3.4.94 \(\int \frac {\arctan (a x)^3}{x^3 (c+a^2 c x^2)} \, dx\) [394]

3.4.94.1 Optimal result

Integrand size = 22, antiderivative size = 262 \[ \int \frac {\arctan (a x)^3}{x^3 \left (c+a^2 c x^2\right )} \, dx=-\frac {3 i a^2 \arctan (a x)^2}{2 c}-\frac {3 a \arctan (a x)^2}{2 c x}-\frac {a^2 \arctan (a x)^3}{2 c}-\frac {\arctan (a x)^3}{2 c x^2}+\frac {i a^2 \arctan (a x)^4}{4 c}+\frac {3 a^2 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{c}-\frac {a^2 \arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )}{c}-\frac {3 i a^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{2 c}+\frac {3 i a^2 \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{2 c}-\frac {3 a^2 \arctan (a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )}{2 c}-\frac {3 i a^2 \operatorname {PolyLog}\left (4,-1+\frac {2}{1-i a x}\right )}{4 c} \]

output

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

3.4.94.2 Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.72 \[ \int \frac {\arctan (a x)^3}{x^3 \left (c+a^2 c x^2\right )} \, dx=\frac {i a^2 \left (\pi ^4-96 \arctan (a x)^2+\frac {96 i \arctan (a x)^2}{a x}+\frac {32 i \left (1+a^2 x^2\right ) \arctan (a x)^3}{a^2 x^2}-16 \arctan (a x)^4+64 i \arctan (a x)^3 \log \left (1-e^{-2 i \arctan (a x)}\right )-192 i \arctan (a x) \log \left (1-e^{2 i \arctan (a x)}\right )-96 \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )-96 \operatorname {PolyLog}\left (2,e^{2 i \arctan (a x)}\right )+96 i \arctan (a x) \operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )+48 \operatorname {PolyLog}\left (4,e^{-2 i \arctan (a x)}\right )\right )}{64 c} \]

input

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

output

((I/64)*a^2*(Pi^4 - 96*ArcTan[a*x]^2 + ((96*I)*ArcTan[a*x]^2)/(a*x) + ((32 
*I)*(1 + a^2*x^2)*ArcTan[a*x]^3)/(a^2*x^2) - 16*ArcTan[a*x]^4 + (64*I)*Arc 
Tan[a*x]^3*Log[1 - E^((-2*I)*ArcTan[a*x])] - (192*I)*ArcTan[a*x]*Log[1 - E 
^((2*I)*ArcTan[a*x])] - 96*ArcTan[a*x]^2*PolyLog[2, E^((-2*I)*ArcTan[a*x]) 
] - 96*PolyLog[2, E^((2*I)*ArcTan[a*x])] + (96*I)*ArcTan[a*x]*PolyLog[3, E 
^((-2*I)*ArcTan[a*x])] + 48*PolyLog[4, E^((-2*I)*ArcTan[a*x])]))/c
 

3.4.94.3 Rubi [A] (verified)

Time = 1.65 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.98, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {5453, 27, 5361, 5453, 5361, 5419, 5459, 5403, 2897, 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.

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

\(\Big \downarrow \) 5453

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5361

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

\(\Big \downarrow \) 5453

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

\(\Big \downarrow \) 5361

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

\(\Big \downarrow \) 5419

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

\(\Big \downarrow \) 5459

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

\(\Big \downarrow \) 5403

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

\(\Big \downarrow \) 2897

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

\(\Big \downarrow \) 5527

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

\(\Big \downarrow \) 5531

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

\(\Big \downarrow \) 7164

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

input

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

output

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

3.4.94.3.1 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 2897

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/ 
D[u, x])]}, Simp[C*PolyLog[2, 1 - u], x] /; FreeQ[C, x]] /; IntegerQ[m] && 
PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponents[u, 
 x][[2]], Expon[Pq, x]]
 

rule 5361

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> 
 Simp[x^(m + 1)*((a + b*ArcTan[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m + 
1))   Int[x^(m + n)*((a + b*ArcTan[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))), x], 
x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1] & 
& IntegerQ[m])) && NeQ[m, -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 5419

