Integrand size = 19, antiderivative size = 289 \[ \int \left (c+a^2 c x^2\right )^2 \arctan (a x)^3 \, dx=-\frac {c^2 \left (1+a^2 x^2\right )}{20 a}+c^2 x \arctan (a x)+\frac {1}{10} c^2 x \left (1+a^2 x^2\right ) \arctan (a x)-\frac {2 c^2 \left (1+a^2 x^2\right ) \arctan (a x)^2}{5 a}-\frac {3 c^2 \left (1+a^2 x^2\right )^2 \arctan (a x)^2}{20 a}+\frac {8 i c^2 \arctan (a x)^3}{15 a}+\frac {8}{15} c^2 x \arctan (a x)^3+\frac {4}{15} c^2 x \left (1+a^2 x^2\right ) \arctan (a x)^3+\frac {1}{5} c^2 x \left (1+a^2 x^2\right )^2 \arctan (a x)^3+\frac {8 c^2 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{5 a}-\frac {c^2 \log \left (1+a^2 x^2\right )}{2 a}+\frac {8 i c^2 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{5 a}+\frac {4 c^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{5 a} \]
-1/20*c^2*(a^2*x^2+1)/a+c^2*x*arctan(a*x)+1/10*c^2*x*(a^2*x^2+1)*arctan(a* x)-2/5*c^2*(a^2*x^2+1)*arctan(a*x)^2/a-3/20*c^2*(a^2*x^2+1)^2*arctan(a*x)^ 2/a+8/15*I*c^2*arctan(a*x)^3/a+8/15*c^2*x*arctan(a*x)^3+4/15*c^2*x*(a^2*x^ 2+1)*arctan(a*x)^3+1/5*c^2*x*(a^2*x^2+1)^2*arctan(a*x)^3+8/5*c^2*arctan(a* x)^2*ln(2/(1+I*a*x))/a-1/2*c^2*ln(a^2*x^2+1)/a+8/5*I*c^2*arctan(a*x)*polyl og(2,1-2/(1+I*a*x))/a+4/5*c^2*polylog(3,1-2/(1+I*a*x))/a
Time = 0.48 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.67 \[ \int \left (c+a^2 c x^2\right )^2 \arctan (a x)^3 \, dx=\frac {c^2 \left (-3-3 a^2 x^2+66 a x \arctan (a x)+6 a^3 x^3 \arctan (a x)-33 \arctan (a x)^2-42 a^2 x^2 \arctan (a x)^2-9 a^4 x^4 \arctan (a x)^2-32 i \arctan (a x)^3+60 a x \arctan (a x)^3+40 a^3 x^3 \arctan (a x)^3+12 a^5 x^5 \arctan (a x)^3+96 \arctan (a x)^2 \log \left (1+e^{2 i \arctan (a x)}\right )-30 \log \left (1+a^2 x^2\right )-96 i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )+48 \operatorname {PolyLog}\left (3,-e^{2 i \arctan (a x)}\right )\right )}{60 a} \]
(c^2*(-3 - 3*a^2*x^2 + 66*a*x*ArcTan[a*x] + 6*a^3*x^3*ArcTan[a*x] - 33*Arc Tan[a*x]^2 - 42*a^2*x^2*ArcTan[a*x]^2 - 9*a^4*x^4*ArcTan[a*x]^2 - (32*I)*A rcTan[a*x]^3 + 60*a*x*ArcTan[a*x]^3 + 40*a^3*x^3*ArcTan[a*x]^3 + 12*a^5*x^ 5*ArcTan[a*x]^3 + 96*ArcTan[a*x]^2*Log[1 + E^((2*I)*ArcTan[a*x])] - 30*Log [1 + a^2*x^2] - (96*I)*ArcTan[a*x]*PolyLog[2, -E^((2*I)*ArcTan[a*x])] + 48 *PolyLog[3, -E^((2*I)*ArcTan[a*x])]))/(60*a)
Time = 1.29 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.08, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.632, Rules used = {5415, 27, 5413, 5345, 240, 5415, 5345, 240, 5455, 5379, 5529, 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.
