\(\int \frac {\text {arctanh}(a x)}{(c+d x^2)^3} \, dx\) [504]

Optimal result

Integrand size = 14, antiderivative size = 657 \[ \int \frac {\text {arctanh}(a x)}{\left (c+d x^2\right )^3} \, dx=\frac {a}{8 c \left (a^2 c+d\right ) \left (c+d x^2\right )}+\frac {x \text {arctanh}(a x)}{4 c \left (c+d x^2\right )^2}+\frac {3 x \text {arctanh}(a x)}{8 c^2 \left (c+d x^2\right )}+\frac {3 \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \text {arctanh}(a x)}{8 c^{5/2} \sqrt {d}}+\frac {3 i \log \left (\frac {\sqrt {d} (1-a x)}{i a \sqrt {c}+\sqrt {d}}\right ) \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{32 c^{5/2} \sqrt {d}}-\frac {3 i \log \left (-\frac {\sqrt {d} (1+a x)}{i a \sqrt {c}-\sqrt {d}}\right ) \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{32 c^{5/2} \sqrt {d}}-\frac {3 i \log \left (-\frac {\sqrt {d} (1-a x)}{i a \sqrt {c}-\sqrt {d}}\right ) \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{32 c^{5/2} \sqrt {d}}+\frac {3 i \log \left (\frac {\sqrt {d} (1+a x)}{i a \sqrt {c}+\sqrt {d}}\right ) \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{32 c^{5/2} \sqrt {d}}+\frac {a \left (5 a^2 c+3 d\right ) \log \left (1-a^2 x^2\right )}{16 c^2 \left (a^2 c+d\right )^2}-\frac {a \left (5 a^2 c+3 d\right ) \log \left (c+d x^2\right )}{16 c^2 \left (a^2 c+d\right )^2}+\frac {3 i \operatorname {PolyLog}\left (2,\frac {a \left (\sqrt {c}-i \sqrt {d} x\right )}{a \sqrt {c}-i \sqrt {d}}\right )}{32 c^{5/2} \sqrt {d}}-\frac {3 i \operatorname {PolyLog}\left (2,\frac {a \left (\sqrt {c}-i \sqrt {d} x\right )}{a \sqrt {c}+i \sqrt {d}}\right )}{32 c^{5/2} \sqrt {d}}+\frac {3 i \operatorname {PolyLog}\left (2,\frac {a \left (\sqrt {c}+i \sqrt {d} x\right )}{a \sqrt {c}-i \sqrt {d}}\right )}{32 c^{5/2} \sqrt {d}}-\frac {3 i \operatorname {PolyLog}\left (2,\frac {a \left (\sqrt {c}+i \sqrt {d} x\right )}{a \sqrt {c}+i \sqrt {d}}\right )}{32 c^{5/2} \sqrt {d}} \] Output:

1/8*a/c/(a^2*c+d)/(d*x^2+c)+1/4*x*arctanh(a*x)/c/(d*x^2+c)^2+3/8*x*arctanh 
(a*x)/c^2/(d*x^2+c)+3/8*arctan(d^(1/2)*x/c^(1/2))*arctanh(a*x)/c^(5/2)/d^( 
1/2)+3/32*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^(5/2)/d^(1/2)-3/32*I*ln(-d^(1/2)*(a*x+1)/(I*a*c^(1/2)-d^(1/2)))*l 
n(1-I*d^(1/2)*x/c^(1/2))/c^(5/2)/d^(1/2)-3/32*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^(5/2)/d^(1/2)+3/32*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^(5/2)/d^(1/ 
2)+1/16*a*(5*a^2*c+3*d)*ln(-a^2*x^2+1)/c^2/(a^2*c+d)^2-1/16*a*(5*a^2*c+3*d 
)*ln(d*x^2+c)/c^2/(a^2*c+d)^2+3/32*I*polylog(2,a*(c^(1/2)-I*d^(1/2)*x)/(a* 
c^(1/2)-I*d^(1/2)))/c^(5/2)/d^(1/2)-3/32*I*polylog(2,a*(c^(1/2)-I*d^(1/2)* 
x)/(a*c^(1/2)+I*d^(1/2)))/c^(5/2)/d^(1/2)+3/32*I*polylog(2,a*(c^(1/2)+I*d^ 
(1/2)*x)/(a*c^(1/2)-I*d^(1/2)))/c^(5/2)/d^(1/2)-3/32*I*polylog(2,a*(c^(1/2 
)+I*d^(1/2)*x)/(a*c^(1/2)+I*d^(1/2)))/c^(5/2)/d^(1/2)
 

