Integrand size = 22, antiderivative size = 227 \[ \int \frac {\arctan (a x)^3}{x^4 \left (c+a^2 c x^2\right )} \, dx=-\frac {a^2 \arctan (a x)}{c x}-\frac {a^3 \arctan (a x)^2}{2 c}-\frac {a \arctan (a x)^2}{2 c x^2}+\frac {4 i a^3 \arctan (a x)^3}{3 c}-\frac {\arctan (a x)^3}{3 c x^3}+\frac {a^2 \arctan (a x)^3}{c x}+\frac {a^3 \arctan (a x)^4}{4 c}+\frac {a^3 \log (x)}{c}-\frac {a^3 \log \left (1+a^2 x^2\right )}{2 c}-\frac {4 a^3 \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c}+\frac {4 i a^3 \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{c}-\frac {2 a^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )}{c} \]
-a^2*arctan(a*x)/c/x-1/2*a^3*arctan(a*x)^2/c-1/2*a*arctan(a*x)^2/c/x^2+4/3 *I*a^3*arctan(a*x)^3/c-1/3*arctan(a*x)^3/c/x^3+a^2*arctan(a*x)^3/c/x+1/4*a ^3*arctan(a*x)^4/c+a^3*ln(x)/c-1/2*a^3*ln(a^2*x^2+1)/c-4*a^3*arctan(a*x)^2 *ln(2-2/(1-I*a*x))/c+4*I*a^3*arctan(a*x)*polylog(2,-1+2/(1-I*a*x))/c-2*a^3 *polylog(3,-1+2/(1-I*a*x))/c
Time = 0.60 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.67 \[ \int \frac {\arctan (a x)^3}{x^4 \left (c+a^2 c x^2\right )} \, dx=\frac {a^3 \left (\frac {1}{12} \left (2 i \pi ^3-\frac {12 \arctan (a x)}{a x}+\left (-16 i-\frac {4}{a^3 x^3}+\frac {12}{a x}\right ) \arctan (a x)^3+3 \arctan (a x)^4+\arctan (a x)^2 \left (-6-\frac {6}{a^2 x^2}-48 \log \left (1-e^{-2 i \arctan (a x)}\right )\right )+12 \log (a x)-6 \log \left (1+a^2 x^2\right )\right )-4 i \arctan (a x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )-2 \operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )\right )}{c} \]
(a^3*(((2*I)*Pi^3 - (12*ArcTan[a*x])/(a*x) + (-16*I - 4/(a^3*x^3) + 12/(a* x))*ArcTan[a*x]^3 + 3*ArcTan[a*x]^4 + ArcTan[a*x]^2*(-6 - 6/(a^2*x^2) - 48 *Log[1 - E^((-2*I)*ArcTan[a*x])]) + 12*Log[a*x] - 6*Log[1 + a^2*x^2])/12 - (4*I)*ArcTan[a*x]*PolyLog[2, E^((-2*I)*ArcTan[a*x])] - 2*PolyLog[3, E^((- 2*I)*ArcTan[a*x])]))/c
Time = 2.31 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.41, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.773, Rules used = {5453, 27, 5361, 5453, 5361, 5419, 5453, 5361, 243, 47, 14, 16, 5419, 5459, 5403, 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.
