\(\int \frac {(c+a^2 c x^2)^3 \arctan (a x)^2}{x^3} \, dx\) [280]

Optimal result

Integrand size = 22, antiderivative size = 299 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^2}{x^3} \, dx=\frac {1}{12} a^4 c^3 x^2-\frac {a c^3 \arctan (a x)}{x}-\frac {5}{2} a^3 c^3 x \arctan (a x)-\frac {1}{6} a^5 c^3 x^3 \arctan (a x)+\frac {3}{4} a^2 c^3 \arctan (a x)^2-\frac {c^3 \arctan (a x)^2}{2 x^2}+\frac {3}{2} a^4 c^3 x^2 \arctan (a x)^2+\frac {1}{4} a^6 c^3 x^4 \arctan (a x)^2+6 a^2 c^3 \arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )+a^2 c^3 \log (x)+\frac {2}{3} a^2 c^3 \log \left (1+a^2 x^2\right )-3 i a^2 c^3 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+3 i a^2 c^3 \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-\frac {3}{2} a^2 c^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\frac {3}{2} a^2 c^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right ) \]

[Out]

1/12*a^4*c^3*x^2-a*c^3*arctan(a*x)/x-5/2*a^3*c^3*x*arctan(a*x)-1/6*a^5*c^3*x^3*arctan(a*x)+3/4*a^2*c^3*arctan(
a*x)^2-1/2*c^3*arctan(a*x)^2/x^2+3/2*a^4*c^3*x^2*arctan(a*x)^2+1/4*a^6*c^3*x^4*arctan(a*x)^2-6*a^2*c^3*arctan(
a*x)^2*arctanh(-1+2/(1+I*a*x))+a^2*c^3*ln(x)+2/3*a^2*c^3*ln(a^2*x^2+1)-3*I*a^2*c^3*arctan(a*x)*polylog(2,1-2/(
1+I*a*x))+3*I*a^2*c^3*arctan(a*x)*polylog(2,-1+2/(1+I*a*x))-3/2*a^2*c^3*polylog(3,1-2/(1+I*a*x))+3/2*a^2*c^3*p
olylog(3,-1+2/(1+I*a*x))

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.00, number of steps used = 31, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {5068, 4946, 5038, 272, 36, 29, 31, 5004, 4942, 5108, 5114, 6745, 5036, 4930, 266, 45} \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^2}{x^3} \, dx=\frac {1}{4} a^6 c^3 x^4 \arctan (a x)^2-\frac {1}{6} a^5 c^3 x^3 \arctan (a x)+\frac {3}{2} a^4 c^3 x^2 \arctan (a x)^2+\frac {1}{12} a^4 c^3 x^2-\frac {5}{2} a^3 c^3 x \arctan (a x)+6 a^2 c^3 \arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-3 i a^2 c^3 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )+3 i a^2 c^3 \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{i a x+1}-1\right )+\frac {3}{4} a^2 c^3 \arctan (a x)^2-\frac {3}{2} a^2 c^3 \operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right )+\frac {3}{2} a^2 c^3 \operatorname {PolyLog}\left (3,\frac {2}{i a x+1}-1\right )+\frac {2}{3} a^2 c^3 \log \left (a^2 x^2+1\right )+a^2 c^3 \log (x)-\frac {c^3 \arctan (a x)^2}{2 x^2}-\frac {a c^3 \arctan (a x)}{x} \]

[In]

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

[Out]

(a^4*c^3*x^2)/12 - (a*c^3*ArcTan[a*x])/x - (5*a^3*c^3*x*ArcTan[a*x])/2 - (a^5*c^3*x^3*ArcTan[a*x])/6 + (3*a^2*
c^3*ArcTan[a*x]^2)/4 - (c^3*ArcTan[a*x]^2)/(2*x^2) + (3*a^4*c^3*x^2*ArcTan[a*x]^2)/2 + (a^6*c^3*x^4*ArcTan[a*x
]^2)/4 + 6*a^2*c^3*ArcTan[a*x]^2*ArcTanh[1 - 2/(1 + I*a*x)] + a^2*c^3*Log[x] + (2*a^2*c^3*Log[1 + a^2*x^2])/3
- (3*I)*a^2*c^3*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)] + (3*I)*a^2*c^3*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 + I
*a*x)] - (3*a^2*c^3*PolyLog[3, 1 - 2/(1 + I*a*x)])/2 + (3*a^2*c^3*PolyLog[3, -1 + 2/(1 + I*a*x)])/2