| \(\displaystyle \int \arctan (a x)^3 \left (a^2 c x^2+c\right )^2 \, dx\) |
| \(\Big \downarrow \) 5415 |
| \(\displaystyle \frac {3}{10} c \int c \left (a^2 x^2+1\right ) \arctan (a x)dx+\frac {4}{5} c \int c \left (a^2 x^2+1\right ) \arctan (a x)^3dx+\frac {1}{5} c^2 x \left (a^2 x^2+1\right )^2 \arctan (a x)^3-\frac {3 c^2 \left (a^2 x^2+1\right )^2 \arctan (a x)^2}{20 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{10} c^2 \int \left (a^2 x^2+1\right ) \arctan (a x)dx+\frac {4}{5} c^2 \int \left (a^2 x^2+1\right ) \arctan (a x)^3dx+\frac {1}{5} c^2 x \left (a^2 x^2+1\right )^2 \arctan (a x)^3-\frac {3 c^2 \left (a^2 x^2+1\right )^2 \arctan (a x)^2}{20 a}\) |
| \(\Big \downarrow \) 5413 |
| \(\displaystyle \frac {3}{10} c^2 \left (\frac {2}{3} \int \arctan (a x)dx+\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)-\frac {a^2 x^2+1}{6 a}\right )+\frac {4}{5} c^2 \int \left (a^2 x^2+1\right ) \arctan (a x)^3dx+\frac {1}{5} c^2 x \left (a^2 x^2+1\right )^2 \arctan (a x)^3-\frac {3 c^2 \left (a^2 x^2+1\right )^2 \arctan (a x)^2}{20 a}\) |
| \(\Big \downarrow \) 5345 |
| \(\displaystyle \frac {3}{10} c^2 \left (\frac {2}{3} \left (x \arctan (a x)-a \int \frac {x}{a^2 x^2+1}dx\right )+\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)-\frac {a^2 x^2+1}{6 a}\right )+\frac {4}{5} c^2 \int \left (a^2 x^2+1\right ) \arctan (a x)^3dx+\frac {1}{5} c^2 x \left (a^2 x^2+1\right )^2 \arctan (a x)^3-\frac {3 c^2 \left (a^2 x^2+1\right )^2 \arctan (a x)^2}{20 a}\) |
| \(\Big \downarrow \) 240 |
| \(\displaystyle \frac {4}{5} c^2 \int \left (a^2 x^2+1\right ) \arctan (a x)^3dx+\frac {1}{5} c^2 x \left (a^2 x^2+1\right )^2 \arctan (a x)^3-\frac {3 c^2 \left (a^2 x^2+1\right )^2 \arctan (a x)^2}{20 a}+\frac {3}{10} c^2 \left (\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)+\frac {2}{3} \left (x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}\right )-\frac {a^2 x^2+1}{6 a}\right )\) |
| \(\Big \downarrow \) 5415 |
| \(\displaystyle \frac {4}{5} c^2 \left (\int \arctan (a x)dx+\frac {2}{3} \int \arctan (a x)^3dx+\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)^3-\frac {\left (a^2 x^2+1\right ) \arctan (a x)^2}{2 a}\right )+\frac {1}{5} c^2 x \left (a^2 x^2+1\right )^2 \arctan (a x)^3-\frac {3 c^2 \left (a^2 x^2+1\right )^2 \arctan (a x)^2}{20 a}+\frac {3}{10} c^2 \left (\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)+\frac {2}{3} \left (x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}\right )-\frac {a^2 x^2+1}{6 a}\right )\) |
| \(\Big \downarrow \) 5345 |
| \(\displaystyle \frac {4}{5} c^2 \left (\frac {2}{3} \left (x \arctan (a x)^3-3 a \int \frac {x \arctan (a x)^2}{a^2 x^2+1}dx\right )-a \int \frac {x}{a^2 x^2+1}dx+\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)^3-\frac {\left (a^2 x^2+1\right ) \arctan (a x)^2}{2 a}+x \arctan (a x)\right )+\frac {1}{5} c^2 x \left (a^2 x^2+1\right )^2 \arctan (a x)^3-\frac {3 c^2 \left (a^2 x^2+1\right )^2 \arctan (a x)^2}{20 a}+\frac {3}{10} c^2 \left (\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)+\frac {2}{3} \left (x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}\right )-\frac {a^2 x^2+1}{6 a}\right )\) |
| \(\Big \downarrow \) 240 |
