Integrand size = 22, antiderivative size = 234 \[ \int \frac {\arctan (a x)^3}{x^2 \left (c+a^2 c x^2\right )^2} \, dx=\frac {3 a}{8 c^2 \left (1+a^2 x^2\right )}+\frac {3 a^2 x \arctan (a x)}{4 c^2 \left (1+a^2 x^2\right )}+\frac {3 a \arctan (a x)^2}{8 c^2}-\frac {3 a \arctan (a x)^2}{4 c^2 \left (1+a^2 x^2\right )}-\frac {i a \arctan (a x)^3}{c^2}-\frac {\arctan (a x)^3}{c^2 x}-\frac {a^2 x \arctan (a x)^3}{2 c^2 \left (1+a^2 x^2\right )}-\frac {3 a \arctan (a x)^4}{8 c^2}+\frac {3 a \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c^2}-\frac {3 i a \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{c^2}+\frac {3 a \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )}{2 c^2} \]
3/8*a/c^2/(a^2*x^2+1)+3/4*a^2*x*arctan(a*x)/c^2/(a^2*x^2+1)+3/8*a*arctan(a *x)^2/c^2-3/4*a*arctan(a*x)^2/c^2/(a^2*x^2+1)-I*a*arctan(a*x)^3/c^2-arctan (a*x)^3/c^2/x-1/2*a^2*x*arctan(a*x)^3/c^2/(a^2*x^2+1)-3/8*a*arctan(a*x)^4/ c^2+3*a*arctan(a*x)^2*ln(2-2/(1-I*a*x))/c^2-3*I*a*arctan(a*x)*polylog(2,-1 +2/(1-I*a*x))/c^2+3/2*a*polylog(3,-1+2/(1-I*a*x))/c^2
Time = 0.31 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.67 \[ \int \frac {\arctan (a x)^3}{x^2 \left (c+a^2 c x^2\right )^2} \, dx=\frac {a \left (-2 i \pi ^3+16 i \arctan (a x)^3-\frac {16 \arctan (a x)^3}{a x}-6 \arctan (a x)^4+3 \cos (2 \arctan (a x))-6 \arctan (a x)^2 \cos (2 \arctan (a x))+48 \arctan (a x)^2 \log \left (1-e^{-2 i \arctan (a x)}\right )+48 i \arctan (a x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )+24 \operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )+6 \arctan (a x) \sin (2 \arctan (a x))-4 \arctan (a x)^3 \sin (2 \arctan (a x))\right )}{16 c^2} \]
(a*((-2*I)*Pi^3 + (16*I)*ArcTan[a*x]^3 - (16*ArcTan[a*x]^3)/(a*x) - 6*ArcT an[a*x]^4 + 3*Cos[2*ArcTan[a*x]] - 6*ArcTan[a*x]^2*Cos[2*ArcTan[a*x]] + 48 *ArcTan[a*x]^2*Log[1 - E^((-2*I)*ArcTan[a*x])] + (48*I)*ArcTan[a*x]*PolyLo g[2, E^((-2*I)*ArcTan[a*x])] + 24*PolyLog[3, E^((-2*I)*ArcTan[a*x])] + 6*A rcTan[a*x]*Sin[2*ArcTan[a*x]] - 4*ArcTan[a*x]^3*Sin[2*ArcTan[a*x]]))/(16*c ^2)
Time = 1.78 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.12, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.591, Rules used = {5501, 27, 5427, 5453, 5361, 5419, 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.
