\(\int \frac {(c+a^2 c x^2)^3 \arctan (a x)^3}{x^4} \, dx\) [386]

Optimal result

Integrand size = 22, antiderivative size = 336 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x^4} \, dx=-\frac {a^2 c^3 \arctan (a x)}{x}+a^4 c^3 x \arctan (a x)-a^3 c^3 \arctan (a x)^2-\frac {a c^3 \arctan (a x)^2}{2 x^2}-\frac {1}{2} a^5 c^3 x^2 \arctan (a x)^2-\frac {c^3 \arctan (a x)^3}{3 x^3}-\frac {3 a^2 c^3 \arctan (a x)^3}{x}+3 a^4 c^3 x \arctan (a x)^3+\frac {1}{3} a^6 c^3 x^3 \arctan (a x)^3+a^3 c^3 \log (x)+8 a^3 c^3 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )-a^3 c^3 \log \left (1+a^2 x^2\right )+8 a^3 c^3 \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )-8 i a^3 c^3 \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )+8 i a^3 c^3 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+4 a^3 c^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )+4 a^3 c^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right ) \]

[Out]

-a^2*c^3*arctan(a*x)/x+a^4*c^3*x*arctan(a*x)-a^3*c^3*arctan(a*x)^2-1/2*a*c^3*arctan(a*x)^2/x^2-1/2*a^5*c^3*x^2
*arctan(a*x)^2-1/3*c^3*arctan(a*x)^3/x^3-3*a^2*c^3*arctan(a*x)^3/x+3*a^4*c^3*x*arctan(a*x)^3+1/3*a^6*c^3*x^3*a
rctan(a*x)^3+a^3*c^3*ln(x)+8*a^3*c^3*arctan(a*x)^2*ln(2/(1+I*a*x))-a^3*c^3*ln(a^2*x^2+1)+8*a^3*c^3*arctan(a*x)
^2*ln(2-2/(1-I*a*x))-8*I*a^3*c^3*arctan(a*x)*polylog(2,-1+2/(1-I*a*x))+8*I*a^3*c^3*arctan(a*x)*polylog(2,1-2/(
1+I*a*x))+4*a^3*c^3*polylog(3,-1+2/(1-I*a*x))+4*a^3*c^3*polylog(3,1-2/(1+I*a*x))

Rubi [A] (verified)

Time = 0.83 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.00, number of steps used = 37, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.818, Rules used = {5068, 4930, 5040, 4964, 5004, 5114, 6745, 4946, 5038, 272, 36, 29, 31, 5044, 4988, 5112, 5036, 266} \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x^4} \, dx=\frac {1}{3} a^6 c^3 x^3 \arctan (a x)^3-\frac {1}{2} a^5 c^3 x^2 \arctan (a x)^2+3 a^4 c^3 x \arctan (a x)^3+a^4 c^3 x \arctan (a x)-8 i a^3 c^3 \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )+8 i a^3 c^3 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )-a^3 c^3 \arctan (a x)^2+8 a^3 c^3 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )+8 a^3 c^3 \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )+4 a^3 c^3 \operatorname {PolyLog}\left (3,\frac {2}{1-i a x}-1\right )+4 a^3 c^3 \operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right )+a^3 c^3 \log (x)-\frac {3 a^2 c^3 \arctan (a x)^3}{x}-\frac {a^2 c^3 \arctan (a x)}{x}-a^3 c^3 \log \left (a^2 x^2+1\right )-\frac {c^3 \arctan (a x)^3}{3 x^3}-\frac {a c^3 \arctan (a x)^2}{2 x^2} \]

[In]

Int[((c + a^2*c*x^2)^3*ArcTan[a*x]^3)/x^4,x]

[Out]

