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

Optimal. Leaf size=429 \[ -\frac {\text {Li}_2\left (-\frac {\sqrt {d} (1-a x)}{a \sqrt {-c}-\sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {\text {Li}_2\left (\frac {\sqrt {d} (1-a x)}{\sqrt {-c} a+\sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}-\frac {\text {Li}_2\left (-\frac {\sqrt {d} (a x+1)}{a \sqrt {-c}-\sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {\text {Li}_2\left (\frac {\sqrt {d} (a x+1)}{\sqrt {-c} a+\sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}-\frac {\log (1-a x) \log \left (\frac {a \left (\sqrt {-c}-\sqrt {d} x\right )}{a \sqrt {-c}-\sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {\log (a x+1) \log \left (\frac {a \left (\sqrt {-c}-\sqrt {d} x\right )}{a \sqrt {-c}+\sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}-\frac {\log (a x+1) \log \left (\frac {a \left (\sqrt {-c}+\sqrt {d} x\right )}{a \sqrt {-c}-\sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {\log (1-a x) \log \left (\frac {a \left (\sqrt {-c}+\sqrt {d} x\right )}{a \sqrt {-c}+\sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}} \]

[Out]

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

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Rubi [A]  time = 0.44, antiderivative size = 429, normalized size of antiderivative = 1.00, 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 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 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]

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*}

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Mathematica [C]  time = 1.46, size = 662, normalized size = 1.54 \[ -\frac {a \left (i \left (\text {Li}_2\left (\frac {\left (-c a^2+d+2 i \sqrt {a^2 c d}\right ) \left (i a c+\sqrt {a^2 c d} x\right )}{\left (c a^2+d\right ) \left (\sqrt {a^2 c d} x-i a c\right )}\right )-\text {Li}_2\left (\frac {\left (-c a^2+d-2 i \sqrt {a^2 c d}\right ) \left (i a c+\sqrt {a^2 c d} x\right )}{\left (c a^2+d\right ) \left (\sqrt {a^2 c d} x-i a c\right )}\right )\right )-2 i \cos ^{-1}\left (\frac {d-a^2 c}{a^2 c+d}\right ) \tan ^{-1}\left (\frac {a d x}{\sqrt {a^2 c d}}\right )+4 \tanh ^{-1}(a x) \tan ^{-1}\left (\frac {a c}{x \sqrt {a^2 c d}}\right )-\log \left (\frac {2 i a c (a x-1) \left (\sqrt {a^2 c d}+i d\right )}{\left (a^2 c+d\right ) \left (a c+i x \sqrt {a^2 c d}\right )}\right ) \left (2 \tan ^{-1}\left (\frac {a d x}{\sqrt {a^2 c d}}\right )+\cos ^{-1}\left (\frac {d-a^2 c}{a^2 c+d}\right )\right )-\log \left (\frac {2 a c (a x+1) \left (d+i \sqrt {a^2 c d}\right )}{\left (a^2 c+d\right ) \left (a c+i x \sqrt {a^2 c d}\right )}\right ) \left (\cos ^{-1}\left (\frac {d-a^2 c}{a^2 c+d}\right )-2 \tan ^{-1}\left (\frac {a d x}{\sqrt {a^2 c d}}\right )\right )+\left (2 \left (\tan ^{-1}\left (\frac {a c}{x \sqrt {a^2 c d}}\right )+\tan ^{-1}\left (\frac {a d x}{\sqrt {a^2 c d}}\right )\right )+\cos ^{-1}\left (\frac {d-a^2 c}{a^2 c+d}\right )\right ) \log \left (\frac {\sqrt {2} \sqrt {a^2 c d} e^{-\tanh ^{-1}(a x)}}{\sqrt {a^2 c+d} \sqrt {\left (a^2 c+d\right ) \cosh \left (2 \tanh ^{-1}(a x)\right )+a^2 c-d}}\right )+\left (\cos ^{-1}\left (\frac {d-a^2 c}{a^2 c+d}\right )-2 \left (\tan ^{-1}\left (\frac {a c}{x \sqrt {a^2 c d}}\right )+\tan ^{-1}\left (\frac {a d x}{\sqrt {a^2 c d}}\right )\right )\right ) \log \left (\frac {\sqrt {2} \sqrt {a^2 c d} e^{\tanh ^{-1}(a x)}}{\sqrt {a^2 c+d} \sqrt {\left (a^2 c+d\right ) \cosh \left (2 \tanh ^{-1}(a x)\right )+a^2 c-d}}\right )\right )}{4 \sqrt {a^2 c d}} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcTanh[a*x]/(c + d*x^2),x]

