3.1.73 \(\int (a+b \tanh ^{-1}(c x^2))^2 \, dx\) [73]

Optimal. Leaf size=958 \[ a^2 x+\frac {2 a b \text {ArcTan}\left (\sqrt {c} x\right )}{\sqrt {c}}+\frac {i b^2 \text {ArcTan}\left (\sqrt {c} x\right )^2}{\sqrt {c}}-\frac {2 a b \tanh ^{-1}\left (\sqrt {c} x\right )}{\sqrt {c}}-\frac {b^2 \tanh ^{-1}\left (\sqrt {c} x\right )^2}{\sqrt {c}}+\frac {2 b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt {c} x}\right )}{\sqrt {c}}-\frac {2 b^2 \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-i \sqrt {c} x}\right )}{\sqrt {c}}+\frac {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 )}{\sqrt {c}}+\frac {2 b^2 \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+i \sqrt {c} x}\right )}{\sqrt {c}}-\frac {2 b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+\sqrt {c} x}\right )}{\sqrt {c}}+\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 )}{\sqrt {c}}+\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 )}{\sqrt {c}}+\frac {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 )}{\sqrt {c}}-a b x \log \left (1-c x^2\right )-\frac {b^2 \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )}{\sqrt {c}}+\frac {b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )}{\sqrt {c}}+\frac {1}{4} b^2 x \log ^2\left (1-c x^2\right )+a b x \log \left (1+c x^2\right )+\frac {b^2 \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )}{\sqrt {c}}-\frac {b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )}{\sqrt {c}}-\frac {1}{2} b^2 x \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac {1}{4} b^2 x \log ^2\left (1+c x^2\right )+\frac {b^2 \text {PolyLog}\left (2,1-\frac {2}{1-\sqrt {c} x}\right )}{\sqrt {c}}+\frac {i b^2 \text {PolyLog}\left (2,1-\frac {2}{1-i \sqrt {c} x}\right )}{\sqrt {c}}-\frac {i b^2 \text {PolyLog}\left (2,1-\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )}{2 \sqrt {c}}+\frac {i b^2 \text {PolyLog}\left (2,1-\frac {2}{1+i \sqrt {c} x}\right )}{\sqrt {c}}+\frac {b^2 \text {PolyLog}\left (2,1-\frac {2}{1+\sqrt {c} x}\right )}{\sqrt {c}}-\frac {b^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 )}{2 \sqrt {c}}-\frac {b^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 )}{2 \sqrt {c}}-\frac {i b^2 \text {PolyLog}\left (2,1-\frac {(1-i) \left (1+\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )}{2 \sqrt {c}} \]

[Out]