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1541\) vs. \(2(657)=1314\).

Time = 9.09 (sec) , antiderivative size = 1541, normalized size of antiderivative = 2.35 \[ \int \frac {\text {arctanh}(a x)}{\left (c+d x^2\right )^3} \, dx =\text {Too large to display} \] Input:

Integrate[ArcTanh[a*x]/(c + d*x^2)^3,x]
 

Output:

(a*(-10*a^2*c*Log[1 + ((a^2*c + d)*Cosh[2*ArcTanh[a*x]])/(a^2*c - d)] - 6* 
d*Log[1 + ((a^2*c + d)*Cosh[2*ArcTanh[a*x]])/(a^2*c - d)] - (3*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 + Sq 
rt[a^2*c*d])*(-1 + a*x))/((a^2*c + d)*(a*c + I*Sqrt[a^2*c*d]*x))] - (ArcCo 
s[(-(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] - (3*Sqrt[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) +...
 

Rubi [A] (verified)

Time = 1.32 (sec) , antiderivative size = 682, normalized size of antiderivative = 1.04, 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.

\(\displaystyle \int \frac {\text {arctanh}(a x)}{\left (c+d x^2\right )^3} \, dx\)

\(\Big \downarrow \) 6538

\(\displaystyle -a \int \frac {\frac {3 d x^3+5 c x}{c^2 \left (d x^2+c\right )^2}+\frac {3 \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{5/2} \sqrt {d}}}{8 \left (1-a^2 x^2\right )}dx+\frac {3 \text {arctanh}(a x) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{5/2} \sqrt {d}}+\frac {3 x \text {arctanh}(a x)}{8 c^2 \left (c+d x^2\right )}+\frac {x \text {arctanh}(a x)}{4 c \left (c+d x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{8} a \int \frac {\frac {3 d x^3+5 c x}{c^2 \left (d x^2+c\right )^2}+\frac {3 \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{5/2} \sqrt {d}}}{1-a^2 x^2}dx+\frac {3 \text {arctanh}(a x) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{5/2} \sqrt {d}}+\frac {3 x \text {arctanh}(a x)}{8 c^2 \left (c+d x^2\right )}+\frac {x \text {arctanh}(a x)}{4 c \left (c+d x^2\right )^2}\)