| \(\displaystyle \frac {4}{5} c^2 \left (\frac {2}{3} \left (x \arctan (a x)^3-3 a \int \frac {x \arctan (a x)^2}{a^2 x^2+1}dx\right )+\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)^3-\frac {\left (a^2 x^2+1\right ) \arctan (a x)^2}{2 a}-\frac {\log \left (a^2 x^2+1\right )}{2 a}+x \arctan (a x)\right )+\frac {1}{5} c^2 x \left (a^2 x^2+1\right )^2 \arctan (a x)^3-\frac {3 c^2 \left (a^2 x^2+1\right )^2 \arctan (a x)^2}{20 a}+\frac {3}{10} c^2 \left (\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)+\frac {2}{3} \left (x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}\right )-\frac {a^2 x^2+1}{6 a}\right )\) |
| \(\Big \downarrow \) 5455 |
| \(\displaystyle \frac {4}{5} c^2 \left (\frac {2}{3} \left (x \arctan (a x)^3-3 a \left (-\frac {\int \frac {\arctan (a x)^2}{i-a x}dx}{a}-\frac {i \arctan (a x)^3}{3 a^2}\right )\right )+\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)^3-\frac {\left (a^2 x^2+1\right ) \arctan (a x)^2}{2 a}-\frac {\log \left (a^2 x^2+1\right )}{2 a}+x \arctan (a x)\right )+\frac {1}{5} c^2 x \left (a^2 x^2+1\right )^2 \arctan (a x)^3-\frac {3 c^2 \left (a^2 x^2+1\right )^2 \arctan (a x)^2}{20 a}+\frac {3}{10} c^2 \left (\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)+\frac {2}{3} \left (x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}\right )-\frac {a^2 x^2+1}{6 a}\right )\) |
| \(\Big \downarrow \) 5379 |
| \(\displaystyle \frac {4}{5} c^2 \left (\frac {2}{3} \left (x \arctan (a x)^3-3 a \left (-\frac {\frac {\arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a}-2 \int \frac {\arctan (a x) \log \left (\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx}{a}-\frac {i \arctan (a x)^3}{3 a^2}\right )\right )+\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)^3-\frac {\left (a^2 x^2+1\right ) \arctan (a x)^2}{2 a}-\frac {\log \left (a^2 x^2+1\right )}{2 a}+x \arctan (a x)\right )+\frac {1}{5} c^2 x \left (a^2 x^2+1\right )^2 \arctan (a x)^3-\frac {3 c^2 \left (a^2 x^2+1\right )^2 \arctan (a x)^2}{20 a}+\frac {3}{10} c^2 \left (\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)+\frac {2}{3} \left (x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}\right )-\frac {a^2 x^2+1}{6 a}\right )\) |
| \(\Big \downarrow \) 5529 |
| \(\displaystyle \frac {4}{5} c^2 \left (\frac {2}{3} \left (x \arctan (a x)^3-3 a \left (-\frac {\frac {\arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a}-2 \left (\frac {1}{2} i \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx-\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{2 a}\right )}{a}-\frac {i \arctan (a x)^3}{3 a^2}\right )\right )+\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)^3-\frac {\left (a^2 x^2+1\right ) \arctan (a x)^2}{2 a}-\frac {\log \left (a^2 x^2+1\right )}{2 a}+x \arctan (a x)\right )+\frac {1}{5} c^2 x \left (a^2 x^2+1\right )^2 \arctan (a x)^3-\frac {3 c^2 \left (a^2 x^2+1\right )^2 \arctan (a x)^2}{20 a}+\frac {3}{10} c^2 \left (\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)+\frac {2}{3} \left (x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}\right )-\frac {a^2 x^2+1}{6 a}\right )\) |
| \(\Big \downarrow \) 7164 |