2*b^2*arctanh(x*c^(1/2))*ln(2/(1-x*c^(1/2)))/c^(1/2)-2*b^2*arctan(x*c^(1/2))*ln(2/(1-I*x*c^(1/2)))/c^(1/2)+2*b
^2*arctan(x*c^(1/2))*ln(2/(1+I*x*c^(1/2)))/c^(1/2)-2*b^2*arctanh(x*c^(1/2))*ln(2/(1+x*c^(1/2)))/c^(1/2)-1/2*I*
b^2*polylog(2,1-(1+I)*(1-x*c^(1/2))/(1-I*x*c^(1/2)))/c^(1/2)-1/2*I*b^2*polylog(2,1+(-1+I)*(1+x*c^(1/2))/(1-I*x
*c^(1/2)))/c^(1/2)-a*b*x*ln(-c*x^2+1)+I*b^2*arctan(x*c^(1/2))^2/c^(1/2)+a*b*x*ln(c*x^2+1)-b^2*arctan(x*c^(1/2)
)*ln(-c*x^2+1)/c^(1/2)+b^2*arctanh(x*c^(1/2))*ln(-c*x^2+1)/c^(1/2)+b^2*arctan(x*c^(1/2))*ln(c*x^2+1)/c^(1/2)-b
^2*arctanh(x*c^(1/2))*ln(c*x^2+1)/c^(1/2)+b^2*arctan(x*c^(1/2))*ln((1+I)*(1-x*c^(1/2))/(1-I*x*c^(1/2)))/c^(1/2
)+b^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)))/c^(1/2)+b^2*arctan
h(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)))/c^(1/2)+b^2*arctan(x*c^(1/2))*l
n((1-I)*(1+x*c^(1/2))/(1-I*x*c^(1/2)))/c^(1/2)+I*b^2*polylog(2,1-2/(1-I*x*c^(1/2)))/c^(1/2)+I*b^2*polylog(2,1-
2/(1+I*x*c^(1/2)))/c^(1/2)-1/2*b^2*x*ln(-c*x^2+1)*ln(c*x^2+1)+2*a*b*arctan(x*c^(1/2))/c^(1/2)-2*a*b*arctanh(x*
c^(1/2))/c^(1/2)+1/4*b^2*x*ln(-c*x^2+1)^2+1/4*b^2*x*ln(c*x^2+1)^2-1/2*b^2*polylog(2,1+2*(1-x*(-c)^(1/2))*c^(1/
2)/((-c)^(1/2)-c^(1/2))/(1+x*c^(1/2)))/c^(1/2)-1/2*b^2*polylog(2,1-2*(1+x*(-c)^(1/2))*c^(1/2)/((-c)^(1/2)+c^(1
/2))/(1+x*c^(1/2)))/c^(1/2)-b^2*arctanh(x*c^(1/2))^2/c^(1/2)+b^2*polylog(2,1-2/(1-x*c^(1/2)))/c^(1/2)+b^2*poly
log(2,1-2/(1+x*c^(1/2)))/c^(1/2)+a^2*x

________________________________________________________________________________________

Rubi [A]
time = 0.99, antiderivative size = 958, normalized size of antiderivative = 1.00, number of steps used = 69, number of rules used = 21, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.750, Rules used = {6023, 2498, 327, 212, 2500, 2526, 2520, 12, 6131, 6055, 2449, 2352, 209, 2636, 6139, 6057, 2497, 5048, 4966, 5040, 4964} \begin {gather*} x a^2+\frac {2 b \text {ArcTan}\left (\sqrt {c} x\right ) a}{\sqrt {c}}-\frac {2 b \tanh ^{-1}\left (\sqrt {c} x\right ) a}{\sqrt {c}}-b x \log \left (1-c x^2\right ) a+b x \log \left (c x^2+1\right ) a+\frac {i b^2 \text {ArcTan}\left (\sqrt {c} x\right )^2}{\sqrt {c}}-\frac {b^2 \tanh ^{-1}\left (\sqrt {c} x\right )^2}{\sqrt {c}}+\frac {1}{4} b^2 x \log ^2\left (1-c x^2\right )+\frac {1}{4} b^2 x \log ^2\left (c x^2+1\right )+\frac {2 b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt {c} x}\right )}{\sqrt {c}}-\frac {2 b^2 \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-i \sqrt {c} x}\right )}{\sqrt {c}}+\frac {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 )}{\sqrt {c}}+\frac {2 b^2 \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (\frac {2}{i \sqrt {c} x+1}\right )}{\sqrt {c}}-\frac {2 b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{\sqrt {c} x+1}\right )}{\sqrt {c}}+\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 (\sqrt {c} x+1\right )}\right )}{\sqrt {c}}+\frac {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 )}{\sqrt {c}}+\frac {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 )}{\sqrt {c}}-\frac {b^2 \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )}{\sqrt {c}}+\frac {b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )}{\sqrt {c}}+\frac {b^2 \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (c x^2+1\right )}{\sqrt {c}}-\frac {b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (c x^2+1\right )}{\sqrt {c}}-\frac {1}{2} b^2 x \log \left (1-c x^2\right ) \log \left (c x^2+1\right )+\frac {b^2 \text {Li}_2\left (1-\frac {2}{1-\sqrt {c} x}\right )}{\sqrt {c}}+\frac {i b^2 \text {Li}_2\left (1-\frac {2}{1-i \sqrt {c} x}\right )}{\sqrt {c}}-\frac {i b^2 \text {Li}_2\left (1-\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )}{2 \sqrt {c}}+\frac {i b^2 \text {Li}_2\left (1-\frac {2}{i \sqrt {c} x+1}\right )}{\sqrt {c}}+\frac {b^2 \text {Li}_2\left (1-\frac {2}{\sqrt {c} x+1}\right )}{\sqrt {c}}-\frac {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 )}{2 \sqrt {c}}-\frac {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 )}{2 \sqrt {c}}-\frac {i b^2 \text {Li}_2\left (1-\frac {(1-i) \left (\sqrt {c} x+1\right )}{1-i \sqrt {c} x}\right )}{2 \sqrt {c}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTanh[c*x^2])^2,x]

