\(\int \frac {\arctan (a x)^3}{x (c+a^2 c x^2)^2} \, dx\) [400]

Optimal result

Integrand size = 22, antiderivative size = 240 \[ \int \frac {\arctan (a x)^3}{x \left (c+a^2 c x^2\right )^2} \, dx=\frac {3 a x}{8 c^2 \left (1+a^2 x^2\right )}+\frac {3 \arctan (a x)}{8 c^2}-\frac {3 \arctan (a x)}{4 c^2 \left (1+a^2 x^2\right )}-\frac {3 a x \arctan (a x)^2}{4 c^2 \left (1+a^2 x^2\right )}-\frac {\arctan (a x)^3}{4 c^2}+\frac {\arctan (a x)^3}{2 c^2 \left (1+a^2 x^2\right )}-\frac {i \arctan (a x)^4}{4 c^2}+\frac {\arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )}{c^2}-\frac {3 i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{2 c^2}+\frac {3 \arctan (a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )}{2 c^2}+\frac {3 i \operatorname {PolyLog}\left (4,-1+\frac {2}{1-i a x}\right )}{4 c^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.65 \[ \int \frac {\arctan (a x)^3}{x \left (c+a^2 c x^2\right )^2} \, dx=\frac {-i \pi ^4+16 i \arctan (a x)^4-24 \arctan (a x) \cos (2 \arctan (a x))+16 \arctan (a x)^3 \cos (2 \arctan (a x))+64 \arctan (a x)^3 \log \left (1-e^{-2 i \arctan (a x)}\right )+96 i \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )+96 \arctan (a x) \operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )-48 i \operatorname {PolyLog}\left (4,e^{-2 i \arctan (a x)}\right )+12 \sin (2 \arctan (a x))-24 \arctan (a x)^2 \sin (2 \arctan (a x))}{64 c^2} \] Input:

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

Output:

((-I)*Pi^4 + (16*I)*ArcTan[a*x]^4 - 24*ArcTan[a*x]*Cos[2*ArcTan[a*x]] + 16 
*ArcTan[a*x]^3*Cos[2*ArcTan[a*x]] + 64*ArcTan[a*x]^3*Log[1 - E^((-2*I)*Arc 
Tan[a*x])] + (96*I)*ArcTan[a*x]^2*PolyLog[2, E^((-2*I)*ArcTan[a*x])] + 96* 
ArcTan[a*x]*PolyLog[3, E^((-2*I)*ArcTan[a*x])] - (48*I)*PolyLog[4, E^((-2* 
I)*ArcTan[a*x])] + 12*Sin[2*ArcTan[a*x]] - 24*ArcTan[a*x]^2*Sin[2*ArcTan[a 
*x]])/(64*c^2)
 

Rubi [A] (verified)

Time = 1.70 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.16, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {5501, 27, 5459, 5403, 5465, 5427, 5465, 215, 216, 5527, 5531, 7164}

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.

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

\(\Big \downarrow \) 5501

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5459

\(\displaystyle -\frac {a^2 \int \frac {x \arctan (a x)^3}{\left (a^2 x^2+1\right )^2}dx}{c^2}+\frac {i \int \frac {\arctan (a x)^3}{x (a x+i)}dx-\frac {1}{4} i \arctan (a x)^4}{c^2}\)

\(\Big \downarrow \) 5403

\(\displaystyle -\frac {a^2 \int \frac {x \arctan (a x)^3}{\left (a^2 x^2+1\right )^2}dx}{c^2}+\frac {i \left (3 i a \int \frac {\arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{4} i \arctan (a x)^4}{c^2}\)

\(\Big \downarrow \) 5465

\(\displaystyle -\frac {a^2 \left (\frac {3 \int \frac {\arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx}{2 a}-\frac {\arctan (a x)^3}{2 a^2 \left (a^2 x^2+1\right )}\right )}{c^2}+\frac {i \left (3 i a \int \frac {\arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{4} i \arctan (a x)^4}{c^2}\)