[Out]

-1/4*(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*ArcT
anh[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*Ar
cTanh[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]

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fricas [F]  time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {artanh}\left (a x\right )}{d x^{2} + c}, x\right ) \]

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)

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

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)

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maple [B]  time = 0.62, size = 833, normalized size = 1.94 \[ \frac {a^{3} \ln \left (1-\frac {\left (a^{2} c +d \right ) \left (a x +1\right )^{2}}{\left (-a^{2} x^{2}+1\right ) \left (-a^{2} c -2 \sqrt {-a^{2} c d}+d \right )}\right ) \arctanh \left (a x \right ) \sqrt {-a^{2} c d}\, c}{2 d \left (a^{4} c^{2}+2 a^{2} c d +d^{2}\right )}+\frac {a \ln \left (1-\frac {\left (a^{2} c +d \right ) \left (a x +1\right )^{2}}{\left (-a^{2} x^{2}+1\right ) \left (-a^{2} c -2 \sqrt {-a^{2} c d}+d \right )}\right ) \arctanh \left (a x \right ) \sqrt {-a^{2} c d}}{a^{4} c^{2}+2 a^{2} c d +d^{2}}-\frac {a^{3} \arctanh \left (a x \right )^{2} \sqrt {-a^{2} c d}\, c}{2 d \left (a^{4} c^{2}+2 a^{2} c d +d^{2}\right )}-\frac {a \arctanh \left (a x \right )^{2} \sqrt {-a^{2} c d}}{a^{4} c^{2}+2 a^{2} c d +d^{2}}+\frac {a^{3} \polylog \left (2, \frac {\left (a^{2} c +d \right ) \left (a x +1\right )^{2}}{\left (-a^{2} x^{2}+1\right ) \left (-a^{2} c -2 \sqrt {-a^{2} c d}+d \right )}\right ) \sqrt {-a^{2} c d}\, c}{4 d \left (a^{4} c^{2}+2 a^{2} c d +d^{2}\right )}+\frac {a \polylog \left (2, \frac {\left (a^{2} c +d \right ) \left (a x +1\right )^{2}}{\left (-a^{2} x^{2}+1\right ) \left (-a^{2} c -2 \sqrt {-a^{2} c d}+d \right )}\right ) \sqrt {-a^{2} c d}}{2 a^{4} c^{2}+4 a^{2} c d +2 d^{2}}+\frac {\ln \left (1-\frac {\left (a^{2} c +d \right ) \left (a x +1\right )^{2}}{\left (-a^{2} x^{2}+1\right ) \left (-a^{2} c -2 \sqrt {-a^{2} c d}+d \right )}\right ) \arctanh \left (a x \right ) \sqrt {-a^{2} c d}\, d}{2 a c \left (a^{4} c^{2}+2 a^{2} c d +d^{2}\right )}-\frac {\arctanh \left (a x \right )^{2} \sqrt {-a^{2} c d}\, d}{2 a c \left (a^{4} c^{2}+2 a^{2} c d +d^{2}\right )}+\frac {\polylog \left (2, \frac {\left (a^{2} c +d \right ) \left (a x +1\right )^{2}}{\left (-a^{2} x^{2}+1\right ) \left (-a^{2} c -2 \sqrt {-a^{2} c d}+d \right )}\right ) \sqrt {-a^{2} c d}\, d}{4 a c \left (a^{4} c^{2}+2 a^{2} c d +d^{2}\right )}-\frac {\sqrt {-a^{2} c d}\, \arctanh \left (a x \right ) \ln \left (1-\frac {\left (a^{2} c +d \right ) \left (a x +1\right )^{2}}{\left (-a^{2} x^{2}+1\right ) \left (-a^{2} c +2 \sqrt {-a^{2} c d}+d \right )}\right )}{2 a c d}+\frac {\sqrt {-a^{2} c d}\, \arctanh \left (a x \right )^{2}}{2 a c d}-\frac {\sqrt {-a^{2} c d}\, \polylog \left (2, \frac {\left (a^{2} c +d \right ) \left (a x +1\right )^{2}}{\left (-a^{2} x^{2}+1\right ) \left (-a^{2} c +2 \sqrt {-a^{2} c d}+d \right )}\right )}{4 a c d} \]

