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

Optimal. Leaf size=429 \[ -\frac{\text{PolyLog}\left (2,-\frac{\sqrt{d} (1-a x)}{a \sqrt{-c}-\sqrt{d}}\right )}{4 \sqrt{-c} \sqrt{d}}+\frac{\text{PolyLog}\left (2,\frac{\sqrt{d} (1-a x)}{a \sqrt{-c}+\sqrt{d}}\right )}{4 \sqrt{-c} \sqrt{d}}-\frac{\text{PolyLog}\left (2,-\frac{\sqrt{d} (a x+1)}{a \sqrt{-c}-\sqrt{d}}\right )}{4 \sqrt{-c} \sqrt{d}}+\frac{\text{PolyLog}\left (2,\frac{\sqrt{d} (a x+1)}{a \sqrt{-c}+\sqrt{d}}\right )}{4 \sqrt{-c} \sqrt{d}}-\frac{\log (1-a x) \log \left (\frac{a \left (\sqrt{-c}-\sqrt{d} x\right )}{a \sqrt{-c}-\sqrt{d}}\right )}{4 \sqrt{-c} \sqrt{d}}+\frac{\log (a x+1) \log \left (\frac{a \left (\sqrt{-c}-\sqrt{d} x\right )}{a \sqrt{-c}+\sqrt{d}}\right )}{4 \sqrt{-c} \sqrt{d}}-\frac{\log (a x+1) \log \left (\frac{a \left (\sqrt{-c}+\sqrt{d} x\right )}{a \sqrt{-c}-\sqrt{d}}\right )}{4 \sqrt{-c} \sqrt{d}}+\frac{\log (1-a x) \log \left (\frac{a \left (\sqrt{-c}+\sqrt{d} x\right )}{a \sqrt{-c}+\sqrt{d}}\right )}{4 \sqrt{-c} \sqrt{d}} \]

[Out]

-(Log[1 - a*x]*Log[(a*(Sqrt[-c] - Sqrt[d]*x))/(a*Sqrt[-c] - Sqrt[d])])/(4*Sqrt[-c]*Sqrt[d]) + (Log[1 + a*x]*Lo
g[(a*(Sqrt[-c] - Sqrt[d]*x))/(a*Sqrt[-c] + Sqrt[d])])/(4*Sqrt[-c]*Sqrt[d]) - (Log[1 + a*x]*Log[(a*(Sqrt[-c] +
Sqrt[d]*x))/(a*Sqrt[-c] - Sqrt[d])])/(4*Sqrt[-c]*Sqrt[d]) + (Log[1 - a*x]*Log[(a*(Sqrt[-c] + Sqrt[d]*x))/(a*Sq
rt[-c] + Sqrt[d])])/(4*Sqrt[-c]*Sqrt[d]) - PolyLog[2, -((Sqrt[d]*(1 - a*x))/(a*Sqrt[-c] - Sqrt[d]))]/(4*Sqrt[-
c]*Sqrt[d]) + PolyLog[2, (Sqrt[d]*(1 - a*x))/(a*Sqrt[-c] + Sqrt[d])]/(4*Sqrt[-c]*Sqrt[d]) - PolyLog[2, -((Sqrt
[d]*(1 + a*x))/(a*Sqrt[-c] - Sqrt[d]))]/(4*Sqrt[-c]*Sqrt[d]) + PolyLog[2, (Sqrt[d]*(1 + a*x))/(a*Sqrt[-c] + Sq
rt[d])]/(4*Sqrt[-c]*Sqrt[d])

