\(\int \frac {(c+a^2 c x^2)^3 \arctan (a x)^2}{x} \, dx\) [278]

Optimal result

Integrand size = 22, antiderivative size = 287 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^2}{x} \, dx=\frac {29}{180} a^2 c^3 x^2+\frac {1}{60} a^4 c^3 x^4-\frac {11}{6} a c^3 x \arctan (a x)-\frac {7}{18} a^3 c^3 x^3 \arctan (a x)-\frac {1}{15} a^5 c^3 x^5 \arctan (a x)+\frac {11}{12} c^3 \arctan (a x)^2+\frac {3}{2} a^2 c^3 x^2 \arctan (a x)^2+\frac {3}{4} a^4 c^3 x^4 \arctan (a x)^2+\frac {1}{6} a^6 c^3 x^6 \arctan (a x)^2+2 c^3 \arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )+\frac {34}{45} c^3 \log \left (1+a^2 x^2\right )-i c^3 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+i c^3 \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-\frac {1}{2} c^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\frac {1}{2} c^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right ) \]

[Out]

29/180*a^2*c^3*x^2+1/60*a^4*c^3*x^4-11/6*a*c^3*x*arctan(a*x)-7/18*a^3*c^3*x^3*arctan(a*x)-1/15*a^5*c^3*x^5*arc
tan(a*x)+11/12*c^3*arctan(a*x)^2+3/2*a^2*c^3*x^2*arctan(a*x)^2+3/4*a^4*c^3*x^4*arctan(a*x)^2+1/6*a^6*c^3*x^6*a
rctan(a*x)^2-2*c^3*arctan(a*x)^2*arctanh(-1+2/(1+I*a*x))+34/45*c^3*ln(a^2*x^2+1)-I*c^3*arctan(a*x)*polylog(2,1
-2/(1+I*a*x))+I*c^3*arctan(a*x)*polylog(2,-1+2/(1+I*a*x))-1/2*c^3*polylog(3,1-2/(1+I*a*x))+1/2*c^3*polylog(3,-
1+2/(1+I*a*x))

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps used = 38, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {5068, 4942, 5108, 5004, 5114, 6745, 4946, 5036, 4930, 266, 272, 45} \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^2}{x} \, dx=\frac {1}{6} a^6 c^3 x^6 \arctan (a x)^2-\frac {1}{15} a^5 c^3 x^5 \arctan (a x)+\frac {3}{4} a^4 c^3 x^4 \arctan (a x)^2+\frac {1}{60} a^4 c^3 x^4-\frac {7}{18} a^3 c^3 x^3 \arctan (a x)+\frac {3}{2} a^2 c^3 x^2 \arctan (a x)^2+\frac {29}{180} a^2 c^3 x^2+\frac {34}{45} c^3 \log \left (a^2 x^2+1\right )+2 c^3 \arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-i c^3 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )+i c^3 \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{i a x+1}-1\right )-\frac {11}{6} a c^3 x \arctan (a x)+\frac {11}{12} c^3 \arctan (a x)^2-\frac {1}{2} c^3 \operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right )+\frac {1}{2} c^3 \operatorname {PolyLog}\left (3,\frac {2}{i a x+1}-1\right ) \]

[In]

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

[Out]

(29*a^2*c^3*x^2)/180 + (a^4*c^3*x^4)/60 - (11*a*c^3*x*ArcTan[a*x])/6 - (7*a^3*c^3*x^3*ArcTan[a*x])/18 - (a^5*c
^3*x^5*ArcTan[a*x])/15 + (11*c^3*ArcTan[a*x]^2)/12 + (3*a^2*c^3*x^2*ArcTan[a*x]^2)/2 + (3*a^4*c^3*x^4*ArcTan[a
*x]^2)/4 + (a^6*c^3*x^6*ArcTan[a*x]^2)/6 + 2*c^3*ArcTan[a*x]^2*ArcTanh[1 - 2/(1 + I*a*x)] + (34*c^3*Log[1 + a^
2*x^2])/45 - I*c^3*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)] + I*c^3*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 + I*a*x)
] - (c^3*PolyLog[3, 1 - 2/(1 + I*a*x)])/2 + (c^3*PolyLog[3, -1 + 2/(1 + I*a*x)])/2

