\(\int \frac {(c+a^2 c x^2)^3 \arctan (a x)^3}{x^2} \, dx\) [384]

Optimal result

Integrand size = 22, antiderivative size = 354 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x^2} \, dx=-\frac {1}{20} a^3 c^3 x^2+\frac {21}{10} a^2 c^3 x \arctan (a x)+\frac {1}{10} a^4 c^3 x^3 \arctan (a x)-\frac {21}{20} a c^3 \arctan (a x)^2-\frac {6}{5} a^3 c^3 x^2 \arctan (a x)^2-\frac {3}{20} a^5 c^3 x^4 \arctan (a x)^2+\frac {6}{5} i a c^3 \arctan (a x)^3-\frac {c^3 \arctan (a x)^3}{x}+3 a^2 c^3 x \arctan (a x)^3+a^4 c^3 x^3 \arctan (a x)^3+\frac {1}{5} a^6 c^3 x^5 \arctan (a x)^3+\frac {33}{5} a c^3 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )-a c^3 \log \left (1+a^2 x^2\right )+3 a c^3 \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )-3 i a c^3 \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )+\frac {33}{5} i a c^3 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\frac {3}{2} a c^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )+\frac {33}{10} a c^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right ) \]

[Out]

-1/20*a^3*c^3*x^2+21/10*a^2*c^3*x*arctan(a*x)+1/10*a^4*c^3*x^3*arctan(a*x)-21/20*a*c^3*arctan(a*x)^2-6/5*a^3*c
^3*x^2*arctan(a*x)^2-3/20*a^5*c^3*x^4*arctan(a*x)^2-3*I*a*c^3*arctan(a*x)*polylog(2,-1+2/(1-I*a*x))-c^3*arctan
(a*x)^3/x+3*a^2*c^3*x*arctan(a*x)^3+a^4*c^3*x^3*arctan(a*x)^3+1/5*a^6*c^3*x^5*arctan(a*x)^3+33/5*a*c^3*arctan(
a*x)^2*ln(2/(1+I*a*x))-a*c^3*ln(a^2*x^2+1)+3*a*c^3*arctan(a*x)^2*ln(2-2/(1-I*a*x))+33/5*I*a*c^3*arctan(a*x)*po
lylog(2,1-2/(1+I*a*x))+6/5*I*a*c^3*arctan(a*x)^3+3/2*a*c^3*polylog(3,-1+2/(1-I*a*x))+33/10*a*c^3*polylog(3,1-2
/(1+I*a*x))

Rubi [A] (verified)

Time = 0.95 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.00, number of steps used = 45, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.682, Rules used = {5068, 4930, 5040, 4964, 5004, 5114, 6745, 4946, 5044, 4988, 5112, 5036, 266, 272, 45} \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x^2} \, dx=\frac {1}{5} a^6 c^3 x^5 \arctan (a x)^3-\frac {3}{20} a^5 c^3 x^4 \arctan (a x)^2+a^4 c^3 x^3 \arctan (a x)^3+\frac {1}{10} a^4 c^3 x^3 \arctan (a x)-\frac {6}{5} a^3 c^3 x^2 \arctan (a x)^2-\frac {1}{20} a^3 c^3 x^2+3 a^2 c^3 x \arctan (a x)^3+\frac {21}{10} a^2 c^3 x \arctan (a x)-a c^3 \log \left (a^2 x^2+1\right )-3 i a c^3 \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )+\frac {33}{5} i a c^3 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )+\frac {6}{5} i a c^3 \arctan (a x)^3-\frac {21}{20} a c^3 \arctan (a x)^2-\frac {c^3 \arctan (a x)^3}{x}+\frac {33}{5} a c^3 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )+3 a c^3 \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )+\frac {3}{2} a c^3 \operatorname {PolyLog}\left (3,\frac {2}{1-i a x}-1\right )+\frac {33}{10} a c^3 \operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right ) \]

[In]

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

[Out]

