\(\int \frac {\text {arctanh}(a x)}{c+d x^2} \, dx\) [502]

Optimal result

Integrand size = 14, antiderivative size = 429 \[ \int \frac {\text {arctanh}(a x)}{c+d x^2} \, dx=-\frac {\log (1-a x) \log \left (\frac {a \left (\sqrt {-c}-\sqrt {d} x\right )}{a \sqrt {-c}-\sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {\log (1+a x) \log \left (\frac {a \left (\sqrt {-c}-\sqrt {d} x\right )}{a \sqrt {-c}+\sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}-\frac {\log (1+a x) \log \left (\frac {a \left (\sqrt {-c}+\sqrt {d} x\right )}{a \sqrt {-c}-\sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {\log (1-a x) \log \left (\frac {a \left (\sqrt {-c}+\sqrt {d} x\right )}{a \sqrt {-c}+\sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}-\frac {\operatorname {PolyLog}\left (2,-\frac {\sqrt {d} (1-a x)}{a \sqrt {-c}-\sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {d} (1-a x)}{a \sqrt {-c}+\sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}-\frac {\operatorname {PolyLog}\left (2,-\frac {\sqrt {d} (1+a x)}{a \sqrt {-c}-\sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {d} (1+a x)}{a \sqrt {-c}+\sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}} \] Output:

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

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 1.22 (sec) , antiderivative size = 662, normalized size of antiderivative = 1.54 \[ \int \frac {\text {arctanh}(a x)}{c+d x^2} \, dx=-\frac {a \left (-2 i \arccos \left (\frac {-a^2 c+d}{a^2 c+d}\right ) \arctan \left (\frac {a d x}{\sqrt {a^2 c d}}\right )+4 \arctan \left (\frac {a c}{\sqrt {a^2 c d} x}\right ) \text {arctanh}(a x)-\left (\arccos \left (\frac {-a^2 c+d}{a^2 c+d}\right )+2 \arctan \left (\frac {a d x}{\sqrt {a^2 c d}}\right )\right ) \log \left (\frac {2 i a c \left (i d+\sqrt {a^2 c d}\right ) (-1+a x)}{\left (a^2 c+d\right ) \left (a c+i \sqrt {a^2 c d} x\right )}\right )-\left (\arccos \left (\frac {-a^2 c+d}{a^2 c+d}\right )-2 \arctan \left (\frac {a d x}{\sqrt {a^2 c d}}\right )\right ) \log \left (\frac {2 a c \left (d+i \sqrt {a^2 c d}\right ) (1+a x)}{\left (a^2 c+d\right ) \left (a c+i \sqrt {a^2 c d} x\right )}\right )+\left (\arccos \left (\frac {-a^2 c+d}{a^2 c+d}\right )+2 \left (\arctan \left (\frac {a c}{\sqrt {a^2 c d} x}\right )+\arctan \left (\frac {a d x}{\sqrt {a^2 c d}}\right )\right )\right ) \log \left (\frac {\sqrt {2} \sqrt {a^2 c d} e^{-\text {arctanh}(a x)}}{\sqrt {a^2 c+d} \sqrt {a^2 c-d+\left (a^2 c+d\right ) \cosh (2 \text {arctanh}(a x))}}\right )+\left (\arccos \left (\frac {-a^2 c+d}{a^2 c+d}\right )-2 \left (\arctan \left (\frac {a c}{\sqrt {a^2 c d} x}\right )+\arctan \left (\frac {a d x}{\sqrt {a^2 c d}}\right )\right )\right ) \log \left (\frac {\sqrt {2} \sqrt {a^2 c d} e^{\text {arctanh}(a x)}}{\sqrt {a^2 c+d} \sqrt {a^2 c-d+\left (a^2 c+d\right ) \cosh (2 \text {arctanh}(a x))}}\right )+i \left (-\operatorname {PolyLog}\left (2,\frac {\left (-a^2 c+d-2 i \sqrt {a^2 c d}\right ) \left (i a c+\sqrt {a^2 c d} x\right )}{\left (a^2 c+d\right ) \left (-i a c+\sqrt {a^2 c d} x\right )}\right )+\operatorname {PolyLog}\left (2,\frac {\left (-a^2 c+d+2 i \sqrt {a^2 c d}\right ) \left (i a c+\sqrt {a^2 c d} x\right )}{\left (a^2 c+d\right ) \left (-i a c+\sqrt {a^2 c d} x\right )}\right )\right )\right )}{4 \sqrt {a^2 c d}} \] Input:

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

Output:

-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])/(Sqr 
t[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^Arc 
Tanh[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 + Sqr 
t[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]
 

Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 439, normalized size of antiderivative = 1.02, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {6534, 2856, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\text {arctanh}(a x)}{c+d x^2} \, dx\)

\(\Big \downarrow \) 6534

\(\displaystyle \frac {1}{2} \int \frac {\log (a x+1)}{d x^2+c}dx-\frac {1}{2} \int \frac {\log (1-a x)}{d x^2+c}dx\)

\(\Big \downarrow \) 2856

\(\displaystyle \frac {1}{2} \int \left (\frac {\sqrt {-c} \log (a x+1)}{2 c \left (\sqrt {-c}-\sqrt {d} x\right )}+\frac {\sqrt {-c} \log (a x+1)}{2 c \left (\sqrt {d} x+\sqrt {-c}\right )}\right )dx-\frac {1}{2} \int \left (\frac {\sqrt {-c} \log (1-a x)}{2 c \left (\sqrt {-c}-\sqrt {d} x\right )}+\frac {\sqrt {-c} \log (1-a x)}{2 c \left (\sqrt {d} x+\sqrt {-c}\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} \left (-\frac {\operatorname {PolyLog}\left (2,-\frac {\sqrt {d} (1-a x)}{a \sqrt {-c}-\sqrt {d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {d} (1-a x)}{\sqrt {-c} a+\sqrt {d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\log (1-a x) \log \left (\frac {a \left (\sqrt {-c}-\sqrt {d} x\right )}{a \sqrt {-c}-\sqrt {d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\log (1-a x) \log \left (\frac {a \left (\sqrt {-c}+\sqrt {d} x\right )}{a \sqrt {-c}+\sqrt {d}}\right )}{2 \sqrt {-c} \sqrt {d}}\right )+\frac {1}{2} \left (-\frac {\operatorname {PolyLog}\left (2,-\frac {\sqrt {d} (a x+1)}{a \sqrt {-c}-\sqrt {d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {d} (a x+1)}{\sqrt {-c} a+\sqrt {d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\log (a x+1) \log \left (\frac {a \left (\sqrt {-c}-\sqrt {d} x\right )}{a \sqrt {-c}+\sqrt {d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\log (a x+1) \log \left (\frac {a \left (\sqrt {-c}+\sqrt {d} x\right )}{a \sqrt {-c}-\sqrt {d}}\right )}{2 \sqrt {-c} \sqrt {d}}\right )\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2856

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_. 
)*(x_)^(r_))^(q_.), x_Symbol] :> Int[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] && I 
GtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[r] && NeQ[r, 1]))
 

rule 6534

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

Maple [A] (verified)

Time = 3.10 (sec) , antiderivative size = 364, normalized size of antiderivative = 0.85

method	result	size
risch	\(\frac {\ln \left (-a x +1\right ) \ln \left (\frac {a \sqrt {-c d}-d \left (-a x +1\right )+d}{a \sqrt {-c d}+d}\right )}{4 \sqrt {-c d}}-\frac {\ln \left (-a x +1\right ) \ln \left (\frac {a \sqrt {-c d}+d \left (-a x +1\right )-d}{a \sqrt {-c d}-d}\right )}{4 \sqrt {-c d}}+\frac {\operatorname {dilog}\left (\frac {a \sqrt {-c d}-d \left (-a x +1\right )+d}{a \sqrt {-c d}+d}\right )}{4 \sqrt {-c d}}-\frac {\operatorname {dilog}\left (\frac {a \sqrt {-c d}+d \left (-a x +1\right )-d}{a \sqrt {-c d}-d}\right )}{4 \sqrt {-c d}}+\frac {\ln \left (a x +1\right ) \ln \left (\frac {a \sqrt {-c d}-d \left (a x +1\right )+d}{a \sqrt {-c d}+d}\right )}{4 \sqrt {-c d}}-\frac {\ln \left (a x +1\right ) \ln \left (\frac {a \sqrt {-c d}+d \left (a x +1\right )-d}{a \sqrt {-c d}-d}\right )}{4 \sqrt {-c d}}+\frac {\operatorname {dilog}\left (\frac {a \sqrt {-c d}-d \left (a x +1\right )+d}{a \sqrt {-c d}+d}\right )}{4 \sqrt {-c d}}-\frac {\operatorname {dilog}\left (\frac {a \sqrt {-c d}+d \left (a x +1\right )-d}{a \sqrt {-c d}-d}\right )}{4 \sqrt {-c d}}\)	\(364\)
derivativedivides	\(\text {Expression too large to display}\)	\(1002\)
default	\(\text {Expression too large to display}\)	\(1002\)