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 4930

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcTan[c*x^n])^p, x] - Dist[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])

Rule 4942

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)/(x_), x_Symbol] :> Simp[2*(a + b*ArcTan[c*x])^p*ArcTanh[1 - 2/(1 +
I*c*x)], x] - Dist[2*b*c*p, Int[(a + b*ArcTan[c*x])^(p - 1)*(ArcTanh[1 - 2/(1 + I*c*x)]/(1 + c^2*x^2)), x], x
] /; FreeQ[{a, b, c}, x] && IGtQ[p, 1]

Rule 4946

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] - Dist[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 5004

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> 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 5036

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[f^2/
e, Int[(f*x)^(m - 2)*(a + b*ArcTan[c*x])^p, x], x] - Dist[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]

Rule 5068

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Int[Ex
pandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*ArcTan[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e,
c^2*d] && IGtQ[p, 0] && IGtQ[q, 1] && (EqQ[p, 1] || IntegerQ[m])

Rule 5108

Int[(ArcTanh[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/2, Int[L
og[1 + u]*((a + b*ArcTan[c*x])^p/(d + e*x^2)), x], x] - Dist[1/2, Int[Log[1 - u]*((a + b*ArcTan[c*x])^p/(d + e
*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[u^2 - (1 - 2*(I/(I - c*x)))^
2, 0]

Rule 5114

Int[(Log[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-I)*(a + b*Ar
cTan[c*x])^p*(PolyLog[2, 1 - u]/(2*c*d)), x] + Dist[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 6745