| \(\displaystyle \int \frac {\arctan (a x)^3}{x^4 \left (a^2 c x^2+c\right )} \, dx\) |
| \(\Big \downarrow \) 5453 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)^3}{x^4}dx}{c}-a^2 \int \frac {\arctan (a x)^3}{c x^2 \left (a^2 x^2+1\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)^3}{x^4}dx}{c}-\frac {a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )}dx}{c}\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {a \int \frac {\arctan (a x)^2}{x^3 \left (a^2 x^2+1\right )}dx-\frac {\arctan (a x)^3}{3 x^3}}{c}-\frac {a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )}dx}{c}\) |
| \(\Big \downarrow \) 5453 |
| \(\displaystyle \frac {a \left (\int \frac {\arctan (a x)^2}{x^3}dx-a^2 \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx\right )-\frac {\arctan (a x)^3}{3 x^3}}{c}-\frac {a^2 \left (\int \frac {\arctan (a x)^3}{x^2}dx-a^2 \int \frac {\arctan (a x)^3}{a^2 x^2+1}dx\right )}{c}\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {a \left (a^2 \left (-\int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx\right )+a \int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )}dx-\frac {\arctan (a x)^2}{2 x^2}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c}-\frac {a^2 \left (3 a \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx-a^2 \int \frac {\arctan (a x)^3}{a^2 x^2+1}dx-\frac {\arctan (a x)^3}{x}\right )}{c}\) |
| \(\Big \downarrow \) 5419 |
| \(\displaystyle \frac {a \left (a^2 \left (-\int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx\right )+a \int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )}dx-\frac {\arctan (a x)^2}{2 x^2}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c}-\frac {a^2 \left (3 a \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )}{c}\) |
| \(\Big \downarrow \) 5453 |
| \(\displaystyle \frac {a \left (a^2 \left (-\int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx\right )+a \left (\int \frac {\arctan (a x)}{x^2}dx-a^2 \int \frac {\arctan (a x)}{a^2 x^2+1}dx\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c}-\frac {a^2 \left (3 a \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )}{c}\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {a \left (a^2 \left (-\int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx\right )+a \left (a^2 \left (-\int \frac {\arctan (a x)}{a^2 x^2+1}dx\right )+a \int \frac {1}{x \left (a^2 x^2+1\right )}dx-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c}-\frac {a^2 \left (3 a \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )}{c}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {a \left (a^2 \left (-\int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx\right )+a \left (a^2 \left (-\int \frac {\arctan (a x)}{a^2 x^2+1}dx\right )+\frac {1}{2} a \int \frac {1}{x^2 \left (a^2 x^2+1\right )}dx^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c}-\frac {a^2 \left (3 a \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )}{c}\) |
| \(\Big \downarrow \) 47 |
| \(\displaystyle \frac {a \left (a^2 \left (-\int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx\right )+a \left (a^2 \left (-\int \frac {\arctan (a x)}{a^2 x^2+1}dx\right )+\frac {1}{2} a \left (\int \frac {1}{x^2}dx^2-a^2 \int \frac {1}{a^2 x^2+1}dx^2\right )-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c}-\frac {a^2 \left (3 a \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )}{c}\) |
| \(\Big \downarrow \) 14 |
| \(\displaystyle \frac {a \left (a^2 \left (-\int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx\right )+a \left (a^2 \left (-\int \frac {\arctan (a x)}{a^2 x^2+1}dx\right )+\frac {1}{2} a \left (\log \left (x^2\right )-a^2 \int \frac {1}{a^2 x^2+1}dx^2\right )-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c}-\frac {a^2 \left (3 a \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )}{c}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {a \left (a^2 \left (-\int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx\right )+a \left (a^2 \left (-\int \frac {\arctan (a x)}{a^2 x^2+1}dx\right )+\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c}-\frac {a^2 \left (3 a \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )}{c}\) |
| \(\Big \downarrow \) 5419 |
| \(\displaystyle \frac {a \left (a^2 \left (-\int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c}-\frac {a^2 \left (3 a \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )}{c}\) |
| \(\Big \downarrow \) 5459 |
| \(\displaystyle \frac {-\frac {\arctan (a x)^3}{3 x^3}+a \left (-\left (a^2 \left (i \int \frac {\arctan (a x)^2}{x (a x+i)}dx-\frac {1}{3} i \arctan (a x)^3\right )\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )}{c}-\frac {a^2 \left (3 a \left (i \int \frac {\arctan (a x)^2}{x (a x+i)}dx-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )}{c}\) |
| \(\Big \downarrow \) 5403 |
| \(\displaystyle \frac {-\frac {\arctan (a x)^3}{3 x^3}+a \left (-\left (a^2 \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )}{c}-\frac {a^2 \left (3 a \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )}{c}\) |
