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

Optimal. Leaf size=657 \[ \frac {a \left (5 a^2 c+3 d\right ) \log \left (1-a^2 x^2\right )}{16 c^2 \left (a^2 c+d\right )^2}-\frac {a \left (5 a^2 c+3 d\right ) \log \left (c+d x^2\right )}{16 c^2 \left (a^2 c+d\right )^2}+\frac {a}{8 c \left (a^2 c+d\right ) \left (c+d x^2\right )}+\frac {3 i \text {Li}_2\left (\frac {a \left (\sqrt {c}-i \sqrt {d} x\right )}{a \sqrt {c}-i \sqrt {d}}\right )}{32 c^{5/2} \sqrt {d}}-\frac {3 i \text {Li}_2\left (\frac {a \left (\sqrt {c}-i \sqrt {d} x\right )}{\sqrt {c} a+i \sqrt {d}}\right )}{32 c^{5/2} \sqrt {d}}+\frac {3 i \text {Li}_2\left (\frac {a \left (i \sqrt {d} x+\sqrt {c}\right )}{a \sqrt {c}-i \sqrt {d}}\right )}{32 c^{5/2} \sqrt {d}}-\frac {3 i \text {Li}_2\left (\frac {a \left (i \sqrt {d} x+\sqrt {c}\right )}{\sqrt {c} a+i \sqrt {d}}\right )}{32 c^{5/2} \sqrt {d}}+\frac {3 i \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {\sqrt {d} (1-a x)}{\sqrt {d}+i a \sqrt {c}}\right )}{32 c^{5/2} \sqrt {d}}-\frac {3 i \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right ) \log \left (-\frac {\sqrt {d} (a x+1)}{-\sqrt {d}+i a \sqrt {c}}\right )}{32 c^{5/2} \sqrt {d}}-\frac {3 i \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right ) \log \left (-\frac {\sqrt {d} (1-a x)}{-\sqrt {d}+i a \sqrt {c}}\right )}{32 c^{5/2} \sqrt {d}}+\frac {3 i \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {\sqrt {d} (a x+1)}{\sqrt {d}+i a \sqrt {c}}\right )}{32 c^{5/2} \sqrt {d}}+\frac {3 \tanh ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{5/2} \sqrt {d}}+\frac {3 x \tanh ^{-1}(a x)}{8 c^2 \left (c+d x^2\right )}+\frac {x \tanh ^{-1}(a x)}{4 c \left (c+d x^2\right )^2} \]

[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 = 0.97, 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 = {199, 205, 5976, 6725, 571, 77, 4908, 2409, 2394, 2393, 2391} \[ \frac {3 i \text {PolyLog}\left (2,\frac {a \left (\sqrt {c}-i \sqrt {d} x\right )}{a \sqrt {c}-i \sqrt {d}}\right )}{32 c^{5/2} \sqrt {d}}-\frac {3 i \text {PolyLog}\left (2,\frac {a \left (\sqrt {c}-i \sqrt {d} x\right )}{a \sqrt {c}+i \sqrt {d}}\right )}{32 c^{5/2} \sqrt {d}}+\frac {3 i \text {PolyLog}\left (2,\frac {a \left (\sqrt {c}+i \sqrt {d} x\right )}{a \sqrt {c}-i \sqrt {d}}\right )}{32 c^{5/2} \sqrt {d}}-\frac {3 i \text {PolyLog}\left (2,\frac {a \left (\sqrt {c}+i \sqrt {d} x\right )}{a \sqrt {c}+i \sqrt {d}}\right )}{32 c^{5/2} \sqrt {d}}+\frac {a \left (5 a^2 c+3 d\right ) \log \left (1-a^2 x^2\right )}{16 c^2 \left (a^2 c+d\right )^2}-\frac {a \left (5 a^2 c+3 d\right ) \log \left (c+d x^2\right )}{16 c^2 \left (a^2 c+d\right )^2}+\frac {a}{8 c \left (a^2 c+d\right ) \left (c+d x^2\right )}+\frac {3 x \tanh ^{-1}(a x)}{8 c^2 \left (c+d x^2\right )}+\frac {3 i \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {\sqrt {d} (1-a x)}{\sqrt {d}+i a \sqrt {c}}\right )}{32 c^{5/2} \sqrt {d}}-\frac {3 i \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right ) \log \left (-\frac {\sqrt {d} (a x+1)}{-\sqrt {d}+i a \sqrt {c}}\right )}{32 c^{5/2} \sqrt {d}}-\frac {3 i \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right ) \log \left (-\frac {\sqrt {d} (1-a x)}{-\sqrt {d}+i a \sqrt {c}}\right )}{32 c^{5/2} \sqrt {d}}+\frac {3 i \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {\sqrt {d} (a x+1)}{\sqrt {d}+i a \sqrt {c}}\right )}{32 c^{5/2} \sqrt {d}}+\frac {3 \tanh ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{5/2} \sqrt {d}}+\frac {x \tanh ^{-1}(a x)}{4 c \left (c+d x^2\right )^2} \]