-((a^2*c^3*ArcTan[a*x])/x) + a^4*c^3*x*ArcTan[a*x] - a^3*c^3*ArcTan[a*x]^2 - (a*c^3*ArcTan[a*x]^2)/(2*x^2) - (
a^5*c^3*x^2*ArcTan[a*x]^2)/2 - (c^3*ArcTan[a*x]^3)/(3*x^3) - (3*a^2*c^3*ArcTan[a*x]^3)/x + 3*a^4*c^3*x*ArcTan[
a*x]^3 + (a^6*c^3*x^3*ArcTan[a*x]^3)/3 + a^3*c^3*Log[x] + 8*a^3*c^3*ArcTan[a*x]^2*Log[2/(1 + I*a*x)] - a^3*c^3
*Log[1 + a^2*x^2] + 8*a^3*c^3*ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*x)] - (8*I)*a^3*c^3*ArcTan[a*x]*PolyLog[2, -1 +
2/(1 - I*a*x)] + (8*I)*a^3*c^3*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)] + 4*a^3*c^3*PolyLog[3, -1 + 2/(1 - I
*a*x)] + 4*a^3*c^3*PolyLog[3, 1 - 2/(1 + I*a*x)]

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 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 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 4964

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] + Dist[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]

Rule 4988

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] - Dist[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]

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 5040

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-I)*((a + b*ArcT
an[c*x])^(p + 1)/(b*e*(p + 1))), x] - Dist[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]

