b x}{5 c^2}+\frac{b x^3 \left (2 a-b \log \left (1-c x^2\right )\right )}{15 c}+\frac{1}{25} b x^5 \left (2 a-b \log \left (1-c x^2\right )\right )-\frac{b \tanh ^{-1}\left (\sqrt{c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )}{5 c^{5/2}}+\frac{1}{20} x^5 \left (2 a-b \log \left (1-c x^2\right )\right )^2-\frac{b^2 x \log \left (1+c x^2\right )}{5 c^2}+\frac{b^2 x^3 \log \left (1+c x^2\right )}{15 c}+\frac{1}{5} a b x^5 \log \left (1+c x^2\right )-\frac{1}{25} b^2 x^5 \log \left (1+c x^2\right )+\frac{b^2 \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1+c x^2\right )}{5 c^{5/2}}-\frac{1}{10} b^2 x^5 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac{1}{20} b^2 x^5 \log ^2\left (1+c x^2\right )-\frac{1}{15} \left (2 b^2\right ) \int \frac{x^4}{1-c x^2} \, dx-\frac{1}{15} \left (2 b^2\right ) \int \frac{x^4}{1+c x^2} \, dx-\frac{b^2 \int \log \left (1-c x^2\right ) \, dx}{5 c^2}+\frac{\left (2 b^2\right ) \int \frac{x^2}{1+c x^2} \, dx}{5 c}-\frac{\left (2 b^2\right ) \int \frac{x \tan ^{-1}\left (\sqrt{c} x\right )}{\sqrt{c} \left (1+c x^2\right )} \, dx}{5 c}+\frac{\left (2 b^2\right ) \int \frac{x \tanh ^{-1}\left (\sqrt{c} x\right )}{\sqrt{c} \left (1-c x^2\right )} \, dx}{5 c}-\frac{1}{5} (2 a b c) \int \left (\frac{1}{c^3}-\frac{x^2}{c^2}+\frac{x^4}{c}-\frac{1}{c^3 \left (1+c x^2\right )}\right ) \, dx-\frac{1}{25} \left (2 b^2 c\right ) \int \frac{x^6}{1-c x^2} \, dx+\frac{1}{25} \left (2 b^2 c\right ) \int \frac{x^6}{1+c x^2} \, dx+\left (b^2 c\right ) \int \frac{x^6 \log \left (1-c x^2\right )}{5+5 c x^2} \, dx+\left (b^2 c\right ) \int \frac{x^6 \log \left (1+c x^2\right )}{-5+5 c x^2} \, dx\\ &=\frac{2 b^2 x}{5 c^2}+\frac{2 a b x^3}{15 c}-\frac{2}{25} a b x^5-\frac{b^2 x \log \left (1-c x^2\right )}{5 c^2}+\frac{b x^3 \left (2 a-b \log \left (1-c x^2\right )\right )}{15 c}+\frac{1}{25} b x^5 \left (2 a-b \log \left (1-c x^2\right )\right )-\frac{b \tanh ^{-1}\left (\sqrt{c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )}{5 c^{5/2}}+\frac{1}{20} x^5 \left (2 a-b \log \left (1-c x^2\right )\right )^2-\frac{b^2 x \log \left (1+c x^2\right )}{5 c^2}+\frac{b^2 x^3 \log \left (1+c x^2\right )}{15 c}+\frac{1}{5} a b x^5 \log \left (1+c x^2\right )-\frac{1}{25} b^2 x^5 \log \left (1+c x^2\right )+\frac{b^2 \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1+c x^2\right )}{5 c^{5/2}}-\frac{1}{10} b^2 x^5 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac{1}{20} b^2 x^5 \log ^2\left (1+c x^2\right )-\frac{1}{15} \left (2 b^2\right ) \int \left (-\frac{1}{c^2}-\frac{x^2}{c}+\frac{1}{c^2 \left (1-c x^2\right )}\right ) \, dx-\frac{1}{15} \left (2 b^2\right ) \int \left (-\frac{1}{c^2}+\frac{x^2}{c}+\frac{1}{c^2 \left (1+c x^2\right )}\right ) \, dx+\frac{(2 a b) \int \frac{1}{1+c x^2} \, dx}{5 c^2}-\frac{\left (2 b^2\right ) \int \frac{1}{1+c x^2} \, dx}{5 c^2}-\frac{\left (2 b^2\right ) \int \frac{x \tan ^{-1}\left (\sqrt{c} x\right )}{1+c x^2} \, dx}{5 c^{3/2}}+\frac{\left (2 b^2\right ) \int \frac{x \tanh ^{-1}\left (\sqrt{c} x\right )}{1-c x^2} \, dx}{5 c^{3/2}}-\frac{\left (2 b^2\right ) \int \frac{x^2}{1-c