\(\Big \downarrow \) 7276

\(\displaystyle -\frac {1}{8} a \int \left (-\frac {x \left (3 d x^2+5 c\right )}{c^2 \left (a^2 x^2-1\right ) \left (d x^2+c\right )^2}-\frac {3 \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{5/2} \sqrt {d} \left (a^2 x^2-1\right )}\right )dx+\frac {3 \text {arctanh}(a x) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{5/2} \sqrt {d}}+\frac {3 x \text {arctanh}(a x)}{8 c^2 \left (c+d x^2\right )}+\frac {x \text {arctanh}(a x)}{4 c \left (c+d x^2\right )^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {1}{8} a \left (-\frac {\left (5 a^2 c+3 d\right ) \log \left (1-a^2 x^2\right )}{2 c^2 \left (a^2 c+d\right )^2}+\frac {\left (5 a^2 c+3 d\right ) \log \left (c+d x^2\right )}{2 c^2 \left (a^2 c+d\right )^2}-\frac {1}{c \left (a^2 c+d\right ) \left (c+d x^2\right )}-\frac {3 i \operatorname {PolyLog}\left (2,\frac {a \left (\sqrt {c}-i \sqrt {d} x\right )}{a \sqrt {c}-i \sqrt {d}}\right )}{4 a c^{5/2} \sqrt {d}}+\frac {3 i \operatorname {PolyLog}\left (2,\frac {a \left (\sqrt {c}-i \sqrt {d} x\right )}{\sqrt {c} a+i \sqrt {d}}\right )}{4 a c^{5/2} \sqrt {d}}-\frac {3 i \operatorname {PolyLog}\left (2,\frac {a \left (i \sqrt {d} x+\sqrt {c}\right )}{a \sqrt {c}-i \sqrt {d}}\right )}{4 a c^{5/2} \sqrt {d}}+\frac {3 i \operatorname {PolyLog}\left (2,\frac {a \left (i \sqrt {d} x+\sqrt {c}\right )}{\sqrt {c} a+i \sqrt {d}}\right )}{4 a c^{5/2} \sqrt {d}}-\frac {3 i \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {\sqrt {d} (1-a x)}{\sqrt {d}+i a \sqrt {c}}\right )}{4 a c^{5/2} \sqrt {d}}+\frac {3 i \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right ) \log \left (-\frac {\sqrt {d} (a x+1)}{-\sqrt {d}+i a \sqrt {c}}\right )}{4 a c^{5/2} \sqrt {d}}+\frac {3 i \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right ) \log \left (-\frac {\sqrt {d} (1-a x)}{-\sqrt {d}+i a \sqrt {c}}\right )}{4 a c^{5/2} \sqrt {d}}-\frac {3 i \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {\sqrt {d} (a x+1)}{\sqrt {d}+i a \sqrt {c}}\right )}{4 a c^{5/2} \sqrt {d}}\right )+\frac {3 \text {arctanh}(a x) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{5/2} \sqrt {d}}+\frac {3 x \text {arctanh}(a x)}{8 c^2 \left (c+d x^2\right )}+\frac {x \text {arctanh}(a x)}{4 c \left (c+d x^2\right )^2}\)

Input:

Int[ArcTanh[a*x]/(c + d*x^2)^3,x]
 

Output:

(x*ArcTanh[a*x])/(4*c*(c + d*x^2)^2) + (3*x*ArcTanh[a*x])/(8*c^2*(c + d*x^ 
2)) + (3*ArcTan[(Sqrt[d]*x)/Sqrt[c]]*ArcTanh[a*x])/(8*c^(5/2)*Sqrt[d]) - ( 
a*(-(1/(c*(a^2*c + d)*(c + d*x^2))) - (((3*I)/4)*Log[(Sqrt[d]*(1 - a*x))/( 
I*a*Sqrt[c] + Sqrt[d])]*Log[1 - (I*Sqrt[d]*x)/Sqrt[c]])/(a*c^(5/2)*Sqrt[d] 
) + (((3*I)/4)*Log[-((Sqrt[d]*(1 + a*x))/(I*a*Sqrt[c] - Sqrt[d]))]*Log[1 - 
 (I*Sqrt[d]*x)/Sqrt[c]])/(a*c^(5/2)*Sqrt[d]) + (((3*I)/4)*Log[-((Sqrt[d]*( 
1 - a*x))/(I*a*Sqrt[c] - Sqrt[d]))]*Log[1 + (I*Sqrt[d]*x)/Sqrt[c]])/(a*c^( 
5/2)*Sqrt[d]) - (((3*I)/4)*Log[(Sqrt[d]*(1 + a*x))/(I*a*Sqrt[c] + Sqrt[d]) 
]*Log[1 + (I*Sqrt[d]*x)/Sqrt[c]])/(a*c^(5/2)*Sqrt[d]) - ((5*a^2*c + 3*d)*L 
og[1 - a^2*x^2])/(2*c^2*(a^2*c + d)^2) + ((5*a^2*c + 3*d)*Log[c + d*x^2])/ 
(2*c^2*(a^2*c + d)^2) - (((3*I)/4)*PolyLog[2, (a*(Sqrt[c] - I*Sqrt[d]*x))/ 
(a*Sqrt[c] - I*Sqrt[d])])/(a*c^(5/2)*Sqrt[d]) + (((3*I)/4)*PolyLog[2, (a*( 
Sqrt[c] - I*Sqrt[d]*x))/(a*Sqrt[c] + I*Sqrt[d])])/(a*c^(5/2)*Sqrt[d]) - (( 
(3*I)/4)*PolyLog[2, (a*(Sqrt[c] + I*Sqrt[d]*x))/(a*Sqrt[c] - I*Sqrt[d])])/ 
(a*c^(5/2)*Sqrt[d]) + (((3*I)/4)*PolyLog[2, (a*(Sqrt[c] + I*Sqrt[d]*x))/(a 
*Sqrt[c] + I*Sqrt[d])])/(a*c^(5/2)*Sqrt[d])))/8
 