[Out]

a^2*x + (2*a*b*ArcTan[Sqrt[c]*x])/Sqrt[c] + (I*b^2*ArcTan[Sqrt[c]*x]^2)/Sqrt[c] - (2*a*b*ArcTanh[Sqrt[c]*x])/S
qrt[c] - (b^2*ArcTanh[Sqrt[c]*x]^2)/Sqrt[c] + (2*b^2*ArcTanh[Sqrt[c]*x]*Log[2/(1 - Sqrt[c]*x)])/Sqrt[c] - (2*b
^2*ArcTan[Sqrt[c]*x]*Log[2/(1 - I*Sqrt[c]*x)])/Sqrt[c] + (b^2*ArcTan[Sqrt[c]*x]*Log[((1 + I)*(1 - Sqrt[c]*x))/
(1 - I*Sqrt[c]*x)])/Sqrt[c] + (2*b^2*ArcTan[Sqrt[c]*x]*Log[2/(1 + I*Sqrt[c]*x)])/Sqrt[c] - (2*b^2*ArcTanh[Sqrt
[c]*x]*Log[2/(1 + Sqrt[c]*x)])/Sqrt[c] + (b^2*ArcTanh[Sqrt[c]*x]*Log[(-2*Sqrt[c]*(1 - Sqrt[-c]*x))/((Sqrt[-c]
- Sqrt[c])*(1 + Sqrt[c]*x))])/Sqrt[c] + (b^2*ArcTanh[Sqrt[c]*x]*Log[(2*Sqrt[c]*(1 + Sqrt[-c]*x))/((Sqrt[-c] +
Sqrt[c])*(1 + Sqrt[c]*x))])/Sqrt[c] + (b^2*ArcTan[Sqrt[c]*x]*Log[((1 - I)*(1 + Sqrt[c]*x))/(1 - I*Sqrt[c]*x)])
/Sqrt[c] - a*b*x*Log[1 - c*x^2] - (b^2*ArcTan[Sqrt[c]*x]*Log[1 - c*x^2])/Sqrt[c] + (b^2*ArcTanh[Sqrt[c]*x]*Log
[1 - c*x^2])/Sqrt[c] + (b^2*x*Log[1 - c*x^2]^2)/4 + a*b*x*Log[1 + c*x^2] + (b^2*ArcTan[Sqrt[c]*x]*Log[1 + c*x^
2])/Sqrt[c] - (b^2*ArcTanh[Sqrt[c]*x]*Log[1 + c*x^2])/Sqrt[c] - (b^2*x*Log[1 - c*x^2]*Log[1 + c*x^2])/2 + (b^2
*x*Log[1 + c*x^2]^2)/4 + (b^2*PolyLog[2, 1 - 2/(1 - Sqrt[c]*x)])/Sqrt[c] + (I*b^2*PolyLog[2, 1 - 2/(1 - I*Sqrt
[c]*x)])/Sqrt[c] - ((I/2)*b^2*PolyLog[2, 1 - ((1 + I)*(1 - Sqrt[c]*x))/(1 - I*Sqrt[c]*x)])/Sqrt[c] + (I*b^2*Po
lyLog[2, 1 - 2/(1 + I*Sqrt[c]*x)])/Sqrt[c] + (b^2*PolyLog[2, 1 - 2/(1 + Sqrt[c]*x)])/Sqrt[c] - (b^2*PolyLog[2,
1 + (2*Sqrt[c]*(1 - Sqrt[-c]*x))/((Sqrt[-c] - Sqrt[c])*(1 + Sqrt[c]*x))])/(2*Sqrt[c]) - (b^2*PolyLog[2, 1 - (
2*Sqrt[c]*(1 + Sqrt[-c]*x))/((Sqrt[-c] + Sqrt[c])*(1 + Sqrt[c]*x))])/(2*Sqrt[c]) - ((I/2)*b^2*PolyLog[2, 1 - (
(1 - I)*(1 + Sqrt[c]*x))/(1 - I*Sqrt[c]*x)])/Sqrt[c]

