3.6.4 \(\int \frac {\tanh ^{-1}(a x)}{(c+d x^2)^3} \, dx\) [504]

Optimal. Leaf size=657 \[ \frac {a}{8 c \left (a^2 c+d\right ) \left (c+d x^2\right )}+\frac {x \tanh ^{-1}(a x)}{4 c \left (c+d x^2\right )^2}+\frac {3 x \tanh ^{-1}(a x)}{8 c^2 \left (c+d x^2\right )}+\frac {3 \text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \tanh ^{-1}(a x)}{8 c^{5/2} \sqrt {d}}+\frac {3 i \log \left (\frac {\sqrt {d} (1-a x)}{i a \sqrt {c}+\sqrt {d}}\right ) \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{32 c^{5/2} \sqrt {d}}-\frac {3 i \log \left (-\frac {\sqrt {d} (1+a x)}{i a \sqrt {c}-\sqrt {d}}\right ) \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{32 c^{5/2} \sqrt {d}}-\frac {3 i \log \left (-\frac {\sqrt {d} (1-a x)}{i a \sqrt {c}-\sqrt {d}}\right ) \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{32 c^{5/2} \sqrt {d}}+\frac {3 i \log \left (\frac {\sqrt {d} (1+a x)}{i a \sqrt {c}+\sqrt {d}}\right ) \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{32 c^{5/2} \sqrt {d}}+\frac {a \left (5 a^2 c+3 d\right ) \log \left (1-a^2 x^2\right )}{16 c^2 \left (a^2 c+d\right )^2}-\frac {a \left (5 a^2 c+3 d\right ) \log \left (c+d x^2\right )}{16 c^2 \left (a^2 c+d\right )^2}+\frac {3 i \text {PolyLog}\left (2,\frac {a \left (\sqrt {c}-i \sqrt {d} x\right )}{a \sqrt {c}-i \sqrt {d}}\right )}{32 c^{5/2} \sqrt {d}}-\frac {3 i \text {PolyLog}\left (2,\frac {a \left (\sqrt {c}-i \sqrt {d} x\right )}{a \sqrt {c}+i \sqrt {d}}\right )}{32 c^{5/2} \sqrt {d}}+\frac {3 i \text {PolyLog}\left (2,\frac {a \left (\sqrt {c}+i \sqrt {d} x\right )}{a \sqrt {c}-i \sqrt {d}}\right )}{32 c^{5/2} \sqrt {d}}-\frac {3 i \text {PolyLog}\left (2,\frac {a \left (\sqrt {c}+i \sqrt {d} x\right )}{a \sqrt {c}+i \sqrt {d}}\right )}{32 c^{5/2} \sqrt {d}} \]

[Out]

1/8*a/c/(a^2*c+d)/(d*x^2+c)+1/4*x*arctanh(a*x)/c/(d*x^2+c)^2+3/8*x*arctanh(a*x)/c^2/(d*x^2+c)+1/16*a*(5*a^2*c+
3*d)*ln(-a^2*x^2+1)/c^2/(a^2*c+d)^2-1/16*a*(5*a^2*c+3*d)*ln(d*x^2+c)/c^2/(a^2*c+d)^2+3/8*arctan(x*d^(1/2)/c^(1
/2))*arctanh(a*x)/c^(5/2)/d^(1/2)-3/32*I*ln(-(a*x+1)*d^(1/2)/(I*a*c^(1/2)-d^(1/2)))*ln(1-I*x*d^(1/2)/c^(1/2))/
c^(5/2)/d^(1/2)+3/32*I*ln((-a*x+1)*d^(1/2)/(I*a*c^(1/2)+d^(1/2)))*ln(1-I*x*d^(1/2)/c^(1/2))/c^(5/2)/d^(1/2)-3/
32*I*ln(-(-a*x+1)*d^(1/2)/(I*a*c^(1/2)-d^(1/2)))*ln(1+I*x*d^(1/2)/c^(1/2))/c^(5/2)/d^(1/2)+3/32*I*ln((a*x+1)*d
^(1/2)/(I*a*c^(1/2)+d^(1/2)))*ln(1+I*x*d^(1/2)/c^(1/2))/c^(5/2)/d^(1/2)+3/32*I*polylog(2,a*(c^(1/2)-I*x*d^(1/2
))/(a*c^(1/2)-I*d^(1/2)))/c^(5/2)/d^(1/2)-3/32*I*polylog(2,a*(c^(1/2)-I*x*d^(1/2))/(a*c^(1/2)+I*d^(1/2)))/c^(5
/2)/d^(1/2)+3/32*I*polylog(2,a*(c^(1/2)+I*x*d^(1/2))/(a*c^(1/2)-I*d^(1/2)))/c^(5/2)/d^(1/2)-3/32*I*polylog(2,a
*(c^(1/2)+I*x*d^(1/2))/(a*c^(1/2)+I*d^(1/2)))/c^(5/2)/d^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.70, antiderivative size = 657, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 11, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.786, Rules used = {205, 211, 6123, 6857, 585, 78, 5028, 2456, 2441, 2440, 2438} \begin {gather*} \frac {a \left (5 a^2 c+3 d\right ) \log \left (1-a^2 x^2\right )}{16 c^2 \left (a^2 c+d\right )^2}-\frac {a \left (5 a^2 c+3 d\right ) \log \left (c+d x^2\right )}{16 c^2 \left (a^2 c+d\right )^2}+\frac {a}{8 c \left (a^2 c+d\right ) \left (c+d x^2\right )}+\frac {3 \tanh ^{-1}(a x) \text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{5/2} \sqrt {d}}+\frac {3 i \text {Li}_2\left (\frac {a \left (\sqrt {c}-i \sqrt {d} x\right )}{a \sqrt {c}-i \sqrt {d}}\right )}{32 c^{5/2} \sqrt {d}}-\frac {3 i \text {Li}_2\left (\frac {a \left (\sqrt {c}-i \sqrt {d} x\right )}{\sqrt {c} a+i \sqrt {d}}\right )}{32 c^{5/2} \sqrt {d}}+\frac {3 i \text {Li}_2\left (\frac {a \left (i \sqrt {d} x+\sqrt {c}\right )}{a \sqrt {c}-i \sqrt {d}}\right )}{32 c^{5/2} \sqrt {d}}-\frac {3 i \text {Li}_2\left (\frac {a \left (i \sqrt {d} x+\sqrt {c}\right )}{\sqrt {c} a+i \sqrt {d}}\right )}{32 c^{5/2} \sqrt {d}}+\frac {3 i \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {\sqrt {d} (1-a x)}{\sqrt {d}+i a \sqrt {c}}\right )}{32 c^{5/2} \sqrt {d}}-\frac {3 i \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right ) \log \left (-\frac {\sqrt {d} (a x+1)}{-\sqrt {d}+i a \sqrt {c}}\right )}{32 c^{5/2} \sqrt {d}}-\frac {3 i \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right ) \log \left (-\frac {\sqrt {d} (1-a x)}{-\sqrt {d}+i a \sqrt {c}}\right )}{32 c^{5/2} \sqrt {d}}+\frac {3 i \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {\sqrt {d} (a x+1)}{\sqrt {d}+i a \sqrt {c}}\right )}{32 c^{5/2} \sqrt {d}}+\frac {3 x \tanh ^{-1}(a x)}{8 c^2 \left (c+d x^2\right )}+\frac {x \tanh ^{-1}(a x)}{4 c \left (c+d x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcTanh[a*x]/(c + d*x^2)^3,x]