x^2} \, dx}{5 c}-\frac{1}{25} \left (2 b^2 c\right ) \int \left (-\frac{1}{c^3}-\frac{x^2}{c^2}-\frac{x^4}{c}+\frac{1}{c^3 \left (1-c x^2\right )}\right ) \, dx+\frac{1}{25} \left (2 b^2 c\right ) \int \left (\frac{1}{c^3}-\frac{x^2}{c^2}+\frac{x^4}{c}-\frac{1}{c^3 \left (1+c x^2\right )}\right ) \, dx+\left (b^2 c\right ) \int \left (\frac{\log \left (1-c x^2\right )}{5 c^3}-\frac{x^2 \log \left (1-c x^2\right )}{5 c^2}+\frac{x^4 \log \left (1-c x^2\right )}{5 c}-\frac{\log \left (1-c x^2\right )}{c^3 \left (5+5 c x^2\right )}\right ) \, dx+\left (b^2 c\right ) \int \left (\frac{\log \left (1+c x^2\right )}{5 c^3}+\frac{x^2 \log \left (1+c x^2\right )}{5 c^2}+\frac{x^4 \log \left (1+c x^2\right )}{5 c}+\frac{\log \left (1+c x^2\right )}{c^3 \left (-5+5 c x^2\right )}\right ) \, dx\\ &=\frac{92 b^2 x}{75 c^2}+\frac{2 a b x^3}{15 c}-\frac{2}{25} a b x^5+\frac{4 b^2 x^5}{125}+\frac{2 a b \tan ^{-1}\left (\sqrt{c} x\right )}{5 c^{5/2}}-\frac{2 b^2 \tan ^{-1}\left (\sqrt{c} x\right )}{5 c^{5/2}}+\frac{i b^2 \tan ^{-1}\left (\sqrt{c} x\right )^2}{5 c^{5/2}}-\frac{b^2 \tanh ^{-1}\left (\sqrt{c} x\right )^2}{5 c^{5/2}}-\frac{b^2 x \log \left (1-c x^2\right )}{5 c^2}+\frac{b x^3 \left (2 a-b \log \left (1-c x^2\right )\right )}{15 c}+\frac{1}{25} b x^5 \left (2 a-b \log \left (1-c x^2\right )\right )-\frac{b \tanh ^{-1}\left (\sqrt{c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )}{5 c^{5/2}}+\frac{1}{20} x^5 \left (2 a-b \log \left (1-c x^2\right )\right )^2-\frac{b^2 x \log \left (1+c x^2\right )}{5 c^2}+\frac{b^2 x^3 \log \left (1+c x^2\right )}{15 c}+\frac{1}{5} a b x^5 \log \left (1+c x^2\right )-\frac{1}{25} b^2 x^5 \log \left (1+c x^2\right )+\frac{b^2 \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1+c x^2\right )}{5 c^{5/2}}-\frac{1}{10} b^2 x^5 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac{1}{20} b^2 x^5 \log ^2\left (1+c x^2\right )+\frac{1}{5} b^2 \int x^4 \log \left (1-c x^2\right ) \, dx+\frac{1}{5} b^2 \int x^4 \log \left (1+c x^2\right ) \, dx-\frac{\left (2 b^2\right ) \int \frac{1}{1-c x^2} \, dx}{25 c^2}-\frac{\left (2 b^2\right ) \int \frac{1}{1+c x^2} \, dx}{25 c^2}-\frac{\left (2 b^2\right ) \int \frac{1}{1-c x^2} \, dx}{15 c^2}-\frac{\left (2 b^2\right ) \int \frac{1}{1+c x^2} \, dx}{15 c^2}+\frac{b^2 \int \log \left (1-c x^2\right ) \, dx}{5 c^2}+\frac{b^2 \int \log \left (1+c x^2\right ) \, dx}{5 c^2}-\frac{\left (2 b^2\right ) \int \frac{1}{1-c x^2} \, dx}{5 c^2}+\frac{\left (2 b^2\right ) \int \frac{\tan ^{-1}\left (\sqrt{c} x\right )}{i-\sqrt{c} x} \, dx}{5 c^2}+\frac{\left (2 b^2\right ) \int \frac{\tanh ^{-1}\left (\sqrt{c} x\right )}{1-\sqrt{c} x} \, dx}{5 c^2}-\frac{b^2 \int \frac{\log \left (1-c x^2\right )}{5+5 c x^2} \, dx}{c^2}+\frac{b^2 \int \frac{\log \left (1+c x^2\right )}{-5+5 c x^2} \, dx}{c^2}-\frac{b^2 \int x^2 \log \left (1-c x^2\right ) \, dx}{5 c}+\frac{b^2 \int x^2 \log \left (1+c x^2\right ) \, dx}{5 c}\\ &=\frac{92 b^2 x}{75 c^2}+\frac{2 a b x^3}{15 c}-\frac{2}{25} a b x^5+\frac{4 b^2 x^5}{125}+\frac{2 a b \tan ^{-1}\left (\sqrt{c} x\right )}{5 c^{5/2}}-\frac{46 b^2 \tan ^{-1}\left (\sqrt{c} x\right )}{75 c^{5/2}}+\frac{i b^2 \tan ^{-1}\left (\sqrt{c} x\right )^2}{5 c^{5/2}}-\frac{46 b^2 \tanh ^{-1}\left (\sqrt{c} x\right )}{75 c^{5/2}}-\frac{b^2 \tanh ^{-1}\left (\sqrt{c} x\right )^2}{5 c^{5/2}}+\frac{2 b^2 \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1-\sqrt{c} x}\right )}{5 c^{5/2}}+\frac{2 b^2 \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1+i \sqrt{c} x}\right )}{5 c^{5/2}}-\frac{b^2 x^3 \log \left (1-c x^2\right )}{15 c}+\frac{1}{25} b^2 x^5 \log \left (1-c x^2\right )-\frac{b^2 \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1-c x^2\right )}{5 c^{5/2}}+\frac{b x^3 \left (2 a-b \log \left (1-c x^2\right )\right )}{15 c}+\frac{1}{25} b x^5 \left (2 a-b \log \left (1-c x^2\right )\right )-\frac{b \tanh ^{-1}\left (\sqrt{c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )}{5 c^{5/2}}+\frac{1}{20} x^5 \left (2 a-b \log \left (1-c x^2\right )\right )^2+\frac{2 b^2 x^3 \log \left (1+c x^2\right )}{15 c}+\frac{1}{5} a b x^5 \log \left (1+c x^2\right )+\frac{b^2 \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1+c x^2\right )}{5 c^{5/2}}-\frac{b^2 \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (1+c x^2\right )}{5 c^{5/2}}-\frac{1}{10} b^2 x^5 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac{1}{20} b^2 x^5 \log ^2\left (1+c x^2\right )-\frac{1}{15} \left (2 b^2\right ) \int \frac{x^4}{1-c x^2} \, dx-\frac{1}{15} \left (2 b^2\right ) \int \frac{x^4}{1+c x^2} \, dx-\frac{\left (2 b^2\right ) \int \frac{\log \left (\frac{2}{1-\sqrt{c} x}\right )}{1-c x^2} \, dx}{5 c^2}-\frac{\left (2 b^2\right ) \int \frac{\log \left (\frac{2}{1+i \sqrt{c} x}\right )}{1+c x^2} \, dx}{5 c^2}+\frac{\left (2 b^2\right ) \int \frac{x^2}{1-c x^2} \, dx}{5 c}-\frac{\left (2 b^2\right ) \int \frac{x^2}{1+c x^2} \, dx}{5 c}-\frac{\left (2 b^2\right ) \int \frac{x \tan ^{-1}\left (\sqrt{c} x\right )}{5 \sqrt{c} \left (1-c x^2\right )} \, dx}{c}-\frac{\left (2 b^2\right ) \int -\frac{x \tanh ^{-1}\left (\sqrt{c} x\right )}{5 \sqrt{c} \left (1+c x^2\right )} \, dx}{c}+\frac{1}{25} \left (2 b^2 c\right ) \int \frac{x^6}{1-c x^2} \, dx-\frac{1}{25} \left (2 b^2 c\right ) \int \frac{x^6}{1+c x^2} \, dx\\ &=\frac{32 b^2 x}{75 c^2}+\frac{2 a b x^3}{15 c}-\frac{2}{25} a b x^5+\frac{4 b^2 x^5}{125}+\frac{2 a b \tan ^{-1}\left (\sqrt{c} x\right )}{5 c^{5/2}}-\frac{46 b^2 \tan ^{-1}\left (\sqrt{c} x\right )}{75 c^{5/2}}+\frac{i b^2 \tan ^{-1}\left (\sqrt{c} x\right )^2}{5 c^{5/2}}-\frac{46 b^2 \tanh ^{-1}\left (\sqrt{c} x\right )}{75 c^{5/2}}-\frac{b^2 \tanh ^{-1}\left (\sqrt{c} x\right )^2}{5 c^{5/2}}+\frac{2 b^2 \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1-\sqrt{c} x}\right )}{5 c^{5/2}}+\frac{2 b^2 \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1+i \sqrt{c} x}\right )}{5 c^{5/2}}-\frac{b^2 x^3 \log \left (1-c x^2\right )}{15 c}+\frac{1}{25} b^2 x^5 \log \left (1-c x^2\right )-\frac{b^2 \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1-c x^2\right )}{5 c^{5/2}}+\frac{b x^3 \left (2 a-b \log \left (1-c x^2\right )\right )}{15 c}+\frac{1}{25} b x^5 \left (2 a-b \log \left (1-c