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

rule 5453

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e 
_.)*(x_)^2), x_Symbol] :> Simp[1/d   Int[(f*x)^m*(a + b*ArcTan[c*x])^p, x], 
 x] - Simp[e/(d*f^2)   Int[(f*x)^(m + 2)*((a + b*ArcTan[c*x])^p/(d + e*x^2) 
), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && LtQ[m, -1]
 

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 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]
 

3.4.94.4 Maple [A] (verified)

Time = 91.86 (sec) , antiderivative size = 441, normalized size of antiderivative = 1.68

method	result	size
derivativedivides	\(a^{2} \left (-\frac {6 i \operatorname {polylog}\left (4, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}-\frac {\arctan \left (a x \right )^{2} \left (-i \arctan \left (a x \right )-3 i a x +x \arctan \left (a x \right ) a \right ) \left (a x +i\right )}{2 c \,a^{2} x^{2}}+\frac {3 i \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}-\frac {\arctan \left (a x \right )^{3} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}-\frac {3 i \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}-\frac {6 \arctan \left (a x \right ) \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}+\frac {3 i \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}-\frac {\arctan \left (a x \right )^{3} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )}{c}-\frac {6 i \operatorname {polylog}\left (4, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}-\frac {6 \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}-\frac {3 i \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}+\frac {i \arctan \left (a x \right )^{4}}{4 c}+\frac {3 \arctan \left (a x \right ) \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}-\frac {3 i \arctan \left (a x \right )^{2}}{c}+\frac {3 \arctan \left (a x \right ) \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )}{c}\right )\)	\(441\)
default	\(a^{2} \left (-\frac {6 i \operatorname {polylog}\left (4, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}-\frac {\arctan \left (a x \right )^{2} \left (-i \arctan \left (a x \right )-3 i a x +x \arctan \left (a x \right ) a \right ) \left (a x +i\right )}{2 c \,a^{2} x^{2}}+\frac {3 i \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}-\frac {\arctan \left (a x \right )^{3} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}-\frac {3 i \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}-\frac {6 \arctan \left (a x \right ) \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}+\frac {3 i \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}-\frac {\arctan \left (a x \right )^{3} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )}{c}-\frac {6 i \operatorname {polylog}\left (4, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}-\frac {6 \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}-\frac {3 i \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}+\frac {i \arctan \left (a x \right )^{4}}{4 c}+\frac {3 \arctan \left (a x \right ) \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}-\frac {3 i \arctan \left (a x \right )^{2}}{c}+\frac {3 \arctan \left (a x \right ) \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )}{c}\right )\)	\(441\)

input

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

output

a^2*(-6*I/c*polylog(4,(1+I*a*x)/(a^2*x^2+1)^(1/2))-1/2/c*arctan(a*x)^2*(-I 
*arctan(a*x)-3*I*a*x+x*arctan(a*x)*a)*(I+a*x)/a^2/x^2+3*I/c*arctan(a*x)^2* 
polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-1/c*arctan(a*x)^3*ln(1-(1+I*a*x)/( 
a^2*x^2+1)^(1/2))-3*I/c*polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-6/c*arctan 
(a*x)*polylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))+3*I/c*arctan(a*x)^2*polylog(2 
,(1+I*a*x)/(a^2*x^2+1)^(1/2))-1/c*arctan(a*x)^3*ln((1+I*a*x)/(a^2*x^2+1)^( 
1/2)+1)-6*I/c*polylog(4,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-6/c*arctan(a*x)*poly 
log(3,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-3*I/c*polylog(2,(1+I*a*x)/(a^2*x^2+1)^ 
(1/2))+1/4*I/c*arctan(a*x)^4+3/c*arctan(a*x)*ln(1-(1+I*a*x)/(a^2*x^2+1)^(1 
/2))-3*I/c*arctan(a*x)^2+3/c*arctan(a*x)*ln((1+I*a*x)/(a^2*x^2+1)^(1/2)+1) 
)
 

3.4.94.5 Fricas [F]

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

input

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

output

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

3.4.94.6 Sympy [F]

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

input

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

output

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

3.4.94.7 Maxima [F]

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

input

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

output

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

3.4.94.8 Giac [F]

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

input

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

output

sage0*x
 

3.4.94.9 Mupad [F(-1)]

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

input

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

output

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