[Out]

a/(8*c*(a^2*c + d)*(c + d*x^2)) + (x*ArcTanh[a*x])/(4*c*(c + d*x^2)^2) + (3*x*ArcTanh[a*x])/(8*c^2*(c + d*x^2)
) + (3*ArcTan[(Sqrt[d]*x)/Sqrt[c]]*ArcTanh[a*x])/(8*c^(5/2)*Sqrt[d]) + (((3*I)/32)*Log[(Sqrt[d]*(1 - a*x))/(I*
a*Sqrt[c] + Sqrt[d])]*Log[1 - (I*Sqrt[d]*x)/Sqrt[c]])/(c^(5/2)*Sqrt[d]) - (((3*I)/32)*Log[-((Sqrt[d]*(1 + a*x)
)/(I*a*Sqrt[c] - Sqrt[d]))]*Log[1 - (I*Sqrt[d]*x)/Sqrt[c]])/(c^(5/2)*Sqrt[d]) - (((3*I)/32)*Log[-((Sqrt[d]*(1
- a*x))/(I*a*Sqrt[c] - Sqrt[d]))]*Log[1 + (I*Sqrt[d]*x)/Sqrt[c]])/(c^(5/2)*Sqrt[d]) + (((3*I)/32)*Log[(Sqrt[d]
*(1 + a*x))/(I*a*Sqrt[c] + Sqrt[d])]*Log[1 + (I*Sqrt[d]*x)/Sqrt[c]])/(c^(5/2)*Sqrt[d]) + (a*(5*a^2*c + 3*d)*Lo
g[1 - a^2*x^2])/(16*c^2*(a^2*c + d)^2) - (a*(5*a^2*c + 3*d)*Log[c + d*x^2])/(16*c^2*(a^2*c + d)^2) + (((3*I)/3
2)*PolyLog[2, (a*(Sqrt[c] - I*Sqrt[d]*x))/(a*Sqrt[c] - I*Sqrt[d])])/(c^(5/2)*Sqrt[d]) - (((3*I)/32)*PolyLog[2,
(a*(Sqrt[c] - I*Sqrt[d]*x))/(a*Sqrt[c] + I*Sqrt[d])])/(c^(5/2)*Sqrt[d]) + (((3*I)/32)*PolyLog[2, (a*(Sqrt[c]
+ I*Sqrt[d]*x))/(a*Sqrt[c] - I*Sqrt[d])])/(c^(5/2)*Sqrt[d]) - (((3*I)/32)*PolyLog[2, (a*(Sqrt[c] + I*Sqrt[d]*x
))/(a*Sqrt[c] + I*Sqrt[d])])/(c^(5/2)*Sqrt[d])

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 205

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

Rule 211

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

Rule 585

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_))^(r_.), x
_Symbol] :> Dist[1/n, Subst[Int[(a + b*x)^p*(c + d*x)^q*(e + f*x)^r, x], x, x^n], x] /; FreeQ[{a, b, c, d, e,
f, m, n, p, q, r}, x] && EqQ[m - n + 1, 0]

Rule 2438

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

Rule 2440

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + c*e*(x/g)])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2441

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

Rule 2456

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_)^(r_))^(q_.), x_Symbol] :> In
t[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (f + g*x^r)^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, r}, x]
&& IGtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[r] && NeQ[r, 1]))

Rule 5028

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

Rule 6123

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> With[{u = IntHide[(d + e*x^
2)^q, x]}, Dist[a + b*ArcTanh[c*x], u, x] - Dist[b*c, Int[u/(1 - c^2*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x
] && (IntegerQ[q] || ILtQ[q + 1/2, 0])