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

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

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

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 5038

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/d,
Int[(f*x)^m*(a + b*ArcTan[c*x])^p, x], x] - Dist[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]

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}+\frac {3 a^2 c^3 \arctan (a x)^2}{x}+3 a^4 c^3 x \arctan (a x)^2+a^6 c^3 x^3 \arctan (a x)^2\right ) \, dx \\ & = c^3 \int \frac {\arctan (a x)^2}{x^3} \, dx+\left (3 a^2 c^3\right ) \int \frac {\arctan (a x)^2}{x} \, dx+\left (3 a^4 c^3\right ) \int x \arctan (a x)^2 \, dx+\left (a^6 c^3\right ) \int x^3 \arctan (a x)^2 \, dx \\ & = -\frac {c^3 \arctan (a x)^2}{2 x^2}+\frac {3}{2} a^4 c^3 x^2 \arctan (a x)^2+\frac {1}{4} a^6 c^3 x^4 \arctan (a x)^2+6 a^2 c^3 \arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )+\left (a c^3\right ) \int \frac {\arctan (a x)}{x^2 \left (1+a^2 x^2\right )} \, dx-\left (12 a^3 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^5 c^3\right ) \int \frac {x^2 \arctan (a x)}{1+a^2 x^2} \, dx-\frac {1}{2} \left (a^7 c^3\right ) \int \frac {x^4 \arctan (a x)}{1+a^2 x^2} \, dx \\ & = -\frac {c^3 \arctan (a x)^2}{2 x^2}+\frac {3}{2} a^4 c^3 x^2 \arctan (a x)^2+\frac {1}{4} a^6 c^3 x^4 \arctan (a x)^2+6 a^2 c^3 \arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )+\left (a c^3\right ) \int \frac {\arctan (a x)}{x^2} \, dx-\left (a^3 c^3\right ) \int \frac {\arctan (a x)}{1+a^2 x^2} \, dx-\left (3 a^3 c^3\right ) \int \arctan (a x) \, dx+\left (3 a^3 c^3\right ) \int \frac {\arctan (a x)}{1+a^2 x^2} \, dx+\left (6 a^3 c^3\right ) \int \frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (6 a^3 c^3\right ) \int \frac {\arctan (a x) \log \left (2-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\frac {1}{2} \left (a^5 c^3\right ) \int x^2 \arctan (a x) \, dx+\frac {1}{2} \left (a^5 c^3\right ) \int \frac {x^2 \arctan (a x)}{1+a^2 x^2} \, dx \\ & = -\frac {a c^3 \arctan (a x)}{x}-3 a^3 c^3 x \arctan (a x)-\frac {1}{6} a^5 c^3 x^3 \arctan (a x)+a^2 c^3 \arctan (a x)^2-\frac {c^3 \arctan (a x)^2}{2 x^2}+\frac {3}{2} a^4 c^3 x^2 \arctan (a x)^2+\frac {1}{4} a^6 c^3 x^4 \arctan (a x)^2+6 a^2 c^3 \arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-3 i a^2 c^3 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+3 i a^2 c^3 \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )+\left (a^2 c^3\right ) \int \frac {1}{x \left (1+a^2 x^2\right )} \, dx+\left (3 i a^3 c^3\right ) \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (3 i a^3 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 (a^3 c^3\right ) \int \arctan (a x) \, dx-\frac {1}{2} \left (a^3 c^3\right ) \int \frac {\arctan (a x)}{1+a^2 x^2} \, dx+\left (3 a^4 c^3\right ) \int \frac {x}{1+a^2 x^2} \, dx+\frac {1}{6} \left (a^6 c^3\right ) \int \frac {x^3}{1+a^2 x^2} \, dx \\ & = -\frac {a c^3 \arctan (a x)}{x}-\frac {5}{2} a^3 c^3 x \arctan (a x)-\frac {1}{6} a^5 c^3 x^3 \arctan (a x)+\frac {3}{4} a^2 c^3 \arctan (a x)^2-\frac {c^3 \arctan (a x)^2}{2 x^2}+\frac {3}{2} a^4 c^3 x^2 \arctan (a x)^2+\frac {1}{4} a^6 c^3 x^4 \arctan (a x)^2+6 a^2 c^3 \arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )+\frac {3}{2} a^2 c^3 \log \left (1+a^2 x^2\right )-3 i a^2 c^3 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+3 i a^2 c^3 \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-\frac {3}{2} a^2 c^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\frac {3}{2} a^2 c^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right )+\frac {1}{2} \left (a^2 c^3\right ) \text {Subst}\left (\int \frac {1}{x \left (1+a^2 x\right )} \, dx,x,x^2\right )-\frac {1}{2} \left (a^4 c^3\right ) \int \frac {x}{1+a^2 x^2} \, dx+\frac {1}{12} \left (a^6 c^3\right ) \text {Subst}\left (\int \frac {x}{1+a^2 x} \, dx,x,x^2\right ) \\ & = -\frac {a c^3 \arctan (a x)}{x}-\frac {5}{2} a^3 c^3 x \arctan (a x)-\frac {1}{6} a^5 c^3 x^3 \arctan (a x)+\frac {3}{4} a^2 c^3 \arctan (a x)^2-\frac {c^3 \arctan (a x)^2}{2 x^2}+\frac {3}{2} a^4 c^3 x^2 \arctan (a x)^2+\frac {1}{4} a^6 c^3 x^4 \arctan (a x)^2+6 a^2 c^3 \arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )+\frac {5}{4} a^2 c^3 \log \left (1+a^2 x^2\right )-3 i a^2 c^3 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+3 i a^2 c^3 \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-\frac {3}{2} a^2 c^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\frac {3}{2} a^2 c^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right )+\frac {1}{2} \left (a^2 c^3\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )-\frac {1}{2} \left (a^4 c^3\right ) \text {Subst}\left (\int \frac {1}{1+a^2 x} \, dx,x,x^2\right )+\frac {1}{12} \left (a^6 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}{12} a^4 c^3 x^2-\frac {a c^3 \arctan (a x)}{x}-\frac {5}{2} a^3 c^3 x \arctan (a x)-\frac {1}{6} a^5 c^3 x^3 \arctan (a x)+\frac {3}{4} a^2 c^3 \arctan (a x)^2-\frac {c^3 \arctan (a x)^2}{2 x^2}+\frac {3}{2} a^4 c^3 x^2 \arctan (a x)^2+\frac {1}{4} a^6 c^3 x^4 \arctan (a x)^2+6 a^2 c^3 \arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )+a^2 c^3 \log (x)+\frac {2}{3} a^2 c^3 \log \left (1+a^2 x^2\right )-3 i a^2 c^3 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+3 i a^2 c^3 \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-\frac {3}{2} a^2 c^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\frac {3}{2} a^2 c^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.11 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^2}{x^3} \, dx=\frac {c^3 \left (2 a^2 x^2-3 i a^2 \pi ^3 x^2+2 a^4 x^4-24 a x \arctan (a x)-60 a^3 x^3 \arctan (a x)-4 a^5 x^5 \arctan (a x)-12 \arctan (a x)^2+18 a^2 x^2 \arctan (a x)^2+36 a^4 x^4 \arctan (a x)^2+6 a^6 x^6 \arctan (a x)^2+48 i a^2 x^2 \arctan (a x)^3+72 a^2 x^2 \arctan (a x)^2 \log \left (1-e^{-2 i \arctan (a x)}\right )-72 a^2 x^2 \arctan (a x)^2 \log \left (1+e^{2 i \arctan (a x)}\right )+24 a^2 x^2 \log \left (\frac {a x}{\sqrt {1+a^2 x^2}}\right )+28 a^2 x^2 \log \left (1+a^2 x^2\right )+72 i a^2 x^2 \arctan (a x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )+72 i a^2 x^2 \arctan (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )+36 a^2 x^2 \operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )-36 a^2 x^2 \operatorname {PolyLog}\left (3,-e^{2 i \arctan (a x)}\right )\right )}{24 x^2} \]

