Integrand size = 22, antiderivative size = 169 \[ \int \frac {x^3 \arctan (a x)^2}{c+a^2 c x^2} \, dx=-\frac {x \arctan (a x)}{a^3 c}+\frac {\arctan (a x)^2}{2 a^4 c}+\frac {x^2 \arctan (a x)^2}{2 a^2 c}+\frac {i \arctan (a x)^3}{3 a^4 c}+\frac {\arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a^4 c}+\frac {\log \left (1+a^2 x^2\right )}{2 a^4 c}+\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{a^4 c}+\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{2 a^4 c} \]
-x*arctan(a*x)/a^3/c+1/2*arctan(a*x)^2/a^4/c+1/2*x^2*arctan(a*x)^2/a^2/c+1 /3*I*arctan(a*x)^3/a^4/c+arctan(a*x)^2*ln(2/(1+I*a*x))/a^4/c+1/2*ln(a^2*x^ 2+1)/a^4/c+I*arctan(a*x)*polylog(2,1-2/(1+I*a*x))/a^4/c+1/2*polylog(3,1-2/ (1+I*a*x))/a^4/c
Time = 0.12 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.73 \[ \int \frac {x^3 \arctan (a x)^2}{c+a^2 c x^2} \, dx=\frac {-a x \arctan (a x)+\frac {1}{2} \left (1+a^2 x^2\right ) \arctan (a x)^2-\frac {1}{3} i \arctan (a x)^3+\arctan (a x)^2 \log \left (1+e^{2 i \arctan (a x)}\right )-\log \left (\frac {1}{\sqrt {1+a^2 x^2}}\right )-i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,-e^{2 i \arctan (a x)}\right )}{a^4 c} \]
(-(a*x*ArcTan[a*x]) + ((1 + a^2*x^2)*ArcTan[a*x]^2)/2 - (I/3)*ArcTan[a*x]^ 3 + ArcTan[a*x]^2*Log[1 + E^((2*I)*ArcTan[a*x])] - Log[1/Sqrt[1 + a^2*x^2] ] - I*ArcTan[a*x]*PolyLog[2, -E^((2*I)*ArcTan[a*x])] + PolyLog[3, -E^((2*I )*ArcTan[a*x])]/2)/(a^4*c)
Time = 1.15 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.04, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5451, 27, 5361, 5451, 5345, 240, 5419, 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.
| \(\displaystyle \int \frac {x^3 \arctan (a x)^2}{a^2 c x^2+c} \, dx\) |
| \(\Big \downarrow \) 5451 |
| \(\displaystyle \frac {\int x \arctan (a x)^2dx}{a^2 c}-\frac {\int \frac {x \arctan (a x)^2}{c \left (a^2 x^2+1\right )}dx}{a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int x \arctan (a x)^2dx}{a^2 c}-\frac {\int \frac {x \arctan (a x)^2}{a^2 x^2+1}dx}{a^2 c}\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {\frac {1}{2} x^2 \arctan (a x)^2-a \int \frac {x^2 \arctan (a x)}{a^2 x^2+1}dx}{a^2 c}-\frac {\int \frac {x \arctan (a x)^2}{a^2 x^2+1}dx}{a^2 c}\) |
| \(\Big \downarrow \) 5451 |
| \(\displaystyle \frac {\frac {1}{2} x^2 \arctan (a x)^2-a \left (\frac {\int \arctan (a x)dx}{a^2}-\frac {\int \frac {\arctan (a x)}{a^2 x^2+1}dx}{a^2}\right )}{a^2 c}-\frac {\int \frac {x \arctan (a x)^2}{a^2 x^2+1}dx}{a^2 c}\) |
| \(\Big \downarrow \) 5345 |
| \(\displaystyle \frac {\frac {1}{2} x^2 \arctan (a x)^2-a \left (\frac {x \arctan (a x)-a \int \frac {x}{a^2 x^2+1}dx}{a^2}-\frac {\int \frac {\arctan (a x)}{a^2 x^2+1}dx}{a^2}\right )}{a^2 c}-\frac {\int \frac {x \arctan (a x)^2}{a^2 x^2+1}dx}{a^2 c}\) |
| \(\Big \downarrow \) 240 |
| \(\displaystyle \frac {\frac {1}{2} x^2 \arctan (a x)^2-a \left (\frac {x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}}{a^2}-\frac {\int \frac {\arctan (a x)}{a^2 x^2+1}dx}{a^2}\right )}{a^2 c}-\frac {\int \frac {x \arctan (a x)^2}{a^2 x^2+1}dx}{a^2 c}\) |
| \(\Big \downarrow \) 5419 |
| \(\displaystyle \frac {\frac {1}{2} x^2 \arctan (a x)^2-a \left (\frac {x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}}{a^2}-\frac {\arctan (a x)^2}{2 a^3}\right )}{a^2 c}-\frac {\int \frac {x \arctan (a x)^2}{a^2 x^2+1}dx}{a^2 c}\) |
| \(\Big \downarrow \) 5455 |
| \(\displaystyle \frac {\frac {1}{2} x^2 \arctan (a x)^2-a \left (\frac {x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}}{a^2}-\frac {\arctan (a x)^2}{2 a^3}\right )}{a^2 c}-\frac {-\frac {\int \frac {\arctan (a x)^2}{i-a x}dx}{a}-\frac {i \arctan (a x)^3}{3 a^2}}{a^2 c}\) |
| \(\Big \downarrow \) 5379 |
| \(\displaystyle \frac {\frac {1}{2} x^2 \arctan (a x)^2-a \left (\frac {x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}}{a^2}-\frac {\arctan (a x)^2}{2 a^3}\right )}{a^2 c}-\frac {-\frac {\frac {\arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a}-2 \int \frac {\arctan (a x) \log \left (\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx}{a}-\frac {i \arctan (a x)^3}{3 a^2}}{a^2 c}\) |
| \(\Big \downarrow \) 5529 |
| \(\displaystyle \frac {\frac {1}{2} x^2 \arctan (a x)^2-a \left (\frac {x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}}{a^2}-\frac {\arctan (a x)^2}{2 a^3}\right )}{a^2 c}-\frac {-\frac {\frac {\arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a}-2 \left (\frac {1}{2} i \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx-\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{2 a}\right )}{a}-\frac {i \arctan (a x)^3}{3 a^2}}{a^2 c}\) |
| \(\Big \downarrow \) 7164 |