| \(\displaystyle \frac {4}{5} c^2 \left (\frac {2}{3} \left (x \arctan (a x)^3-3 a \left (-\frac {i \arctan (a x)^3}{3 a^2}-\frac {\frac {\arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a}-2 \left (-\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{2 a}-\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right )}{4 a}\right )}{a}\right )\right )+\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)^3-\frac {\left (a^2 x^2+1\right ) \arctan (a x)^2}{2 a}-\frac {\log \left (a^2 x^2+1\right )}{2 a}+x \arctan (a x)\right )+\frac {1}{5} c^2 x \left (a^2 x^2+1\right )^2 \arctan (a x)^3-\frac {3 c^2 \left (a^2 x^2+1\right )^2 \arctan (a x)^2}{20 a}+\frac {3}{10} c^2 \left (\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)+\frac {2}{3} \left (x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}\right )-\frac {a^2 x^2+1}{6 a}\right )\) |
(-3*c^2*(1 + a^2*x^2)^2*ArcTan[a*x]^2)/(20*a) + (c^2*x*(1 + a^2*x^2)^2*Arc Tan[a*x]^3)/5 + (3*c^2*(-1/6*(1 + a^2*x^2)/a + (x*(1 + a^2*x^2)*ArcTan[a*x ])/3 + (2*(x*ArcTan[a*x] - Log[1 + a^2*x^2]/(2*a)))/3))/10 + (4*c^2*(x*Arc Tan[a*x] - ((1 + a^2*x^2)*ArcTan[a*x]^2)/(2*a) + (x*(1 + a^2*x^2)*ArcTan[a *x]^3)/3 - Log[1 + a^2*x^2]/(2*a) + (2*(x*ArcTan[a*x]^3 - 3*a*(((-1/3*I)*A rcTan[a*x]^3)/a^2 - ((ArcTan[a*x]^2*Log[2/(1 + I*a*x)])/a - 2*(((-1/2*I)*A rcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)])/a - PolyLog[3, 1 - 2/(1 + I*a*x) ]/(4*a)))/a)))/3))/5
3.4.74.3.1 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), x_Symbol] :> Simp[Log[RemoveContent[a + b*x ^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcTan[c*x^n])^p, x] - Simp[b*c*n*p Int[x^n*((a + b*ArcTan[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0] && (EqQ[n, 1] || EqQ[p, 1])
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTan[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] + Simp[b*c*( p/e) Int[(a + b*ArcTan[c*x])^(p - 1)*(Log[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_.))*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbo l] :> Simp[(-b)*((d + e*x^2)^q/(2*c*q*(2*q + 1))), x] + (Simp[x*(d + e*x^2) ^q*((a + b*ArcTan[c*x])/(2*q + 1)), x] + Simp[2*d*(q/(2*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] && GtQ[q, 0]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_)^2)^(q_.), x_ Symbol] :> Simp[(-b)*p*(d + e*x^2)^q*((a + b*ArcTan[c*x])^(p - 1)/(2*c*q*(2 *q + 1))), x] + (Simp[x*(d + e*x^2)^q*((a + b*ArcTan[c*x])^p/(2*q + 1)), x] + Simp[2*d*(q/(2*q + 1)) Int[(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x])^p, x], x] + Simp[b^2*d*p*((p - 1)/(2*q*(2*q + 1))) Int[(d + e*x^2)^(q - 1)*( a + b*ArcTan[c*x])^(p - 2), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[q, 0] && GtQ[p, 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*e*(p + 1))), x] - Si mp[1/(c*d) Int[(a + b*ArcTan[c*x])^p/(I - c*x), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[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 = 14.29 (sec) , antiderivative size = 953, normalized size of antiderivative = 3.30
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(953\) |
| default | \(\text {Expression too large to display}\) | \(953\) |
| parts | \(\text {Expression too large to display}\) | \(955\) |