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. \(4046\) vs. \(2(493)=986\).

Time = 2.29 (sec) , antiderivative size = 4047, normalized size of antiderivative = 6.16

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(4047\)
default	\(\text {Expression too large to display}\)	\(4047\)
risch	\(\text {Expression too large to display}\)	\(4564\)

Input:

int(arctanh(a*x)/(d*x^2+c)^3,x,method=_RETURNVERBOSE)
 

Output:

1/a*(-3/32*((-a^2*c*d)^(1/2)*a^2*c+2*a^2*c*d-(-a^2*c*d)^(1/2)*d)/c^3/(a^4* 
c^2+2*a^2*c*d+d^2)^2*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))-3/16*(-a^2*c*d)^(1/2)/c*a^4/d/(a^4*c^2+2*a^2*c*d+ 
d^2)*arctanh(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))-3/16*((-a^2*c*d)^(1/2)*a^2*c+2*a^2*c*d-(-a^2*c*d)^(1/2)*d)/c^ 
2/(a^4*c^2+2*a^2*c*d+d^2)^2*a^2*d*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)+3/16*(c*d)^(1/2)*d^2/c^3*a*arcta 
n(1/4*(2*(a^2*c+d)*(a*x+1)^2/(-a^2*x^2+1)+2*a^2*c-2*d)/a/(c*d)^(1/2))/(a^2 
*c+d)/(a^4*c^2+2*a^2*c*d+d^2)+5/16*(c*d)^(1/2)/c^2*d*a^3*arctan(1/4*(2*(a^ 
2*c+d)*(a*x+1)^2/(-a^2*x^2+1)+2*a^2*c-2*d)/a/(c*d)^(1/2))/(a^2*c+d)/(a^4*c 
^2+2*a^2*c*d+d^2)-3/8*(a^2*c-2*(-a^2*c*d)^(1/2)-d)/c^2/(a^4*c^2+2*a^2*c*d+ 
d^2)^2*a^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))*d^2*arctanh(a*x)-3/4*(a^2*c-2*(-a^2*c*d)^(1/2)-d)/c/(a^4*c^2+2*a^2*c 
*d+d^2)^2*a^4*d*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)+3/8*(a^2*c-2*(-a^2*c*d)^(1/2)-d)/(a^4*c^2+2*a^2*c* 
d+d^2)^2*a^6*arctanh(a*x)^2-5/16/(a^4*c^2+2*a^2*c*d+d^2)*a^6/(a^2*c+d)*ln( 
a^2*c*(a*x+1)^4/(-a^2*x^2+1)^2+2*(a*x+1)^2/(-a^2*x^2+1)*a^2*c+d*(a*x+1)^4/ 
(-a^2*x^2+1)^2+a^2*c-2*(a*x+1)^2/(-a^2*x^2+1)*d+d)+5/4/(a^4*c^2+2*a^2*c*d+ 
d^2)*a^6/(a^2*c+d)*ln((a*x+1)/(-a^2*x^2+1)^(1/2))-3/16*(a^2*c-2*(-a^2*c*d) 
^(1/2)-d)/(a^4*c^2+2*a^2*c*d+d^2)^2*a^6*polylog(2,(a^2*c+d)*(a*x+1)^2/(...
 