Rule 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {\tanh ^{-1}(a x)}{\left (c+d x^2\right )^3} \, dx &=\frac {x \tanh ^{-1}(a x)}{4 c \left (c+d x^2\right )^2}+\frac {3 x \tanh ^{-1}(a x)}{8 c^2 \left (c+d x^2\right )}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \tanh ^{-1}(a x)}{8 c^{5/2} \sqrt {d}}-a \int \frac {\frac {x}{4 c \left (c+d x^2\right )^2}+\frac {3 x}{8 c^2 \left (c+d x^2\right )}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{5/2} \sqrt {d}}}{1-a^2 x^2} \, dx\\ &=\frac {x \tanh ^{-1}(a x)}{4 c \left (c+d x^2\right )^2}+\frac {3 x \tanh ^{-1}(a x)}{8 c^2 \left (c+d x^2\right )}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \tanh ^{-1}(a x)}{8 c^{5/2} \sqrt {d}}-a \int \left (-\frac {x \left (5 c+3 d x^2\right )}{8 c^2 \left (-1+a^2 x^2\right ) \left (c+d x^2\right )^2}-\frac {3 \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{5/2} \sqrt {d} \left (-1+a^2 x^2\right )}\right ) \, dx\\ &=\frac {x \tanh ^{-1}(a x)}{4 c \left (c+d x^2\right )^2}+\frac {3 x \tanh ^{-1}(a x)}{8 c^2 \left (c+d x^2\right )}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \tanh ^{-1}(a x)}{8 c^{5/2} \sqrt {d}}+\frac {a \int \frac {x \left (5 c+3 d x^2\right )}{\left (-1+a^2 x^2\right ) \left (c+d x^2\right )^2} \, dx}{8 c^2}+\frac {(3 a) \int \frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{-1+a^2 x^2} \, dx}{8 c^{5/2} \sqrt {d}}\\ &=\frac {x \tanh ^{-1}(a x)}{4 c \left (c+d x^2\right )^2}+\frac {3 x \tanh ^{-1}(a x)}{8 c^2 \left (c+d x^2\right )}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \tanh ^{-1}(a x)}{8 c^{5/2} \sqrt {d}}+\frac {a \text {Subst}\left (\int \frac {5 c+3 d x}{\left (-1+a^2 x\right ) (c+d x)^2} \, dx,x,x^2\right )}{16 c^2}+\frac {(3 i a) \int \frac {\log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{-1+a^2 x^2} \, dx}{16 c^{5/2} \sqrt {d}}-\frac {(3 i a) \int \frac {\log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{-1+a^2 x^2} \, dx}{16 c^{5/2} \sqrt {d}}\\ &=\frac {x \tanh ^{-1}(a x)}{4 c \left (c+d x^2\right )^2}+\frac {3 x \tanh ^{-1}(a x)}{8 c^2 \left (c+d x^2\right )}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \tanh ^{-1}(a x)}{8 c^{5/2} \sqrt {d}}+\frac {a \text {Subst}\left (\int \left (\frac {a^2 \left (5 a^2 c+3 d\right )}{\left (a^2 c+d\right )^2 \left (-1+a^2 x\right )}-\frac {2 c d}{\left (a^2 c+d\right ) (c+d x)^2}-\frac {d \left (5 a^2 c+3 d\right )}{\left (a^2 c+d\right )^2 (c+d x)}\right ) \, dx,x,x^2\right )}{16 c^2}+\frac {(3 i a) \int \left (-\frac {\log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{2 (1-a x)}-\frac {\log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{2 (1+a x)}\right ) \, dx}{16 c^{5/2} \sqrt {d}}-\frac {(3 i a) \int \left (-\frac {\log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{2 (1-a x)}-\frac {\log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{2 (1+a x)}\right ) \, dx}{16 c^{5/2} \sqrt {d}}\\ &=\frac {a}{8 c \left (a^2 c+d\right ) \left (c+d x^2\right )}+\frac {x \tanh ^{-1}(a x)}{4 c \left (c+d x^2\right )^2}+\frac {3 x \tanh ^{-1}(a x)}{8 c^2 \left (c+d x^2\right )}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \tanh ^{-1}(a x)}{8 c^{5/2} \sqrt {d}}+\frac {a \left (5 a^2 c+3 d\right ) \log \left (1-a^2 x^2\right )}{16 c^2 \left (a^2 c+d\right )^2}-\frac {a \left (5 a^2 c+3 d\right ) \log \left (c+d x^2\right )}{16 c^2 \left (a^2 c+d\right )^2}-\frac {(3 i a) \int \frac {\log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{1-a x} \, dx}{32 c^{5/2} \sqrt {d}}-\frac {(3 i a) \int \frac {\log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{1+a x} \, dx}{32 c^{5/2} \sqrt {d}}+\frac {(3 i a) \int \frac {\log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{1-a x} \, dx}{32 c^{5/2} \sqrt {d}}+\frac {(3 i a) \int \frac {\log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{1+a x} \, dx}{32 c^{5/2} \sqrt {d}}\\ &=\frac {a}{8 c \left (a^2 c+d\right ) \left (c+d x^2\right )}+\frac {x \tanh ^{-1}(a x)}{4 c \left (c+d x^2\right )^2}+\frac {3 x \tanh ^{-1}(a x)}{8 c^2 \left (c+d x^2\right )}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \tanh ^{-1}(a x)}{8 c^{5/2} \sqrt {d}}+\frac {3 i \log \left (\frac {\sqrt {d} (1-a x)}{i a \sqrt {c}+\sqrt {d}}\right ) \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{32 c^{5/2} \sqrt {d}}-\frac {3 i \log \left (-\frac {\sqrt {d} (1+a x)}{i a \sqrt {c}-\sqrt {d}}\right ) \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{32 c^{5/2} \sqrt {d}}-\frac {3 i \log \left (-\frac {\sqrt {d} (1-a x)}{i a \sqrt {c}-\sqrt {d}}\right ) \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{32 c^{5/2} \sqrt {d}}+\frac {3 i \log \left (\frac {\sqrt {d} (1+a x)}{i a \sqrt {c}+\sqrt {d}}\right ) \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{32 c^{5/2} \sqrt {d}}+\frac {a \left (5 a^2 c+3 d\right ) \log \left (1-a^2 x^2\right )}{16 c^2 \left (a^2 c+d\right )^2}-\frac {a \left (5 a^2 c+3 d\right ) \log \left (c+d x^2\right )}{16 c^2 \left (a^2 c+d\right )^2}-\frac {3 \int \frac {\log \left (-\frac {i \sqrt {d} (1-a x)}{\sqrt {c} \left (a-\frac {i \sqrt {d}}{\sqrt {c}}\right )}\right )}{1-\frac {i \sqrt {d} x}{\sqrt {c}}} \, dx}{32 c^3}-\frac {3 \int \frac {\log \left (\frac {i \sqrt {d} (1-a x)}{\sqrt {c} \left (a+\frac {i \sqrt {d}}{\sqrt {c}}\right )}\right )}{1+\frac {i \sqrt {d} x}{\sqrt {c}}} \, dx}{32 c^3}+\frac {3 \int \frac {\log \left (-\frac {i \sqrt {d} (1+a x)}{\sqrt {c} \left (-a-\frac {i \sqrt {d}}{\sqrt {c}}\right )}\right )}{1-\frac {i \sqrt {d} x}{\sqrt {c}}} \, dx}{32 c^3}+\frac {3 \int \frac {\log \left (\frac {i \sqrt {d} (1+a x)}{\sqrt {c} \left (-a+\frac {i \sqrt {d}}{\sqrt {c}}\right )}\right )}{1+\frac {i \sqrt {d} x}{\sqrt {c}}} \, dx}{32 c^3}\\ &=\frac {a}{8 c \left (a^2 c+d\right ) \left (c+d x^2\right )}+\frac {x \tanh ^{-1}(a x)}{4 c \left (c+d x^2\right )^2}+\frac {3 x \tanh ^{-1}(a x)}{8 c^2 \left (c+d x^2\right )}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \tanh ^{-1}(a x)}{8 c^{5/2} \sqrt {d}}+\frac {3 i \log \left (\frac {\sqrt {d} (1-a x)}{i a \sqrt {c}+\sqrt {d}}\right ) \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{32 c^{5/2} \sqrt {d}}-\frac {3 i \log \left (-\frac {\sqrt {d} (1+a x)}{i a \sqrt {c}-\sqrt {d}}\right ) \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{32 c^{5/2} \sqrt {d}}-\frac {3 i \log \left (-\frac {\sqrt {d} (1-a x)}{i a \sqrt {c}-\sqrt {d}}\right ) \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{32 c^{5/2} \sqrt {d}}+\frac {3 i \log \left (\frac {\sqrt {d} (1+a x)}{i a \sqrt {c}+\sqrt {d}}\right ) \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{32 c^{5/2} \sqrt {d}}+\frac {a \left (5 a^2 c+3 d\right ) \log \left (1-a^2 x^2\right )}{16 c^2 \left (a^2 c+d\right )^2}-\frac {a \left (5 a^2 c+3 d\right ) \log \left (c+d x^2\right )}{16 c^2 \left (a^2 c+d\right )^2}+\frac {(3 i) \text {Subst}\left (\int \frac {\log \left (1+\frac {a x}{-a-\frac {i \sqrt {d}}{\sqrt {c}}}\right )}{x} \, dx,x,1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{32 c^{5/2} \sqrt {d}}-\frac {(3 i) \text {Subst}\left (\int \frac {\log \left (1-\frac {a x}{a-\frac {i \sqrt {d}}{\sqrt {c}}}\right )}{x} \, dx,x,1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{32 c^{5/2} \sqrt {d}}-\frac {(3 i) \text {Subst}\left (\int \frac {\log \left (1+\frac {a x}{-a+\frac {i \sqrt {d}}{\sqrt {c}}}\right )}{x} \, dx,x,1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{32 c^{5/2} \sqrt {d}}+\frac {(3 i) \text {Subst}\left (\int \frac {\log \left (1-\frac {a x}{a+\frac {i \sqrt {d}}{\sqrt {c}}}\right )}{x} \, dx,x,1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{32 c^{5/2} \sqrt {d}}\\ &=\frac {a}{8 c \left (a^2 c+d\right ) \left (c+d x^2\right )}+\frac {x \tanh ^{-1}(a x)}{4 c \left (c+d x^2\right )^2}+\frac {3 x \tanh ^{-1}(a x)}{8 c^2 \left (c+d x^2\right )}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \tanh ^{-1}(a x)}{8 c^{5/2} \sqrt {d}}+\frac {3 i \log \left (\frac {\sqrt {d} (1-a x)}{i a \sqrt {c}+\sqrt {d}}\right ) \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{32 c^{5/2} \sqrt {d}}-\frac {3 i \log \left (-\frac {\sqrt {d} (1+a x)}{i a \sqrt {c}-\sqrt {d}}\right ) \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{32 c^{5/2} \sqrt {d}}-\frac {3 i \log \left (-\frac {\sqrt {d} (1-a x)}{i a \sqrt {c}-\sqrt {d}}\right ) \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{32 c^{5/2} \sqrt {d}}+\frac {3 i \log \left (\frac {\sqrt {d} (1+a x)}{i a \sqrt {c}+\sqrt {d}}\right ) \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{32 c^{5/2} \sqrt {d}}+\frac {a \left (5 a^2 c+3 d\right ) \log \left (1-a^2 x^2\right )}{16 c^2 \left (a^2 c+d\right )^2}-\frac {a \left (5 a^2 c+3 d\right ) \log \left (c+d x^2\right )}{16 c^2 \left (a^2 c+d\right )^2}+\frac {3 i \text {Li}_2\left (\frac {a \left (\sqrt {c}-i \sqrt {d} x\right )}{a \sqrt {c}-i \sqrt {d}}\right )}{32 c^{5/2} \sqrt {d}}-\frac {3 i \text {Li}_2\left (\frac {a \left (\sqrt {c}-i \sqrt {d} x\right )}{a \sqrt {c}+i \sqrt {d}}\right )}{32 c^{5/2} \sqrt {d}}+\frac {3 i \text {Li}_2\left (\frac {a \left (\sqrt {c}+i \sqrt {d} x\right )}{a \sqrt {c}-i \sqrt {d}}\right )}{32 c^{5/2} \sqrt {d}}-\frac {3 i \text {Li}_2\left (\frac {a \left (\sqrt {c}+i \sqrt {d} x\right )}{a \sqrt {c}+i \sqrt {d}}\right )}{32 c^{5/2} \sqrt {d}}\\ \end {align*}

