3.4.3 \(\int \frac {\arctan (a x)^2}{x (c+a^2 c x^2)^3} \, dx\) [303]

3.4.3.1 Optimal result

Integrand size = 22, antiderivative size = 236 \[ \int \frac {\arctan (a x)^2}{x \left (c+a^2 c x^2\right )^3} \, dx=-\frac {1}{32 c^3 \left (1+a^2 x^2\right )^2}-\frac {11}{32 c^3 \left (1+a^2 x^2\right )}-\frac {a x \arctan (a x)}{8 c^3 \left (1+a^2 x^2\right )^2}-\frac {11 a x \arctan (a x)}{16 c^3 \left (1+a^2 x^2\right )}-\frac {11 \arctan (a x)^2}{32 c^3}+\frac {\arctan (a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {\arctan (a x)^2}{2 c^3 \left (1+a^2 x^2\right )}-\frac {i \arctan (a x)^3}{3 c^3}+\frac {\arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c^3}-\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{c^3}+\frac {\operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )}{2 c^3} \]

output

-1/32/c^3/(a^2*x^2+1)^2-11/32/c^3/(a^2*x^2+1)-1/8*a*x*arctan(a*x)/c^3/(a^2 
*x^2+1)^2-11/16*a*x*arctan(a*x)/c^3/(a^2*x^2+1)-11/32*arctan(a*x)^2/c^3+1/ 
4*arctan(a*x)^2/c^3/(a^2*x^2+1)^2+1/2*arctan(a*x)^2/c^3/(a^2*x^2+1)-1/3*I* 
arctan(a*x)^3/c^3+arctan(a*x)^2*ln(2-2/(1-I*a*x))/c^3-I*arctan(a*x)*polylo 
g(2,-1+2/(1-I*a*x))/c^3+1/2*polylog(3,-1+2/(1-I*a*x))/c^3
 

3.4.3.2 Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.66 \[ \int \frac {\arctan (a x)^2}{x \left (c+a^2 c x^2\right )^3} \, dx=\frac {-32 i \pi ^3+256 i \arctan (a x)^3-144 \cos (2 \arctan (a x))+288 \arctan (a x)^2 \cos (2 \arctan (a x))-3 \cos (4 \arctan (a x))+24 \arctan (a x)^2 \cos (4 \arctan (a x))+768 \arctan (a x)^2 \log \left (1-e^{-2 i \arctan (a x)}\right )+768 i \arctan (a x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )+384 \operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )-288 \arctan (a x) \sin (2 \arctan (a x))-12 \arctan (a x) \sin (4 \arctan (a x))}{768 c^3} \]

input

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

output

((-32*I)*Pi^3 + (256*I)*ArcTan[a*x]^3 - 144*Cos[2*ArcTan[a*x]] + 288*ArcTa 
n[a*x]^2*Cos[2*ArcTan[a*x]] - 3*Cos[4*ArcTan[a*x]] + 24*ArcTan[a*x]^2*Cos[ 
4*ArcTan[a*x]] + 768*ArcTan[a*x]^2*Log[1 - E^((-2*I)*ArcTan[a*x])] + (768* 
I)*ArcTan[a*x]*PolyLog[2, E^((-2*I)*ArcTan[a*x])] + 384*PolyLog[3, E^((-2* 
I)*ArcTan[a*x])] - 288*ArcTan[a*x]*Sin[2*ArcTan[a*x]] - 12*ArcTan[a*x]*Sin 
[4*ArcTan[a*x]])/(768*c^3)
 

3.4.3.3 Rubi [A] (verified)

Time = 1.94 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.39, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {5501, 27, 5465, 5431, 5427, 241, 5501, 5459, 5403, 5465, 5427, 241, 5527, 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.

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

\(\Big \downarrow \) 5501

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5465

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

\(\Big \downarrow \) 5431

\(\displaystyle \frac {\int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )^2}dx}{c^3}-\frac {a^2 \left (\frac {\frac {3}{4} \int \frac {\arctan (a x)}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {1}{16 a \left (a^2 x^2+1\right )^2}}{2 a}-\frac {\arctan (a x)^2}{4 a^2 \left (a^2 x^2+1\right )^2}\right )}{c^3}\)

\(\Big \downarrow \) 5427

\(\displaystyle \frac {\int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )^2}dx}{c^3}-\frac {a^2 \left (\frac {\frac {3}{4} \left (-\frac {1}{2} a \int \frac {x}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {1}{16 a \left (a^2 x^2+1\right )^2}}{2 a}-\frac {\arctan (a x)^2}{4 a^2 \left (a^2 x^2+1\right )^2}\right )}{c^3}\)

\(\Big \downarrow \) 241

\(\displaystyle \frac {\int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )^2}dx}{c^3}-\frac {a^2 \left (\frac {\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}}{2 a}-\frac {\arctan (a x)^2}{4 a^2 \left (a^2 x^2+1\right )^2}\right )}{c^3}\)

\(\Big \downarrow \) 5501

\(\displaystyle \frac {\int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx-a^2 \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx}{c^3}-\frac {a^2 \left (\frac {\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}}{2 a}-\frac {\arctan (a x)^2}{4 a^2 \left (a^2 x^2+1\right )^2}\right )}{c^3}\)

\(\Big \downarrow \) 5459

\(\displaystyle -\frac {a^2 \left (\frac {\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}}{2 a}-\frac {\arctan (a x)^2}{4 a^2 \left (a^2 x^2+1\right )^2}\right )}{c^3}+\frac {-a^2 \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+i \int \frac {\arctan (a x)^2}{x (a x+i)}dx-\frac {1}{3} i \arctan (a x)^3}{c^3}\)