\(\Big \downarrow \) 5427

\(\displaystyle -\frac {a^2 \left (\frac {3 \left (-a \int \frac {x \arctan (a x)}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^3}{6 a}\right )}{2 a}-\frac {\arctan (a x)^3}{2 a^2 \left (a^2 x^2+1\right )}\right )}{c^2}+\frac {i \left (3 i a \int \frac {\arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{4} i \arctan (a x)^4}{c^2}\)

\(\Big \downarrow \) 5465

\(\displaystyle -\frac {a^2 \left (\frac {3 \left (-a \left (\frac {\int \frac {1}{\left (a^2 x^2+1\right )^2}dx}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^3}{6 a}\right )}{2 a}-\frac {\arctan (a x)^3}{2 a^2 \left (a^2 x^2+1\right )}\right )}{c^2}+\frac {i \left (3 i a \int \frac {\arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{4} i \arctan (a x)^4}{c^2}\)

\(\Big \downarrow \) 215

\(\displaystyle -\frac {a^2 \left (\frac {3 \left (-a \left (\frac {\frac {1}{2} \int \frac {1}{a^2 x^2+1}dx+\frac {x}{2 \left (a^2 x^2+1\right )}}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^3}{6 a}\right )}{2 a}-\frac {\arctan (a x)^3}{2 a^2 \left (a^2 x^2+1\right )}\right )}{c^2}+\frac {i \left (3 i a \int \frac {\arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{4} i \arctan (a x)^4}{c^2}\)

\(\Big \downarrow \) 216

\(\displaystyle -\frac {a^2 \left (\frac {3 \left (\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}-a \left (\frac {\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^3}{6 a}\right )}{2 a}-\frac {\arctan (a x)^3}{2 a^2 \left (a^2 x^2+1\right )}\right )}{c^2}+\frac {i \left (3 i a \int \frac {\arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{4} i \arctan (a x)^4}{c^2}\)

\(\Big \downarrow \) 5527

\(\displaystyle -\frac {a^2 \left (\frac {3 \left (\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}-a \left (\frac {\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^3}{6 a}\right )}{2 a}-\frac {\arctan (a x)^3}{2 a^2 \left (a^2 x^2+1\right )}\right )}{c^2}+\frac {i \left (3 i a \left (\frac {i \arctan (a x)^2 \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{2 a}-i \int \frac {\arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{a^2 x^2+1}dx\right )-i \arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{4} i \arctan (a x)^4}{c^2}\)

\(\Big \downarrow \) 5531

\(\displaystyle -\frac {a^2 \left (\frac {3 \left (\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}-a \left (\frac {\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^3}{6 a}\right )}{2 a}-\frac {\arctan (a x)^3}{2 a^2 \left (a^2 x^2+1\right )}\right )}{c^2}+\frac {i \left (3 i a \left (\frac {i \arctan (a x)^2 \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{2 a}-i \left (\frac {1}{2} i \int \frac {\operatorname {PolyLog}\left (3,\frac {2}{1-i a x}-1\right )}{a^2 x^2+1}dx-\frac {i \arctan (a x) \operatorname {PolyLog}\left (3,\frac {2}{1-i a x}-1\right )}{2 a}\right )\right )-i \arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{4} i \arctan (a x)^4}{c^2}\)

\(\Big \downarrow \) 7164

\(\displaystyle -\frac {a^2 \left (\frac {3 \left (\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}-a \left (\frac {\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^3}{6 a}\right )}{2 a}-\frac {\arctan (a x)^3}{2 a^2 \left (a^2 x^2+1\right )}\right )}{c^2}+\frac {i \left (3 i a \left (\frac {i \arctan (a x)^2 \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{2 a}-i \left (\frac {\operatorname {PolyLog}\left (4,\frac {2}{1-i a x}-1\right )}{4 a}-\frac {i \arctan (a x) \operatorname {PolyLog}\left (3,\frac {2}{1-i a x}-1\right )}{2 a}\right )\right )-i \arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{4} i \arctan (a x)^4}{c^2}\)

Input:

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

Output:

-((a^2*(-1/2*ArcTan[a*x]^3/(a^2*(1 + a^2*x^2)) + (3*((x*ArcTan[a*x]^2)/(2* 
(1 + a^2*x^2)) + ArcTan[a*x]^3/(6*a) - a*(-1/2*ArcTan[a*x]/(a^2*(1 + a^2*x 
^2)) + (x/(2*(1 + a^2*x^2)) + ArcTan[a*x]/(2*a))/(2*a))))/(2*a)))/c^2) + ( 
(-1/4*I)*ArcTan[a*x]^4 + I*((-I)*ArcTan[a*x]^3*Log[2 - 2/(1 - I*a*x)] + (3 
*I)*a*(((I/2)*ArcTan[a*x]^2*PolyLog[2, -1 + 2/(1 - I*a*x)])/a - I*(((-1/2* 
I)*ArcTan[a*x]*PolyLog[3, -1 + 2/(1 - I*a*x)])/a + PolyLog[4, -1 + 2/(1 - 
I*a*x)]/(4*a)))))/c^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 215

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) 
/(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1))   Int[(a + b*x^2)^(p + 1 
), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 
*p])
 

rule 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 5403

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x_ 
Symbol] :> Simp[(a + b*ArcTan[c*x])^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] - Si 
mp[b*c*(p/d)   Int[(a + b*ArcTan[c*x])^(p - 1)*(Log[2 - 2/(1 + e*(x/d))]/(1 
 + c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2* 
d^2 + e^2, 0]
 

rule 5427

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2)^2, x_Sym 
bol] :> Simp[x*((a + b*ArcTan[c*x])^p/(2*d*(d + e*x^2))), x] + (Simp[(a + b 
*ArcTan[c*x])^(p + 1)/(2*b*c*d^2*(p + 1)), x] - Simp[b*c*(p/2)   Int[x*((a 
+ b*ArcTan[c*x])^(p - 1)/(d + e*x^2)^2), x], x]) /; FreeQ[{a, b, c, d, e}, 
x] && EqQ[e, c^2*d] && GtQ[p, 0]
 

rule 5459

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^2)), 
x_Symbol] :> Simp[(-I)*((a + b*ArcTan[c*x])^(p + 1)/(b*d*(p + 1))), x] + Si 
mp[I/d   Int[(a + b*ArcTan[c*x])^p/(x*(I + c*x)), x], x] /; FreeQ[{a, b, c, 
 d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 0]
 

rule 5465

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)*((d_) + (e_.)*(x_)^2)^(q_ 
.), x_Symbol] :> Simp[(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^p/(2*e*(q + 
1))), x] - Simp[b*(p/(2*c*(q + 1)))   Int[(d + e*x^2)^q*(a + b*ArcTan[c*x]) 
^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && EqQ[e, c^2*d] && GtQ[p, 
 0] && NeQ[q, -1]
 

rule 5501

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_)*((d_) + (e_.)*(x_)^2 
)^(q_), x_Symbol] :> Simp[1/d   Int[x^m*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c 
*x])^p, x], x] - Simp[e/d   Int[x^(m + 2)*(d + e*x^2)^q*(a + b*ArcTan[c*x]) 
^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IntegersQ[p, 2* 
q] && LtQ[q, -1] && ILtQ[m, 0] && NeQ[p, -1]
 

rule 5527

Int[(Log[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2 
), x_Symbol] :> Simp[I*(a + b*ArcTan[c*x])^p*(PolyLog[2, 1 - u]/(2*c*d)), x 
] - Simp[b*p*(I/2)   Int[(a + b*ArcTan[c*x])^(p - 1)*(PolyLog[2, 1 - u]/(d 
+ e*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c^2* 
d] && EqQ[(1 - u)^2 - (1 - 2*(I/(I + c*x)))^2, 0]
 

rule 5531

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*PolyLog[k_, u_])/((d_) + (e_. 
)*(x_)^2), x_Symbol] :> Simp[(-I)*(a + b*ArcTan[c*x])^p*(PolyLog[k + 1, u]/ 
(2*c*d)), x] + Simp[b*p*(I/2)   Int[(a + b*ArcTan[c*x])^(p - 1)*(PolyLog[k 
+ 1, u]/(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, k}, x] && IGtQ[p, 0] 
&& EqQ[e, c^2*d] && EqQ[u^2 - (1 - 2*(I/(I + c*x)))^2, 0]
 

rule 7164

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, 
x]}, Simp[w*PolyLog[n + 1, v], x] /;  !FalseQ[w]] /; FreeQ[n, x]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 7.50 (sec) , antiderivative size = 1787, normalized size of antiderivative = 7.45

\[\text {Expression too large to display}\]