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1541\) vs. \(2(657)=1314\).
time = 9.22, size = 1541, normalized size = 2.35 \begin {gather*} \frac {a \left (-10 a^2 c \log \left (1+\frac {\left (a^2 c+d\right ) \cosh \left (2 \tanh ^{-1}(a x)\right )}{a^2 c-d}\right )-6 d \log \left (1+\frac {\left (a^2 c+d\right ) \cosh \left (2 \tanh ^{-1}(a x)\right )}{a^2 c-d}\right )-\frac {3 d \left (a^2 c+d\right ) \left (-2 i \text {ArcCos}\left (\frac {-a^2 c+d}{a^2 c+d}\right ) \text {ArcTan}\left (\frac {a d x}{\sqrt {a^2 c d}}\right )+4 \text {ArcTan}\left (\frac {a c}{\sqrt {a^2 c d} x}\right ) \tanh ^{-1}(a x)-\left (\text {ArcCos}\left (\frac {-a^2 c+d}{a^2 c+d}\right )+2 \text {ArcTan}\left (\frac {a d x}{\sqrt {a^2 c d}}\right )\right ) \log \left (\frac {2 i a c \left (i d+\sqrt {a^2 c d}\right ) (-1+a x)}{\left (a^2 c+d\right ) \left (a c+i \sqrt {a^2 c d} x\right )}\right )-\left (\text {ArcCos}\left (\frac {-a^2 c+d}{a^2 c+d}\right )-2 \text {ArcTan}\left (\frac {a d x}{\sqrt {a^2 c d}}\right )\right ) \log \left (\frac {2 a c \left (d+i \sqrt {a^2 c d}\right ) (1+a x)}{\left (a^2 c+d\right ) \left (a c+i \sqrt {a^2 c d} x\right )}\right )+\left (\text {ArcCos}\left (\frac {-a^2 c+d}{a^2 c+d}\right )+2 \left (\text {ArcTan}\left (\frac {a c}{\sqrt {a^2 c d} x}\right )+\text {ArcTan}\left (\frac {a d x}{\sqrt {a^2 c d}}\right )\right )\right ) \log \left (\frac {\sqrt {2} \sqrt {a^2 c d} e^{-\tanh ^{-1}(a x)}}{\sqrt {a^2 c+d} \sqrt {a^2 c-d+\left (a^2 c+d\right ) \cosh \left (2 \tanh ^{-1}(a x)\right )}}\right )+\left (\text {ArcCos}\left (\frac {-a^2 c+d}{a^2 c+d}\right )-2 \left (\text {ArcTan}\left (\frac {a c}{\sqrt {a^2 c d} x}\right )+\text {ArcTan}\left (\frac {a d x}{\sqrt {a^2 c d}}\right )\right )\right ) \log \left (\frac {\sqrt {2} \sqrt {a^2 c d} e^{\tanh ^{-1}(a x)}}{\sqrt {a^2 c+d} \sqrt {a^2 c-d+\left (a^2 c+d\right ) \cosh \left (2 \tanh ^{-1}(a x)\right )}}\right )+i \left (-\text {PolyLog}\left (2,\frac {\left (-a^2 c+d-2 i \sqrt {a^2 c d}\right ) \left (i a c+\sqrt {a^2 c d} x\right )}{\left (a^2 c+d\right ) \left (-i a c+\sqrt {a^2 c d} x\right )}\right )+\text {PolyLog}\left (2,\frac {\left (-a^2 c+d+2 i \sqrt {a^2 c d}\right ) \left (i a c+\sqrt {a^2 c d} x\right )}{\left (a^2 c+d\right ) \left (-i a c+\sqrt {a^2 c d} x\right )}\right )\right )\right )}{\sqrt {a^2 c d}}-\frac {3 \sqrt {a^2 c d} \left (a^2 c+d\right ) \left (-2 i \text {ArcCos}\left (\frac {-a^2 c+d}{a^2 c+d}\right ) \text {ArcTan}\left (\frac {a d x}{\sqrt {a^2 c d}}\right )+4 \text {ArcTan}\left (\frac {a c}{\sqrt {a^2 c d} x}\right ) \tanh ^{-1}(a x)-\left (\text {ArcCos}\left (\frac {-a^2 c+d}{a^2 c+d}\right )+2 \text {ArcTan}\left (\frac {a d x}{\sqrt {a^2 c d}}\right )\right ) \log \left (\frac {2 i a c \left (i d+\sqrt {a^2 c d}\right ) (-1+a x)}{\left (a^2 c+d\right ) \left (a c+i \sqrt {a^2 c d} x\right )}\right )-\left (\text {ArcCos}\left (\frac {-a^2 c+d}{a^2 c+d}\right )-2 \text {ArcTan}\left (\frac {a d x}{\sqrt {a^2 c d}}\right )\right ) \log \left (\frac {2 a c \left (d+i \sqrt {a^2 c d}\right ) (1+a x)}{\left (a^2 c+d\right ) \left (a c+i \sqrt {a^2 c d} x\right )}\right )+\left (\text {ArcCos}\left (\frac {-a^2 c+d}{a^2 c+d}\right )+2 \left (\text {ArcTan}\left (\frac {a c}{\sqrt {a^2 c d} x}\right )+\text {ArcTan}\left (\frac {a d x}{\sqrt {a^2 c d}}\right )\right )\right ) \log \left (\frac {\sqrt {2} \sqrt {a^2 c d} e^{-\tanh ^{-1}(a x)}}{\sqrt {a^2 c+d} \sqrt {a^2 c-d+\left (a^2 c+d\right ) \cosh \left (2 \tanh ^{-1}(a x)\right )}}\right )+\left (\text {ArcCos}\left (\frac {-a^2 c+d}{a^2 c+d}\right )-2 \left (\text {ArcTan}\left (\frac {a c}{\sqrt {a^2 c d} x}\right )+\text {ArcTan}\left (\frac {a d x}{\sqrt {a^2 c d}}\right )\right )\right ) \log \left (\frac {\sqrt {2} \sqrt {a^2 c d} e^{\tanh ^{-1}(a x)}}{\sqrt {a^2 c+d} \sqrt {a^2 c-d+\left (a^2 c+d\right ) \cosh \left (2 \tanh ^{-1}(a x)\right )}}\right )+i \left (-\text {PolyLog}\left (2,\frac {\left (-a^2 c+d-2 i \sqrt {a^2 c d}\right ) \left (i a c+\sqrt {a^2 c d} x\right )}{\left (a^2 c+d\right ) \left (-i a c+\sqrt {a^2 c d} x\right )}\right )+\text {PolyLog}\left (2,\frac {\left (-a^2 c+d+2 i \sqrt {a^2 c d}\right ) \left (i a c+\sqrt {a^2 c d} x\right )}{\left (a^2 c+d\right ) \left (-i a c+\sqrt {a^2 c d} x\right )}\right )\right )\right )}{d}+\frac {16 a^2 c d \left (a^2 c+d\right ) \tanh ^{-1}(a x) \sinh \left (2 \tanh ^{-1}(a x)\right )}{\left (a^2 c-d+\left (a^2 c+d\right ) \cosh \left (2 \tanh ^{-1}(a x)\right )\right )^2}+\frac {8 a^2 c d+4 \left (5 a^4 c^2+8 a^2 c d+3 d^2\right ) \tanh ^{-1}(a x) \sinh \left (2 \tanh ^{-1}(a x)\right )}{a^2 c-d+\left (a^2 c+d\right ) \cosh \left (2 \tanh ^{-1}(a x)\right )}\right )}{32 c^2 \left (a^2 c+d\right )^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcTanh[a*x]/(c + d*x^2)^3,x]