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]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {c^3 \arctan (a x)^2}{x}+3 a^2 c^3 x \arctan (a x)^2+3 a^4 c^3 x^3 \arctan (a x)^2+a^6 c^3 x^5 \arctan (a x)^2\right ) \, dx \\ & = c^3 \int \frac {\arctan (a x)^2}{x} \, dx+\left (3 a^2 c^3\right ) \int x \arctan (a x)^2 \, dx+\left (3 a^4 c^3\right ) \int x^3 \arctan (a x)^2 \, dx+\left (a^6 c^3\right ) \int x^5 \arctan (a x)^2 \, dx \\ & = \frac {3}{2} a^2 c^3 x^2 \arctan (a x)^2+\frac {3}{4} a^4 c^3 x^4 \arctan (a x)^2+\frac {1}{6} a^6 c^3 x^6 \arctan (a x)^2+2 c^3 \arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-\left (4 a c^3\right ) \int \frac {\arctan (a x) \text {arctanh}\left (1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (3 a^3 c^3\right ) \int \frac {x^2 \arctan (a x)}{1+a^2 x^2} \, dx-\frac {1}{2} \left (3 a^5 c^3\right ) \int \frac {x^4 \arctan (a x)}{1+a^2 x^2} \, dx-\frac {1}{3} \left (a^7 c^3\right ) \int \frac {x^6 \arctan (a x)}{1+a^2 x^2} \, dx \\ & = \frac {3}{2} a^2 c^3 x^2 \arctan (a x)^2+\frac {3}{4} a^4 c^3 x^4 \arctan (a x)^2+\frac {1}{6} a^6 c^3 x^6 \arctan (a x)^2+2 c^3 \arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )+\left (2 a c^3\right ) \int \frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (2 a c^3\right ) \int \frac {\arctan (a x) \log \left (2-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (3 a c^3\right ) \int \arctan (a x) \, dx+\left (3 a c^3\right ) \int \frac {\arctan (a x)}{1+a^2 x^2} \, dx-\frac {1}{2} \left (3 a^3 c^3\right ) \int x^2 \arctan (a x) \, dx+\frac {1}{2} \left (3 a^3 c^3\right ) \int \frac {x^2 \arctan (a x)}{1+a^2 x^2} \, dx-\frac {1}{3} \left (a^5 c^3\right ) \int x^4 \arctan (a x) \, dx+\frac {1}{3} \left (a^5 c^3\right ) \int \frac {x^4 \arctan (a x)}{1+a^2 x^2} \, dx \\ & = -3 a c^3 x \arctan (a x)-\frac {1}{2} a^3 c^3 x^3 \arctan (a x)-\frac {1}{15} a^5 c^3 x^5 \arctan (a x)+\frac {3}{2} c^3 \arctan (a x)^2+\frac {3}{2} a^2 c^3 x^2 \arctan (a x)^2+\frac {3}{4} a^4 c^3 x^4 \arctan (a x)^2+\frac {1}{6} a^6 c^3 x^6 \arctan (a x)^2+2 c^3 \arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-i c^3 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+i c^3 \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )+\left (i a c^3\right ) \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (i a c^3\right ) \int \frac {\operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx+\frac {1}{2} \left (3 a c^3\right ) \int \arctan (a x) \, dx-\frac {1}{2} \left (3 a c^3\right ) \int \frac {\arctan (a x)}{1+a^2 x^2} \, dx+\left (3 a^2 c^3\right ) \int \frac {x}{1+a^2 x^2} \, dx+\frac {1}{3} \left (a^3 c^3\right ) \int x^2 \arctan (a x) \, dx-\frac {1}{3} \left (a^3 c^3\right ) \int \frac {x^2 \arctan (a x)}{1+a^2 x^2} \, dx+\frac {1}{2} \left (a^4 c^3\right ) \int \frac {x^3}{1+a^2 x^2} \, dx+\frac {1}{15} \left (a^6 c^3\right ) \int \frac {x^5}{1+a^2 x^2} \, dx \\ & = -\frac {3}{2} a c^3 x \arctan (a x)-\frac {7}{18} a^3 c^3 x^3 \arctan (a x)-\frac {1}{15} a^5 c^3 x^5 \arctan (a x)+\frac {3}{4} c^3 \arctan (a x)^2+\frac {3}{2} a^2 c^3 x^2 \arctan (a x)^2+\frac {3}{4} a^4 c^3 x^4 \arctan (a x)^2+\frac {1}{6} a^6 c^3 x^6 \arctan (a x)^2+2 c^3 \arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )+\frac {3}{2} c^3 \log \left (1+a^2 x^2\right )-i c^3 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+i c^3 \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-\frac {1}{2} c^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\frac {1}{2} c^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right )-\frac {1}{3} \left (a c^3\right ) \int \arctan (a x) \, dx+\frac {1}{3} \left (a c^3\right ) \int \frac {\arctan (a x)}{1+a^2 x^2} \, dx-\frac {1}{2} \left (3 a^2 c^3\right ) \int \frac {x}{1+a^2 x^2} \, dx-\frac {1}{9} \left (a^4 c^3\right ) \int \frac {x^3}{1+a^2 x^2} \, dx+\frac {1}{4} \left (a^4 c^3\right ) \text {Subst}\left (\int \frac {x}{1+a^2 x} \, dx,x,x^2\right )+\frac {1}{30} \left (a^6 c^3\right ) \text {Subst}\left (\int \frac {x^2}{1+a^2 x} \, dx,x,x^2\right ) \\ & = -\frac {11}{6} a c^3 x \arctan (a x)-\frac {7}{18} a^3 c^3 x^3 \arctan (a x)-\frac {1}{15} a^5 c^3 x^5 \arctan (a x)+\frac {11}{12} c^3 \arctan (a x)^2+\frac {3}{2} a^2 c^3 x^2 \arctan (a x)^2+\frac {3}{4} a^4 c^3 x^4 \arctan (a x)^2+\frac {1}{6} a^6 c^3 x^6 \arctan (a x)^2+2 c^3 \arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )+\frac {3}{4} c^3 \log \left (1+a^2 x^2\right )-i c^3 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+i c^3 \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-\frac {1}{2} c^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\frac {1}{2} c^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right )+\frac {1}{3} \left (a^2 c^3\right ) \int \frac {x}{1+a^2 x^2} \, dx-\frac {1}{18} \left (a^4 c^3\right ) \text {Subst}\left (\int \frac {x}{1+a^2 x} \, dx,x,x^2\right )+\frac {1}{4} \left (a^4 c^3\right ) \text {Subst}\left (\int \left (\frac {1}{a^2}-\frac {1}{a^2 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )+\frac {1}{30} \left (a^6 c^3\right ) \text {Subst}\left (\int \left (-\frac {1}{a^4}+\frac {x}{a^2}+\frac {1}{a^4 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right ) \\ & = \frac {13}{60} a^2 c^3 x^2+\frac {1}{60} a^4 c^3 x^4-\frac {11}{6} a c^3 x \arctan (a x)-\frac {7}{18} a^3 c^3 x^3 \arctan (a x)-\frac {1}{15} a^5 c^3 x^5 \arctan (a x)+\frac {11}{12} c^3 \arctan (a x)^2+\frac {3}{2} a^2 c^3 x^2 \arctan (a x)^2+\frac {3}{4} a^4 c^3 x^4 \arctan (a x)^2+\frac {1}{6} a^6 c^3 x^6 \arctan (a x)^2+2 c^3 \arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )+\frac {7}{10} c^3 \log \left (1+a^2 x^2\right )-i c^3 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+i c^3 \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-\frac {1}{2} c^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\frac {1}{2} c^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right )-\frac {1}{18} \left (a^4 c^3\right ) \text {Subst}\left (\int \left (\frac {1}{a^2}-\frac {1}{a^2 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right ) \\ & = \frac {29}{180} a^2 c^3 x^2+\frac {1}{60} a^4 c^3 x^4-\frac {11}{6} a c^3 x \arctan (a x)-\frac {7}{18} a^3 c^3 x^3 \arctan (a x)-\frac {1}{15} a^5 c^3 x^5 \arctan (a x)+\frac {11}{12} c^3 \arctan (a x)^2+\frac {3}{2} a^2 c^3 x^2 \arctan (a x)^2+\frac {3}{4} a^4 c^3 x^4 \arctan (a x)^2+\frac {1}{6} a^6 c^3 x^6 \arctan (a x)^2+2 c^3 \arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )+\frac {34}{45} c^3 \log \left (1+a^2 x^2\right )-i c^3 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+i c^3 \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-\frac {1}{2} c^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\frac {1}{2} c^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.46 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.88 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^2}{x} \, dx=\frac {1}{360} c^3 \left (52-15 i \pi ^3+58 a^2 x^2+6 a^4 x^4-660 a x \arctan (a x)-140 a^3 x^3 \arctan (a x)-24 a^5 x^5 \arctan (a x)+330 \arctan (a x)^2+540 a^2 x^2 \arctan (a x)^2+270 a^4 x^4 \arctan (a x)^2+60 a^6 x^6 \arctan (a x)^2+240 i \arctan (a x)^3+360 \arctan (a x)^2 \log \left (1-e^{-2 i \arctan (a x)}\right )-360 \arctan (a x)^2 \log \left (1+e^{2 i \arctan (a x)}\right )+272 \log \left (1+a^2 x^2\right )+360 i \arctan (a x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )+360 i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )+180 \operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )-180 \operatorname {PolyLog}\left (3,-e^{2 i \arctan (a x)}\right )\right ) \]

