3.1.41 \(\int \frac {\coth ^{-1}(a x)}{(c+d x^2)^3} \, dx\) [41]

3.1.41.1 Optimal result

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

output

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

3.1.41.2 Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1559\) vs. \(2(657)=1314\).

Time = 9.81 (sec) , antiderivative size = 1559, normalized size of antiderivative = 2.37 \[ \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^3} \, dx =\text {Too large to display} \]

input

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

output

-1/32*(a*(10*a^2*c*Log[1 - ((a^2*c + d)*Cosh[2*ArcCoth[a*x]])/(a^2*c - d)] 
 + 6*d*Log[1 - ((a^2*c + d)*Cosh[2*ArcCoth[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*c)/(Sqrt[a^2*c*d 
]*x)] + 4*ArcCoth[a*x]*ArcTan[(a*d*x)/Sqrt[a^2*c*d]] - (ArcCos[(a^2*c - d) 
/(a^2*c + d)] + 2*ArcTan[(a*c)/(Sqrt[a^2*c*d]*x)])*Log[(2*d*(a^2*c - I*Sqr 
t[a^2*c*d])*(-1 + a*x))/((a^2*c + d)*(I*Sqrt[a^2*c*d] + a*d*x))] - (ArcCos 
[(a^2*c - d)/(a^2*c + d)] - 2*ArcTan[(a*c)/(Sqrt[a^2*c*d]*x)])*Log[(2*d*(a 
^2*c + I*Sqrt[a^2*c*d])*(1 + a*x))/((a^2*c + d)*(I*Sqrt[a^2*c*d] + a*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^ArcCoth[a*x]*Sqrt[-(a^2*c) + d + (a^2*c + d)*Cosh[2*ArcCoth[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^ArcCoth[a*x 
])/(Sqrt[a^2*c + d]*Sqrt[-(a^2*c) + d + (a^2*c + d)*Cosh[2*ArcCoth[a*x]]]) 
] + I*(-PolyLog[2, ((a^2*c - d - (2*I)*Sqrt[a^2*c*d])*(Sqrt[a^2*c*d] + I*a 
*d*x))/((a^2*c + d)*(Sqrt[a^2*c*d] - I*a*d*x))] + PolyLog[2, ((a^2*c - d + 
 (2*I)*Sqrt[a^2*c*d])*(Sqrt[a^2*c*d] + I*a*d*x))/((a^2*c + d)*(Sqrt[a^2*c* 
d] - I*a*d*x))])))/Sqrt[a^2*c*d] - (3*Sqrt[a^2*c*d]*(a^2*c + d)*((-2*I)*Ar 
cCos[(a^2*c - d)/(a^2*c + d)]*ArcTan[(a*c)/(Sqrt[a^2*c*d]*x)] + 4*ArcCoth[ 
a*x]*ArcTan[(a*d*x)/Sqrt[a^2*c*d]] - (ArcCos[(a^2*c - d)/(a^2*c + d)] +...
 

3.1.41.3 Rubi [A] (verified)

Time = 1.34 (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 = {6539, 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.

\(\displaystyle \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^3} \, dx\)

\(\Big \downarrow \) 6539

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

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

input

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

output

(x*ArcCoth[a*x])/(4*c*(c + d*x^2)^2) + (3*x*ArcCoth[a*x])/(8*c^2*(c + d*x^ 
2)) + (3*ArcCoth[a*x]*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(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
 

3.1.41.3.1 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 6539

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_)^2)^(q_.), x_Sym 
bol] :> With[{u = IntHide[(d + e*x^2)^q, x]}, Simp[(a + b*ArcCoth[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]
 

3.1.41.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3790\) vs. \(2(493)=986\).

Time = 1.48 (sec) , antiderivative size = 3791, normalized size of antiderivative = 5.77