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 77

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 199

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] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rule 205

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

Rule 571

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 2391

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 2393

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 2394

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 2409

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 4908

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 5976

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 6725

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 \operatorname {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 \operatorname {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) \operatorname {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) \operatorname {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) \operatorname {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) \operatorname {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]  time = 12.94, size = 1828, normalized size = 2.78 \[ \text {result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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fricas [F]  time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {artanh}\left (a x\right )}{d^{3} x^{6} + 3 \, c d^{2} x^{4} + 3 \, c^{2} d x^{2} + c^{3}}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {artanh}\left (a x\right )}{{\left (d x^{2} + c\right )}^{3}}\,{d x} \]

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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maple [B]  time = 1.41, size = 4311, normalized size = 6.56 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*a^5/(a^4*c^2+2*a^2*c*d+d^2)/(a^2*d*x^2+a^2*c)^2*d+5/4*a^5/(a^4*c^2+2*a^2*c*d+d^2)/(a^2*c+d)*ln((a*x+1)/(-a
^2*x^2+1)^(1/2))+3/8*a^5/(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*
c*d)^(1/2)+d))*(-a^2*c*d)^(1/2)-3/4*a^5/(a^4*c^2+2*a^2*c*d+d^2)^2*arctanh(a*x)^2*(-a^2*c*d)^(1/2)-5/16*a^5/(a^
4*c^2+2*a^2*c*d+d^2)/(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*(a*x+1)^2/(-a^2*x^2+1)*d+d)+5/16*a^4*(c*d)^(1/2)/c/d/(a^4*c^2+2*a^2*c*d+d^2)*arctan(a/d*(
c*d)^(1/2))-3/16*a^4*(c*d)^(1/2)/c/(a^2*c+d)/(a^4*c^2+2*a^2*c*d+d^2)*arctan(a/d*(c*d)^(1/2))-5/16*a^6*(c*d)^(1
/2)/d/(a^2*c+d)/(a^4*c^2+2*a^2*c*d+d^2)*arctan(a/d*(c*d)^(1/2))+3/16*(c*d)^(1/2)/c^3*d^2/(a^2*c+d)/(a^4*c^2+2*
a^2*c*d+d^2)*arctan(a/d*(c*d)^(1/2))-1/8*a^7/(a^4*c^2+2*a^2*c*d+d^2)/(a^2*d*x^2+a^2*c)^2/c*d^2*x^4+1/8*a^5/(a^
4*c^2+2*a^2*c*d+d^2)/(a^2*d*x^2+a^2*c)^2/c*d^2*x^2+9/16*a^3/c/(a^4*c^2+2*a^2*c*d+d^2)^2*d*polylog(2,(a^2*c+d)*
(a*x+1)^2/(-a^2*x^2+1)/(-a^2*c-2*(-a^2*c*d)^(1/2)+d))*(-a^2*c*d)^(1/2)+5/16*a^4*(c*d)^(1/2)/c/d/(a^4*c^2+2*a^2
*c*d+d^2)*arctan(1/(a^2*c+d)*d^2/(c*d)^(1/2)*x+1/(a^2*c+d)*d/(c*d)^(1/2)*a*c+a^2/(a^2*c+d)*(c*d)^(1/2)*x-a/(a^