________________________________________________________________________________________

Rubi [A]  time = 0.441711, antiderivative size = 429, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {5972, 2409, 2394, 2393, 2391} \[ -\frac{\text{PolyLog}\left (2,-\frac{\sqrt{d} (1-a x)}{a \sqrt{-c}-\sqrt{d}}\right )}{4 \sqrt{-c} \sqrt{d}}+\frac{\text{PolyLog}\left (2,\frac{\sqrt{d} (1-a x)}{a \sqrt{-c}+\sqrt{d}}\right )}{4 \sqrt{-c} \sqrt{d}}-\frac{\text{PolyLog}\left (2,-\frac{\sqrt{d} (a x+1)}{a \sqrt{-c}-\sqrt{d}}\right )}{4 \sqrt{-c} \sqrt{d}}+\frac{\text{PolyLog}\left (2,\frac{\sqrt{d} (a x+1)}{a \sqrt{-c}+\sqrt{d}}\right )}{4 \sqrt{-c} \sqrt{d}}-\frac{\log (1-a x) \log \left (\frac{a \left (\sqrt{-c}-\sqrt{d} x\right )}{a \sqrt{-c}-\sqrt{d}}\right )}{4 \sqrt{-c} \sqrt{d}}+\frac{\log (a x+1) \log \left (\frac{a \left (\sqrt{-c}-\sqrt{d} x\right )}{a \sqrt{-c}+\sqrt{d}}\right )}{4 \sqrt{-c} \sqrt{d}}-\frac{\log (a x+1) \log \left (\frac{a \left (\sqrt{-c}+\sqrt{d} x\right )}{a \sqrt{-c}-\sqrt{d}}\right )}{4 \sqrt{-c} \sqrt{d}}+\frac{\log (1-a x) \log \left (\frac{a \left (\sqrt{-c}+\sqrt{d} x\right )}{a \sqrt{-c}+\sqrt{d}}\right )}{4 \sqrt{-c} \sqrt{d}} \]

Antiderivative was successfully verified.

[In]

Int[ArcTanh[a*x]/(c + d*x^2),x]

[Out]

-(Log[1 - a*x]*Log[(a*(Sqrt[-c] - Sqrt[d]*x))/(a*Sqrt[-c] - Sqrt[d])])/(4*Sqrt[-c]*Sqrt[d]) + (Log[1 + a*x]*Lo
g[(a*(Sqrt[-c] - Sqrt[d]*x))/(a*Sqrt[-c] + Sqrt[d])])/(4*Sqrt[-c]*Sqrt[d]) - (Log[1 + a*x]*Log[(a*(Sqrt[-c] +
Sqrt[d]*x))/(a*Sqrt[-c] - Sqrt[d])])/(4*Sqrt[-c]*Sqrt[d]) + (Log[1 - a*x]*Log[(a*(Sqrt[-c] + Sqrt[d]*x))/(a*Sq
rt[-c] + Sqrt[d])])/(4*Sqrt[-c]*Sqrt[d]) - PolyLog[2, -((Sqrt[d]*(1 - a*x))/(a*Sqrt[-c] - Sqrt[d]))]/(4*Sqrt[-
c]*Sqrt[d]) + PolyLog[2, (Sqrt[d]*(1 - a*x))/(a*Sqrt[-c] + Sqrt[d])]/(4*Sqrt[-c]*Sqrt[d]) - PolyLog[2, -((Sqrt
[d]*(1 + a*x))/(a*Sqrt[-c] - Sqrt[d]))]/(4*Sqrt[-c]*Sqrt[d]) + PolyLog[2, (Sqrt[d]*(1 + a*x))/(a*Sqrt[-c] + Sq
rt[d])]/(4*Sqrt[-c]*Sqrt[d])

Rule 5972

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

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 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 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 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]