[Out]

(a*(-10*a^2*c*Log[1 + ((a^2*c + d)*Cosh[2*ArcTanh[a*x]])/(a^2*c - d)] - 6*d*Log[1 + ((a^2*c + d)*Cosh[2*ArcTan
h[a*x]])/(a^2*c - d)] - (3*d*(a^2*c + d)*((-2*I)*ArcCos[(-(a^2*c) + d)/(a^2*c + d)]*ArcTan[(a*d*x)/Sqrt[a^2*c*
d]] + 4*ArcTan[(a*c)/(Sqrt[a^2*c*d]*x)]*ArcTanh[a*x] - (ArcCos[(-(a^2*c) + d)/(a^2*c + d)] + 2*ArcTan[(a*d*x)/
Sqrt[a^2*c*d]])*Log[((2*I)*a*c*(I*d + Sqrt[a^2*c*d])*(-1 + a*x))/((a^2*c + d)*(a*c + I*Sqrt[a^2*c*d]*x))] - (A
rcCos[(-(a^2*c) + d)/(a^2*c + d)] - 2*ArcTan[(a*d*x)/Sqrt[a^2*c*d]])*Log[(2*a*c*(d + I*Sqrt[a^2*c*d])*(1 + a*x
))/((a^2*c + d)*(a*c + I*Sqrt[a^2*c*d]*x))] + (ArcCos[(-(a^2*c) + d)/(a^2*c + d)] + 2*(ArcTan[(a*c)/(Sqrt[a^2*
c*d]*x)] + ArcTan[(a*d*x)/Sqrt[a^2*c*d]]))*Log[(Sqrt[2]*Sqrt[a^2*c*d])/(Sqrt[a^2*c + d]*E^ArcTanh[a*x]*Sqrt[a^
2*c - d + (a^2*c + d)*Cosh[2*ArcTanh[a*x]]])] + (ArcCos[(-(a^2*c) + d)/(a^2*c + d)] - 2*(ArcTan[(a*c)/(Sqrt[a^
2*c*d]*x)] + ArcTan[(a*d*x)/Sqrt[a^2*c*d]]))*Log[(Sqrt[2]*Sqrt[a^2*c*d]*E^ArcTanh[a*x])/(Sqrt[a^2*c + d]*Sqrt[
a^2*c - d + (a^2*c + d)*Cosh[2*ArcTanh[a*x]]])] + I*(-PolyLog[2, ((-(a^2*c) + d - (2*I)*Sqrt[a^2*c*d])*(I*a*c
+ Sqrt[a^2*c*d]*x))/((a^2*c + d)*((-I)*a*c + Sqrt[a^2*c*d]*x))] + PolyLog[2, ((-(a^2*c) + d + (2*I)*Sqrt[a^2*c
*d])*(I*a*c + Sqrt[a^2*c*d]*x))/((a^2*c + d)*((-I)*a*c + Sqrt[a^2*c*d]*x))])))/Sqrt[a^2*c*d] - (3*Sqrt[a^2*c*d
]*(a^2*c + d)*((-2*I)*ArcCos[(-(a^2*c) + d)/(a^2*c + d)]*ArcTan[(a*d*x)/Sqrt[a^2*c*d]] + 4*ArcTan[(a*c)/(Sqrt[
a^2*c*d]*x)]*ArcTanh[a*x] - (ArcCos[(-(a^2*c) + d)/(a^2*c + d)] + 2*ArcTan[(a*d*x)/Sqrt[a^2*c*d]])*Log[((2*I)*
a*c*(I*d + Sqrt[a^2*c*d])*(-1 + a*x))/((a^2*c + d)*(a*c + I*Sqrt[a^2*c*d]*x))] - (ArcCos[(-(a^2*c) + d)/(a^2*c
+ d)] - 2*ArcTan[(a*d*x)/Sqrt[a^2*c*d]])*Log[(2*a*c*(d + I*Sqrt[a^2*c*d])*(1 + a*x))/((a^2*c + d)*(a*c + I*Sq
rt[a^2*c*d]*x))] + (ArcCos[(-(a^2*c) + d)/(a^2*c + d)] + 2*(ArcTan[(a*c)/(Sqrt[a^2*c*d]*x)] + ArcTan[(a*d*x)/S
qrt[a^2*c*d]]))*Log[(Sqrt[2]*Sqrt[a^2*c*d])/(Sqrt[a^2*c + d]*E^ArcTanh[a*x]*Sqrt[a^2*c - d + (a^2*c + d)*Cosh[
2*ArcTanh[a*x]]])] + (ArcCos[(-(a^2*c) + d)/(a^2*c + d)] - 2*(ArcTan[(a*c)/(Sqrt[a^2*c*d]*x)] + ArcTan[(a*d*x)
/Sqrt[a^2*c*d]]))*Log[(Sqrt[2]*Sqrt[a^2*c*d]*E^ArcTanh[a*x])/(Sqrt[a^2*c + d]*Sqrt[a^2*c - d + (a^2*c + d)*Cos
h[2*ArcTanh[a*x]]])] + I*(-PolyLog[2, ((-(a^2*c) + d - (2*I)*Sqrt[a^2*c*d])*(I*a*c + Sqrt[a^2*c*d]*x))/((a^2*c
+ d)*((-I)*a*c + Sqrt[a^2*c*d]*x))] + PolyLog[2, ((-(a^2*c) + d + (2*I)*Sqrt[a^2*c*d])*(I*a*c + Sqrt[a^2*c*d]
*x))/((a^2*c + d)*((-I)*a*c + Sqrt[a^2*c*d]*x))])))/d + (16*a^2*c*d*(a^2*c + d)*ArcTanh[a*x]*Sinh[2*ArcTanh[a*
x]])/(a^2*c - d + (a^2*c + d)*Cosh[2*ArcTanh[a*x]])^2 + (8*a^2*c*d + 4*(5*a^4*c^2 + 8*a^2*c*d + 3*d^2)*ArcTanh
[a*x]*Sinh[2*ArcTanh[a*x]])/(a^2*c - d + (a^2*c + d)*Cosh[2*ArcTanh[a*x]])))/(32*c^2*(a^2*c + d)^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(4046\) vs. \(2(493)=986\).
time = 4.70, size = 4047, normalized size = 6.16

