\(\int \frac {\text {arctanh}(a x)}{(c+d x^2)^2} \, dx\) [503]

Optimal result

Integrand size = 14, antiderivative size = 590 \[ \int \frac {\text {arctanh}(a x)}{\left (c+d x^2\right )^2} \, dx=\frac {x \text {arctanh}(a x)}{2 c \left (c+d x^2\right )}+\frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \text {arctanh}(a x)}{2 c^{3/2} \sqrt {d}}+\frac {i \log \left (\frac {\sqrt {d} (1-a x)}{i a \sqrt {c}+\sqrt {d}}\right ) \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 c^{3/2} \sqrt {d}}-\frac {i \log \left (-\frac {\sqrt {d} (1+a x)}{i a \sqrt {c}-\sqrt {d}}\right ) \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 c^{3/2} \sqrt {d}}-\frac {i \log \left (-\frac {\sqrt {d} (1-a x)}{i a \sqrt {c}-\sqrt {d}}\right ) \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 c^{3/2} \sqrt {d}}+\frac {i \log \left (\frac {\sqrt {d} (1+a x)}{i a \sqrt {c}+\sqrt {d}}\right ) \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 c^{3/2} \sqrt {d}}+\frac {a \log \left (1-a^2 x^2\right )}{4 c \left (a^2 c+d\right )}-\frac {a \log \left (c+d x^2\right )}{4 c \left (a^2 c+d\right )}+\frac {i \operatorname {PolyLog}\left (2,\frac {a \left (\sqrt {c}-i \sqrt {d} x\right )}{a \sqrt {c}-i \sqrt {d}}\right )}{8 c^{3/2} \sqrt {d}}-\frac {i \operatorname {PolyLog}\left (2,\frac {a \left (\sqrt {c}-i \sqrt {d} x\right )}{a \sqrt {c}+i \sqrt {d}}\right )}{8 c^{3/2} \sqrt {d}}+\frac {i \operatorname {PolyLog}\left (2,\frac {a \left (\sqrt {c}+i \sqrt {d} x\right )}{a \sqrt {c}-i \sqrt {d}}\right )}{8 c^{3/2} \sqrt {d}}-\frac {i \operatorname {PolyLog}\left (2,\frac {a \left (\sqrt {c}+i \sqrt {d} x\right )}{a \sqrt {c}+i \sqrt {d}}\right )}{8 c^{3/2} \sqrt {d}} \] Output:

1/2*x*arctanh(a*x)/c/(d*x^2+c)+1/2*arctan(d^(1/2)*x/c^(1/2))*arctanh(a*x)/ 
c^(3/2)/d^(1/2)+1/8*I*ln(d^(1/2)*(-a*x+1)/(I*a*c^(1/2)+d^(1/2)))*ln(1-I*d^ 
(1/2)*x/c^(1/2))/c^(3/2)/d^(1/2)-1/8*I*ln(-d^(1/2)*(a*x+1)/(I*a*c^(1/2)-d^ 
(1/2)))*ln(1-I*d^(1/2)*x/c^(1/2))/c^(3/2)/d^(1/2)-1/8*I*ln(-d^(1/2)*(-a*x+ 
1)/(I*a*c^(1/2)-d^(1/2)))*ln(1+I*d^(1/2)*x/c^(1/2))/c^(3/2)/d^(1/2)+1/8*I* 
ln(d^(1/2)*(a*x+1)/(I*a*c^(1/2)+d^(1/2)))*ln(1+I*d^(1/2)*x/c^(1/2))/c^(3/2 
)/d^(1/2)+1/4*a*ln(-a^2*x^2+1)/c/(a^2*c+d)-1/4*a*ln(d*x^2+c)/c/(a^2*c+d)+1 
/8*I*polylog(2,a*(c^(1/2)-I*d^(1/2)*x)/(a*c^(1/2)-I*d^(1/2)))/c^(3/2)/d^(1 
/2)-1/8*I*polylog(2,a*(c^(1/2)-I*d^(1/2)*x)/(a*c^(1/2)+I*d^(1/2)))/c^(3/2) 
/d^(1/2)+1/8*I*polylog(2,a*(c^(1/2)+I*d^(1/2)*x)/(a*c^(1/2)-I*d^(1/2)))/c^ 
(3/2)/d^(1/2)-1/8*I*polylog(2,a*(c^(1/2)+I*d^(1/2)*x)/(a*c^(1/2)+I*d^(1/2) 
))/c^(3/2)/d^(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 4.81 (sec) , antiderivative size = 746, normalized size of antiderivative = 1.26 \[ \int \frac {\text {arctanh}(a x)}{\left (c+d x^2\right )^2} \, dx=\frac {a \left (-\frac {2 \log \left (1+\frac {\left (a^2 c+d\right ) \cosh (2 \text {arctanh}(a x))}{a^2 c-d}\right )}{a^2 c+d}+\frac {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 )}{\sqrt {a^2 c d}}+\frac {4 \text {arctanh}(a x) \sinh (2 \text {arctanh}(a x))}{a^2 c-d+\left (a^2 c+d\right ) \cosh (2 \text {arctanh}(a x))}\right )}{8 c} \] Input:

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

Output:

(a*((-2*Log[1 + ((a^2*c + d)*Cosh[2*ArcTanh[a*x]])/(a^2*c - d)])/(a^2*c + 
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[((2*I)*a*c*(I*d + S 
qrt[a^2*c*d])*(-1 + a*x))/((a^2*c + d)*(a*c + I*Sqrt[a^2*c*d]*x))] + (ArcC 
os[(-(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*ArcTanh[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] + (4*ArcTanh[a*x]*Sinh[2*ArcTa 
nh[a*x]])/(a^2*c - d + (a^2*c + d)*Cosh[2*ArcTanh[a*x]])))/(8*c)
 

Rubi [A] (verified)

Time = 1.31 (sec) , antiderivative size = 618, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6538, 27, 7276, 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.

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

\(\Big \downarrow \) 6538

\(\displaystyle -a \int \frac {\frac {x}{c \left (d x^2+c\right )}+\frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{3/2} \sqrt {d}}}{2 \left (1-a^2 x^2\right )}dx+\frac {\text {arctanh}(a x) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} \sqrt {d}}+\frac {x \text {arctanh}(a x)}{2 c \left (c+d x^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{2} a \int \frac {\frac {x}{c \left (d x^2+c\right )}+\frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{3/2} \sqrt {d}}}{1-a^2 x^2}dx+\frac {\text {arctanh}(a x) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} \sqrt {d}}+\frac {x \text {arctanh}(a x)}{2 c \left (c+d x^2\right )}\)