x^2\right )\right )-\frac{b \tanh ^{-1}\left (\sqrt{c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )}{5 c^{5/2}}+\frac{1}{20} x^5 \left (2 a-b \log \left (1-c x^2\right )\right )^2+\frac{2 b^2 x^3 \log \left (1+c x^2\right )}{15 c}+\frac{1}{5} a b x^5 \log \left (1+c x^2\right )+\frac{b^2 \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1+c x^2\right )}{5 c^{5/2}}-\frac{b^2 \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (1+c x^2\right )}{5 c^{5/2}}-\frac{1}{10} b^2 x^5 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac{1}{20} b^2 x^5 \log ^2\left (1+c x^2\right )-\frac{1}{15} \left (2 b^2\right ) \int \left (-\frac{1}{c^2}-\frac{x^2}{c}+\frac{1}{c^2 \left (1-c x^2\right )}\right ) \, dx-\frac{1}{15} \left (2 b^2\right ) \int \left (-\frac{1}{c^2}+\frac{x^2}{c}+\frac{1}{c^2 \left (1+c x^2\right )}\right ) \, dx+\frac{\left (2 i b^2\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i \sqrt{c} x}\right )}{5 c^{5/2}}+\frac{\left (2 b^2\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-\sqrt{c} x}\right )}{5 c^{5/2}}+\frac{\left (2 b^2\right ) \int \frac{1}{1-c x^2} \, dx}{5 c^2}+\frac{\left (2 b^2\right ) \int \frac{1}{1+c x^2} \, dx}{5 c^2}-\frac{\left (2 b^2\right ) \int \frac{x \tan ^{-1}\left (\sqrt{c} x\right )}{1-c x^2} \, dx}{5 c^{3/2}}+\frac{\left (2 b^2\right ) \int \frac{x \tanh ^{-1}\left (\sqrt{c} x\right )}{1+c x^2} \, dx}{5 c^{3/2}}+\frac{1}{25} \left (2 b^2 c\right ) \int \left (-\frac{1}{c^3}-\frac{x^2}{c^2}-\frac{x^4}{c}+\frac{1}{c^3 \left (1-c x^2\right )}\right ) \, dx-\frac{1}{25} \left (2 b^2 c\right ) \int \left (\frac{1}{c^3}-\frac{x^2}{c^2}+\frac{x^4}{c}-\frac{1}{c^3 \left (1+c x^2\right )}\right ) \, dx\\ &=\frac{8 b^2 x}{15 c^2}+\frac{2 a b x^3}{15 c}-\frac{2}{25} a b x^5+\frac{2 a b \tan ^{-1}\left (\sqrt{c} x\right )}{5 c^{5/2}}-\frac{16 b^2 \tan ^{-1}\left (\sqrt{c} x\right )}{75 c^{5/2}}+\frac{i b^2 \tan ^{-1}\left (\sqrt{c} x\right )^2}{5 c^{5/2}}-\frac{16 b^2 \tanh ^{-1}\left (\sqrt{c} x\right )}{75 c^{5/2}}-\frac{b^2 \tanh ^{-1}\left (\sqrt{c} x\right )^2}{5 c^{5/2}}+\frac{2 b^2 \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1-\sqrt{c} x}\right )}{5 c^{5/2}}+\frac{2 b^2 \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1+i \sqrt{c} x}\right )}{5 c^{5/2}}-\frac{b^2 x^3 \log \left (1-c x^2\right )}{15 c}+\frac{1}{25} b^2 x^5 \log \left (1-c x^2\right )-\frac{b^2 \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1-c x^2\right )}{5 c^{5/2}}+\frac{b x^3 \left (2 a-b \log \left (1-c x^2\right )\right )}{15 c}+\frac{1}{25} b x^5 \left (2 a-b \log \left (1-c x^2\right )\right )-\frac{b \tanh ^{-1}\left (\sqrt{c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )}{5 c^{5/2}}+\frac{1}{20} x^5 \left (2 a-b \log \left (1-c x^2\right )\right )^2+\frac{2 b^2 x^3 \log \left (1+c x^2\right )}{15 c}+\frac{1}{5} a b x^5 \log \left (1+c x^2\right )+\frac{b^2 \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1+c x^2\right )}{5 c^{5/2}}-\frac{b^2 \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (1+c x^2\right )}{5 c^{5/2}}-\frac{1}{10} b^2 x^5 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac{1}{20} b^2 