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

Optimal. Leaf size=481 \[ -\frac{\text{PolyLog}\left (2,-\frac{\sqrt{d} (-a-b x+1)}{b \sqrt{-c}-(1-a) \sqrt{d}}\right )}{4 \sqrt{-c} \sqrt{d}}+\frac{\text{PolyLog}\left (2,\frac{\sqrt{d} (-a-b x+1)}{(1-a) \sqrt{d}+b \sqrt{-c}}\right )}{4 \sqrt{-c} \sqrt{d}}-\frac{\text{PolyLog}\left (2,-\frac{\sqrt{d} (a+b x+1)}{b \sqrt{-c}-(a+1) \sqrt{d}}\right )}{4 \sqrt{-c} \sqrt{d}}+\frac{\text{PolyLog}\left (2,\frac{\sqrt{d} (a+b x+1)}{(a+1) \sqrt{d}+b \sqrt{-c}}\right )}{4 \sqrt{-c} \sqrt{d}}-\frac{\log (-a-b x+1) \log \left (\frac{b \left (\sqrt{-c}-\sqrt{d} x\right )}{b \sqrt{-c}-(1-a) \sqrt{d}}\right )}{4 \sqrt{-c} \sqrt{d}}+\frac{\log (a+b x+1) \log \left (\frac{b \left (\sqrt{-c}-\sqrt{d} x\right )}{(a+1) \sqrt{d}+b \sqrt{-c}}\right )}{4 \sqrt{-c} \sqrt{d}}+\frac{\log (-a-b x+1) \log \left (\frac{b \left (\sqrt{-c}+\sqrt{d} x\right )}{(1-a) \sqrt{d}+b \sqrt{-c}}\right )}{4 \sqrt{-c} \sqrt{d}}-\frac{\log (a+b x+1) \log \left (\frac{b \left (\sqrt{-c}+\sqrt{d} x\right )}{b \sqrt{-c}-(a+1) \sqrt{d}}\right )}{4 \sqrt{-c} \sqrt{d}} \]

[Out]

-(Log[1 - a - b*x]*Log[(b*(Sqrt[-c] - Sqrt[d]*x))/(b*Sqrt[-c] - (1 - a)*Sqrt[d])])/(4*Sqrt[-c]*Sqrt[d]) + (Log
[1 + a + b*x]*Log[(b*(Sqrt[-c] - Sqrt[d]*x))/(b*Sqrt[-c] + (1 + a)*Sqrt[d])])/(4*Sqrt[-c]*Sqrt[d]) + (Log[1 -
a - b*x]*Log[(b*(Sqrt[-c] + Sqrt[d]*x))/(b*Sqrt[-c] + (1 - a)*Sqrt[d])])/(4*Sqrt[-c]*Sqrt[d]) - (Log[1 + a + b
*x]*Log[(b*(Sqrt[-c] + Sqrt[d]*x))/(b*Sqrt[-c] - (1 + a)*Sqrt[d])])/(4*Sqrt[-c]*Sqrt[d]) - PolyLog[2, -((Sqrt[
d]*(1 - a - b*x))/(b*Sqrt[-c] - (1 - a)*Sqrt[d]))]/(4*Sqrt[-c]*Sqrt[d]) + PolyLog[2, (Sqrt[d]*(1 - a - b*x))/(
b*Sqrt[-c] + (1 - a)*Sqrt[d])]/(4*Sqrt[-c]*Sqrt[d]) - PolyLog[2, -((Sqrt[d]*(1 + a + b*x))/(b*Sqrt[-c] - (1 +
a)*Sqrt[d]))]/(4*Sqrt[-c]*Sqrt[d]) + PolyLog[2, (Sqrt[d]*(1 + a + b*x))/(b*Sqrt[-c] + (1 + a)*Sqrt[d])]/(4*Sqr
t[-c]*Sqrt[d])