-1/20*(a^3*c^3*x^2) + (21*a^2*c^3*x*ArcTan[a*x])/10 + (a^4*c^3*x^3*ArcTan[a*x])/10 - (21*a*c^3*ArcTan[a*x]^2)/
20 - (6*a^3*c^3*x^2*ArcTan[a*x]^2)/5 - (3*a^5*c^3*x^4*ArcTan[a*x]^2)/20 + ((6*I)/5)*a*c^3*ArcTan[a*x]^3 - (c^3
*ArcTan[a*x]^3)/x + 3*a^2*c^3*x*ArcTan[a*x]^3 + a^4*c^3*x^3*ArcTan[a*x]^3 + (a^6*c^3*x^5*ArcTan[a*x]^3)/5 + (3
3*a*c^3*ArcTan[a*x]^2*Log[2/(1 + I*a*x)])/5 - a*c^3*Log[1 + a^2*x^2] + 3*a*c^3*ArcTan[a*x]^2*Log[2 - 2/(1 - I*
a*x)] - (3*I)*a*c^3*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)] + ((33*I)/5)*a*c^3*ArcTan[a*x]*PolyLog[2, 1 - 2
/(1 + I*a*x)] + (3*a*c^3*PolyLog[3, -1 + 2/(1 - I*a*x)])/2 + (33*a*c^3*PolyLog[3, 1 - 2/(1 + I*a*x)])/10