| \(\displaystyle \int \frac {\arctan (a x)^3}{x^2 \left (a^2 c x^2+c\right )^2} \, dx\) |
| \(\Big \downarrow \) 5501 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)^3}{c x^2 \left (a^2 x^2+1\right )}dx}{c}-a^2 \int \frac {\arctan (a x)^3}{c^2 \left (a^2 x^2+1\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )}dx}{c^2}-\frac {a^2 \int \frac {\arctan (a x)^3}{\left (a^2 x^2+1\right )^2}dx}{c^2}\) |
| \(\Big \downarrow \) 5427 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )}dx}{c^2}-\frac {a^2 \left (-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )}{c^2}\) |
| \(\Big \downarrow \) 5453 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)^3}{x^2}dx-a^2 \int \frac {\arctan (a x)^3}{a^2 x^2+1}dx}{c^2}-\frac {a^2 \left (-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )}{c^2}\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {3 a \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx-a^2 \int \frac {\arctan (a x)^3}{a^2 x^2+1}dx-\frac {\arctan (a x)^3}{x}}{c^2}-\frac {a^2 \left (-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )}{c^2}\) |
| \(\Big \downarrow \) 5419 |
| \(\displaystyle \frac {3 a \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}}{c^2}-\frac {a^2 \left (-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )}{c^2}\) |
| \(\Big \downarrow \) 5459 |
| \(\displaystyle -\frac {a^2 \left (-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )}{c^2}+\frac {3 a \left (i \int \frac {\arctan (a x)^2}{x (a x+i)}dx-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}}{c^2}\) |
| \(\Big \downarrow \) 5403 |
| \(\displaystyle -\frac {a^2 \left (-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )}{c^2}+\frac {3 a \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}}{c^2}\) |
| \(\Big \downarrow \) 5465 |
| \(\displaystyle -\frac {a^2 \left (-\frac {3}{2} a \left (\frac {\int \frac {\arctan (a x)}{\left (a^2 x^2+1\right )^2}dx}{a}-\frac {\arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )}{c^2}+\frac {3 a \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}}{c^2}\) |
| \(\Big \downarrow \) 5427 |
| \(\displaystyle -\frac {a^2 \left (-\frac {3}{2} a \left (\frac {-\frac {1}{2} a \int \frac {x}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}}{a}-\frac {\arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )}{c^2}+\frac {3 a \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}}{c^2}\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle -\frac {a^2 \left (\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}-\frac {3}{2} a \left (\frac {\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}}{a}-\frac {\arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^4}{8 a}\right )}{c^2}+\frac {3 a \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}}{c^2}\) |
| \(\Big \downarrow \) 5527 |
| \(\displaystyle -\frac {a^2 \left (\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}-\frac {3}{2} a \left (\frac {\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}}{a}-\frac {\arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^4}{8 a}\right )}{c^2}+\frac {3 a \left (i \left (2 i a \left (\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{2 a}-\frac {1}{2} i \int \frac {\operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{a^2 x^2+1}dx\right )-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}}{c^2}\) |
| \(\Big \downarrow \) 7164 |
| \(\displaystyle -\frac {a^2 \left (\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}-\frac {3}{2} a \left (\frac {\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}}{a}-\frac {\arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^4}{8 a}\right )}{c^2}+\frac {3 a \left (i \left (2 i a \left (\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{2 a}-\frac {\operatorname {PolyLog}\left (3,\frac {2}{1-i a x}-1\right )}{4 a}\right )-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}}{c^2}\) |
-((a^2*((x*ArcTan[a*x]^3)/(2*(1 + a^2*x^2)) + ArcTan[a*x]^4/(8*a) - (3*a*( -1/2*ArcTan[a*x]^2/(a^2*(1 + a^2*x^2)) + (1/(4*a*(1 + a^2*x^2)) + (x*ArcTa n[a*x])/(2*(1 + a^2*x^2)) + ArcTan[a*x]^2/(4*a))/a))/2))/c^2) + (-(ArcTan[ a*x]^3/x) - (a*ArcTan[a*x]^4)/4 + 3*a*((-1/3*I)*ArcTan[a*x]^3 + I*((-I)*Ar cTan[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 - PolyLog[3, -1 + 2/(1 - I*a*x)]/(4*a)))))/c^2
3.5.1.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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_.)/((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]
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[((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]
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]
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 = 77.72 (sec) , antiderivative size = 1731, normalized size of antiderivative = 7.40
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1731\) |
| default | \(\text {Expression too large to display}\) | \(1731\) |
| parts | \(\text {Expression too large to display}\) | \(1735\) |
a*(-1/c^2*arctan(a*x)^3/a/x-1/2/c^2*arctan(a*x)^3*a*x/(a^2*x^2+1)-3/2/c^2* arctan(a*x)^4-3/2/c^2*(-3/4*arctan(a*x)^4-2*arctan(a*x)^2*ln(a*x)+1/2*arct an(a*x)^2/(a^2*x^2+1)+arctan(a*x)^2*ln(a^2*x^2+1)-2*arctan(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+1/16*(I+a*x)/(a*x-I)+I*arctan(a*x)*(I+a*x)/(8*a*x -8*I)+1/16*(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+I*a*x)/(a^2*x^2+1)^(1/2)+1)-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))-2*arctan(a*x)^2*ln(1-(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))-4*polylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))-1/4*(2*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+4*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))^3-2*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+2*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)-4*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)^2)^2+2*I*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2) ^3-4*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))^2- 4*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+4*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))^2-2*I*Pi*csgn(I*(1+I*a*x)/(a^2*x^2+...