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(3791\)
default	\(\text {Expression too large to display}\)	\(3791\)
risch	\(\text {Expression too large to display}\)	\(4508\)

input

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

output

1/a*(3/8*(a^2*c+2*(-a^2*c*d)^(1/2)-d)*a^2*d^2*arccoth(a*x)^2/c^2/(a^4*c^2+ 
2*a^2*c*d+d^2)^2-3/8/c^2/(a^4*c^2+2*a^2*c*d+d^2)*a^2*d^2/(a^2*c+d)*ln((a*x 
-1)/(a*x+1))-1/2/c/(a^4*c^2+2*a^2*c*d+d^2)*a^4*d/(a^2*c+d)*ln(a^2*c/(a*x-1 
)^2*(a*x+1)^2-2*a^2*c/(a*x-1)*(a*x+1)+d/(a*x-1)^2*(a*x+1)^2+a^2*c+2/(a*x-1 
)*(a*x+1)*d+d)+1/8*a^2*(5*arccoth(a*x)*a^6*c^3+3*arccoth(a*x)*a^6*c^2*d*x^ 
2-7*arccoth(a*x)*a^5*c^2*d*x-5*arccoth(a*x)*a^5*c*d^2*x^3+3*arccoth(a*x)*a 
^4*c^2*d+arccoth(a*x)*a^4*c*d^2*x^2-a^5*c^2*d*x-a^5*c*d^2*x^3-5*arccoth(a* 
x)*a^3*c*d^2*x-3*arccoth(a*x)*d^3*a^3*x^3-a^4*c^2*d-c*a^4*d^2*x^2)*(a*x-1) 
/(a^4*c^2+2*a^2*c*d+d^2)/(a^2*d*x^2+a^2*c)^2/c^2+3/16*(-a^2*c*d)^(1/2)/c^3 
*d/(a^4*c^2+2*a^2*c*d+d^2)*arccoth(a*x)*ln(1-(a^2*c+d)/(a*x-1)*(a*x+1)/(a^ 
2*c+2*(-a^2*c*d)^(1/2)-d))+5/16*(c*d)^(1/2)/d*a^7*arctan(1/4*(2*(a^2*c+d)/ 
(a*x-1)*(a*x+1)-2*a^2*c+2*d)/a/(c*d)^(1/2))/(a^4*c^2+2*a^2*c*d+d^2)/(a^2*c 
+d)-3/32*(-(-a^2*c*d)^(1/2)*a^2*c+2*a^2*c*d+(-a^2*c*d)^(1/2)*d)/c^2*a^2*d/ 
(a^4*c^2+2*a^2*c*d+d^2)^2*polylog(2,(a^2*c+d)/(a*x-1)*(a*x+1)/(a^2*c-2*(-a 
^2*c*d)^(1/2)-d))+3/8*(-a^2*c*d)^(1/2)/c^2*a^2/(a^4*c^2+2*a^2*c*d+d^2)*arc 
coth(a*x)*ln(1-(a^2*c+d)/(a*x-1)*(a*x+1)/(a^2*c+2*(-a^2*c*d)^(1/2)-d))+3/1 
6*(c*d)^(1/2)/c*a^5*arctan(1/4*(2*(a^2*c+d)/(a*x-1)*(a*x+1)-2*a^2*c+2*d)/a 
/(c*d)^(1/2))/(a^4*c^2+2*a^2*c*d+d^2)/(a^2*c+d)+3/16*(-(-a^2*c*d)^(1/2)*a^ 
2*c+2*a^2*c*d+(-a^2*c*d)^(1/2)*d)*a^6/d/(a^4*c^2+2*a^2*c*d+d^2)^2*ln(1-(a^ 
2*c+d)/(a*x-1)*(a*x+1)/(a^2*c-2*(-a^2*c*d)^(1/2)-d))*arccoth(a*x)+3/16*...
 

3.1.41.5 Fricas [F]

\[ \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^3} \, dx=\int { \frac {\operatorname {arcoth}\left (a x\right )}{{\left (d x^{2} + c\right )}^{3}} \,d x } \]

input

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

output

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

3.1.41.6 Sympy [F(-1)]

Timed out. \[ \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^3} \, dx=\text {Timed out} \]

input

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

output

Timed out
 

3.1.41.7 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 = 0.42 (sec) , antiderivative size = 1087, normalized size of antiderivative = 1.65 \[ \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^3} \, dx=\text {Too large to display} \]

input

integrate(arccoth(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))*arccoth(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...
 

3.1.41.8 Giac [F]

\[ \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^3} \, dx=\int { \frac {\operatorname {arcoth}\left (a x\right )}{{\left (d x^{2} + c\right )}^{3}} \,d x } \]

input

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

output

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

3.1.41.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^3} \, dx=\int \frac {\mathrm {acoth}\left (a\,x\right )}{{\left (d\,x^2+c\right )}^3} \,d x \]

input

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

output

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