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Rubi [A]  time = 0.605511, antiderivative size = 481, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {6115, 2409, 2394, 2393, 2391} \[ -\frac{\text{PolyLog}\left (2,-\frac{\sqrt{d} (-a-b x+1)}{b \sqrt{-c}-(1-a) \sqrt{d}}\right )}{4 \sqrt{-c} \sqrt{d}}+\frac{\text{PolyLog}\left (2,\frac{\sqrt{d} (-a-b x+1)}{(1-a) \sqrt{d}+b \sqrt{-c}}\right )}{4 \sqrt{-c} \sqrt{d}}-\frac{\text{PolyLog}\left (2,-\frac{\sqrt{d} (a+b x+1)}{b \sqrt{-c}-(a+1) \sqrt{d}}\right )}{4 \sqrt{-c} \sqrt{d}}+\frac{\text{PolyLog}\left (2,\frac{\sqrt{d} (a+b x+1)}{(a+1) \sqrt{d}+b \sqrt{-c}}\right )}{4 \sqrt{-c} \sqrt{d}}-\frac{\log (-a-b x+1) \log \left (\frac{b \left (\sqrt{-c}-\sqrt{d} x\right )}{b \sqrt{-c}-(1-a) \sqrt{d}}\right )}{4 \sqrt{-c} \sqrt{d}}+\frac{\log (a+b x+1) \log \left (\frac{b \left (\sqrt{-c}-\sqrt{d} x\right )}{(a+1) \sqrt{d}+b \sqrt{-c}}\right )}{4 \sqrt{-c} \sqrt{d}}+\frac{\log (-a-b x+1) \log \left (\frac{b \left (\sqrt{-c}+\sqrt{d} x\right )}{(1-a) \sqrt{d}+b \sqrt{-c}}\right )}{4 \sqrt{-c} \sqrt{d}}-\frac{\log (a+b x+1) \log \left (\frac{b \left (\sqrt{-c}+\sqrt{d} x\right )}{b \sqrt{-c}-(a+1) \sqrt{d}}\right )}{4 \sqrt{-c} \sqrt{d}} \]

Antiderivative was successfully verified.

[In]

Int[ArcTanh[a + b*x]/(c + d*x^2),x]

[Out]

-(Log[1 - a - b*x]*Log[(b*(Sqrt[-c] - Sqrt[d]*x))/(b*Sqrt[-c] - (1 - a)*Sqrt[d])])/(4*Sqrt[-c]*Sqrt[d]) + (Log
[1 + a + b*x]*Log[(b*(Sqrt[-c] - Sqrt[d]*x))/(b*Sqrt[-c] + (1 + a)*Sqrt[d])])/(4*Sqrt[-c]*Sqrt[d]) + (Log[1 -
a - b*x]*Log[(b*(Sqrt[-c] + Sqrt[d]*x))/(b*Sqrt[-c] + (1 - a)*Sqrt[d])])/(4*Sqrt[-c]*Sqrt[d]) - (Log[1 + a + b
*x]*Log[(b*(Sqrt[-c] + Sqrt[d]*x))/(b*Sqrt[-c] - (1 + a)*Sqrt[d])])/(4*Sqrt[-c]*Sqrt[d]) - PolyLog[2, -((Sqrt[
d]*(1 - a - b*x))/(b*Sqrt[-c] - (1 - a)*Sqrt[d]))]/(4*Sqrt[-c]*Sqrt[d]) + PolyLog[2, (Sqrt[d]*(1 - a - b*x))/(
b*Sqrt[-c] + (1 - a)*Sqrt[d])]/(4*Sqrt[-c]*Sqrt[d]) - PolyLog[2, -((Sqrt[d]*(1 + a + b*x))/(b*Sqrt[-c] - (1 +
a)*Sqrt[d]))]/(4*Sqrt[-c]*Sqrt[d]) + PolyLog[2, (Sqrt[d]*(1 + a + b*x))/(b*Sqrt[-c] + (1 + a)*Sqrt[d])]/(4*Sqr
t[-c]*Sqrt[d])