[In]

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

[Out]

(c^3*(52 - (15*I)*Pi^3 + 58*a^2*x^2 + 6*a^4*x^4 - 660*a*x*ArcTan[a*x] - 140*a^3*x^3*ArcTan[a*x] - 24*a^5*x^5*A
rcTan[a*x] + 330*ArcTan[a*x]^2 + 540*a^2*x^2*ArcTan[a*x]^2 + 270*a^4*x^4*ArcTan[a*x]^2 + 60*a^6*x^6*ArcTan[a*x
]^2 + (240*I)*ArcTan[a*x]^3 + 360*ArcTan[a*x]^2*Log[1 - E^((-2*I)*ArcTan[a*x])] - 360*ArcTan[a*x]^2*Log[1 + E^
((2*I)*ArcTan[a*x])] + 272*Log[1 + a^2*x^2] + (360*I)*ArcTan[a*x]*PolyLog[2, E^((-2*I)*ArcTan[a*x])] + (360*I)
*ArcTan[a*x]*PolyLog[2, -E^((2*I)*ArcTan[a*x])] + 180*PolyLog[3, E^((-2*I)*ArcTan[a*x])] - 180*PolyLog[3, -E^(
(2*I)*ArcTan[a*x])]))/360

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 68.15 (sec) , antiderivative size = 1405, normalized size of antiderivative = 4.90

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1405\)
default	\(\text {Expression too large to display}\)	\(1405\)
parts	\(\text {Expression too large to display}\)	\(1948\)

[In]

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

[Out]

1/6*a^6*c^3*x^6*arctan(a*x)^2+3/4*a^4*c^3*x^4*arctan(a*x)^2+3/2*a^2*c^3*x^2*arctan(a*x)^2+c^3*arctan(a*x)^2*ln
(a*x)-1/6*c^3*(-3*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))*csgn(I*((1+I*a*
x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))*arctan(a*x)^2+2*arctan(a*x)*(a*x-I)*(I+a*x)^4+3*I*arctan(a*x)
*(a*x-I)^2+4*arctan(a*x)*(a*x-I)^3*(I+a*x)^2-11/2*arctan(a*x)^2-5/3*arctan(a*x)*(a*x-I)^3+23/15*I*(I+a*x)+2/5*
I*(I+a*x)^3+6*arctan(a*x)^2*ln((1+I*a*x)^2/(a^2*x^2+1)-1)-6*arctan(a*x)^2*ln(1-(1+I*a*x)/(a^2*x^2+1)^(1/2))-6*
arctan(a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2)+1)+6*arctan(a*x)*(a*x-I)+2/5*arctan(a*x)*(a*x-I)^5-12*polylog(3,-
(1+I*a*x)/(a^2*x^2+1)^(1/2))-12*polylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))-1/10*(I+a*x)^4-4*arctan(a*x)*(a*x-I)^2*
(I+a*x)^3+3*polylog(3,-(1+I*a*x)^2/(a^2*x^2+1))+136/15*ln((1+I*a*x)^2/(a^2*x^2+1)+1)-2*arctan(a*x)*(a*x-I)^4*(
I+a*x)-5*arctan(a*x)*(a*x-I)*(I+a*x)^2-8*I*arctan(a*x)*(a*x-I)*(I+a*x)^3+3*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*arctan(a*x)^2-3*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))^3*arctan(a*x)^2-3*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*arctan(a*x)^2-
8*I*arctan(a*x)*(a*x-I)^3*(I+a*x)-6*I*arctan(a*x)*(a*x-I)*(I+a*x)+12*I*arctan(a*x)*(a*x-I)^2*(I+a*x)^2+3*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*arctan(a
*x)^2+3*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))*csgn(((1+I*a*x)^2/(a^2*x^2+1)-1)/
((1+I*a*x)^2/(a^2*x^2+1)+1))^2*arctan(a*x)^2-11/30*(I+a*x)^2+5*arctan(a*x)*(a*x-I)^2*(I+a*x)-3*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))*csgn(((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1
)+1))*arctan(a*x)^2+3*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*arctan(a*x)^2+2*I*arctan(a*x)*(a*x-I)^4-3*I*Pi*arctan(a*x)^2+12*I*arctan(a*x)*polylog(2,(1+
I*a*x)/(a^2*x^2+1)^(1/2))+12*I*arctan(a*x)*polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-6*I*arctan(a*x)*polylog(2,-
(1+I*a*x)^2/(a^2*x^2+1)))