Rule 5044

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

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 5112

Int[(Log[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[I*(a + b*ArcTa
n[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 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 (3 a^4 c^3 \arctan (a x)^3+\frac {c^3 \arctan (a x)^3}{x^4}+\frac {3 a^2 c^3 \arctan (a x)^3}{x^2}+a^6 c^3 x^2 \arctan (a x)^3\right ) \, dx \\ & = c^3 \int \frac {\arctan (a x)^3}{x^4} \, dx+\left (3 a^2 c^3\right ) \int \frac {\arctan (a x)^3}{x^2} \, dx+\left (3 a^4 c^3\right ) \int \arctan (a x)^3 \, dx+\left (a^6 c^3\right ) \int x^2 \arctan (a x)^3 \, dx \\ & = -\frac {c^3 \arctan (a x)^3}{3 x^3}-\frac {3 a^2 c^3 \arctan (a x)^3}{x}+3 a^4 c^3 x \arctan (a x)^3+\frac {1}{3} a^6 c^3 x^3 \arctan (a x)^3+\left (a c^3\right ) \int \frac {\arctan (a x)^2}{x^3 \left (1+a^2 x^2\right )} \, dx+\left (9 a^3 c^3\right ) \int \frac {\arctan (a x)^2}{x \left (1+a^2 x^2\right )} \, dx-\left (9 a^5 c^3\right ) \int \frac {x \arctan (a x)^2}{1+a^2 x^2} \, dx-\left (a^7 c^3\right ) \int \frac {x^3 \arctan (a x)^2}{1+a^2 x^2} \, dx \\ & = -\frac {c^3 \arctan (a x)^3}{3 x^3}-\frac {3 a^2 c^3 \arctan (a x)^3}{x}+3 a^4 c^3 x \arctan (a x)^3+\frac {1}{3} a^6 c^3 x^3 \arctan (a x)^3+\left (a c^3\right ) \int \frac {\arctan (a x)^2}{x^3} \, dx+\left (9 i a^3 c^3\right ) \int \frac {\arctan (a x)^2}{x (i+a x)} \, dx-\left (a^3 c^3\right ) \int \frac {\arctan (a x)^2}{x \left (1+a^2 x^2\right )} \, dx+\left (9 a^4 c^3\right ) \int \frac {\arctan (a x)^2}{i-a x} \, dx-\left (a^5 c^3\right ) \int x \arctan (a x)^2 \, dx+\left (a^5 c^3\right ) \int \frac {x \arctan (a x)^2}{1+a^2 x^2} \, dx \\ & = -\frac {a c^3 \arctan (a x)^2}{2 x^2}-\frac {1}{2} a^5 c^3 x^2 \arctan (a x)^2-\frac {c^3 \arctan (a x)^3}{3 x^3}-\frac {3 a^2 c^3 \arctan (a x)^3}{x}+3 a^4 c^3 x \arctan (a x)^3+\frac {1}{3} a^6 c^3 x^3 \arctan (a x)^3+9 a^3 c^3 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )+9 a^3 c^3 \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )+\left (a^2 c^3\right ) \int \frac {\arctan (a x)}{x^2 \left (1+a^2 x^2\right )} \, dx-\left (i a^3 c^3\right ) \int \frac {\arctan (a x)^2}{x (i+a x)} \, dx-\left (a^4 c^3\right ) \int \frac {\arctan (a x)^2}{i-a x} \, dx-\left (18 a^4 c^3\right ) \int \frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (18 a^4 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 (a^6 c^3\right ) \int \frac {x^2 \arctan (a x)}{1+a^2 x^2} \, dx \\ & = -\frac {a c^3 \arctan (a x)^2}{2 x^2}-\frac {1}{2} a^5 c^3 x^2 \arctan (a x)^2-\frac {c^3 \arctan (a x)^3}{3 x^3}-\frac {3 a^2 c^3 \arctan (a x)^3}{x}+3 a^4 c^3 x \arctan (a x)^3+\frac {1}{3} a^6 c^3 x^3 \arctan (a x)^3+8 a^3 c^3 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )+8 a^3 c^3 \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )-9 i a^3 c^3 \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )+9 i a^3 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 {\arctan (a x)}{x^2} \, dx+\left (9 i a^4 c^3\right ) \int \frac {\operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx-\left (9 i a^4 c^3\right ) \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx+\left (a^4 c^3\right ) \int \arctan (a x) \, dx-2 \left (\left (a^4 c^3\right ) \int \frac {\arctan (a x)}{1+a^2 x^2} \, dx\right )+\left (2 a^4 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^4 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 {a^2 c^3 \arctan (a x)}{x}+a^4 c^3 x \arctan (a x)-a^3 c^3 \arctan (a x)^2-\frac {a c^3 \arctan (a x)^2}{2 x^2}-\frac {1}{2} a^5 c^3 x^2 \arctan (a x)^2-\frac {c^3 \arctan (a x)^3}{3 x^3}-\frac {3 a^2 c^3 \arctan (a x)^3}{x}+3 a^4 c^3 x \arctan (a x)^3+\frac {1}{3} a^6 c^3 x^3 \arctan (a x)^3+8 a^3 c^3 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )+8 a^3 c^3 \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )-8 i a^3 c^3 \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )+8 