2*c+d)*(c*d)^(1/2))-3/4*a/(a^4*c^2+2*a^2*c*d+d^2)^2/c^2*d^2*arctanh(a*x)^2*(-a^2*c*d)^(1/2)-3/16*a/c^2/(a^4*c^
2+2*a^2*c*d+d^2)*d^2/(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*(a*x+1)^2/(-a^2*x^2+1)*d+d)-1/2*a^3/c/(a^4*c^2+2*a^2*c*d+d^2)*d/(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*(a*x+1)^2/(-a^2*x^2+1)*d+
d)+3/4*a/c^2/(a^4*c^2+2*a^2*c*d+d^2)*d^2/(a^2*c+d)*ln((a*x+1)/(-a^2*x^2+1)^(1/2))-3/16*a^4*(c*d)^(1/2)/c/(a^2*
c+d)/(a^4*c^2+2*a^2*c*d+d^2)*arctan(1/(a^2*c+d)*d^2/(c*d)^(1/2)*x+1/(a^2*c+d)*d/(c*d)^(1/2)*a*c+a^2/(a^2*c+d)*
(c*d)^(1/2)*x-a/(a^2*c+d)*(c*d)^(1/2))+5/8*a^8/(a^4*c^2+2*a^2*c*d+d^2)/(a^2*d*x^2+a^2*c)^2*c*arctanh(a*x)*x-3/
8*a*(-a^2*c*d)^(1/2)/c^2/(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*c*d)^(1/2)+d))+5/4*a^6/(a^4*c^2+2*a^2*c*d+d^2)/(a^2*d*x^2+a^2*c)^2*arctanh(a*x)*x*d+3/8*a^8/(a^4*c^2+2*a
^2*c*d+d^2)/(a^2*d*x^2+a^2*c)^2*arctanh(a*x)*x^3*d-5/4*a^7/(a^4*c^2+2*a^2*c*d+d^2)/(a^2*d*x^2+a^2*c)^2*arctanh
(a*x)*x^2*d-3/16/a/c^3*d^3/(a^4*c^2+2*a^2*c*d+d^2)^2*arctanh(a*x)^2*(-a^2*c*d)^(1/2)-3/32/a*(-a^2*c*d)^(1/2)/c
^3*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*c*d)^(1/2)+d))+3/16/a*
(-a^2*c*d)^(1/2)/c^3*d/(a^4*c^2+2*a^2*c*d+d^2)*arctanh(a*x)^2+3/32/a/c^3*d^3/(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*c*d)^(1/2)+d))*(-a^2*c*d)^(1/2)-3/16*a^7/d/(a^4*c^2+2*a^2*
c*d+d^2)^2*arctanh(a*x)^2*(-a^2*c*d)^(1/2)*c-9/8*a^3/c/(a^4*c^2+2*a^2*c*d+d^2)^2*d*arctanh(a*x)^2*(-a^2*c*d)^(
1/2)-3/32*a^3*(-a^2*c*d)^(1/2)/c/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*c*d)^(1/2)+d))+3/16*a^3*(-a^2*c*d)^(1/2)/c/d/(a^4*c^2+2*a^2*c*d+d^2)*arctanh(a*x)^2+3/32*a^7/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*c*d)^(1/2)+d))*(-a^2*c*d)^(1/2)*
c+2*a^3/c/(a^4*c^2+2*a^2*c*d+d^2)*d/(a^2*c+d)*ln((a*x+1)/(-a^2*x^2+1)^(1/2))-5/16*a^6*(c*d)^(1/2)/d/(a^2*c+d)/
(a^4*c^2+2*a^2*c*d+d^2)*arctan(1/(a^2*c+d)*d^2/(c*d)^(1/2)*x+1/(a^2*c+d)*d/(c*d)^(1/2)*a*c+a^2/(a^2*c+d)*(c*d)
^(1/2)*x-a/(a^2*c+d)*(c*d)^(1/2))+3/8*a/(a^4*c^2+2*a^2*c*d+d^2)^2/c^2*d^2*polylog(2,(a^2*c+d)*(a*x+1)^2/(-a^2*
x^2+1)/(-a^2*c-2*(-a^2*c*d)^(1/2)+d))*(-a^2*c*d)^(1/2)+3/16*(c*d)^(1/2)/c^3*d^2/(a^2*c+d)/(a^4*c^2+2*a^2*c*d+d
^2)*arctan(1/(a^2*c+d)*d^2/(c*d)^(1/2)*x+1/(a^2*c+d)*d/(c*d)^(1/2)*a*c+a^2/(a^2*c+d)*(c*d)^(1/2)*x-a/(a^2*c+d)
*(c*d)^(1/2))-5/8*a^7/(a^4*c^2+2*a^2*c*d+d^2)/(a^2*d*x^2+a^2*c)^2*c*arctanh(a*x)-1/8*a^2*(c*d)^(1/2)/c^2/(a^4*
c^2+2*a^2*c*d+d^2)*arctan(1/(a^2*c+d)*d^2/(c*d)^(1/2)*x+1/(a^2*c+d)*d/(c*d)^(1/2)*a*c+a^2/(a^2*c+d)*(c*d)^(1/2
)*x-a/(a^2*c+d)*(c*d)^(1/2))-1/8*a^7/(a^4*c^2+2*a^2*c*d+d^2)/(a^2*d*x^2+a^2*c)^2*d*x^2+5/16*a^2*(c*d)^(1/2)/c^
2*d/(a^2*c+d)/(a^4*c^2+2*a^2*c*d+d^2)*arctan(1/(a^2*c+d)*d^2/(c*d)^(1/2)*x+1/(a^2*c+d)*d/(c*d)^(1/2)*a*c+a^2/(
a^2*c+d)*(c*d)^(1/2)*x-a/(a^2*c+d)*(c*d)^(1/2))-3/4*a^5/(a^4*c^2+2*a^2*c*d+d^2)/(a^2*d*x^2+a^2*c)^2/c*arctanh(
a*x)*x^2*d^2+5/8*a^4/(a^4*c^2+2*a^2*c*d+d^2)/(a^2*d*x^2+a^2*c)^2/c*arctanh(a*x)*x*d^2+3/4*a/c^2*d^2/(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*c*d)^(1/2)+d))*arctanh(a*x)*(-a^2*c*d)^(