Rule 12

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

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 327

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 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 2498

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 2500

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

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 2636

Int[Log[v_]*Log[w_], x_Symbol] :> Simp[x*Log[v]*Log[w], x] + (-Int[SimplifyIntegrand[x*Log[w]*(D[v, x]/v), x],
x] - Int[SimplifyIntegrand[x*Log[v]*(D[w, x]/w), x], x]) /; InverseFunctionFreeQ[v, x] && InverseFunctionFree
Q[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 6023

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_)]*(b_.))^(p_), x_Symbol] :> Int[ExpandIntegrand[(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]

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])

Rubi steps

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

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Mathematica [A]
time = 1.79, size = 566, normalized size = 0.59 \begin {gather*} \frac {1}{2} x \left (2 a^2+4 a b \tanh ^{-1}\left (c x^2\right )+\frac {4 a b \left (\text {ArcTan}\left (\sqrt {c x^2}\right )-\tanh ^{-1}\left (\sqrt {c x^2}\right )\right )}{\sqrt {c x^2}}+\frac {b^2 \left (-2 i \text {ArcTan}\left (\sqrt {c x^2}\right )^2+4 \text {ArcTan}\left (\sqrt {c x^2}\right ) \tanh ^{-1}\left (c x^2\right )+2 \sqrt {c x^2} \tanh ^{-1}\left (c x^2\right )^2+2 \text {ArcTan}\left (\sqrt {c x^2}\right ) \log \left (1+e^{4 i \text {ArcTan}\left (\sqrt {c x^2}\right )}\right )+2 \tanh ^{-1}\left (c x^2\right ) \log \left (1-\sqrt {c x^2}\right )-\log (2) \log \left (1-\sqrt {c x^2}\right )+\frac {1}{2} \log ^2\left (1-\sqrt {c x^2}\right )-\log \left (1-\sqrt {c x^2}\right ) \log \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-i+\sqrt {c x^2}\right )\right )-2 \tanh ^{-1}\left (c x^2\right ) \log \left (1+\sqrt {c x^2}\right )+\log (2) \log \left (1+\sqrt {c x^2}\right )+\log \left (\frac {1}{2} \left ((1+i)-(1-i) \sqrt {c x^2}\right )\right ) \log \left (1+\sqrt {c x^2}\right )+\log \left (\left (-\frac {1}{2}-\frac {i}{2}\right ) \left (i+\sqrt {c x^2}\right )\right ) \log \left (1+\sqrt {c x^2}\right )-\frac {1}{2} \log ^2\left (1+\sqrt {c x^2}\right )-\log \left (1-\sqrt {c x^2}\right ) \log \left (\frac {1}{2} \left ((1+i)+(1-i) \sqrt {c x^2}\right )\right )-\frac {1}{2} i \text {PolyLog}\left (2,-e^{4 i \text {ArcTan}\left (\sqrt {c x^2}\right )}\right )+\text {PolyLog}\left (2,\frac {1}{2} \left (1-\sqrt {c x^2}\right )\right )-\text {PolyLog}\left (2,\left (-\frac {1}{2}-\frac {i}{2}\right ) \left (-1+\sqrt {c x^2}\right )\right )-\text {PolyLog}\left (2,\left (-\frac {1}{2}+\frac {i}{2}\right ) \left (-1+\sqrt {c x^2}\right )\right )-\text {PolyLog}\left (2,\frac {1}{2} \left (1+\sqrt {c x^2}\right )\right )+\text {PolyLog}\left (2,\left (\frac {1}{2}-\frac {i}{2}\right ) \left (1+\sqrt {c x^2}\right )\right )+\text {PolyLog}\left (2,\left (\frac {1}{2}+\frac {i}{2}\right ) \left (1+\sqrt {c x^2}\right )\right )\right )}{\sqrt {c x^2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcTanh[c*x^2])^2,x]