Rule 6115

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

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

Mathematica [A]  time = 0.29092, size = 365, normalized size = 0.76 \[ \frac{\text{PolyLog}\left (2,-\frac{\sqrt{d} (a+b x-1)}{b \sqrt{-c}-(a-1) \sqrt{d}}\right )-\text{PolyLog}\left (2,\frac{\sqrt{d} (a+b x-1)}{(a-1) \sqrt{d}+b \sqrt{-c}}\right )-\text{PolyLog}\left (2,-\frac{\sqrt{d} (a+b x+1)}{b \sqrt{-c}-(a+1) \sqrt{d}}\right )+\text{PolyLog}\left (2,\frac{\sqrt{d} (a+b x+1)}{(a+1) \sqrt{d}+b \sqrt{-c}}\right )-\log (-a-b x+1) \log \left (\frac{b \left (\sqrt{-c}-\sqrt{d} x\right )}{(a-1) \sqrt{d}+b \sqrt{-c}}\right )+\log (a+b x+1) \log \left (\frac{b \left (\sqrt{-c}-\sqrt{d} x\right )}{(a+1) \sqrt{d}+b \sqrt{-c}}\right )+\log (-a-b x+1) \log \left (\frac{b \left (\sqrt{-c}+\sqrt{d} x\right )}{b \sqrt{-c}-(a-1) \sqrt{d}}\right )-\log (a+b x+1) \log \left (\frac{b \left (\sqrt{-c}+\sqrt{d} x\right )}{b \sqrt{-c}-(a+1) \sqrt{d}}\right )}{4 \sqrt{-c} \sqrt{d}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(Log[1 - a - b*x]*Log[(b*(Sqrt[-c] - Sqrt[d]*x))/(b*Sqrt[-c] + (-1 + a)*Sqrt[d])]) + Log[1 + a + b*x]*Log[(b
*(Sqrt[-c] - Sqrt[d]*x))/(b*Sqrt[-c] + (1 + a)*Sqrt[d])] + Log[1 - a - b*x]*Log[(b*(Sqrt[-c] + Sqrt[d]*x))/(b*
Sqrt[-c] - (-1 + a)*Sqrt[d])] - Log[1 + a + b*x]*Log[(b*(Sqrt[-c] + Sqrt[d]*x))/(b*Sqrt[-c] - (1 + a)*Sqrt[d])
] + PolyLog[2, -((Sqrt[d]*(-1 + a + b*x))/(b*Sqrt[-c] - (-1 + a)*Sqrt[d]))] - PolyLog[2, (Sqrt[d]*(-1 + a + b*
x))/(b*Sqrt[-c] + (-1 + a)*Sqrt[d])] - PolyLog[2, -((Sqrt[d]*(1 + a + b*x))/(b*Sqrt[-c] - (1 + a)*Sqrt[d]))] +
PolyLog[2, (Sqrt[d]*(1 + a + b*x))/(b*Sqrt[-c] + (1 + a)*Sqrt[d])])/(4*Sqrt[-c]*Sqrt[d])