\(\Big \downarrow \) 7276

\(\displaystyle -\frac {1}{2} a \int \left (-\frac {x}{c (a x-1) (a x+1) \left (d x^2+c\right )}-\frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{3/2} \sqrt {d} \left (a^2 x^2-1\right )}\right )dx+\frac {\text {arctanh}(a x) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} \sqrt {d}}+\frac {x \text {arctanh}(a x)}{2 c \left (c+d x^2\right )}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {1}{2} a \left (-\frac {\log \left (1-a^2 x^2\right )}{2 c \left (a^2 c+d\right )}+\frac {\log \left (c+d x^2\right )}{2 c \left (a^2 c+d\right )}-\frac {i \operatorname {PolyLog}\left (2,\frac {a \left (\sqrt {c}-i \sqrt {d} x\right )}{a \sqrt {c}-i \sqrt {d}}\right )}{4 a c^{3/2} \sqrt {d}}+\frac {i \operatorname {PolyLog}\left (2,\frac {a \left (\sqrt {c}-i \sqrt {d} x\right )}{\sqrt {c} a+i \sqrt {d}}\right )}{4 a c^{3/2} \sqrt {d}}-\frac {i \operatorname {PolyLog}\left (2,\frac {a \left (i \sqrt {d} x+\sqrt {c}\right )}{a \sqrt {c}-i \sqrt {d}}\right )}{4 a c^{3/2} \sqrt {d}}+\frac {i \operatorname {PolyLog}\left (2,\frac {a \left (i \sqrt {d} x+\sqrt {c}\right )}{\sqrt {c} a+i \sqrt {d}}\right )}{4 a c^{3/2} \sqrt {d}}-\frac {i \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {\sqrt {d} (1-a x)}{\sqrt {d}+i a \sqrt {c}}\right )}{4 a c^{3/2} \sqrt {d}}+\frac {i \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right ) \log \left (-\frac {\sqrt {d} (a x+1)}{-\sqrt {d}+i a \sqrt {c}}\right )}{4 a c^{3/2} \sqrt {d}}+\frac {i \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right ) \log \left (-\frac {\sqrt {d} (1-a x)}{-\sqrt {d}+i a \sqrt {c}}\right )}{4 a c^{3/2} \sqrt {d}}-\frac {i \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {\sqrt {d} (a x+1)}{\sqrt {d}+i a \sqrt {c}}\right )}{4 a c^{3/2} \sqrt {d}}\right )+\frac {\text {arctanh}(a x) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} \sqrt {d}}+\frac {x \text {arctanh}(a x)}{2 c \left (c+d x^2\right )}\)

Input:

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

Output:

(x*ArcTanh[a*x])/(2*c*(c + d*x^2)) + (ArcTan[(Sqrt[d]*x)/Sqrt[c]]*ArcTanh[ 
a*x])/(2*c^(3/2)*Sqrt[d]) - (a*(((-1/4*I)*Log[(Sqrt[d]*(1 - a*x))/(I*a*Sqr 
t[c] + Sqrt[d])]*Log[1 - (I*Sqrt[d]*x)/Sqrt[c]])/(a*c^(3/2)*Sqrt[d]) + ((I 
/4)*Log[-((Sqrt[d]*(1 + a*x))/(I*a*Sqrt[c] - Sqrt[d]))]*Log[1 - (I*Sqrt[d] 
*x)/Sqrt[c]])/(a*c^(3/2)*Sqrt[d]) + ((I/4)*Log[-((Sqrt[d]*(1 - a*x))/(I*a* 
Sqrt[c] - Sqrt[d]))]*Log[1 + (I*Sqrt[d]*x)/Sqrt[c]])/(a*c^(3/2)*Sqrt[d]) - 
 ((I/4)*Log[(Sqrt[d]*(1 + a*x))/(I*a*Sqrt[c] + Sqrt[d])]*Log[1 + (I*Sqrt[d 
]*x)/Sqrt[c]])/(a*c^(3/2)*Sqrt[d]) - Log[1 - a^2*x^2]/(2*c*(a^2*c + d)) + 
Log[c + d*x^2]/(2*c*(a^2*c + d)) - ((I/4)*PolyLog[2, (a*(Sqrt[c] - I*Sqrt[ 
d]*x))/(a*Sqrt[c] - I*Sqrt[d])])/(a*c^(3/2)*Sqrt[d]) + ((I/4)*PolyLog[2, ( 
a*(Sqrt[c] - I*Sqrt[d]*x))/(a*Sqrt[c] + I*Sqrt[d])])/(a*c^(3/2)*Sqrt[d]) - 
 ((I/4)*PolyLog[2, (a*(Sqrt[c] + I*Sqrt[d]*x))/(a*Sqrt[c] - I*Sqrt[d])])/( 
a*c^(3/2)*Sqrt[d]) + ((I/4)*PolyLog[2, (a*(Sqrt[c] + I*Sqrt[d]*x))/(a*Sqrt 
[c] + I*Sqrt[d])])/(a*c^(3/2)*Sqrt[d])))/2
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