[Out]

(x*(2*a^2 + 4*a*b*ArcTanh[c*x^2] + (4*a*b*(ArcTan[Sqrt[c*x^2]] - ArcTanh[Sqrt[c*x^2]]))/Sqrt[c*x^2] + (b^2*((-
2*I)*ArcTan[Sqrt[c*x^2]]^2 + 4*ArcTan[Sqrt[c*x^2]]*ArcTanh[c*x^2] + 2*Sqrt[c*x^2]*ArcTanh[c*x^2]^2 + 2*ArcTan[
Sqrt[c*x^2]]*Log[1 + E^((4*I)*ArcTan[Sqrt[c*x^2]])] + 2*ArcTanh[c*x^2]*Log[1 - Sqrt[c*x^2]] - Log[2]*Log[1 - S
qrt[c*x^2]] + Log[1 - Sqrt[c*x^2]]^2/2 - Log[1 - Sqrt[c*x^2]]*Log[(1/2 + I/2)*(-I + Sqrt[c*x^2])] - 2*ArcTanh[
c*x^2]*Log[1 + Sqrt[c*x^2]] + Log[2]*Log[1 + Sqrt[c*x^2]] + Log[((1 + I) - (1 - I)*Sqrt[c*x^2])/2]*Log[1 + Sqr
t[c*x^2]] + Log[(-1/2 - I/2)*(I + Sqrt[c*x^2])]*Log[1 + Sqrt[c*x^2]] - Log[1 + Sqrt[c*x^2]]^2/2 - Log[1 - Sqrt
[c*x^2]]*Log[((1 + I) + (1 - I)*Sqrt[c*x^2])/2] - (I/2)*PolyLog[2, -E^((4*I)*ArcTan[Sqrt[c*x^2]])] + PolyLog[2
, (1 - Sqrt[c*x^2])/2] - PolyLog[2, (-1/2 - I/2)*(-1 + Sqrt[c*x^2])] - PolyLog[2, (-1/2 + I/2)*(-1 + Sqrt[c*x^
2])] - PolyLog[2, (1 + Sqrt[c*x^2])/2] + PolyLog[2, (1/2 - I/2)*(1 + Sqrt[c*x^2])] + PolyLog[2, (1/2 + I/2)*(1
+ Sqrt[c*x^2])]))/Sqrt[c*x^2]))/2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

(c*(2*arctan(sqrt(c)*x)/c^(3/2) + log((c*x - sqrt(c))/(c*x + sqrt(c)))/c^(3/2)) + 2*x*arctanh(c*x^2))*a*b + 1/
4*(x*log(-c*x^2 + 1)^2 - integrate(-((c*x^2 - 1)*log(c*x^2 + 1)^2 - 2*(2*c*x^2 + (c*x^2 - 1)*log(c*x^2 + 1))*l
og(-c*x^2 + 1))/(c*x^2 - 1), x))*b^2 + a^2*x

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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, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atanh(c*x^2))^2,x)

[Out]

int((a + b*atanh(c*x^2))^2, x)

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