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Maple [B]  time = 0.352, size = 1300, normalized size = 2.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/b*(-b^2*c*d)^(1/2)/c*ln(1-(a^2*d+b^2*c-2*a*d+d)*(b*x+a+1)^2/(1-(b*x+a)^2)/(-a^2*d-b^2*c-2*(-b^2*c*d)^(1/2)
+d))*arctanh(b*x+a)*a^2/(a^2*d+b^2*c+2*(-b^2*c*d)^(1/2)-d)+1/2*b*(-b^2*c*d)^(1/2)/d*ln(1-(a^2*d+b^2*c-2*a*d+d)
*(b*x+a+1)^2/(1-(b*x+a)^2)/(-a^2*d-b^2*c-2*(-b^2*c*d)^(1/2)+d))*arctanh(b*x+a)/(a^2*d+b^2*c+2*(-b^2*c*d)^(1/2)
-d)-1/2/b*(-b^2*c*d)^(1/2)/c*arctanh(b*x+a)^2*a^2/(a^2*d+b^2*c+2*(-b^2*c*d)^(1/2)-d)-1/2*b*(-b^2*c*d)^(1/2)/d*
arctanh(b*x+a)^2/(a^2*d+b^2*c+2*(-b^2*c*d)^(1/2)-d)+1/4/b*(-b^2*c*d)^(1/2)/c*polylog(2,(a^2*d+b^2*c-2*a*d+d)*(
b*x+a+1)^2/(1-(b*x+a)^2)/(-a^2*d-b^2*c-2*(-b^2*c*d)^(1/2)+d))*a^2/(a^2*d+b^2*c+2*(-b^2*c*d)^(1/2)-d)+1/4*b*(-b
^2*c*d)^(1/2)/d*polylog(2,(a^2*d+b^2*c-2*a*d+d)*(b*x+a+1)^2/(1-(b*x+a)^2)/(-a^2*d-b^2*c-2*(-b^2*c*d)^(1/2)+d))
/(a^2*d+b^2*c+2*(-b^2*c*d)^(1/2)-d)-b/(a^2*d+b^2*c+2*(-b^2*c*d)^(1/2)-d)*ln(1-(a^2*d+b^2*c-2*a*d+d)*(b*x+a+1)^
2/(1-(b*x+a)^2)/(-a^2*d-b^2*c-2*(-b^2*c*d)^(1/2)+d))*arctanh(b*x+a)-1/2/b*(-b^2*c*d)^(1/2)/c*ln(1-(a^2*d+b^2*c
-2*a*d+d)*(b*x+a+1)^2/(1-(b*x+a)^2)/(-a^2*d-b^2*c-2*(-b^2*c*d)^(1/2)+d))*arctanh(b*x+a)/(a^2*d+b^2*c+2*(-b^2*c
*d)^(1/2)-d)+b/(a^2*d+b^2*c+2*(-b^2*c*d)^(1/2)-d)*arctanh(b*x+a)^2+1/2/b*(-b^2*c*d)^(1/2)/c*arctanh(b*x+a)^2/(
a^2*d+b^2*c+2*(-b^2*c*d)^(1/2)-d)-1/2*b/(a^2*d+b^2*c+2*(-b^2*c*d)^(1/2)-d)*polylog(2,(a^2*d+b^2*c-2*a*d+d)*(b*
x+a+1)^2/(1-(b*x+a)^2)/(-a^2*d-b^2*c-2*(-b^2*c*d)^(1/2)+d))-1/4/b*(-b^2*c*d)^(1/2)/c*polylog(2,(a^2*d+b^2*c-2*
a*d+d)*(b*x+a+1)^2/(1-(b*x+a)^2)/(-a^2*d-b^2*c-2*(-b^2*c*d)^(1/2)+d))/(a^2*d+b^2*c+2*(-b^2*c*d)^(1/2)-d)-1/2/b
*(-b^2*c*d)^(1/2)/c/d*arctanh(b*x+a)*ln(1-(a^2*d+b^2*c-2*a*d+d)*(b*x+a+1)^2/(1-(b*x+a)^2)/(-a^2*d-b^2*c+2*(-b^
2*c*d)^(1/2)+d))+1/2/b*(-b^2*c*d)^(1/2)/c/d*arctanh(b*x+a)^2-1/4/b*(-b^2*c*d)^(1/2)/c/d*polylog(2,(a^2*d+b^2*c
-2*a*d+d)*(b*x+a+1)^2/(1-(b*x+a)^2)/(-a^2*d-b^2*c+2*(-b^2*c*d)^(1/2)+d))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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