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 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 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^2 c^3 \arctan (a x)^3+\frac {c^3 \arctan (a x)^3}{x^2}+3 a^4 c^3 x^2 \arctan (a x)^3+a^6 c^3 x^4 \arctan (a x)^3\right ) \, dx \\ & = c^3 \int \frac {\arctan (a x)^3}{x^2} \, dx+\left (3 a^2 c^3\right ) \int \arctan (a x)^3 \, dx+\left (3 a^4 c^3\right ) \int x^2 \arctan (a x)^3 \, dx+\left (a^6 c^3\right ) \int x^4 \arctan (a x)^3 \, dx \\ & = -\frac {c^3 \arctan (a x)^3}{x}+3 a^2 c^3 x \arctan (a x)^3+a^4 c^3 x^3 \arctan (a x)^3+\frac {1}{5} a^6 c^3 x^5 \arctan (a x)^3+\left (3 a c^3\right ) \int \frac {\arctan (a x)^2}{x \left (1+a^2 x^2\right )} \, dx-\left (9 a^3 c^3\right ) \int \frac {x \arctan (a x)^2}{1+a^2 x^2} \, dx-\left (3 a^5 c^3\right ) \int \frac {x^3 \arctan (a x)^2}{1+a^2 x^2} \, dx-\frac {1}{5} \left (3 a^7 c^3\right ) \int \frac {x^5 \arctan (a x)^2}{1+a^2 x^2} \, dx \\ & = 2 i a c^3 \arctan (a x)^3-\frac {c^3 \arctan (a x)^3}{x}+3 a^2 c^3 x \arctan (a x)^3+a^4 c^3 x^3 \arctan (a x)^3+\frac {1}{5} a^6 c^3 x^5 \arctan (a x)^3+\left (3 i a c^3\right ) \int \frac {\arctan (a x)^2}{x (i+a x)} \, dx+\left (9 a^2 c^3\right ) \int \frac {\arctan (a x)^2}{i-a x} \, dx-\left (3 a^3 c^3\right ) \int x \arctan (a x)^2 \, dx+\left (3 a^3 c^3\right ) \int \frac {x \arctan (a x)^2}{1+a^2 x^2} \, dx-\frac {1}{5} \left (3 a^5 c^3\right ) \int x^3 \arctan (a x)^2 \, dx+\frac {1}{5} \left (3 a^5 c^3\right ) \int \frac {x^3 \arctan (a x)^2}{1+a^2 x^2} \, dx \\ & = -\frac {3}{2} a^3 c^3 x^2 \arctan (a x)^2-\frac {3}{20} a^5 c^3 x^4 \arctan (a x)^2+i a c^3 \arctan (a x)^3-\frac {c^3 \arctan (a x)^3}{x}+3 a^2 c^3 x \arctan (a x)^3+a^4 c^3 x^3 \arctan (a x)^3+\frac {1}{5} a^6 c^3 x^5 \arctan (a x)^3+9 a c^3 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )+3 a c^3 \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )-\left (3 a^2 c^3\right ) \int \frac {\arctan (a x)^2}{i-a x} \, dx-\left (6 a^2 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 (18 a^2 c^3\right ) \int \frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx+\frac {1}{5} \left (3 a^3 c^3\right ) \int x \arctan (a x)^2 \, dx-\frac {1}{5} \left (3 a^3 c^3\right ) \int \frac {x \arctan (a x)^2}{1+a^2 x^2} \, dx+\left (3 a^4 c^3\right ) \int \frac {x^2 \arctan (a x)}{1+a^2 x^2} \, dx+\frac {1}{10} \left (3 a^6 c^3\right ) \int \frac {x^4 \arctan (a x)}{1+a^2 x^2} \, dx \\ & = -\frac {6}{5} a^3 c^3 x^2 \arctan (a x)^2-\frac {3}{20} a^5 c^3 x^4 \arctan (a x)^2+\frac {6}{5} i a c^3 \arctan (a x)^3-\frac {c^3 \arctan (a x)^3}{x}+3 a^2 c^3 x \arctan (a x)^3+a^4 c^3 x^3 \arctan (a x)^3+\frac {1}{5} a^6 c^3 x^5 \arctan (a x)^3+6 a c^3 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )+3 a c^3 \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )-3 i a c^3 \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )+9 i a c^3 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\left (3 i a^2 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^2 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}{5} \left (3 a^2 c^3\right ) \int \frac {\arctan (a x)^2}{i-a x} \, dx+\left (3 a^2 c^3\right ) \int \arctan (a x) \, dx-\left (3 a^2 c^3\right ) \int \frac {\arctan (a x)}{1+a^2 x^2} \, dx+\left (6 a^2 c^3\right ) \int \frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx+\frac {1}{10} \left (3 a^4 c^3\right ) \int x^2 \arctan (a x) \, dx-\frac {1}{10} \left (3 a^4 c^3\right ) \int \frac {x^2 \arctan (a x)}{1+a^2 x^2} \, dx-\frac {1}{5} \left (3 a^4 c^3\right ) \int \frac {x^2 \arctan (a x)}{1+a^2 x^2} \, dx \\ & = 3 a^2 c^3 x \arctan (a x)+\frac {1}{10} a^4 c^3 x^3 \arctan (a x)-\frac {3}{2} a c^3 \arctan (a x)^2-\frac {6}{5} a^3 c^3 x^2 \arctan (a x)^2-\frac {3}{20} a^5 c^3 x^4 \arctan (a x)^2+\frac {6}{5} i a c^3 \arctan (a x)^3-\frac {c^3 \arctan (a x)^3}{x}+3 a^2 c^3 x \arctan (a x)^3+a^4 c^3 x^3 \arctan (a x)^3+\frac {1}{5} a^6 c^3 x^5 \arctan (a x)^3+\frac {33}{5} a c^3 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )+3 a c^3 \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )-3 i a c^3 \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )+6 i a c^3 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\frac {3}{2} a c^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )+\frac {9}{2} a c^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\left (3 i a^2 