Fricas [F]

\[ \int \frac {\text {arctanh}(a x)}{\left (c+d x^2\right )^3} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )}{{\left (d x^{2} + c\right )}^{3}} \,d x } \] Input:

integrate(arctanh(a*x)/(d*x^2+c)^3,x, algorithm="fricas")
 

Output:

integral(arctanh(a*x)/(d^3*x^6 + 3*c*d^2*x^4 + 3*c^2*d*x^2 + c^3), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\text {arctanh}(a x)}{\left (c+d x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate(atanh(a*x)/(d*x**2+c)**3,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1087 vs. \(2 (463) = 926\).

Time = 0.22 (sec) , antiderivative size = 1087, normalized size of antiderivative = 1.65 \[ \int \frac {\text {arctanh}(a x)}{\left (c+d x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(arctanh(a*x)/(d*x^2+c)^3,x, algorithm="maxima")
 

Output:

1/8*((3*d*x^3 + 5*c*x)/(c^2*d^2*x^4 + 2*c^3*d*x^2 + c^4) + 3*arctan(d*x/sq 
rt(c*d))/(sqrt(c*d)*c^2))*arctanh(a*x) + 1/32*(4*a^3*c^3*d + 4*a*c^2*d^2 - 
 3*((a^4*c^3 + 2*a^2*c^2*d + c*d^2 + (a^4*c^2*d + 2*a^2*c*d^2 + d^3)*x^2)* 
arctan(sqrt(d)*x/sqrt(c))*log((a^2*d*x^2 + 2*a*d*x + d)/(a^2*c + d)) - (a^ 
4*c^3 + 2*a^2*c^2*d + c*d^2 + (a^4*c^2*d + 2*a^2*c*d^2 + d^3)*x^2)*arctan( 
sqrt(d)*x/sqrt(c))*log((a^2*d*x^2 - 2*a*d*x + d)/(a^2*c + d)) - (-I*a^4*c^ 
3 - 2*I*a^2*c^2*d - I*c*d^2 + (-I*a^4*c^2*d - 2*I*a^2*c*d^2 - I*d^3)*x^2)* 
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*a^4*c^3 - 2*I*a^2*c^2*d - I*c*d^2 + (-I*a^4*c^2*d 
 - 2*I*a^2*c*d^2 - I*d^3)*x^2)*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^4*c^3 + 2*I*a^2*c 
^2*d + I*c*d^2 + (I*a^4*c^2*d + 2*I*a^2*c*d^2 + I*d^3)*x^2)*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^4*c^3 + 2*I*a^2*c^2*d + I*c*d^2 + (I*a^4*c^2*d + 2*I*a^2*c*d^2 
 + I*d^3)*x^2)*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^4*c^3 + 2*a^2*c^2*d + c*d^2 + (a^4 
*c^2*d + 2*a^2*c*d^2 + d^3)*x^2)*arctan2((a^2*x + a)*sqrt(c)*sqrt(d)/(a^2* 
c + d), (a*d*x + d)/(a^2*c + d)) - (a^4*c^3 + 2*a^2*c^2*d + c*d^2 + (a^4*c 
^2*d + 2*a^2*c*d^2 + d^3)*x^2)*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))*sqrt(c)*sqrt(d) - 2*(5...
 