| \(\Big \downarrow \) 5527 |
| \(\displaystyle \frac {-\frac {\arctan (a x)^3}{3 x^3}+a \left (-\left (a^2 \left (i \left (2 i a \left (\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{2 a}-\frac {1}{2} i \int \frac {\operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{a^2 x^2+1}dx\right )-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )}{c}-\frac {a^2 \left (3 a \left (i \left (2 i a \left (\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{2 a}-\frac {1}{2} i \int \frac {\operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{a^2 x^2+1}dx\right )-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )}{c}\) |
| \(\Big \downarrow \) 7164 |
| \(\displaystyle \frac {-\frac {\arctan (a x)^3}{3 x^3}+a \left (-\left (a^2 \left (i \left (2 i a \left (\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{2 a}-\frac {\operatorname {PolyLog}\left (3,\frac {2}{1-i a x}-1\right )}{4 a}\right )-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )}{c}-\frac {a^2 \left (3 a \left (i \left (2 i a \left (\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{2 a}-\frac {\operatorname {PolyLog}\left (3,\frac {2}{1-i a x}-1\right )}{4 a}\right )-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )}{c}\) |
-((a^2*(-(ArcTan[a*x]^3/x) - (a*ArcTan[a*x]^4)/4 + 3*a*((-1/3*I)*ArcTan[a* x]^3 + I*((-I)*ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*x)] + (2*I)*a*(((I/2)*ArcT an[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)])/a - PolyLog[3, -1 + 2/(1 - I*a*x)] /(4*a))))))/c) + (-1/3*ArcTan[a*x]^3/x^3 + a*(-1/2*ArcTan[a*x]^2/x^2 + a*( -(ArcTan[a*x]/x) - (a*ArcTan[a*x]^2)/2 + (a*(Log[x^2] - Log[1 + a^2*x^2])) /2) - a^2*((-1/3*I)*ArcTan[a*x]^3 + I*((-I)*ArcTan[a*x]^2*Log[2 - 2/(1 - I *a*x)] + (2*I)*a*(((I/2)*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)])/a - P olyLog[3, -1 + 2/(1 - I*a*x)]/(4*a))))))/c
3.4.95.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Simp[b/(b*c - a*d) Int[1/(a + b*x), x], x] - Simp[d/(b*c - a*d) Int[1/(c + d*x), x ], x] /; FreeQ[{a, b, c, d}, x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
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]
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]
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]
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]
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]
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]
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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 77.33 (sec) , antiderivative size = 1831, normalized size of antiderivative = 8.07
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1831\) |
| default | \(\text {Expression too large to display}\) | \(1831\) |
| parts | \(\text {Expression too large to display}\) | \(1930\) |
a^3*(1/c*arctan(a*x)^4-1/3/c*arctan(a*x)^3/a^3/x^3+1/c*arctan(a*x)^3/a/x-1 /c*(-2*arctan(a*x)^2*ln(a^2*x^2+1)+1/2*arctan(a*x)^2/a^2/x^2+4*arctan(a*x) ^2*ln(a*x)+4*arctan(a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2))-4*arctan(a*x)^2 *ln((1+I*a*x)^2/(a^2*x^2+1)-1)+1/6*arctan(a*x)*(6*I*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*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))* Pi*arctan(a*x)*a*x+6*I*a*x+12*I*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*Pi*arctan(a*x)*a*x+12*I*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*Pi*arctan(a*x)*a*x+6*I*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*Pi*arctan(a*x)*a*x+12*I*csgn(((1+I*a*x)^2/(a^2*x^2+1) -1)/((1+I*a*x)^2/(a^2*x^2+1)+1))^3*Pi*arctan(a*x)*a*x-6*I*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)*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))*Pi*arctan(a*x)*a*x+12*I*csgn(I*( (1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))*csgn(((1+I*a*x)^2/ (a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))*Pi*arctan(a*x)*a*x-12*I*csgn(( (1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))^2*Pi*arctan(a*x)*a *x-12*I*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)^2)^2*Pi*arctan(a*x)*a*x-8*I*arctan(a*x)^2*a*x-12*I*csgn(I*((1+I*a*x)^ 2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))*csgn(((1+I*a*x)^2/(a^2*x^2+1 )-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))^2*Pi*arctan(a*x)*a*x+12*I*csgn(I*((1+...
\[ \int \frac {\arctan (a x)^3}{x^4 \left (c+a^2 c x^2\right )} \, dx=\int { \frac {\arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )} x^{4}} \,d x } \]
\[ \int \frac {\arctan (a x)^3}{x^4 \left (c+a^2 c x^2\right )} \, dx=\frac {\int \frac {\operatorname {atan}^{3}{\left (a x \right )}}{a^{2} x^{6} + x^{4}}\, dx}{c} \]
Timed out. \[ \int \frac {\arctan (a x)^3}{x^4 \left (c+a^2 c x^2\right )} \, dx=\text {Timed out} \]
\[ \int \frac {\arctan (a x)^3}{x^4 \left (c+a^2 c x^2\right )} \, dx=\int { \frac {\arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )} x^{4}} \,d x } \]
Timed out. \[ \int \frac {\arctan (a x)^3}{x^4 \left (c+a^2 c x^2\right )} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^3}{x^4\,\left (c\,a^2\,x^2+c\right )} \,d x \]