Input:

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

Output:

-1/2/c^2*arctan(a*x)^3*ln(a^2*x^2+1)+1/2*arctan(a*x)^3/c^2/(a^2*x^2+1)+1/c 
^2*arctan(a*x)^3*ln(a*x)-3/2/c^2*(-2/3*arctan(a*x)^3*ln((1+I*a*x)/(a^2*x^2 
+1)^(1/2))-4*I*polylog(4,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-4*I*polylog(4,(1+I* 
a*x)/(a^2*x^2+1)^(1/2))-1/8*arctan(a*x)*(a*x+I)/(a*x-I)-I*(a*x-I)/(16*a*x+ 
16*I)+1/6*I*arctan(a*x)^4-1/8*arctan(a*x)*(a*x-I)/(a*x+I)+I*arctan(a*x)^2* 
(a*x-I)/(8*a*x+8*I)+2/3*arctan(a*x)^3*ln((1+I*a*x)^2/(a^2*x^2+1)-1)-2/3*ar 
ctan(a*x)^3*ln(1+(1+I*a*x)/(a^2*x^2+1)^(1/2))+I*(a*x+I)/(16*a*x-16*I)-4*ar 
ctan(a*x)*polylog(3,-(1+I*a*x)/(a^2*x^2+1)^(1/2))+2*I*arctan(a*x)^2*polylo 
g(2,(1+I*a*x)/(a^2*x^2+1)^(1/2))-2/3*arctan(a*x)^3*ln(1-(1+I*a*x)/(a^2*x^2 
+1)^(1/2))-I*arctan(a*x)^2*(a*x+I)/(8*a*x-8*I)-4*arctan(a*x)*polylog(3,(1+ 
I*a*x)/(a^2*x^2+1)^(1/2))+2*I*arctan(a*x)^2*polylog(2,-(1+I*a*x)/(a^2*x^2+ 
1)^(1/2))-1/6*(I*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))*csgn(I*(1+I*a*x)^2/(a^ 
2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^2-2*I*Pi*csgn(I/((1+I*a*x)^2/(a^2* 
x^2+1)+1))*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1)) 
^2+2*I*Pi*csgn(((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))^3- 
I*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))^3+I*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1 
)+1))^2*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)-I*Pi*csgn(I*(1+I*a*x)^2/(a^2 
*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^3-I*Pi*csgn(I*(1+I*a*x)/(a^2*x^2+1) 
^(1/2))^2*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))+I*Pi*csgn(I/((1+I*a*x)^2/(a^2*x^ 
2+1)+1)^2)*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2...
 

Fricas [F]

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

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

Output:

integral(arctan(a*x)^3/(a^4*c^2*x^5 + 2*a^2*c^2*x^3 + c^2*x), x)
 

Sympy [F]

\[ \int \frac {\arctan (a x)^3}{x \left (c+a^2 c x^2\right )^2} \, dx=\frac {\int \frac {\operatorname {atan}^{3}{\left (a x \right )}}{a^{4} x^{5} + 2 a^{2} x^{3} + x}\, dx}{c^{2}} \] Input:

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

Output:

Integral(atan(a*x)**3/(a**4*x**5 + 2*a**2*x**3 + x), x)/c**2
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\arctan (a x)^3}{x \left (c+a^2 c x^2\right )^2} \, dx=\frac {\int \frac {\mathit {atan} \left (a x \right )^{3}}{a^{4} x^{5}+2 a^{2} x^{3}+x}d x}{c^{2}} \] Input:

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

Output:

int(atan(a*x)**3/(a**4*x**5 + 2*a**2*x**3 + x),x)/c**2