i a^3 c^3 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\frac {9}{2} a^3 c^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )+\frac {9}{2} a^3 c^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\left (a^3 c^3\right ) \int \frac {1}{x \left (1+a^2 x^2\right )} \, dx-\left (i a^4 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^4 c^3\right ) \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (a^5 c^3\right ) \int \frac {x}{1+a^2 x^2} \, dx \\ & = -\frac {a^2 c^3 \arctan (a x)}{x}+a^4 c^3 x \arctan (a x)-a^3 c^3 \arctan (a x)^2-\frac {a c^3 \arctan (a x)^2}{2 x^2}-\frac {1}{2} a^5 c^3 x^2 \arctan (a x)^2-\frac {c^3 \arctan (a x)^3}{3 x^3}-\frac {3 a^2 c^3 \arctan (a x)^3}{x}+3 a^4 c^3 x \arctan (a x)^3+\frac {1}{3} a^6 c^3 x^3 \arctan (a x)^3+8 a^3 c^3 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )-\frac {1}{2} a^3 c^3 \log \left (1+a^2 x^2\right )+8 a^3 c^3 \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )-8 i a^3 c^3 \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )+8 i a^3 c^3 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+4 a^3 c^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )+4 a^3 c^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\frac {1}{2} \left (a^3 c^3\right ) \text {Subst}\left (\int \frac {1}{x \left (1+a^2 x\right )} \, dx,x,x^2\right ) \\ & = -\frac {a^2 c^3 \arctan (a x)}{x}+a^4 c^3 x \arctan (a x)-a^3 c^3 \arctan (a x)^2-\frac {a c^3 \arctan (a x)^2}{2 x^2}-\frac {1}{2} a^5 c^3 x^2 \arctan (a x)^2-\frac {c^3 \arctan (a x)^3}{3 x^3}-\frac {3 a^2 c^3 \arctan (a x)^3}{x}+3 a^4 c^3 x \arctan (a x)^3+\frac {1}{3} a^6 c^3 x^3 \arctan (a x)^3+8 a^3 c^3 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )-\frac {1}{2} a^3 c^3 \log \left (1+a^2 x^2\right )+8 a^3 c^3 \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )-8 i a^3 c^3 \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )+8 i a^3 c^3 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+4 a^3 c^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )+4 a^3 c^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\frac {1}{2} \left (a^3 c^3\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )-\frac {1}{2} \left (a^5 c^3\right ) \text {Subst}\left (\int \frac {1}{1+a^2 x} \, dx,x,x^2\right ) \\ & = -\frac {a^2 c^3 \arctan (a x)}{x}+a^4 c^3 x \arctan (a x)-a^3 c^3 \arctan (a x)^2-\frac {a c^3 \arctan (a x)^2}{2 x^2}-\frac {1}{2} a^5 c^3 x^2 \arctan (a x)^2-\frac {c^3 \arctan (a x)^3}{3 x^3}-\frac {3 a^2 c^3 \arctan (a x)^3}{x}+3 a^4 c^3 x \arctan (a x)^3+\frac {1}{3} a^6 c^3 x^3 \arctan (a x)^3+a^3 c^3 \log (x)+8 a^3 c^3 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )-a^3 c^3 \log \left (1+a^2 x^2\right )+8 a^3 c^3 \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )-8 i a^3 c^3 \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )+8 i a^3 c^3 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+4 a^3 c^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )+4 a^3 c^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.66 (sec) , antiderivative size = 318, normalized size of antiderivative = 0.95 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x^4} \, dx=\frac {c^3 \left (-2 i a^3 \pi ^3 x^3-6 a^2 x^2 \arctan (a x)+6 a^4 x^4 \arctan (a x)-3 a x \arctan (a x)^2-6 a^3 x^3 \arctan (a x)^2-3 a^5 x^5 \arctan (a x)^2-2 \arctan (a x)^3-18 a^2 x^2 \arctan (a x)^3+18 a^4 x^4 \arctan (a x)^3+2 a^6 x^6 \arctan (a x)^3+48 a^3 x^3 \arctan (a x)^2 \log \left (1-e^{-2 i \arctan (a x)}\right )+48 a^3 x^3 \arctan (a x)^2 \log \left (1+e^{2 i \arctan (a x)}\right )+6 a^3 x^3 \log (a x)-6 a^3 x^3 \log \left (1+a^2 x^2\right )+48 i a^3 x^3 \arctan (a x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )-48 i a^3 x^3 \arctan (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )+24 a^3 x^3 \operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )+24 a^3 x^3 \operatorname {PolyLog}\left (3,-e^{2 i \arctan (a x)}\right )\right )}{6 x^3} \]