x^5 \log ^2\left (1+c x^2\right )+\frac{b^2 \text{Li}_2\left (1-\frac{2}{1-\sqrt{c} x}\right )}{5 c^{5/2}}+\frac{i b^2 \text{Li}_2\left (1-\frac{2}{1+i \sqrt{c} x}\right )}{5 c^{5/2}}+\frac{\left (2 b^2\right ) \int \frac{1}{1-c x^2} \, dx}{25 c^2}+\frac{\left (2 b^2\right ) \int \frac{1}{1+c x^2} \, dx}{25 c^2}-\frac{\left (2 b^2\right ) \int \frac{1}{1-c x^2} \, dx}{15 c^2}-\frac{\left (2 b^2\right ) \int \frac{1}{1+c x^2} \, dx}{15 c^2}-\frac{\left (2 b^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}{5 c^{3/2}}+\frac{\left (2 b^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}{5 c^{3/2}}\\ &=\frac{8 b^2 x}{15 c^2}+\frac{2 a b x^3}{15 c}-\frac{2}{25} a b x^5+\frac{2 a b \tan ^{-1}\left (\sqrt{c} x\right )}{5 c^{5/2}}-\frac{4 b^2 \tan ^{-1}\left (\sqrt{c} x\right )}{15 c^{5/2}}+\frac{i b^2 \tan ^{-1}\left (\sqrt{c} x\right )^2}{5 c^{5/2}}-\frac{4 b^2 \tanh ^{-1}\left (\sqrt{c} x\right )}{15 c^{5/2}}-\frac{b^2 \tanh ^{-1}\left (\sqrt{c} x\right )^2}{5 c^{5/2}}+\frac{2 b^2 \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1-\sqrt{c} x}\right )}{5 c^{5/2}}+\frac{2 b^2 \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1+i \sqrt{c} x}\right )}{5 c^{5/2}}-\frac{b^2 x^3 \log \left (1-c x^2\right )}{15 c}+\frac{1}{25} b^2 x^5 \log \left (1-c x^2\right )-\frac{b^2 \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1-c x^2\right )}{5 c^{5/2}}+\frac{b x^3 \left (2 a-b \log \left (1-c x^2\right )\right )}{15 c}+\frac{1}{25} b x^5 \left (2 a-b \log \left (1-c x^2\right )\right )-\frac{b \tanh ^{-1}\left (\sqrt{c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )}{5 c^{5/2}}+\frac{1}{20} x^5 \left (2 a-b \log \left (1-c x^2\right )\right )^2+\frac{2 b^2 x^3 \log \left (1+c x^2\right )}{15 c}+\frac{1}{5} a b x^5 \log \left (1+c x^2\right )+\frac{b^2 \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1+c x^2\right )}{5 c^{5/2}}-\frac{b^2 \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (1+c x^2\right )}{5 c^{5/2}}-\frac{1}{10} b^2 x^5 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac{1}{20} b^2 x^5 \log ^2\left (1+c x^2\right )+\frac{b^2 \text{Li}_2\left (1-\frac{2}{1-\sqrt{c} x}\right )}{5 c^{5/2}}+\frac{i b^2 \text{Li}_2\left (1-\frac{2}{1+i \sqrt{c} x}\right )}{5 c^{5/2}}-\frac{b^2 \int \frac{\tan ^{-1}\left (\sqrt{c} x\right )}{1-\sqrt{c} x} \, dx}{5 c^2}+\frac{b^2 \int \frac{\tan ^{-1}\left (\sqrt{c} x\right )}{1+\sqrt{c} x} \, dx}{5 c^2}+\frac{b^2 \int \frac{\tanh ^{-1}\left (\sqrt{c} x\right )}{1-\sqrt{-c} x} \, dx}{5 \sqrt{-c} c^{3/2}}-\frac{b^2 \int \frac{\tanh ^{-1}\left (\sqrt{c} x\right )}{1+\sqrt{-c} x} \, dx}{5 \sqrt{-c} c^{3/2}}\\ &=\frac{8 b^2 x}{15 c^2}+\frac{2 a b x^3}{15 c}-\frac{2}{25} a b x^5+\frac{2 a b \tan ^{-1}\left (\sqrt{c} x\right )}{5 c^{5/2}}-\frac{4 b^2 \tan ^{-1}\left (\sqrt{c} x\right )}{15 c^{5/2}}+\frac{i b^2 \tan ^{-1}\left (\sqrt{c} x\right )^2}{5 c^{5/2}}-\frac{4 b^2 \tanh ^{-1}\left (\sqrt{c} x\right )}{15 c^{5/2}}-\frac{b^2 \tanh ^{-1}\left (\sqrt{c} x\right )^2}{5 c^{5/2}}+\frac{2 b^2 \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1-\sqrt{c} x}\right )}{5 c^{5/2}}-\frac{2 b^2 \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1-i \sqrt{c} x}\right )}{5 c^{5/2}}+\frac{b^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 )}{5 c^{5/2}}+\frac{2 b^2 \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1+i \sqrt{c} x}\right )}{5 c^{5/2}}-\frac{2 b^2 \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1+\sqrt{c} x}\right )}{5 c^{5/2}}+\frac{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 (1+\sqrt{c} x\right )}\right )}{5 c^{5/2}}+\frac{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 (1+\sqrt{c} x\right )}\right )}{5 c^{5/2}}+\frac{b^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 )}{5 c^{5/2}}-\frac{b^2 x^3 \log \left (1-c x^2\right )}{15 c}+\frac{1}{25} b^2 x^5 \log \left (1-c x^2\right )-\frac{b^2 \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1-c x^2\right )}{5 c^{5/2}}+\frac{b x^3 \left (2 a-b \log \left (1-c x^2\right )\right )}{15 c}+\frac{1}{25} b x^5 \left (2 a-b \log \left (1-c x^2\right )\right )-\frac{b \tanh ^{-1}\left (\sqrt{c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )}{5 c^{5/2}}+\frac{1}{20} x^5 \left (2 a-b \log \left (1-c x^2\right )\right )^2+\frac{2 b^2 x^3 \log \left (1+c x^2\right )}{15 c}+\frac{1}{5} a b x^5 \log \left (1+c x^2\right )+\frac{b^2 \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1+c x^2\right )}{5 c^{5/2}}-\frac{b^2 \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (1+c x^2\right )}{5 c^{5/2}}-\frac{1}{10} b^2 x^5 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac{1}{20} b^2 x^5 \log ^2\left (1+c x^2\right )+\frac{b^2 \text{Li}_2\left (1-\frac{2}{1-\sqrt{c} x}\right )}{5 c^{5/2}}+\frac{i b^2 \text{Li}_2\left (1-\frac{2}{1+i \sqrt{c} x}\right )}{5 c^{5/2}}+2 \frac{b^2 \int \frac{\log \left (\frac{2}{1-i \sqrt{c} x}\right )}{1+c x^2} \, dx}{5 c^2}-\frac{b^2 \int \frac{\log \left (\frac{(1+i) \left (1-\sqrt{c} x\right )}{1-i \sqrt{c} x}\right )}{1+c x^2} \, dx}{5 c^2}+2 \frac{b^2 \int \frac{\log \left (\frac{2}{1+\sqrt{c} x}\right )}{1-c x^2} \, dx}{5 c^2}-\frac{b^2 \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}{5 c^2}-\frac{b^2 \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}{5 c^2}-\frac{b^2 \int \frac{\log \left (\frac{(1-i) \left (1+\sqrt{c} x\right )}{1-i \sqrt{c} x}\right )}{1+c x^2} \, dx}{5 c^2}\\ &=\frac{8 b^2 x}{15 c^2}+\frac{2 a b x^3}{15 c}-\frac{2}{25} a b x^5+\frac{2 a b \tan ^{-1}\left (\sqrt{c} x\right )}{5 c^{5/2}}-\frac{4 b^2 \tan ^{-1}\left (\sqrt{c} x\right )}{15 c^{5/2}}+\frac{i b^2 \tan ^{-1}\left (\sqrt{c} x\right )^2}{5 c^{5/2}}-\frac{4 b^2 \tanh ^{-1}\left (\sqrt{c} x\right )}{15 c^{5/2}}-\frac{b^2 \tanh ^{-1}\left (\sqrt{c} x\right )^2}{5 c^{5/2}}+\frac{2 b^2 \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1-\sqrt{c} x}\right )}{5 c^{5/2}}-\frac{2 b^2 \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1-i \sqrt{c} x}\right )}{5 c^{5/2}}+\frac{b^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 )}{5 c^{5/2}}+\frac{2 b^2 \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1+i \sqrt{c} x}\right )}{5 c^{5/2}}-\frac{2 b^2 \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1+\sqrt{c} x}\right )}{5 c^{5/2}}+\frac{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 (1+\sqrt{c} x\right )}\right )}{5 c^{5/2}}+\frac{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 (1+\sqrt{c} x\right )}\right )}{5 c^{5/2}}+\frac{b^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 )}{5 c^{5/2}}-\frac{b^2 x^3 \log \left (1-c x^2\right )}{15 c}+\frac{1}{25} b^2 x^5 \log \left (1-c x^2\right )-\frac{b^2 \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1-c x^2\right )}{5 c^{5/2}}+\frac{b x^3 \left (2 a-b \log \left (1-c x^2\right )\right )}{15 c}+\frac{1}{25} b x^5 \left (2 a-b \log \left (1-c x^2\right )\right )-\frac{b \tanh ^{-1}\left (\sqrt{c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )}{5 c^{5/2}}+\frac{1}{20} x^5 \left (2 a-b \log \left (1-c x^2\right )\right )^2+\frac{2 b^2 x^3 \log \left (1+c x^2\right )}{15 c}+\frac{1}{5} a b x^5 \log \left (1+c x^2\right )+\frac{b^2 \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1+c x^2\right )}{5 c^{5/2}}-\frac{b^2 \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (1+c x^2\right )}{5 c^{5/2}}-\frac{1}{10} b^2 x^5 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac{1}{20} b^2 x^5 \log ^2\left (1+c x^2\right )+\frac{b^2 \text{Li}_2\left (1-\frac{2}{1-\sqrt{c} x}\right )}{5 c^{5/2}}-\frac{i b^2 \text{Li}_2\left (1-\frac{(1+i) \left (1-\sqrt{c} x\right )}{1-i \sqrt{c} x}\right )}{10 c^{5/2}}+\frac{i b^2 \text{Li}_2\left (1-\frac{2}{1+i \sqrt{c} x}\right )}{5 c^{5/2}}-\frac{b^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 )}{10 c^{5/2}}-\frac{b^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 )}{10 c^{5/2}}-\frac{i b^2 \text{Li}_2\left (1-\frac{(1-i) \left (1+\sqrt{c} x\right )}{1-i \sqrt{c} x}\right )}{10 c^{5/2}}+2 \frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-i \sqrt{c} x}\right )}{5 c^{5/2}}+2 \frac{b^2 \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+\sqrt{c} x}\right )}{5 c^{5/2}}\\ &=\frac{8 b^2 x}{15 c^2}+\frac{2 a b x^3}{15 c}-\frac{2}{25} a b x^5+\frac{2 a b \tan ^{-1}\left (\sqrt{c} x\right )}{5 c^{5/2}}-\frac{4 b^2 \tan ^{-1}\left (\sqrt{c} x\right )}{15 c^{5/2}}+\frac{i b^2 \tan ^{-1}\left (\sqrt{c} x\right )^2}{5 c^{5/2}}-\frac{4 b^2 \tanh ^{-1}\left (\sqrt{c} x\right )}{15 c^{5/2}}-\frac{b^2 \tanh ^{-1}\left (\sqrt{c} x\right )^2}{5 c^{5/2}}+\frac{2 b^2 \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1-\sqrt{c} x}\right )}{5 c^{5/2}}-\frac{2 b^2 \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1-i \sqrt{c} x}\right )}{5 c^{5/2}}+\frac{b^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 )}{5 c^{5/2}}+\frac{2 b^2 \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1+i \sqrt{c} x}\right )}{5 c^{5/2}}-\frac{2 b^2 \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1+\sqrt{c} x}\right )}{5 c^{5/2}}+\frac{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 (1+\sqrt{c} x\right )}\right )}{5 c^{5/2}}+\frac{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 (1+\sqrt{c} x\right )}\right )}{5 c^{5/2}}+\frac{b^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 )}{5 