Fricas [F]

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

[In]

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

[Out]

integral((a^6*c^3*x^6 + 3*a^4*c^3*x^4 + 3*a^2*c^3*x^2 + c^3)*arctan(a*x)^2/x, x)

Sympy [F]

\[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^2}{x} \, dx=c^{3} \left (\int \frac {\operatorname {atan}^{2}{\left (a x \right )}}{x}\, dx + \int 3 a^{2} x \operatorname {atan}^{2}{\left (a x \right )}\, dx + \int 3 a^{4} x^{3} \operatorname {atan}^{2}{\left (a x \right )}\, dx + \int a^{6} x^{5} \operatorname {atan}^{2}{\left (a x \right )}\, dx\right ) \]

[In]

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

[Out]

c**3*(Integral(atan(a*x)**2/x, x) + Integral(3*a**2*x*atan(a*x)**2, x) + Integral(3*a**4*x**3*atan(a*x)**2, x)
+ Integral(a**6*x**5*atan(a*x)**2, x))

Maxima [F]

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

[In]

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

[Out]

36*a^8*c^3*integrate(1/48*x^8*arctan(a*x)^2/(a^2*x^3 + x), x) + 3*a^8*c^3*integrate(1/48*x^8*log(a^2*x^2 + 1)^
2/(a^2*x^3 + x), x) + 2*a^8*c^3*integrate(1/48*x^8*log(a^2*x^2 + 1)/(a^2*x^3 + x), x) - 4*a^7*c^3*integrate(1/
48*x^7*arctan(a*x)/(a^2*x^3 + x), x) + 144*a^6*c^3*integrate(1/48*x^6*arctan(a*x)^2/(a^2*x^3 + x), x) + 12*a^6
*c^3*integrate(1/48*x^6*log(a^2*x^2 + 1)^2/(a^2*x^3 + x), x) + 9*a^6*c^3*integrate(1/48*x^6*log(a^2*x^2 + 1)/(
a^2*x^3 + x), x) - 18*a^5*c^3*integrate(1/48*x^5*arctan(a*x)/(a^2*x^3 + x), x) + 216*a^4*c^3*integrate(1/48*x^
4*arctan(a*x)^2/(a^2*x^3 + x), x) + 18*a^4*c^3*integrate(1/48*x^4*log(a^2*x^2 + 1)^2/(a^2*x^3 + x), x) + 18*a^
4*c^3*integrate(1/48*x^4*log(a^2*x^2 + 1)/(a^2*x^3 + x), x) - 36*a^3*c^3*integrate(1/48*x^3*arctan(a*x)/(a^2*x
^3 + x), x) + 144*a^2*c^3*integrate(1/48*x^2*arctan(a*x)^2/(a^2*x^3 + x), x) + 1/24*c^3*log(a^2*x^2 + 1)^3 + 3
6*c^3*integrate(1/48*arctan(a*x)^2/(a^2*x^3 + x), x) + 3*c^3*integrate(1/48*log(a^2*x^2 + 1)^2/(a^2*x^3 + x),
x) + 1/48*(2*a^6*c^3*x^6 + 9*a^4*c^3*x^4 + 18*a^2*c^3*x^2)*arctan(a*x)^2 - 1/192*(2*a^6*c^3*x^6 + 9*a^4*c^3*x^
4 + 18*a^2*c^3*x^2)*log(a^2*x^2 + 1)^2

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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