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}{10} \left (3 a^2 c^3\right ) \int \arctan (a x) \, dx+\frac {1}{10} \left (3 a^2 c^3\right ) \int \frac {\arctan (a x)}{1+a^2 x^2} \, dx-\frac {1}{5} \left (3 a^2 c^3\right ) \int \arctan (a x) \, dx+\frac {1}{5} \left (3 a^2 c^3\right ) \int \frac {\arctan (a x)}{1+a^2 x^2} \, dx-\frac {1}{5} \left (6 a^2 c^3\right ) \int \frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (3 a^3 c^3\right ) \int \frac {x}{1+a^2 x^2} \, dx-\frac {1}{10} \left (a^5 c^3\right ) \int \frac {x^3}{1+a^2 x^2} \, dx \\ & = \frac {21}{10} a^2 c^3 x \arctan (a x)+\frac {1}{10} a^4 c^3 x^3 \arctan (a x)-\frac {21}{20} a c^3 \arctan (a x)^2-\frac {6}{5} a^3 c^3 x^2 \arctan (a x)^2-\frac {3}{20} a^5 c^3 x^4 \arctan (a x)^2+\frac {6}{5} i a c^3 \arctan (a x)^3-\frac {c^3 \arctan (a x)^3}{x}+3 a^2 c^3 x \arctan (a x)^3+a^4 c^3 x^3 \arctan (a x)^3+\frac {1}{5} a^6 c^3 x^5 \arctan (a x)^3+\frac {33}{5} a c^3 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )-\frac {3}{2} a c^3 \log \left (1+a^2 x^2\right )+3 a c^3 \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )-3 i a c^3 \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )+\frac {33}{5} i a c^3 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\frac {3}{2} a c^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )+3 a c^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )-\frac {1}{5} \left (3 i a^2 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}{10} \left (3 a^3 c^3\right ) \int \frac {x}{1+a^2 x^2} \, dx+\frac {1}{5} \left (3 a^3 c^3\right ) \int \frac {x}{1+a^2 x^2} \, dx-\frac {1}{20} \left (a^5 c^3\right ) \text {Subst}\left (\int \frac {x}{1+a^2 x} \, dx,x,x^2\right ) \\ & = \frac {21}{10} a^2 c^3 x \arctan (a x)+\frac {1}{10} a^4 c^3 x^3 \arctan (a x)-\frac {21}{20} a c^3 \arctan (a x)^2-\frac {6}{5} a^3 c^3 x^2 \arctan (a x)^2-\frac {3}{20} a^5 c^3 x^4 \arctan (a x)^2+\frac {6}{5} i a c^3 \arctan (a x)^3-\frac {c^3 \arctan (a x)^3}{x}+3 a^2 c^3 x \arctan (a x)^3+a^4 c^3 x^3 \arctan (a x)^3+\frac {1}{5} a^6 c^3 x^5 \arctan (a x)^3+\frac {33}{5} a c^3 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )-\frac {21}{20} a c^3 \log \left (1+a^2 x^2\right )+3 a c^3 \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )-3 i a c^3 \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )+\frac {33}{5} i a c^3 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\frac {3}{2} a c^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )+\frac {33}{10} a c^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )-\frac {1}{20} \left (a^5 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}{20} a^3 c^3 x^2+\frac {21}{10} a^2 c^3 x \arctan (a x)+\frac {1}{10} a^4 c^3 x^3 \arctan (a x)-\frac {21}{20} a c^3 \arctan (a x)^2-\frac {6}{5} a^3 c^3 x^2 \arctan (a x)^2-\frac {3}{20} a^5 c^3 x^4 \arctan (a x)^2+\frac {6}{5} i a c^3 \arctan (a x)^3-\frac {c^3 \arctan (a x)^3}{x}+3 a^2 c^3 x \arctan (a x)^3+a^4 c^3 x^3 \arctan (a x)^3+\frac {1}{5} a^6 c^3 x^5 \arctan (a x)^3+\frac {33}{5} a c^3 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )-a c^3 \log \left (1+a^2 x^2\right )+3 a c^3 \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )-3 i a c^3 \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )+\frac {33}{5} i a c^3 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\frac {3}{2} a c^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )+\frac {33}{10} a c^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.60 (sec) , antiderivative size = 298, normalized size of antiderivative = 0.84 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x^2} \, dx=\frac {c^3 \left (-2 a x-5 i a \pi ^3 x-2 a^3 x^3+84 a^2 x^2 \arctan (a x)+4 a^4 x^4 \arctan (a x)-42 a x \arctan (a x)^2-48 a^3 x^3 \arctan (a x)^2-6 a^5 x^5 \arctan (a x)^2-40 \arctan (a x)^3-48 i a x \arctan (a x)^3+120 a^2 x^2 \arctan (a x)^3+40 a^4 x^4 \arctan (a x)^3+8 a^6 x^6 \arctan (a x)^3+120 a x \arctan (a x)^2 \log \left (1-e^{-2 i \arctan (a x)}\right )+264 a x \arctan (a x)^2 \log \left (1+e^{2 i \arctan (a x)}\right )-40 a x \log \left (1+a^2 x^2\right )+120 i a x \arctan (a x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )-264 i a x \arctan (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )+60 a x \operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )+132 a x \operatorname {PolyLog}\left (3,-e^{2 i \arctan (a x)}\right )\right )}{40 x} \]