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))

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maxima [C]  time = 0.57, size = 406, normalized size = 0.95 \[ \frac {\arctan \left (\frac {d x}{\sqrt {c d}}\right ) \operatorname {artanh}\left (a x\right )}{\sqrt {c d}} + \frac {{\left (\arctan \left (\frac {{\left (a^{2} x + a\right )} \sqrt {c} \sqrt {d}}{a^{2} c + d}, \frac {a d x + d}{a^{2} c + d}\right ) - \arctan \left (\frac {{\left (a^{2} x - a\right )} \sqrt {c} \sqrt {d}}{a^{2} c + d}, -\frac {a d x - d}{a^{2} c + d}\right )\right )} \log \left (d x^{2} + c\right ) - \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {a^{2} d x^{2} + 2 \, a d x + d}{a^{2} c + d}\right ) + \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {a^{2} d x^{2} - 2 \, a d x + d}{a^{2} c + d}\right ) - i \, {\rm Li}_2\left (\frac {a^{2} c + a d x - {\left (i \, a^{2} x - i \, a\right )} \sqrt {c} \sqrt {d}}{a^{2} c + 2 i \, a \sqrt {c} \sqrt {d} - d}\right ) - i \, {\rm Li}_2\left (\frac {a^{2} c - a d x + {\left (i \, a^{2} x + i \, a\right )} \sqrt {c} \sqrt {d}}{a^{2} c + 2 i \, a \sqrt {c} \sqrt {d} - d}\right ) + i \, {\rm Li}_2\left (\frac {a^{2} c + a d x + {\left (i \, a^{2} x - i \, a\right )} \sqrt {c} \sqrt {d}}{a^{2} c - 2 i \, a \sqrt {c} \sqrt {d} - d}\right ) + i \, {\rm Li}_2\left (\frac {a^{2} c - a d x - {\left (i \, a^{2} x + i \, a\right )} \sqrt {c} \sqrt {d}}{a^{2} c - 2 i \, a \sqrt {c} \sqrt {d} - d}\right )}{4 \, \sqrt {c d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(d*x/sqrt(c*d))*arctanh(a*x)/sqrt(c*d) + 1/4*((arctan2((a^2*x + a)*sqrt(c)*sqrt(d)/(a^2*c + d), (a*d*x +
d)/(a^2*c + d)) - 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)
- arctan(sqrt(d)*x/sqrt(c))*log((a^2*d*x^2 + 2*a*d*x + d)/(a^2*c + d)) + arctan(sqrt(d)*x/sqrt(c))*log((a^2*d*
x^2 - 2*a*d*x + d)/(a^2*c + d)) - I*dilog((a^2*c + a*d*x - (I*a^2*x - I*a)*sqrt(c)*sqrt(d))/(a^2*c + 2*I*a*sqr
t(c)*sqrt(d) - d)) - I*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*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*dil
og((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)))/sqrt(c*d)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(atanh(a*x)/(c + d*x^2),x)

[Out]

int(atanh(a*x)/(c + d*x^2), x)

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

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)

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