[In]

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

[Out]

(c^3*(2*a^2*x^2 - (3*I)*a^2*Pi^3*x^2 + 2*a^4*x^4 - 24*a*x*ArcTan[a*x] - 60*a^3*x^3*ArcTan[a*x] - 4*a^5*x^5*Arc
Tan[a*x] - 12*ArcTan[a*x]^2 + 18*a^2*x^2*ArcTan[a*x]^2 + 36*a^4*x^4*ArcTan[a*x]^2 + 6*a^6*x^6*ArcTan[a*x]^2 +
(48*I)*a^2*x^2*ArcTan[a*x]^3 + 72*a^2*x^2*ArcTan[a*x]^2*Log[1 - E^((-2*I)*ArcTan[a*x])] - 72*a^2*x^2*ArcTan[a*
x]^2*Log[1 + E^((2*I)*ArcTan[a*x])] + 24*a^2*x^2*Log[(a*x)/Sqrt[1 + a^2*x^2]] + 28*a^2*x^2*Log[1 + a^2*x^2] +
(72*I)*a^2*x^2*ArcTan[a*x]*PolyLog[2, E^((-2*I)*ArcTan[a*x])] + (72*I)*a^2*x^2*ArcTan[a*x]*PolyLog[2, -E^((2*I
)*ArcTan[a*x])] + 36*a^2*x^2*PolyLog[3, E^((-2*I)*ArcTan[a*x])] - 36*a^2*x^2*PolyLog[3, -E^((2*I)*ArcTan[a*x])
]))/(24*x^2)