c^{5/2}}-\frac{b^2 x^3 \log \left (1-c x^2\right )}{15 c}+\frac{1}{25} b^2 x^5 \log \left (1-c x^2\right )-\frac{b^2 \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1-c x^2\right )}{5 c^{5/2}}+\frac{b x^3 \left (2 a-b \log \left (1-c x^2\right )\right )}{15 c}+\frac{1}{25} b x^5 \left (2 a-b \log \left (1-c x^2\right )\right )-\frac{b \tanh ^{-1}\left (\sqrt{c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )}{5 c^{5/2}}+\frac{1}{20} x^5 \left (2 a-b \log \left (1-c x^2\right )\right )^2+\frac{2 b^2 x^3 \log \left (1+c x^2\right )}{15 c}+\frac{1}{5} a b x^5 \log \left (1+c x^2\right )+\frac{b^2 \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1+c x^2\right )}{5 c^{5/2}}-\frac{b^2 \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (1+c x^2\right )}{5 c^{5/2}}-\frac{1}{10} b^2 x^5 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac{1}{20} b^2 x^5 \log ^2\left (1+c x^2\right )+\frac{b^2 \text{Li}_2\left (1-\frac{2}{1-\sqrt{c} x}\right )}{5 c^{5/2}}+\frac{i b^2 \text{Li}_2\left (1-\frac{2}{1-i \sqrt{c} x}\right )}{5 c^{5/2}}-\frac{i b^2 \text{Li}_2\left (1-\frac{(1+i) \left (1-\sqrt{c} x\right )}{1-i \sqrt{c} x}\right )}{10 c^{5/2}}+\frac{i b^2 \text{Li}_2\left (1-\frac{2}{1+i \sqrt{c} x}\right )}{5 c^{5/2}}+\frac{b^2 \text{Li}_2\left (1-\frac{2}{1+\sqrt{c} x}\right )}{5 c^{5/2}}-\frac{b^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 )}{10 c^{5/2}}-\frac{b^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 )}{10 c^{5/2}}-\frac{i b^2 \text{Li}_2\left (1-\frac{(1-i) \left (1+\sqrt{c} x\right )}{1-i \sqrt{c} x}\right )}{10 c^{5/2}}\\ \end{align*}

Mathematica [F]  time = 9.30766, size = 0, normalized size = 0. \[ \int x^4 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2 \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[x^4*(a + b*ArcTanh[c*x^2])^2,x]

[Out]

Integrate[x^4*(a + b*ArcTanh[c*x^2])^2, x]

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Maple [F]  time = 0.216, size = 0, normalized size = 0. \begin{align*} \int{x}^{4} \left ( a+b{\it Artanh} \left ( c{x}^{2} \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a+b*arctanh(c*x^2))^2,x)

[Out]

int(x^4*(a+b*arctanh(c*x^2))^2,x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*arctanh(c*x^2))^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (b^{2} x^{4} \operatorname{artanh}\left (c x^{2}\right )^{2} + 2 \, a b x^{4} \operatorname{artanh}\left (c x^{2}\right ) + a^{2} x^{4}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*arctanh(c*x^2))^2,x, algorithm="fricas")

[Out]

integral(b^2*x^4*arctanh(c*x^2)^2 + 2*a*b*x^4*arctanh(c*x^2) + a^2*x^4, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{4} \left (a + b \operatorname{atanh}{\left (c x^{2} \right )}\right )^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(a+b*atanh(c*x**2))**2,x)

[Out]

Integral(x**4*(a + b*atanh(c*x**2))**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{artanh}\left (c x^{2}\right ) + a\right )}^{2} x^{4}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*arctanh(c*x^2))^2,x, algorithm="giac")

[Out]

integrate((b*arctanh(c*x^2) + a)^2*x^4, x)