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(4047\)
default	\(\text {Expression too large to display}\)	\(4047\)
risch	\(\text {Expression too large to display}\)	\(4564\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctanh(a*x)/(d*x^2+c)^3,x,method=_RETURNVERBOSE)

[Out]

1/a*(-3/16*((-a^2*d*c)^(1/2)*a^2*c+2*a^2*d*c-(-a^2*d*c)^(1/2)*d)/c^2/(a^4*c^2+2*a^2*c*d+d^2)^2*a^2*d*ln(1-(a^2
*c+d)*(a*x+1)^2/(-a^2*x^2+1)/(-a^2*c-2*(-a^2*d*c)^(1/2)+d))*arctanh(a*x)+5/16*(d*c)^(1/2)/c^2*d*a^3*arctan(1/4
*(2*(a^2*c+d)*(a*x+1)^2/(-a^2*x^2+1)+2*a^2*c-2*d)/a/(d*c)^(1/2))/(a^2*c+d)/(a^4*c^2+2*a^2*c*d+d^2)-3/8*(a^2*c-
2*(-a^2*d*c)^(1/2)-d)/c^2/(a^4*c^2+2*a^2*c*d+d^2)^2*a^2*ln(1-(a^2*c+d)*(a*x+1)^2/(-a^2*x^2+1)/(-a^2*c-2*(-a^2*
d*c)^(1/2)+d))*d^2*arctanh(a*x)+3/16*(d*c)^(1/2)*d^2/c^3*a*arctan(1/4*(2*(a^2*c+d)*(a*x+1)^2/(-a^2*x^2+1)+2*a^
2*c-2*d)/a/(d*c)^(1/2))/(a^2*c+d)/(a^4*c^2+2*a^2*c*d+d^2)-3/4*(a^2*c-2*(-a^2*d*c)^(1/2)-d)/c/(a^4*c^2+2*a^2*c*
d+d^2)^2*a^4*d*ln(1-(a^2*c+d)*(a*x+1)^2/(-a^2*x^2+1)/(-a^2*c-2*(-a^2*d*c)^(1/2)+d))*arctanh(a*x)-3/16*(-a^2*d*
c)^(1/2)/c*a^4/d/(a^4*c^2+2*a^2*c*d+d^2)*arctanh(a*x)*ln(1-(a^2*c+d)*(a*x+1)^2/(-a^2*x^2+1)/(-a^2*c+2*(-a^2*d*
c)^(1/2)+d))+3/16*((-a^2*d*c)^(1/2)*a^2*c+2*a^2*d*c-(-a^2*d*c)^(1/2)*d)/c^3/(a^4*c^2+2*a^2*c*d+d^2)^2*d^2*arct
anh(a*x)^2-3/8*(a^2*c-2*(-a^2*d*c)^(1/2)-d)/(a^4*c^2+2*a^2*c*d+d^2)^2*a^6*ln(1-(a^2*c+d)*(a*x+1)^2/(-a^2*x^2+1
)/(-a^2*c-2*(-a^2*d*c)^(1/2)+d))*arctanh(a*x)+3/32*((-a^2*d*c)^(1/2)*a^2*c+2*a^2*d*c-(-a^2*d*c)^(1/2)*d)*a^6/d
/(a^4*c^2+2*a^2*c*d+d^2)^2*polylog(2,(a^2*c+d)*(a*x+1)^2/(-a^2*x^2+1)/(-a^2*c-2*(-a^2*d*c)^(1/2)+d))-3/16*((-a
^2*d*c)^(1/2)*a^2*c+2*a^2*d*c-(-a^2*d*c)^(1/2)*d)*a^6/d/(a^4*c^2+2*a^2*c*d+d^2)^2*arctanh(a*x)^2-3/16*(-a^2*d*
c)^(1/2)/c^2/(a^4*c^2+2*a^2*c*d+d^2)*a^2*polylog(2,(a^2*c+d)*(a*x+1)^2/(-a^2*x^2+1)/(-a^2*c+2*(-a^2*d*c)^(1/2)
+d))+3/8*(-a^2*d*c)^(1/2)/c^2/(a^4*c^2+2*a^2*c*d+d^2)*a^2*arctanh(a*x)^2-1/8*(d*c)^(1/2)/c^2*a^3*arctan(1/4*(2
*(a^2*c+d)*(a*x+1)^2/(-a^2*x^2+1)+2*a^2*c-2*d)/a/(d*c)^(1/2))/(a^4*c^2+2*a^2*c*d+d^2)+3/32*((-a^2*d*c)^(1/2)*a
^2*c+2*a^2*d*c-(-a^2*d*c)^(1/2)*d)/c*a^4/(a^4*c^2+2*a^2*c*d+d^2)^2*polylog(2,(a^2*c+d)*(a*x+1)^2/(-a^2*x^2+1)/
(-a^2*c-2*(-a^2*d*c)^(1/2)+d))-3/16*((-a^2*d*c)^(1/2)*a^2*c+2*a^2*d*c-(-a^2*d*c)^(1/2)*d)/c*a^4/(a^4*c^2+2*a^2
*c*d+d^2)^2*arctanh(a*x)^2-3/32*(-a^2*d*c)^(1/2)/c^3/(a^4*c^2+2*a^2*c*d+d^2)*d*polylog(2,(a^2*c+d)*(a*x+1)^2/(
-a^2*x^2+1)/(-a^2*c+2*(-a^2*d*c)^(1/2)+d))+3/16*(-a^2*d*c)^(1/2)/c^3/(a^4*c^2+2*a^2*c*d+d^2)*d*arctanh(a*x)^2-
3/16*(a^2*c-2*(-a^2*d*c)^(1/2)-d)/(a^4*c^2+2*a^2*c*d+d^2)^2*a^6*polylog(2,(a^2*c+d)*(a*x+1)^2/(-a^2*x^2+1)/(-a
^2*c-2*(-a^2*d*c)^(1/2)+d))+3/8*(a^2*c-2*(-a^2*d*c)^(1/2)-d)/(a^4*c^2+2*a^2*c*d+d^2)^2*a^6*arctanh(a*x)^2-5/16
/(a^4*c^2+2*a^2*c*d+d^2)*a^6/(a^2*c+d)*ln(a^2*c*(a*x+1)^4/(-a^2*x^2+1)^2+2*a^2*c*(a*x+1)^2/(-a^2*x^2+1)+d*(a*x
+1)^4/(-a^2*x^2+1)^2+a^2*c-2*d*(a*x+1)^2/(-a^2*x^2+1)+d)+5/4/(a^4*c^2+2*a^2*c*d+d^2)*a^6/(a^2*c+d)*ln((a*x+1)/
(-a^2*x^2+1)^(1/2))-3/16*(-a^2*d*c)^(1/2)/c^3/(a^4*c^2+2*a^2*c*d+d^2)*d*arctanh(a*x)*ln(1-(a^2*c+d)*(a*x+1)^2/
(-a^2*x^2+1)/(-a^2*c+2*(-a^2*d*c)^(1/2)+d))+3/16*((-a^2*d*c)^(1/2)*a^2*c+2*a^2*d*c-(-a^2*d*c)^(1/2)*d)/c*a^4/(
a^4*c^2+2*a^2*c*d+d^2)^2*ln(1-(a^2*c+d)*(a*x+1)^2/(-a^2*x^2+1)/(-a^2*c-2*(-a^2*d*c)^(1/2)+d))*arctanh(a*x)-3/3
2*((-a^2*d*c)^(1/2)*a^2*c+2*a^2*d*c-(-a^2*d*c)^(1/2)*d)/c^2/(a^4*c^2+2*a^2*c*d+d^2)^2*a^2*d*polylog(2,(a^2*c+d
)*(a*x+1)^2/(-a^2*x^2+1)/(-a^2*c-2*(-a^2*d*c)^(1/2)+d))+3/16*((-a^2*d*c)^(1/2)*a^2*c+2*a^2*d*c-(-a^2*d*c)^(1/2
)*d)/c^2/(a^4*c^2+2*a^2*c*d+d^2)^2*a^2*d*arctanh(a*x)^2-3/8*(-a^2*d*c)^(1/2)/c^2/(a^4*c^2+2*a^2*c*d+d^2)*a^2*a
rctanh(a*x)*ln(1-(a^2*c+d)*(a*x+1)^2/(-a^2*x^2+1)/(-a^2*c+2*(-a^2*d*c)^(1/2)+d))+3/16*((-a^2*d*c)^(1/2)*a^2*c+
2*a^2*d*c-(-a^2*d*c)^(1/2)*d)*a^6/d/(a^4*c^2+2*a^2*c*d+d^2)^2*ln(1-(a^2*c+d)*(a*x+1)^2/(-a^2*x^2+1)/(-a^2*c-2*
(-a^2*d*c)^(1/2)+d))*arctanh(a*x)+1/8*a^2*(5*a^6*c^3*arctanh(a*x)+3*arctanh(a*x)*a^6*c^2*d*x^2-7*arctanh(a*x)*
a^5*c^2*d*x-5*arctanh(a*x)*a^5*c*d^2*x^3+3*a^4*c^2*d*arctanh(a*x)+arctanh(a*x)*a^4*c*d^2*x^2-c^2*a^5*d*x-a^5*c
*d^2*x^3-5*arctanh(a*x)*a^3*c*d^2*x-3*arctanh(a*x)*d^3*a^3*x^3-a^4*c^2*d-a^4*c*d^2*x^2)*(a*x-1)/(a^4*c^2+2*a^2
*c*d+d^2)/(a^2*d*x^2+a^2*c)^2/c^2-3/16*(a^2*c-2*(-a^2*d*c)^(1/2)-d)/c^2/(a^4*c^2+2*a^2*c*d+d^2)^2*a^2*polylog(
2,(a^2*c+d)*(a*x+1)^2/(-a^2*x^2+1)/(-a^2*c-2*(-a^2*d*c)^(1/2)+d))*d^2+3/8*(a^2*c-2*(-a^2*d*c)^(1/2)-d)/c^2/(a^
4*c^2+2*a^2*c*d+d^2)^2*a^2*d^2*arctanh(a*x)^2-3/8*(a^2*c-2*(-a^2*d*c)^(1/2)-d)/c/(a^4*c^2+2*a^2*c*d+d^2)^2*a^4
*d*polylog(2,(a^2*c+d)*(a*x+1)^2/(-a^2*x^2+1)/(-a^2*c-2*(-a^2*d*c)^(1/2)+d))+3/4*(a^2*c-2*(-a^2*d*c)^(1/2)-d)/
c/(a^4*c^2+2*a^2*c*d+d^2)^2*a^4*d*arctanh(a*x)^2-5/16*(d*c)^(1/2)/d*a^7*arctan(1/4*(2*(a^2*c+d)*(a*x+1)^2/(-a^
2*x^2+1)+2*a^2*c-2*d)/a/(d*c)^(1/2))/(a^2*c+d)/(a^4*c^2+2*a^2*c*d+d^2)-3/16*(d*c)^(1/2)*d/c^3*a*arctan(1/4*(2*
(a^2*c+d)*(a*x+1)^2/(-a^2*x^2+1)+2*a^2*c-2*d)/a/(d*c)^(1/2))/(a^4*c^2+2*a^2*c*d+d^2)-3/16*(d*c)^(1/2)/c*a^5*ar
ctan(1/4*(2*(a^2*c+d)*(a*x+1)^2/(-a^2*x^2+1)+2*a^2*c-2*d)/a/(d*c)^(1/2))/(a^2*c+d)/(a^4*c^2+2*a^2*c*d+d^2)-3/3
2*(-a^2*d*c)^(1/2)/c*a^4/d/(a^4*c^2+2*a^2*c*d+d^2)*polylog(2,(a^2*c+d)*(a*x+1)^2/(-a^2*x^2+1)/(-a^2*c+2*(-a^2*
d*c)^(1/2)+d))+3/16*(-a^2*d*c)^(1/2)/c*a^4/d/(a^4*c^2+2*a^2*c*d+d^2)*arctanh(a*x)^2+5/16*(d*c)^(1/2)/d/c*a^5*a
rctan(1/4*(2*(a^2*c+d)*(a*x+1)^2/(-a^2*x^2+1)+2...