Maple [C] (warning: unable to verify)

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

Time = 67.32 (sec) , antiderivative size = 1318, normalized size of antiderivative = 4.41

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1318\)
default	\(\text {Expression too large to display}\)	\(1318\)
parts	\(\text {Expression too large to display}\)	\(1754\)

[In]

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

[Out]

a^2*(1/4*a^4*c^3*x^4*arctan(a*x)^2+3/2*a^2*c^3*x^2*arctan(a*x)^2-1/2*c^3*arctan(a*x)^2/a^2/x^2+3*c^3*arctan(a*
x)^2*ln(a*x)-1/2*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)^2-3/2*arctan(a*x)^2+1/3*arctan(a*x)*(a*x-I
)^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)+4*arctan(a*x)*(a*x-I)-12*polylog(3,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-12*po
lylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))-2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2)-1)-2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2)+1)+3
*polylog(3,-(1+I*a*x)^2/(a^2*x^2+1))+14/3*ln((1+I*a*x)^2/(a^2*x^2+1)+1)+arctan(a*x)*(a*x-I)*(I+a*x)^2+3*I*Pi*c
sgn(((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+arctan(a*x)*(I*a*x-(a^2*x^2+1)^(1/2)+1)/a/x-2*I*arctan(a*x)*(a*x-I)*(I+a*x)+arcta
n(a*x)*(I*a*x+(a^2*x^2+1)^(1/2)+1)/a/x+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+I*arctan(a*x)*(a*x-
I)^2-1/6*(I+a*x)^2-arctan(a*x)*(a*x-I)^2*(I+a*x)+1/3*I*(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*csg
n(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*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]

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

[In]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

1/64*(4*(192*a^8*c^3*integrate(1/16*x^8*arctan(a*x)^2/(a^2*x^5 + x^3), x) + 16*a^8*c^3*integrate(1/16*x^8*log(
a^2*x^2 + 1)^2/(a^2*x^5 + x^3), x) + 16*a^8*c^3*integrate(1/16*x^8*log(a^2*x^2 + 1)/(a^2*x^5 + x^3), x) - 32*a
^7*c^3*integrate(1/16*x^7*arctan(a*x)/(a^2*x^5 + x^3), x) + 768*a^6*c^3*integrate(1/16*x^6*arctan(a*x)^2/(a^2*
x^5 + x^3), x) + 64*a^6*c^3*integrate(1/16*x^6*log(a^2*x^2 + 1)^2/(a^2*x^5 + x^3), x) + 96*a^6*c^3*integrate(1
/16*x^6*log(a^2*x^2 + 1)/(a^2*x^5 + x^3), x) - 192*a^5*c^3*integrate(1/16*x^5*arctan(a*x)/(a^2*x^5 + x^3), x)
+ 1152*a^4*c^3*integrate(1/16*x^4*arctan(a*x)^2/(a^2*x^5 + x^3), x) + a^2*c^3*log(a^2*x^2 + 1)^3 + 768*a^2*c^3
*integrate(1/16*x^2*arctan(a*x)^2/(a^2*x^5 + x^3), x) + 64*a^2*c^3*integrate(1/16*x^2*log(a^2*x^2 + 1)^2/(a^2*
x^5 + x^3), x) - 32*a^2*c^3*integrate(1/16*x^2*log(a^2*x^2 + 1)/(a^2*x^5 + x^3), x) + 64*a*c^3*integrate(1/16*
x*arctan(a*x)/(a^2*x^5 + x^3), x) + 192*c^3*integrate(1/16*arctan(a*x)^2/(a^2*x^5 + x^3), x) + 16*c^3*integrat
e(1/16*log(a^2*x^2 + 1)^2/(a^2*x^5 + x^3), x))*x^2 + 4*(a^6*c^3*x^6 + 6*a^4*c^3*x^4 - 2*c^3)*arctan(a*x)^2 - (
a^6*c^3*x^6 + 6*a^4*c^3*x^4 - 2*c^3)*log(a^2*x^2 + 1)^2)/x^2

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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