[In]

Integrate[((c + a^2*c*x^2)^3*ArcTan[a*x]^3)/x^4,x]

[Out]

(c^3*((-2*I)*a^3*Pi^3*x^3 - 6*a^2*x^2*ArcTan[a*x] + 6*a^4*x^4*ArcTan[a*x] - 3*a*x*ArcTan[a*x]^2 - 6*a^3*x^3*Ar
cTan[a*x]^2 - 3*a^5*x^5*ArcTan[a*x]^2 - 2*ArcTan[a*x]^3 - 18*a^2*x^2*ArcTan[a*x]^3 + 18*a^4*x^4*ArcTan[a*x]^3
+ 2*a^6*x^6*ArcTan[a*x]^3 + 48*a^3*x^3*ArcTan[a*x]^2*Log[1 - E^((-2*I)*ArcTan[a*x])] + 48*a^3*x^3*ArcTan[a*x]^
2*Log[1 + E^((2*I)*ArcTan[a*x])] + 6*a^3*x^3*Log[a*x] - 6*a^3*x^3*Log[1 + a^2*x^2] + (48*I)*a^3*x^3*ArcTan[a*x
]*PolyLog[2, E^((-2*I)*ArcTan[a*x])] - (48*I)*a^3*x^3*ArcTan[a*x]*PolyLog[2, -E^((2*I)*ArcTan[a*x])] + 24*a^3*
x^3*PolyLog[3, E^((-2*I)*ArcTan[a*x])] + 24*a^3*x^3*PolyLog[3, -E^((2*I)*ArcTan[a*x])]))/(6*x^3)

Maple [C] (warning: unable to verify)

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

Time = 138.35 (sec) , antiderivative size = 1948, normalized size of antiderivative = 5.80

method	result	size
parts	\(\text {Expression too large to display}\)	\(1948\)
derivativedivides	\(\text {Expression too large to display}\)	\(1949\)
default	\(\text {Expression too large to display}\)	\(1949\)

[In]

int((a^2*c*x^2+c)^3*arctan(a*x)^3/x^4,x,method=_RETURNVERBOSE)

[Out]

1/3*a^6*c^3*x^3*arctan(a*x)^3+3*a^4*c^3*x*arctan(a*x)^3-3*a^2*c^3*arctan(a*x)^3/x-1/3*c^3*arctan(a*x)^3/x^3-c^
3*(1/2*a^5*arctan(a*x)^2*x^2+1/2*a*arctan(a*x)^2/x^2-8*a^3*arctan(a*x)^2*ln(a*x)+8*a^3*arctan(a*x)^2*ln(a^2*x^
2+1)-a^3*(16*arctan(a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2))-8*arctan(a*x)^2*ln((1+I*a*x)^2/(a^2*x^2+1)-1)+8*arc
tan(a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2)+1)-16*I*arctan(a*x)*polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))+16*polyl
og(3,-(1+I*a*x)/(a^2*x^2+1)^(1/2))+8*arctan(a*x)^2*ln(1-(1+I*a*x)/(a^2*x^2+1)^(1/2))-16*I*arctan(a*x)*polylog(
2,(1+I*a*x)/(a^2*x^2+1)^(1/2))+16*polylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))-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))-1/3*arctan(a*x)*(3-12*I*arctan(a*x)*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*a*x-12*I*arctan(a*x)*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*a*x+12*I*arctan(a*x)*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*a*x+12*I*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*Pi*arctan(a*x)*a*x-12*I*csgn(I*((
1+I*a*x)^2/(a^2*x^2+1)+1)^2)^3*Pi*arctan(a*x)*a*x+12*I*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))^3*Pi*arctan(a*x)*a*x+6*
I*a*x-3*a^2*x^2+3*x*arctan(a*x)*a-12*I*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*Pi*arctan(a*x)*a*x-12*I*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*Pi*arctan(a*x)*a*x-12*I*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*Pi*arctan(a*x)*a*x+12*I*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))*Pi*arctan(a*x)*a*x+24*I*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))*Pi*arctan(a*x)*a*x-24*I*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*Pi*arctan(a
*x)*a*x-48*ln(2)*arctan(a*x)*a*x+16*I*arctan(a*x)^2*a*x-12*I*arctan(a*x)*Pi*a*x+12*I*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)*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)*Pi*
arctan(a*x)*a*x-12*I*arctan(a*x)*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))*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1))*a*x+12*I*arctan(a*x)*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*a*x+12*I*arctan(a*x)*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))^2*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1))*a*x-12*I*arctan(
a*x)*Pi*csgn(((1+I*a*x)^2/(a^2*x^2+1)-1)/((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))*a*x+12*I*arctan(a*x)*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*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))*a*x)/a/x+ln((1+I*a*x)/(a^2*x^2+1)^(1/2)-1)+l
n((1+I*a*x)/(a^2*x^2+1)^(1/2)+1)+ln((1+I*a*x)^2/(a^2*x^2+1)+1)))