Rubi steps

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

Mathematica [C]  time = 1.34531, size = 662, normalized size = 1.54 \[ -\frac{a \left (i \left (\text{PolyLog}\left (2,\frac{\left (2 i \sqrt{a^2 c d}+a^2 (-c)+d\right ) \left (x \sqrt{a^2 c d}+i a c\right )}{\left (a^2 c+d\right ) \left (x \sqrt{a^2 c d}-i a c\right )}\right )-\text{PolyLog}\left (2,\frac{\left (-2 i \sqrt{a^2 c d}+a^2 (-c)+d\right ) \left (x \sqrt{a^2 c d}+i a c\right )}{\left (a^2 c+d\right ) \left (x \sqrt{a^2 c d}-i a c\right )}\right )\right )-2 i \cos ^{-1}\left (\frac{d-a^2 c}{a^2 c+d}\right ) \tan ^{-1}\left (\frac{a d x}{\sqrt{a^2 c d}}\right )+4 \tanh ^{-1}(a x) \tan ^{-1}\left (\frac{a c}{x \sqrt{a^2 c d}}\right )-\log \left (\frac{2 i a c (a x-1) \left (\sqrt{a^2 c d}+i d\right )}{\left (a^2 c+d\right ) \left (a c+i x \sqrt{a^2 c d}\right )}\right ) \left (2 \tan ^{-1}\left (\frac{a d x}{\sqrt{a^2 c d}}\right )+\cos ^{-1}\left (\frac{d-a^2 c}{a^2 c+d}\right )\right )-\log \left (\frac{2 a c (a x+1) \left (d+i \sqrt{a^2 c d}\right )}{\left (a^2 c+d\right ) \left (a c+i x \sqrt{a^2 c d}\right )}\right ) \left (\cos ^{-1}\left (\frac{d-a^2 c}{a^2 c+d}\right )-2 \tan ^{-1}\left (\frac{a d x}{\sqrt{a^2 c d}}\right )\right )+\left (2 \left (\tan ^{-1}\left (\frac{a c}{x \sqrt{a^2 c d}}\right )+\tan ^{-1}\left (\frac{a d x}{\sqrt{a^2 c d}}\right )\right )+\cos ^{-1}\left (\frac{d-a^2 c}{a^2 c+d}\right )\right ) \log \left (\frac{\sqrt{2} \sqrt{a^2 c d} e^{-\tanh ^{-1}(a x)}}{\sqrt{a^2 c+d} \sqrt{\left (a^2 c+d\right ) \cosh \left (2 \tanh ^{-1}(a x)\right )+a^2 c-d}}\right )+\left (\cos ^{-1}\left (\frac{d-a^2 c}{a^2 c+d}\right )-2 \left (\tan ^{-1}\left (\frac{a c}{x \sqrt{a^2 c d}}\right )+\tan ^{-1}\left (\frac{a d x}{\sqrt{a^2 c d}}\right )\right )\right ) \log \left (\frac{\sqrt{2} \sqrt{a^2 c d} e^{\tanh ^{-1}(a x)}}{\sqrt{a^2 c+d} \sqrt{\left (a^2 c+d\right ) \cosh \left (2 \tanh ^{-1}(a x)\right )+a^2 c-d}}\right )\right )}{4 \sqrt{a^2 c d}} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcTanh[a*x]/(c + d*x^2),x]

[Out]

-(a*((-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*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*ArcTan
h[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))])))/(4*Sqrt[a^2*c*d])

________________________________________________________________________________________

Maple [B]  time = 0.197, size = 833, normalized size = 1.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a^3/d/(a^4*c^2+2*a^2*c*d+d^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+a/(a^4*c^2+2*a^2*c*d+d^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)-1/2*a^3/d/(a^4*c^2+2*a^2*c*d+d^2)*arctanh(a*x)^2*(-a^2*c*d)^(1/2)*c-a/
(a^4*c^2+2*a^2*c*d+d^2)*arctanh(a*x)^2*(-a^2*c*d)^(1/2)+1/4*a^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))*(-a^2*c*d)^(1/2)*c+1/2*a/(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))*(-a^2*c*d)^(1/2)+1/2/a/c/(a^4*c^2+2*a^2*c*d
+d^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-1/2
/a/c/(a^4*c^2+2*a^2*c*d+d^2)*arctanh(a*x)^2*(-a^2*c*d)^(1/2)*d+1/4/a/c/(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))*(-a^2*c*d)^(1/2)*d-1/2/a*(-a^2*c*d)^(1/2)/c/d*arcta
nh(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))+1/2/a*(-a^2*c*d)^(1/2)/c/d*arctan
h(a*x)^2-1/4/a*(-a^2*c*d)^(1/2)/c/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))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(arctanh(a*x)/(d*x^2 + c), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(atanh(a*x)/(c + d*x**2), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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