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1087 vs. \(2 (463) = 926\).
time = 0.57, size = 1087, normalized size = 1.65 \begin {gather*} \frac {1}{8} \, {\left (\frac {3 \, d x^{3} + 5 \, c x}{c^{2} d^{2} x^{4} + 2 \, c^{3} d x^{2} + c^{4}} + \frac {3 \, \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d} c^{2}}\right )} \operatorname {artanh}\left (a x\right ) + \frac {{\left (4 \, a^{3} c^{3} d + 4 \, a c^{2} d^{2} - 3 \, {\left ({\left (a^{4} c^{3} + 2 \, a^{2} c^{2} d + c d^{2} + {\left (a^{4} c^{2} d + 2 \, a^{2} c d^{2} + d^{3}\right )} x^{2}\right )} \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {a^{2} d x^{2} + 2 \, a d x + d}{a^{2} c + d}\right ) - {\left (a^{4} c^{3} + 2 \, a^{2} c^{2} d + c d^{2} + {\left (a^{4} c^{2} d + 2 \, a^{2} c d^{2} + d^{3}\right )} x^{2}\right )} \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {a^{2} d x^{2} - 2 \, a d x + d}{a^{2} c + d}\right ) - {\left (-i \, a^{4} c^{3} - 2 i \, a^{2} c^{2} d - i \, c d^{2} + {\left (-i \, a^{4} c^{2} d - 2 i \, a^{2} c d^{2} - i \, d^{3}\right )} x^{2}\right )} {\rm Li}_2\left (\frac {a^{2} c + a d x - {\left (i \, a^{2} x - i \, a\right )} \sqrt {c} \sqrt {d}}{a^{2} c + 2 i \, a \sqrt {c} \sqrt {d} - d}\right ) - {\left (-i \, a^{4} c^{3} - 2 i \, a^{2} c^{2} d - i \, c d^{2} + {\left (-i \, a^{4} c^{2} d - 2 i \, a^{2} c d^{2} - i \, d^{3}\right )} x^{2}\right )} {\rm Li}_2\left (\frac {a^{2} c - a d x + {\left (i \, a^{2} x + i \, a\right )} \sqrt {c} \sqrt {d}}{a^{2} c + 2 i \, a \sqrt {c} \sqrt {d} - d}\right ) - {\left (i \, a^{4} c^{3} + 2 i \, a^{2} c^{2} d + i \, c d^{2} + {\left (i \, a^{4} c^{2} d + 2 i \, a^{2} c d^{2} + i \, d^{3}\right )} x^{2}\right )} {\rm Li}_2\left (\frac {a^{2} c + a d x + {\left (i \, a^{2} x - i \, a\right )} \sqrt {c} \sqrt {d}}{a^{2} c - 2 i \, a \sqrt {c} \sqrt {d} - d}\right ) - {\left (i \, a^{4} c^{3} + 2 i \, a^{2} c^{2} d + i \, c d^{2} + {\left (i \, a^{4} c^{2} d + 2 i \, a^{2} c d^{2} + i \, d^{3}\right )} x^{2}\right )} {\rm Li}_2\left (\frac {a^{2} c - a d x - {\left (i \, a^{2} x + i \, a\right )} \sqrt {c} \sqrt {d}}{a^{2} c - 2 i \, a \sqrt {c} \sqrt {d} - d}\right ) - {\left ({\left (a^{4} c^{3} + 2 \, a^{2} c^{2} d + c d^{2} + {\left (a^{4} c^{2} d + 2 \, a^{2} c d^{2} + d^{3}\right )} x^{2}\right )} \arctan \left (\frac {{\left (a^{2} x + a\right )} \sqrt {c} \sqrt {d}}{a^{2} c + d}, \frac {a d x + d}{a^{2} c + d}\right ) - {\left (a^{4} c^{3} + 2 \, a^{2} c^{2} d + c d^{2} + {\left (a^{4} c^{2} d + 2 \, a^{2} c d^{2} + d^{3}\right )} x^{2}\right )} \arctan \left (\frac {{\left (a^{2} x - a\right )} \sqrt {c} \sqrt {d}}{a^{2} c + d}, -\frac {a d x - d}{a^{2} c + d}\right )\right )} \log \left (d x^{2} + c\right )\right )} \sqrt {c} \sqrt {d} - 2 \, {\left (5 \, a^{3} c^{3} d + 3 \, a c^{2} d^{2} + {\left (5 \, a^{3} c^{2} d^{2} + 3 \, a c d^{3}\right )} x^{2}\right )} \log \left (d x^{2} + c\right ) + 2 \, {\left (5 \, a^{3} c^{3} d + 3 \, a c^{2} d^{2} + {\left (5 \, a^{3} c^{2} d^{2} + 3 \, a c d^{3}\right )} x^{2}\right )} \log \left (a x + 1\right ) + 2 \, {\left (5 \, a^{3} c^{3} d + 3 \, a c^{2} d^{2} + {\left (5 \, a^{3} c^{2} d^{2} + 3 \, a c d^{3}\right )} x^{2}\right )} \log \left (a x - 1\right )\right )} a}{32 \, {\left (a^{5} c^{6} d + 2 \, a^{3} c^{5} d^{2} + a c^{4} d^{3} + {\left (a^{5} c^{5} d^{2} + 2 \, a^{3} c^{4} d^{3} + a c^{3} d^{4}\right )} x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(a*x)/(d*x^2+c)^3,x, algorithm="maxima")