\(\Big \downarrow \) 5403

\(\displaystyle -\frac {a^2 \left (\frac {\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}}{2 a}-\frac {\arctan (a x)^2}{4 a^2 \left (a^2 x^2+1\right )^2}\right )}{c^3}+\frac {-a^2 \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3}{c^3}\)

\(\Big \downarrow \) 5465

\(\displaystyle -\frac {a^2 \left (\frac {\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}}{2 a}-\frac {\arctan (a x)^2}{4 a^2 \left (a^2 x^2+1\right )^2}\right )}{c^3}+\frac {-a^2 \left (\frac {\int \frac {\arctan (a x)}{\left (a^2 x^2+1\right )^2}dx}{a}-\frac {\arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )+i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3}{c^3}\)

\(\Big \downarrow \) 5427

\(\displaystyle -\frac {a^2 \left (\frac {\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}}{2 a}-\frac {\arctan (a x)^2}{4 a^2 \left (a^2 x^2+1\right )^2}\right )}{c^3}+\frac {-a^2 \left (\frac {-\frac {1}{2} a \int \frac {x}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}}{a}-\frac {\arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )+i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3}{c^3}\)

\(\Big \downarrow \) 241

\(\displaystyle -\frac {a^2 \left (\frac {\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}}{2 a}-\frac {\arctan (a x)^2}{4 a^2 \left (a^2 x^2+1\right )^2}\right )}{c^3}+\frac {i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-a^2 \left (\frac {\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}}{a}-\frac {\arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )-\frac {1}{3} i \arctan (a x)^3}{c^3}\)

\(\Big \downarrow \) 5527

\(\displaystyle -\frac {a^2 \left (\frac {\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}}{2 a}-\frac {\arctan (a x)^2}{4 a^2 \left (a^2 x^2+1\right )^2}\right )}{c^3}+\frac {i \left (2 i a \left (\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{2 a}-\frac {1}{2} i \int \frac {\operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{a^2 x^2+1}dx\right )-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-a^2 \left (\frac {\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}}{a}-\frac {\arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )-\frac {1}{3} i \arctan (a x)^3}{c^3}\)

\(\Big \downarrow \) 7164

\(\displaystyle -\frac {a^2 \left (\frac {\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}}{2 a}-\frac {\arctan (a x)^2}{4 a^2 \left (a^2 x^2+1\right )^2}\right )}{c^3}+\frac {-a^2 \left (\frac {\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}}{a}-\frac {\arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )+i \left (2 i a \left (\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{2 a}-\frac {\operatorname {PolyLog}\left (3,\frac {2}{1-i a x}-1\right )}{4 a}\right )-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3}{c^3}\)

input

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

output

-((a^2*(-1/4*ArcTan[a*x]^2/(a^2*(1 + a^2*x^2)^2) + (1/(16*a*(1 + a^2*x^2)^ 
2) + (x*ArcTan[a*x])/(4*(1 + a^2*x^2)^2) + (3*(1/(4*a*(1 + a^2*x^2)) + (x* 
ArcTan[a*x])/(2*(1 + a^2*x^2)) + ArcTan[a*x]^2/(4*a)))/4)/(2*a)))/c^3) + ( 
(-1/3*I)*ArcTan[a*x]^3 - a^2*(-1/2*ArcTan[a*x]^2/(a^2*(1 + a^2*x^2)) + (1/ 
(4*a*(1 + a^2*x^2)) + (x*ArcTan[a*x])/(2*(1 + a^2*x^2)) + ArcTan[a*x]^2/(4 
*a))/a) + I*((-I)*ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*x)] + (2*I)*a*(((I/2)*A 
rcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)])/a - PolyLog[3, -1 + 2/(1 - I*a* 
x)]/(4*a))))/c^3
 

3.4.3.3.1 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 241

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

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 5431

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol 
] :> Simp[b*((d + e*x^2)^(q + 1)/(4*c*d*(q + 1)^2)), x] + (-Simp[x*(d + e*x 
^2)^(q + 1)*((a + b*ArcTan[c*x])/(2*d*(q + 1))), x] + Simp[(2*q + 3)/(2*d*( 
q + 1))   Int[(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x]), x], x]) /; FreeQ[{a, 
 b, c, d, e}, x] && EqQ[e, c^2*d] && LtQ[q, -1] && NeQ[q, -3/2]
 

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 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]
 

3.4.3.4 Maple [C] (warning: unable to verify)

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

Time = 19.06 (sec) , antiderivative size = 1722, normalized size of antiderivative = 7.30

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1722\)
default	\(\text {Expression too large to display}\)	\(1722\)
parts	\(\text {Expression too large to display}\)	\(2148\)

input

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

output

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

3.4.3.5 Fricas [F]

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

input

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

output

integral(arctan(a*x)^2/(a^6*c^3*x^7 + 3*a^4*c^3*x^5 + 3*a^2*c^3*x^3 + c^3* 
x), x)
 

3.4.3.6 Sympy [F]

\[ \int \frac {\arctan (a x)^2}{x \left (c+a^2 c x^2\right )^3} \, dx=\frac {\int \frac {\operatorname {atan}^{2}{\left (a x \right )}}{a^{6} x^{7} + 3 a^{4} x^{5} + 3 a^{2} x^{3} + x}\, dx}{c^{3}} \]

input

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

output

Integral(atan(a*x)**2/(a**6*x**7 + 3*a**4*x**5 + 3*a**2*x**3 + x), x)/c**3
 

3.4.3.7 Maxima [F]

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

input

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

output

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

3.4.3.8 Giac [F]

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

input

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

output

sage0*x
 

3.4.3.9 Mupad [F(-1)]

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

input

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

output

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