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

rule 6538

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_)^2)^(q_.), x_Sym 
bol] :> With[{u = IntHide[(d + e*x^2)^q, x]}, Simp[(a + b*ArcTanh[c*x])   u 
, x] - Simp[b*c   Int[SimplifyIntegrand[u/(1 - c^2*x^2), x], x], x]] /; Fre 
eQ[{a, b, c, d, e}, x] && (IntegerQ[q] || ILtQ[q + 1/2, 0])
 

rule 7276

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE 
xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ 
[n, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1950\) vs. \(2(430)=860\).

Time = 8.62 (sec) , antiderivative size = 1951, normalized size of antiderivative = 3.31

method	result	size
risch	\(\text {Expression too large to display}\)	\(1951\)
derivativedivides	\(\text {Expression too large to display}\)	\(2227\)
default	\(\text {Expression too large to display}\)	\(2227\)

Input:

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

Output:

1/8*a^2*ln(a*x+1)/c/(a^2*c+d)/(a^2*d*x^2+a^2*c)/(-c*d)^(1/2)*ln((a*(-c*d)^ 
(1/2)-d*(a*x+1)+d)/(a*(-c*d)^(1/2)+d))*d^2*x^2-1/8*a^2*ln(a*x+1)/c/(a^2*c+ 
d)/(a^2*d*x^2+a^2*c)/(-c*d)^(1/2)*ln((a*(-c*d)^(1/2)+d*(a*x+1)-d)/(a*(-c*d 
)^(1/2)-d))*d^2*x^2+1/8*a^2*ln(-a*x+1)/c/(a^2*c+d)/(a^2*d*x^2+a^2*c)/(-c*d 
)^(1/2)*ln((a*(-c*d)^(1/2)-d*(-a*x+1)+d)/(a*(-c*d)^(1/2)+d))*d^2*x^2-1/8*a 
^2*ln(-a*x+1)/c/(a^2*c+d)/(a^2*d*x^2+a^2*c)/(-c*d)^(1/2)*ln((a*(-c*d)^(1/2 
)+d*(-a*x+1)-d)/(a*(-c*d)^(1/2)-d))*d^2*x^2+1/8*a^4*ln(a*x+1)/(a^2*c+d)/(a 
^2*d*x^2+a^2*c)/(-c*d)^(1/2)*ln((a*(-c*d)^(1/2)-d*(a*x+1)+d)/(a*(-c*d)^(1/ 
2)+d))*d*x^2-1/8*a^4*ln(a*x+1)/(a^2*c+d)/(a^2*d*x^2+a^2*c)/(-c*d)^(1/2)*ln 
((a*(-c*d)^(1/2)+d*(a*x+1)-d)/(a*(-c*d)^(1/2)-d))*d*x^2+1/8*a^4*ln(-a*x+1) 
/(a^2*c+d)/(a^2*d*x^2+a^2*c)/(-c*d)^(1/2)*ln((a*(-c*d)^(1/2)-d*(-a*x+1)+d) 
/(a*(-c*d)^(1/2)+d))*d*x^2-1/8*a^4*ln(-a*x+1)/(a^2*c+d)/(a^2*d*x^2+a^2*c)/ 
(-c*d)^(1/2)*ln((a*(-c*d)^(1/2)+d*(-a*x+1)-d)/(a*(-c*d)^(1/2)-d))*d*x^2+1/ 
4*a^4*ln(a*x+1)/(a^2*c+d)/(a^2*d*x^2+a^2*c)*x-1/4*a^4*ln(-a*x+1)/(a^2*c+d) 
/(a^2*d*x^2+a^2*c)*x-1/8*a/c/(a^2*c+d)*ln((-a*x+1)^2*d+a^2*c-2*d*(-a*x+1)+ 
d)-1/4*a^2/(a^2*c+d)/(c*d)^(1/2)*arctan(1/2*(2*d*(-a*x+1)-2*d)/a/(c*d)^(1/ 
2))+1/4*a^3*ln(-a*x+1)/(a^2*c+d)/(a^2*d*x^2+a^2*c)-1/8*a/c/(a^2*c+d)*ln((a 
*x+1)^2*d+a^2*c-2*d*(a*x+1)+d)-1/4*a^2/(a^2*c+d)/(c*d)^(1/2)*arctan(1/2*(2 
*d*(a*x+1)-2*d)/a/(c*d)^(1/2))+1/4*a^3*ln(a*x+1)/(a^2*c+d)/(a^2*d*x^2+a^2* 
c)+1/8/c/(-c*d)^(1/2)*dilog((a*(-c*d)^(1/2)-d*(-a*x+1)+d)/(a*(-c*d)^(1/...
 