[Out]

1/8*((3*d*x^3 + 5*c*x)/(c^2*d^2*x^4 + 2*c^3*d*x^2 + c^4) + 3*arctan(d*x/sqrt(c*d))/(sqrt(c*d)*c^2))*arctanh(a*
x) + 1/32*(4*a^3*c^3*d + 4*a*c^2*d^2 - 3*((a^4*c^3 + 2*a^2*c^2*d + c*d^2 + (a^4*c^2*d + 2*a^2*c*d^2 + d^3)*x^2
)*arctan(sqrt(d)*x/sqrt(c))*log((a^2*d*x^2 + 2*a*d*x + d)/(a^2*c + d)) - (a^4*c^3 + 2*a^2*c^2*d + c*d^2 + (a^4
*c^2*d + 2*a^2*c*d^2 + d^3)*x^2)*arctan(sqrt(d)*x/sqrt(c))*log((a^2*d*x^2 - 2*a*d*x + d)/(a^2*c + d)) - (-I*a^
4*c^3 - 2*I*a^2*c^2*d - I*c*d^2 + (-I*a^4*c^2*d - 2*I*a^2*c*d^2 - I*d^3)*x^2)*dilog((a^2*c + a*d*x - (I*a^2*x
- I*a)*sqrt(c)*sqrt(d))/(a^2*c + 2*I*a*sqrt(c)*sqrt(d) - d)) - (-I*a^4*c^3 - 2*I*a^2*c^2*d - I*c*d^2 + (-I*a^4
*c^2*d - 2*I*a^2*c*d^2 - I*d^3)*x^2)*dilog((a^2*c - a*d*x + (I*a^2*x + I*a)*sqrt(c)*sqrt(d))/(a^2*c + 2*I*a*sq
rt(c)*sqrt(d) - d)) - (I*a^4*c^3 + 2*I*a^2*c^2*d + I*c*d^2 + (I*a^4*c^2*d + 2*I*a^2*c*d^2 + I*d^3)*x^2)*dilog(
(a^2*c + a*d*x + (I*a^2*x - I*a)*sqrt(c)*sqrt(d))/(a^2*c - 2*I*a*sqrt(c)*sqrt(d) - d)) - (I*a^4*c^3 + 2*I*a^2*
c^2*d + I*c*d^2 + (I*a^4*c^2*d + 2*I*a^2*c*d^2 + I*d^3)*x^2)*dilog((a^2*c - a*d*x - (I*a^2*x + I*a)*sqrt(c)*sq
rt(d))/(a^2*c - 2*I*a*sqrt(c)*sqrt(d) - d)) - ((a^4*c^3 + 2*a^2*c^2*d + c*d^2 + (a^4*c^2*d + 2*a^2*c*d^2 + d^3
)*x^2)*arctan2((a^2*x + a)*sqrt(c)*sqrt(d)/(a^2*c + d), (a*d*x + d)/(a^2*c + d)) - (a^4*c^3 + 2*a^2*c^2*d + c*
d^2 + (a^4*c^2*d + 2*a^2*c*d^2 + d^3)*x^2)*arctan2((a^2*x - a)*sqrt(c)*sqrt(d)/(a^2*c + d), -(a*d*x - d)/(a^2*
c + d)))*log(d*x^2 + c))*sqrt(c)*sqrt(d) - 2*(5*a^3*c^3*d + 3*a*c^2*d^2 + (5*a^3*c^2*d^2 + 3*a*c*d^3)*x^2)*log
(d*x^2 + c) + 2*(5*a^3*c^3*d + 3*a*c^2*d^2 + (5*a^3*c^2*d^2 + 3*a*c*d^3)*x^2)*log(a*x + 1) + 2*(5*a^3*c^3*d +
3*a*c^2*d^2 + (5*a^3*c^2*d^2 + 3*a*c*d^3)*x^2)*log(a*x - 1))*a/(a^5*c^6*d + 2*a^3*c^5*d^2 + a*c^4*d^3 + (a^5*c
^5*d^2 + 2*a^3*c^4*d^3 + a*c^3*d^4)*x^2)

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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(arctanh(a*x)/(d*x^2+c)^3,x, algorithm="fricas")

[Out]

integral(arctanh(a*x)/(d^3*x^6 + 3*c*d^2*x^4 + 3*c^2*d*x^2 + c^3), x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atanh(a*x)/(d*x**2+c)**3,x)

[Out]

Timed out

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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(arctanh(a*x)/(d*x^2+c)^3,x, algorithm="giac")

[Out]

integrate(arctanh(a*x)/(d*x^2 + c)^3, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\mathrm {atanh}\left (a\,x\right )}{{\left (d\,x^2+c\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(atanh(a*x)/(c + d*x^2)^3,x)

[Out]

int(atanh(a*x)/(c + d*x^2)^3, x)

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