Fricas [F]

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

[In]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

integrate((a^2*c*x^2+c)^3*arctan(a*x)^3/x^4,x, algorithm="maxima")

[Out]

1/192*(3*(1792*a^8*c^3*integrate(1/32*x^8*arctan(a*x)^3/(a^2*x^6 + x^4), x) + 192*a^8*c^3*integrate(1/32*x^8*a
rctan(a*x)*log(a^2*x^2 + 1)^2/(a^2*x^6 + x^4), x) + 256*a^8*c^3*integrate(1/32*x^8*arctan(a*x)*log(a^2*x^2 + 1
)/(a^2*x^6 + x^4), x) - 256*a^7*c^3*integrate(1/32*x^7*arctan(a*x)^2/(a^2*x^6 + x^4), x) + 64*a^7*c^3*integrat
e(1/32*x^7*log(a^2*x^2 + 1)^2/(a^2*x^6 + x^4), x) + 84*a^3*c^3*arctan(a*x)^4 + 7168*a^6*c^3*integrate(1/32*x^6
*arctan(a*x)^3/(a^2*x^6 + x^4), x) + 768*a^6*c^3*integrate(1/32*x^6*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^2*x^6 +
x^4), x) + 2304*a^6*c^3*integrate(1/32*x^6*arctan(a*x)*log(a^2*x^2 + 1)/(a^2*x^6 + x^4), x) - 2304*a^5*c^3*int
egrate(1/32*x^5*arctan(a*x)^2/(a^2*x^6 + x^4), x) + 3*a^3*c^3*log(a^2*x^2 + 1)^3 + 1152*a^4*c^3*integrate(1/32
*x^4*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^2*x^6 + x^4), x) - 2304*a^4*c^3*integrate(1/32*x^4*arctan(a*x)*log(a^2*
x^2 + 1)/(a^2*x^6 + x^4), x) + 2304*a^3*c^3*integrate(1/32*x^3*arctan(a*x)^2/(a^2*x^6 + x^4), x) - 576*a^3*c^3
*integrate(1/32*x^3*log(a^2*x^2 + 1)^2/(a^2*x^6 + x^4), x) + 7168*a^2*c^3*integrate(1/32*x^2*arctan(a*x)^3/(a^
2*x^6 + x^4), x) + 768*a^2*c^3*integrate(1/32*x^2*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^2*x^6 + x^4), x) - 256*a^2
*c^3*integrate(1/32*x^2*arctan(a*x)*log(a^2*x^2 + 1)/(a^2*x^6 + x^4), x) + 256*a*c^3*integrate(1/32*x*arctan(a
*x)^2/(a^2*x^6 + x^4), x) - 64*a*c^3*integrate(1/32*x*log(a^2*x^2 + 1)^2/(a^2*x^6 + x^4), x) + 1792*c^3*integr
ate(1/32*arctan(a*x)^3/(a^2*x^6 + x^4), x) + 192*c^3*integrate(1/32*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^2*x^6 +
x^4), x))*x^3 + 8*(a^6*c^3*x^6 + 9*a^4*c^3*x^4 - 9*a^2*c^3*x^2 - c^3)*arctan(a*x)^3 - 6*(a^6*c^3*x^6 + 9*a^4*c
^3*x^4 - 9*a^2*c^3*x^2 - c^3)*arctan(a*x)*log(a^2*x^2 + 1)^2)/x^3

Giac [F(-1)]

Timed out. \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x^4} \, dx=\text {Timed out} \]

[In]

integrate((a^2*c*x^2+c)^3*arctan(a*x)^3/x^4,x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x^4} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^3}{x^4} \,d x \]

[In]

int((atan(a*x)^3*(c + a^2*c*x^2)^3)/x^4,x)

[Out]

int((atan(a*x)^3*(c + a^2*c*x^2)^3)/x^4, x)