Input:

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

Output:

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

Fricas [F]

\[ \int \frac {\text {arctanh}(a x)}{c+d x^2} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )}{d x^{2} + c} \,d x } \] Input:

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

Output:

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

Sympy [F]

\[ \int \frac {\text {arctanh}(a x)}{c+d x^2} \, dx=\int \frac {\operatorname {atanh}{\left (a x \right )}}{c + d x^{2}}\, dx \] Input:

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

Output:

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

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.20 (sec) , antiderivative size = 406, normalized size of antiderivative = 0.95 \[ \int \frac {\text {arctanh}(a x)}{c+d x^2} \, dx=\frac {\arctan \left (\frac {d x}{\sqrt {c d}}\right ) \operatorname {artanh}\left (a x\right )}{\sqrt {c d}} + \frac {{\left (\arctan \left (\frac {{\left (a^{2} x + a\right )} \sqrt {c} \sqrt {d}}{a^{2} c + d}, \frac {a d x + d}{a^{2} c + d}\right ) - \arctan \left (\frac {{\left (a^{2} x - a\right )} \sqrt {c} \sqrt {d}}{a^{2} c + d}, -\frac {a d x - d}{a^{2} c + d}\right )\right )} \log \left (d x^{2} + c\right ) - \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {a^{2} d x^{2} + 2 \, a d x + d}{a^{2} c + d}\right ) + \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {a^{2} d x^{2} - 2 \, a d x + d}{a^{2} c + d}\right ) - i \, {\rm Li}_2\left (\frac {a^{2} c + a d x - {\left (i \, a^{2} x - i \, a\right )} \sqrt {c} \sqrt {d}}{a^{2} c + 2 i \, a \sqrt {c} \sqrt {d} - d}\right ) - i \, {\rm Li}_2\left (\frac {a^{2} c - a d x + {\left (i \, a^{2} x + i \, a\right )} \sqrt {c} \sqrt {d}}{a^{2} c + 2 i \, a \sqrt {c} \sqrt {d} - d}\right ) + i \, {\rm Li}_2\left (\frac {a^{2} c + a d x + {\left (i \, a^{2} x - i \, a\right )} \sqrt {c} \sqrt {d}}{a^{2} c - 2 i \, a \sqrt {c} \sqrt {d} - d}\right ) + i \, {\rm Li}_2\left (\frac {a^{2} c - a d x - {\left (i \, a^{2} x + i \, a\right )} \sqrt {c} \sqrt {d}}{a^{2} c - 2 i \, a \sqrt {c} \sqrt {d} - d}\right )}{4 \, \sqrt {c d}} \] Input:

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

Output:

arctan(d*x/sqrt(c*d))*arctanh(a*x)/sqrt(c*d) + 1/4*((arctan2((a^2*x + a)*s 
qrt(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)) + arc 
tan(sqrt(d)*x/sqrt(c))*log((a^2*d*x^2 - 2*a*d*x + d)/(a^2*c + d)) - I*dilo 
g((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*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)))/sqrt(c*d)
 

Giac [F]

\[ \int \frac {\text {arctanh}(a x)}{c+d x^2} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )}{d x^{2} + c} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arctanh}(a x)}{c+d x^2} \, dx=\int \frac {\mathrm {atanh}\left (a\,x\right )}{d\,x^2+c} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\text {arctanh}(a x)}{c+d x^2} \, dx=\frac {-\mathit {atanh} \left (a x \right )^{2} a -2 \left (\int \frac {\mathit {atanh} \left (a x \right )}{a^{2} d \,x^{4}+a^{2} c \,x^{2}-d \,x^{2}-c}d x \right ) a^{2} c -2 \left (\int \frac {\mathit {atanh} \left (a x \right )}{a^{2} d \,x^{4}+a^{2} c \,x^{2}-d \,x^{2}-c}d x \right ) d}{2 d} \] Input:

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

Output:

( - atanh(a*x)**2*a - 2*int(atanh(a*x)/(a**2*c*x**2 + a**2*d*x**4 - c - d* 
x**2),x)*a**2*c - 2*int(atanh(a*x)/(a**2*c*x**2 + a**2*d*x**4 - c - d*x**2 
),x)*d)/(2*d)