| \(\displaystyle \frac {\frac {1}{2} x^2 \arctan (a x)^2-a \left (\frac {x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}}{a^2}-\frac {\arctan (a x)^2}{2 a^3}\right )}{a^2 c}-\frac {-\frac {i \arctan (a x)^3}{3 a^2}-\frac {\frac {\arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a}-2 \left (-\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{2 a}-\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right )}{4 a}\right )}{a}}{a^2 c}\) |
((x^2*ArcTan[a*x]^2)/2 - a*(-1/2*ArcTan[a*x]^2/a^3 + (x*ArcTan[a*x] - Log[ 1 + a^2*x^2]/(2*a))/a^2))/(a^2*c) - (((-1/3*I)*ArcTan[a*x]^3)/a^2 - ((ArcT an[a*x]^2*Log[2/(1 + I*a*x)])/a - 2*(((-1/2*I)*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)])/a - PolyLog[3, 1 - 2/(1 + I*a*x)]/(4*a)))/a)/(a^2*c)
3.3.83.3.1 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_)^(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_.)/((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_.))^(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[f^2/e Int[(f*x)^(m - 2)*(a + b*ArcTan[c*x] )^p, x], x] - Simp[d*(f^2/e) 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] && GtQ[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*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 = 19.90 (sec) , antiderivative size = 844, normalized size of antiderivative = 4.99
| method | result | size |
| derivativedivides | \(\frac {\frac {\arctan \left (a x \right )^{2} a^{2} x^{2}}{2 c}-\frac {\arctan \left (a x \right )^{2} \ln \left (a^{2} x^{2}+1\right )}{2 c}-\frac {-\arctan \left (a x \right )^{2} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+i \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )-\frac {\operatorname {polylog}\left (3, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}+\frac {i \arctan \left (a x \right ) \left (3 \arctan \left (a x \right ) \pi \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )^{3}-3 \arctan \left (a x \right ) \pi \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )^{2} \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )-3 \arctan \left (a x \right ) \pi \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )^{2} \operatorname {csgn}\left (\frac {i}{\left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )+3 \arctan \left (a x \right ) \pi \,\operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {i}{\left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )-3 \arctan \left (a x \right ) \pi {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )}^{3}+6 \arctan \left (a x \right ) \pi {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )}^{2} \operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )\right )-3 \arctan \left (a x \right ) \pi \,\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right ) {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )\right )}^{2}+3 \arctan \left (a x \right ) \pi {\operatorname {csgn}\left (\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )}^{2} \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )-6 \arctan \left (a x \right ) \pi \,\operatorname {csgn}\left (\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )^{2}+3 \arctan \left (a x \right ) \pi \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )^{3}+4 \arctan \left (a x \right )^{2}+12 i \arctan \left (a x \right ) \ln \left (2\right )+6 i \arctan \left (a x \right )-12-12 i a x \right )}{12}+\ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )}{c}}{a^{4}}\) | \(844\) |
| default | \(\frac {\frac {\arctan \left (a x \right )^{2} a^{2} x^{2}}{2 c}-\frac {\arctan \left (a x \right )^{2} \ln \left (a^{2} x^{2}+1\right )}{2 c}-\frac {-\arctan \left (a x \right )^{2} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+i \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )-\frac {\operatorname {polylog}\left (3, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}+\frac {i \arctan \left (a x \right ) \left (3 \arctan \left (a x \right ) \pi \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )^{3}-3 \arctan \left (a x \right ) \pi \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )^{2} \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )-3 \arctan \left (a x \right ) \pi \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )^{2} \operatorname {csgn}\left (\frac {i}{\left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )+3 \arctan \left (a x \right ) \pi \,\operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {i}{\left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )-3 \arctan \left (a x \right ) \pi {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )}^{3}+6 \arctan \left (a x \right ) \pi {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )}^{2} \operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )\right )-3 \arctan \left (a x \right ) \pi \,\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right ) {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )\right )}^{2}+3 \arctan \left (a x \right ) \pi {\operatorname {csgn}\left (\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )}^{2} \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )-6 \arctan \left (a x \right ) \pi \,\operatorname {csgn}\left (\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )^{2}+3 \arctan \left (a x \right ) \pi \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )^{3}+4 \arctan \left (a x \right )^{2}+12 i \arctan \left (a x \right ) \ln \left (2\right )+6 i \arctan \left (a x \right )-12-12 i a x \right )}{12}+\ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )}{c}}{a^{4}}\) | \(844\) |
| parts | \(\frac {x^{2} \arctan \left (a x \right )^{2}}{2 a^{2} c}-\frac {\arctan \left (a x \right )^{2} \ln \left (a^{2} x^{2}+1\right )}{2 c \,a^{4}}-\frac {a \left (-\frac {\arctan \left (a x \right )^{2} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{a^{5}}+\frac {i \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{a^{5}}-\frac {\operatorname {polylog}\left (3, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2 a^{5}}+\frac {i \arctan \left (a x \right ) \left (3 \arctan \left (a x \right ) \pi \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )^{3}-3 \arctan \left (a x \right ) \pi \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )^{2} \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )-3 \arctan \left (a x \right ) \pi \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )^{2} \operatorname {csgn}\left (\frac {i}{\left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )+3 \arctan \left (a x \right ) \pi \,\operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {i}{\left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )-3 \arctan \left (a x \right ) \pi {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )}^{3}+6 \arctan \left (a x \right ) \pi {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )}^{2} \operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )\right )-3 \arctan \left (a x \right ) \pi \,\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right ) {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )\right )}^{2}+3 \arctan \left (a x \right ) \pi {\operatorname {csgn}\left (\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )}^{2} \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )-6 \arctan \left (a x \right ) \pi \,\operatorname {csgn}\left (\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )^{2}+3 \arctan \left (a x \right ) \pi \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )^{3}+4 \arctan \left (a x \right )^{2}+12 i \arctan \left (a x \right ) \ln \left (2\right )+6 i \arctan \left (a x \right )-12-12 i a x \right )}{12 a^{5}}+\frac {\ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )}{a^{5}}\right )}{c}\) | \(860\) |
1/a^4*(1/2/c*arctan(a*x)^2*a^2*x^2-1/2/c*arctan(a*x)^2*ln(a^2*x^2+1)-1/c*( -arctan(a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2))+I*arctan(a*x)*polylog(2,-(1 +I*a*x)^2/(a^2*x^2+1))-1/2*polylog(3,-(1+I*a*x)^2/(a^2*x^2+1))+1/12*I*arct an(a*x)*(3*arctan(a*x)*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)^3-3*arctan(a*x)*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))-3*arctan(a*x)*Pi*cs gn(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)+3*arctan(a*x)*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)*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)^2)-3*arctan(a*x)*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1 )+1)^2)^3+6*arctan(a*x)*Pi*csgn(I*((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))-3*arctan(a*x)*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+3*arctan(a*x)*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))-6*arctan(a*x) *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+ 3*arctan(a*x)*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))^3+4*arctan(a*x)^2+12*I*ar ctan(a*x)*ln(2)+6*I*arctan(a*x)-12-12*I*a*x)+ln((1+I*a*x)^2/(a^2*x^2+1)+1) ))
\[ \int \frac {x^3 \arctan (a x)^2}{c+a^2 c x^2} \, dx=\int { \frac {x^{3} \arctan \left (a x\right )^{2}}{a^{2} c x^{2} + c} \,d x } \]
\[ \int \frac {x^3 \arctan (a x)^2}{c+a^2 c x^2} \, dx=\frac {\int \frac {x^{3} \operatorname {atan}^{2}{\left (a x \right )}}{a^{2} x^{2} + 1}\, dx}{c} \]
\[ \int \frac {x^3 \arctan (a x)^2}{c+a^2 c x^2} \, dx=\int { \frac {x^{3} \arctan \left (a x\right )^{2}}{a^{2} c x^{2} + c} \,d x } \]
\[ \int \frac {x^3 \arctan (a x)^2}{c+a^2 c x^2} \, dx=\int { \frac {x^{3} \arctan \left (a x\right )^{2}}{a^{2} c x^{2} + c} \,d x } \]
Timed out. \[ \int \frac {x^3 \arctan (a x)^2}{c+a^2 c x^2} \, dx=\int \frac {x^3\,{\mathrm {atan}\left (a\,x\right )}^2}{c\,a^2\,x^2+c} \,d x \]