1/a*(1/5*c^2*arctan(a*x)^3*a^5*x^5+2/3*c^2*arctan(a*x)^3*a^3*x^3+c^2*arcta n(a*x)^3*a*x-1/5*c^2*(3/4*a^4*arctan(a*x)^2*x^4+7/2*x^2*arctan(a*x)^2*a^2+ 4*arctan(a*x)^2*ln(a^2*x^2+1)-8*arctan(a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/ 2))+1/12*I*(24*arctan(a*x)^2*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))^3-48*arcta n(a*x)^2*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))^2*csgn(I*(1+I*a*x)/(a^2*x^2+1) ^(1/2))+24*arctan(a*x)^2*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))*csgn(I*(1+I*a* x)/(a^2*x^2+1)^(1/2))^2-24*arctan(a*x)^2*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+24*arcta n(a*x)^2*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)*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)+24*a rctan(a*x)^2*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)^3-24*arctan(a*x)^2*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)^2*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)-24*arctan(a*x)^2*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 )+48*arctan(a*x)^2*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)^2-24*arctan(a*x)^2*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1 )+1)^2)^3+32*arctan(a*x)^3-3*I*a^2*x^2+66*I*arctan(a*x)*a*x-33*I*arctan(a* x)^2+60*arctan(a*x)+6*I*arctan(a*x)*a^3*x^3+96*I*arctan(a*x)^2*ln(2)-3*I)- 5*ln((1+I*a*x)^2/(a^2*x^2+1)+1)+8*I*arctan(a*x)*polylog(2,-(1+I*a*x)^2/(a^ 2*x^2+1))-4*polylog(3,-(1+I*a*x)^2/(a^2*x^2+1))))
\[ \int \left (c+a^2 c x^2\right )^2 \arctan (a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right )^{3} \,d x } \]
\[ \int \left (c+a^2 c x^2\right )^2 \arctan (a x)^3 \, dx=c^{2} \left (\int 2 a^{2} x^{2} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int a^{4} x^{4} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int \operatorname {atan}^{3}{\left (a x \right )}\, dx\right ) \]
c**2*(Integral(2*a**2*x**2*atan(a*x)**3, x) + Integral(a**4*x**4*atan(a*x) **3, x) + Integral(atan(a*x)**3, x))
\[ \int \left (c+a^2 c x^2\right )^2 \arctan (a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right )^{3} \,d x } \]
140*a^6*c^2*integrate(1/160*x^6*arctan(a*x)^3/(a^2*x^2 + 1), x) + 15*a^6*c ^2*integrate(1/160*x^6*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^2*x^2 + 1), x) + 12*a^6*c^2*integrate(1/160*x^6*arctan(a*x)*log(a^2*x^2 + 1)/(a^2*x^2 + 1), x) - 12*a^5*c^2*integrate(1/160*x^5*arctan(a*x)^2/(a^2*x^2 + 1), x) + 3*a ^5*c^2*integrate(1/160*x^5*log(a^2*x^2 + 1)^2/(a^2*x^2 + 1), x) + 420*a^4* c^2*integrate(1/160*x^4*arctan(a*x)^3/(a^2*x^2 + 1), x) + 45*a^4*c^2*integ rate(1/160*x^4*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^2*x^2 + 1), x) + 40*a^4*c ^2*integrate(1/160*x^4*arctan(a*x)*log(a^2*x^2 + 1)/(a^2*x^2 + 1), x) - 40 *a^3*c^2*integrate(1/160*x^3*arctan(a*x)^2/(a^2*x^2 + 1), x) + 10*a^3*c^2* integrate(1/160*x^3*log(a^2*x^2 + 1)^2/(a^2*x^2 + 1), x) + 7/32*c^2*arctan (a*x)^4/a + 420*a^2*c^2*integrate(1/160*x^2*arctan(a*x)^3/(a^2*x^2 + 1), x ) + 45*a^2*c^2*integrate(1/160*x^2*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^2*x^2 + 1), x) + 60*a^2*c^2*integrate(1/160*x^2*arctan(a*x)*log(a^2*x^2 + 1)/(a ^2*x^2 + 1), x) + 1/120*(3*a^4*c^2*x^5 + 10*a^2*c^2*x^3 + 15*c^2*x)*arctan (a*x)^3 - 60*a*c^2*integrate(1/160*x*arctan(a*x)^2/(a^2*x^2 + 1), x) + 15* a*c^2*integrate(1/160*x*log(a^2*x^2 + 1)^2/(a^2*x^2 + 1), x) - 1/160*(3*a^ 4*c^2*x^5 + 10*a^2*c^2*x^3 + 15*c^2*x)*arctan(a*x)*log(a^2*x^2 + 1)^2 + 15 *c^2*integrate(1/160*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^2*x^2 + 1), x)
\[ \int \left (c+a^2 c x^2\right )^2 \arctan (a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right )^{3} \,d x } \]
Timed out. \[ \int \left (c+a^2 c x^2\right )^2 \arctan (a x)^3 \, dx=\int {\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^2 \,d x \]