Fricas [F]

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

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

Output:

integral(arctanh(a*x)/(d^2*x^4 + 2*c*d*x^2 + c^2), x)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 550, normalized size of antiderivative = 0.93 \[ \int \frac {\text {arctanh}(a x)}{\left (c+d x^2\right )^2} \, dx=\frac {1}{2} \, {\left (\frac {x}{c d x^{2} + c^{2}} + \frac {\arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d} c}\right )} \operatorname {artanh}\left (a x\right ) - \frac {{\left (2 \, a c d \log \left (d x^{2} + c\right ) - 2 \, a c d \log \left (a x + 1\right ) - 2 \, a c d \log \left (a x - 1\right ) + {\left ({\left (a^{2} c + d\right )} \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {a^{2} d x^{2} + 2 \, a d x + d}{a^{2} c + d}\right ) - {\left (a^{2} c + d\right )} \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {a^{2} d x^{2} - 2 \, a d x + d}{a^{2} c + d}\right ) + {\left (i \, a^{2} c + i \, d\right )} {\rm Li}_2\left (\frac {a^{2} c + a d x - {\left (i \, a^{2} x - i \, a\right )} \sqrt {c} \sqrt {d}}{a^{2} c + 2 i \, a \sqrt {c} \sqrt {d} - d}\right ) + {\left (i \, a^{2} c + i \, d\right )} {\rm Li}_2\left (\frac {a^{2} c - a d x + {\left (i \, a^{2} x + i \, a\right )} \sqrt {c} \sqrt {d}}{a^{2} c + 2 i \, a \sqrt {c} \sqrt {d} - d}\right ) + {\left (-i \, a^{2} c - i \, d\right )} {\rm Li}_2\left (\frac {a^{2} c + a d x + {\left (i \, a^{2} x - i \, a\right )} \sqrt {c} \sqrt {d}}{a^{2} c - 2 i \, a \sqrt {c} \sqrt {d} - d}\right ) + {\left (-i \, a^{2} c - i \, d\right )} {\rm Li}_2\left (\frac {a^{2} c - a d x - {\left (i \, a^{2} x + i \, a\right )} \sqrt {c} \sqrt {d}}{a^{2} c - 2 i \, a \sqrt {c} \sqrt {d} - d}\right ) - {\left ({\left (a^{2} c + d\right )} \arctan \left (\frac {{\left (a^{2} x + a\right )} \sqrt {c} \sqrt {d}}{a^{2} c + d}, \frac {a d x + d}{a^{2} c + d}\right ) - {\left (a^{2} c + d\right )} \arctan \left (\frac {{\left (a^{2} x - a\right )} \sqrt {c} \sqrt {d}}{a^{2} c + d}, -\frac {a d x - d}{a^{2} c + d}\right )\right )} \log \left (d x^{2} + c\right )\right )} \sqrt {c} \sqrt {d}\right )} a}{8 \, {\left (a^{3} c^{3} d + a c^{2} d^{2}\right )}} \] Input:

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

Output:

1/2*(x/(c*d*x^2 + c^2) + arctan(d*x/sqrt(c*d))/(sqrt(c*d)*c))*arctanh(a*x) 
 - 1/8*(2*a*c*d*log(d*x^2 + c) - 2*a*c*d*log(a*x + 1) - 2*a*c*d*log(a*x - 
1) + ((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)) - (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*a^2*c + I*d)*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*a^2*c + I* 
d)*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*a^2*c - I*d)*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*a^2*c - 
I*d)*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)) - ((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)) - (a^2*c + d)*arctan2((a^2*x - a)*s 
qrt(c)*sqrt(d)/(a^2*c + d), -(a*d*x - d)/(a^2*c + d)))*log(d*x^2 + c))*sqr 
t(c)*sqrt(d))*a/(a^3*c^3*d + a*c^2*d^2)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(atanh(a*x)**2*a**3*c**2 + atanh(a*x)**2*a**3*c*d*x**2 + atanh(a*x)**2*a*c 
*d + atanh(a*x)**2*a*d**2*x**2 - 2*atanh(a*x)*a**2*c*d*x - 2*atanh(a*x)*d* 
*2*x + 2*int((atanh(a*x)*x**2)/(a**4*c**3*x**2 + 2*a**4*c**2*d*x**4 + a**4 
*c*d**2*x**6 - a**2*c**3 - 3*a**2*c**2*d*x**2 - 3*a**2*c*d**2*x**4 - a**2* 
d**3*x**6 + c**2*d + 2*c*d**2*x**2 + d**3*x**4),x)*a**8*c**5 + 2*int((atan 
h(a*x)*x**2)/(a**4*c**3*x**2 + 2*a**4*c**2*d*x**4 + a**4*c*d**2*x**6 - a** 
2*c**3 - 3*a**2*c**2*d*x**2 - 3*a**2*c*d**2*x**4 - a**2*d**3*x**6 + c**2*d 
 + 2*c*d**2*x**2 + d**3*x**4),x)*a**8*c**4*d*x**2 + 4*int((atanh(a*x)*x**2 
)/(a**4*c**3*x**2 + 2*a**4*c**2*d*x**4 + a**4*c*d**2*x**6 - a**2*c**3 - 3* 
a**2*c**2*d*x**2 - 3*a**2*c*d**2*x**4 - a**2*d**3*x**6 + c**2*d + 2*c*d**2 
*x**2 + d**3*x**4),x)*a**6*c**4*d + 4*int((atanh(a*x)*x**2)/(a**4*c**3*x** 
2 + 2*a**4*c**2*d*x**4 + a**4*c*d**2*x**6 - a**2*c**3 - 3*a**2*c**2*d*x**2 
 - 3*a**2*c*d**2*x**4 - a**2*d**3*x**6 + c**2*d + 2*c*d**2*x**2 + d**3*x** 
4),x)*a**6*c**3*d**2*x**2 - 4*int((atanh(a*x)*x**2)/(a**4*c**3*x**2 + 2*a* 
*4*c**2*d*x**4 + a**4*c*d**2*x**6 - a**2*c**3 - 3*a**2*c**2*d*x**2 - 3*a** 
2*c*d**2*x**4 - a**2*d**3*x**6 + c**2*d + 2*c*d**2*x**2 + d**3*x**4),x)*a* 
*2*c**2*d**3 - 4*int((atanh(a*x)*x**2)/(a**4*c**3*x**2 + 2*a**4*c**2*d*x** 
4 + a**4*c*d**2*x**6 - a**2*c**3 - 3*a**2*c**2*d*x**2 - 3*a**2*c*d**2*x**4 
 - a**2*d**3*x**6 + c**2*d + 2*c*d**2*x**2 + d**3*x**4),x)*a**2*c*d**4*x** 
2 - 2*int((atanh(a*x)*x**2)/(a**4*c**3*x**2 + 2*a**4*c**2*d*x**4 + a**4...