Giac [F]

\[ \int \frac {\text {arctanh}(a x)}{\left (c+d x^2\right )^3} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )}{{\left (d x^{2} + c\right )}^{3}} \,d x } \] Input:

integrate(arctanh(a*x)/(d*x^2+c)^3,x, algorithm="giac")
 

Output:

integrate(arctanh(a*x)/(d*x^2 + c)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arctanh}(a x)}{\left (c+d x^2\right )^3} \, dx=\int \frac {\mathrm {atanh}\left (a\,x\right )}{{\left (d\,x^2+c\right )}^3} \,d x \] Input:

int(atanh(a*x)/(c + d*x^2)^3,x)
 

Output:

int(atanh(a*x)/(c + d*x^2)^3, x)
 

Reduce [F]

\[ \int \frac {\text {arctanh}(a x)}{\left (c+d x^2\right )^3} \, dx=\text {too large to display} \] Input:

int(atanh(a*x)/(d*x^2+c)^3,x)
 

Output:

(6*atanh(a*x)**2*a**7*c**5 + 12*atanh(a*x)**2*a**7*c**4*d*x**2 + 6*atanh(a 
*x)**2*a**7*c**3*d**2*x**4 + 12*atanh(a*x)**2*a**5*c**4*d + 24*atanh(a*x)* 
*2*a**5*c**3*d**2*x**2 + 12*atanh(a*x)**2*a**5*c**2*d**3*x**4 + 6*atanh(a* 
x)**2*a**3*c**3*d**2 + 12*atanh(a*x)**2*a**3*c**2*d**3*x**2 + 6*atanh(a*x) 
**2*a**3*c*d**4*x**4 - 24*atanh(a*x)*a**6*c**4*d*x - 12*atanh(a*x)*a**6*c* 
*3*d**2*x**3 - 52*atanh(a*x)*a**4*c**3*d**2*x - 24*atanh(a*x)*a**4*c**2*d* 
*3*x**3 - 32*atanh(a*x)*a**2*c**2*d**3*x - 12*atanh(a*x)*a**2*c*d**4*x**3 
- 4*atanh(a*x)*c*d**4*x + 36*int((atanh(a*x)*x**2)/(3*a**6*c**5*x**2 + 9*a 
**6*c**4*d*x**4 + 9*a**6*c**3*d**2*x**6 + 3*a**6*c**2*d**3*x**8 - 3*a**4*c 
**5 - 15*a**4*c**4*d*x**2 - 27*a**4*c**3*d**2*x**4 - 21*a**4*c**2*d**3*x** 
6 - 6*a**4*c*d**4*x**8 + 6*a**2*c**4*d + 17*a**2*c**3*d**2*x**2 + 15*a**2* 
c**2*d**3*x**4 + 3*a**2*c*d**4*x**6 - a**2*d**5*x**8 + c**3*d**2 + 3*c**2* 
d**3*x**2 + 3*c*d**4*x**4 + d**5*x**6),x)*a**14*c**10 + 72*int((atanh(a*x) 
*x**2)/(3*a**6*c**5*x**2 + 9*a**6*c**4*d*x**4 + 9*a**6*c**3*d**2*x**6 + 3* 
a**6*c**2*d**3*x**8 - 3*a**4*c**5 - 15*a**4*c**4*d*x**2 - 27*a**4*c**3*d** 
2*x**4 - 21*a**4*c**2*d**3*x**6 - 6*a**4*c*d**4*x**8 + 6*a**2*c**4*d + 17* 
a**2*c**3*d**2*x**2 + 15*a**2*c**2*d**3*x**4 + 3*a**2*c*d**4*x**6 - a**2*d 
**5*x**8 + c**3*d**2 + 3*c**2*d**3*x**2 + 3*c*d**4*x**4 + d**5*x**6),x)*a* 
*14*c**9*d*x**2 + 36*int((atanh(a*x)*x**2)/(3*a**6*c**5*x**2 + 9*a**6*c**4 
*d*x**4 + 9*a**6*c**3*d**2*x**6 + 3*a**6*c**2*d**3*x**8 - 3*a**4*c**5 -...