\[ \int \frac {\arctan (a x)^3}{x^2 \left (c+a^2 c x^2\right )^2} \, dx=\int { \frac {\arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{2} x^{2}} \,d x } \]
\[ \int \frac {\arctan (a x)^3}{x^2 \left (c+a^2 c x^2\right )^2} \, dx=\frac {\int \frac {\operatorname {atan}^{3}{\left (a x \right )}}{a^{4} x^{6} + 2 a^{2} x^{4} + x^{2}}\, dx}{c^{2}} \]
\[ \int \frac {\arctan (a x)^3}{x^2 \left (c+a^2 c x^2\right )^2} \, dx=\int { \frac {\arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{2} x^{2}} \,d x } \]
-1/2048*(240*(a^3*x^3 + a*x)*arctan(a*x)^4 - 9*(a^3*x^3 + a*x)*log(a^2*x^2 + 1)^4 + 128*(3*a^2*x^2 + 2)*arctan(a*x)^3 - 24*(3*(a^3*x^3 + a*x)*arctan (a*x)^2 + 4*(3*a^2*x^2 + 2)*arctan(a*x))*log(a^2*x^2 + 1)^2 - 4*(a^2*c^2*x ^3 + c^2*x)*(72*a^5*(a^2/(a^8*c^2*x^2 + a^6*c^2) + log(a^2*x^2 + 1)/(a^6*c ^2*x^2 + a^4*c^2)) - 18432*a^5*integrate(1/256*x^5*arctan(a*x)^2*log(a^2*x ^2 + 1)/(a^4*c^2*x^6 + 2*a^2*c^2*x^4 + c^2*x^2), x) - 4608*a^5*integrate(1 /256*x^5*log(a^2*x^2 + 1)^3/(a^4*c^2*x^6 + 2*a^2*c^2*x^4 + c^2*x^2), x) + 36864*a^4*integrate(1/256*x^4*arctan(a*x)^3/(a^4*c^2*x^6 + 2*a^2*c^2*x^4 + c^2*x^2), x) + 9216*a^4*integrate(1/256*x^4*arctan(a*x)*log(a^2*x^2 + 1)^ 2/(a^4*c^2*x^6 + 2*a^2*c^2*x^4 + c^2*x^2), x) - 73728*a^4*integrate(1/256* x^4*arctan(a*x)*log(a^2*x^2 + 1)/(a^4*c^2*x^6 + 2*a^2*c^2*x^4 + c^2*x^2), x) + 9*a^3*log(a^2*x^2 + 1)^3/(a^4*c^2*x^2 + a^2*c^2) + 27*(2*a^4*(a^2/(a^ 10*c^2*x^2 + a^8*c^2) + log(a^2*x^2 + 1)/(a^8*c^2*x^2 + a^6*c^2)) + a^2*lo g(a^2*x^2 + 1)^2/(a^6*c^2*x^2 + a^4*c^2))*a^3 - 18432*a^3*integrate(1/256* x^3*arctan(a*x)^2*log(a^2*x^2 + 1)/(a^4*c^2*x^6 + 2*a^2*c^2*x^4 + c^2*x^2) , x) + 73728*a^3*integrate(1/256*x^3*arctan(a*x)^2/(a^4*c^2*x^6 + 2*a^2*c^ 2*x^4 + c^2*x^2), x) + 36*a^3*log(a^2*x^2 + 1)^2/(a^4*c^2*x^2 + a^2*c^2) + 36864*a^2*integrate(1/256*x^2*arctan(a*x)^3/(a^4*c^2*x^6 + 2*a^2*c^2*x^4 + c^2*x^2), x) + 9216*a^2*integrate(1/256*x^2*arctan(a*x)*log(a^2*x^2 + 1) ^2/(a^4*c^2*x^6 + 2*a^2*c^2*x^4 + c^2*x^2), x) - 49152*a^2*integrate(1/...
\[ \int \frac {\arctan (a x)^3}{x^2 \left (c+a^2 c x^2\right )^2} \, dx=\int { \frac {\arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{2} x^{2}} \,d x } \]
Timed out. \[ \int \frac {\arctan (a x)^3}{x^2 \left (c+a^2 c x^2\right )^2} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^3}{x^2\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]