1/2)+3/16*a^7/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*c*d)^(1/2)+d))
*arctanh(a*x)*(-a^2*c*d)^(1/2)*c-3/8*a^5/(a^4*c^2+2*a^2*c*d+d^2)/(a^2*d*x^2+a^2*c)^2/c^2*arctanh(a*x)*x^4*d^3+
3/4*a^6/(a^4*c^2+2*a^2*c*d+d^2)/(a^2*d*x^2+a^2*c)^2/c*arctanh(a*x)*x^3*d^2-3/16*a^3*(-a^2*c*d)^(1/2)/c/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*c*d)^(1/2)+d))+3/8*a^4/(
a^4*c^2+2*a^2*c*d+d^2)/(a^2*d*x^2+a^2*c)^2/c^2*arctanh(a*x)*x^3*d^3+3/16/a/c^3*d^3/(a^4*c^2+2*a^2*c*d+d^2)^2*l
n(1-(a^2*c+d)*(a*x+1)^2/(-a^2*x^2+1)/(-a^2*c-2*(-a^2*c*d)^(1/2)+d))*arctanh(a*x)*(-a^2*c*d)^(1/2)-3/16/a*(-a^2
*c*d)^(1/2)/c^3*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*c
*d)^(1/2)+d))+9/8*a^3/c/(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*c*d)^(
1/2)+d))*arctanh(a*x)*(-a^2*c*d)^(1/2)*d-5/8*a^7/(a^4*c^2+2*a^2*c*d+d^2)/(a^2*d*x^2+a^2*c)^2/c*arctanh(a*x)*x^
4*d^2+5/16*a^2*(c*d)^(1/2)/c^2*d/(a^2*c+d)/(a^4*c^2+2*a^2*c*d+d^2)*arctan(a/d*(c*d)^(1/2))-3/16*(c*d)^(1/2)/c^
3*d/(a^4*c^2+2*a^2*c*d+d^2)*arctan(a/d*(c*d)^(1/2))-1/8*a^2*(c*d)^(1/2)/c^2/(a^4*c^2+2*a^2*c*d+d^2)*arctan(a/d
*(c*d)^(1/2))-3/16*(c*d)^(1/2)/c^3*d/(a^4*c^2+2*a^2*c*d+d^2)*arctan(1/(a^2*c+d)*d^2/(c*d)^(1/2)*x+1/(a^2*c+d)*
d/(c*d)^(1/2)*a*c+a^2/(a^2*c+d)*(c*d)^(1/2)*x-a/(a^2*c+d)*(c*d)^(1/2))+3/8*a*(-a^2*c*d)^(1/2)/c^2/(a^4*c^2+2*a
^2*c*d+d^2)*arctanh(a*x)^2-3/16*a*(-a^2*c*d)^(1/2)/c^2/(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*c*d)^(1/2)+d))+3/4*a^5/(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*c*d)^(1/2)+d))*arctanh(a*x)*(-a^2*c*d)^(1/2)-3/8*a^5/(a^4*c^2+2*a^2*c*d+d^2)/(a^2*d*x^2+a
^2*c)^2*d*arctanh(a*x)

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maxima [B]  time = 0.60, size = 1084, normalized size = 1.65 \[ \text {result too large to display} \]

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)) - 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)) + (3*I
*a^4*c^3 + 6*I*a^2*c^2*d + 3*I*c*d^2 + (3*I*a^4*c^2*d + 6*I*a^2*c*d^2 + 3*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)) + (3*I*a^4*c^3 + 6*I*a^2*c^2*d + 3*I*c*d^
2 + (3*I*a^4*c^2*d + 6*I*a^2*c*d^2 + 3*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)) + (-3*I*a^4*c^3 - 6*I*a^2*c^2*d - 3*I*c*d^2 + (-3*I*a^4*c^2*d - 6*I*a^2*c*d^
2 - 3*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))
+ (-3*I*a^4*c^3 - 6*I*a^2*c^2*d - 3*I*c*d^2 + (-3*I*a^4*c^2*d - 6*I*a^2*c*d^2 - 3*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)) - 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)*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)*sqr
t(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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\mathrm {atanh}\left (a\,x\right )}{{\left (d\,x^2+c\right )}^3} \,d x \]

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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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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