[In]

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

[Out]

(c^3*(-2*a*x - (5*I)*a*Pi^3*x - 2*a^3*x^3 + 84*a^2*x^2*ArcTan[a*x] + 4*a^4*x^4*ArcTan[a*x] - 42*a*x*ArcTan[a*x
]^2 - 48*a^3*x^3*ArcTan[a*x]^2 - 6*a^5*x^5*ArcTan[a*x]^2 - 40*ArcTan[a*x]^3 - (48*I)*a*x*ArcTan[a*x]^3 + 120*a
^2*x^2*ArcTan[a*x]^3 + 40*a^4*x^4*ArcTan[a*x]^3 + 8*a^6*x^6*ArcTan[a*x]^3 + 120*a*x*ArcTan[a*x]^2*Log[1 - E^((
-2*I)*ArcTan[a*x])] + 264*a*x*ArcTan[a*x]^2*Log[1 + E^((2*I)*ArcTan[a*x])] - 40*a*x*Log[1 + a^2*x^2] + (120*I)
*a*x*ArcTan[a*x]*PolyLog[2, E^((-2*I)*ArcTan[a*x])] - (264*I)*a*x*ArcTan[a*x]*PolyLog[2, -E^((2*I)*ArcTan[a*x]
)] + 60*a*x*PolyLog[3, E^((-2*I)*ArcTan[a*x])] + 132*a*x*PolyLog[3, -E^((2*I)*ArcTan[a*x])]))/(40*x)

Maple [C] (warning: unable to verify)

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

Time = 163.40 (sec) , antiderivative size = 1894, normalized size of antiderivative = 5.35

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1894\)
default	\(\text {Expression too large to display}\)	\(1894\)
parts	\(\text {Expression too large to display}\)	\(1897\)

[In]

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

[Out]

a*(1/5*c^3*arctan(a*x)^3*a^5*x^5+c^3*arctan(a*x)^3*a^3*x^3+3*c^3*arctan(a*x)^3*a*x-c^3*arctan(a*x)^3/a/x-3/5*c
^3*(1/4*a^4*arctan(a*x)^2*x^4+2*x^2*arctan(a*x)^2*a^2+8*arctan(a*x)^2*ln(a^2*x^2+1)-5*arctan(a*x)^2*ln(a*x)-16
*arctan(a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2))+10*I*arctan(a*x)*polylog(2,(1+I*a*x)/(a^2*x^2+1)^(1/2))-10/3*ln
((1+I*a*x)^2/(a^2*x^2+1)+1)+5*arctan(a*x)^2*ln((1+I*a*x)^2/(a^2*x^2+1)-1)-5*arctan(a*x)^2*ln((1+I*a*x)/(a^2*x^
2+1)^(1/2)+1)-1/12*I*(I-64*arctan(a*x)^3+I*a^2*x^2-40*arctan(a*x)-48*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*arctan(a*x)^2*Pi+48*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*arctan(a*x)^2*Pi+48*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*arctan(a*x)^2*Pi-96*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^2*csg
n(I*((1+I*a*x)^2/(a^2*x^2+1)+1))*arctan(a*x)^2*Pi+48*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*arctan(a*x)^2*Pi+96*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))*a
rctan(a*x)^2*Pi-48*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)*arctan(a*x)^2*Pi+30*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))*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1))*arctan(a*x)^2*Pi+30*csg
n(((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*Pi+30*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*Pi-192*I*arctan(a*x)^2*ln(2)+48*csgn(I*((1+I*a*x)^2/(a^2*x^
2+1)+1)^2)^3*arctan(a*x)^2*Pi-48*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))^3*arctan(a*x)^2*Pi-48*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*arctan(a*x)^2*Pi-30*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*Pi+30*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))*arctan(a*x)^2*Pi-30*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))*arctan(a*x)^2*Pi-30*c
sgn(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))*arctan(a*
x)^2*Pi-30*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
))*arctan(a*x)^2*Pi-2*I*arctan(a*x)*a^3*x^3-42*I*arctan(a*x)*a*x+30*arctan(a*x)^2*Pi+21*I*arctan(a*x)^2)-10*po
lylog(3,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-5*arctan(a*x)^2*ln(1-(1+I*a*x)/(a^2*x^2+1)^(1/2))+11*I*arctan(a*x)*polyl
og(2,-(1+I*a*x)^2/(a^2*x^2+1))-10*polylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))+10*I*arctan(a*x)*polylog(2,-(1+I*a*x)
/(a^2*x^2+1)^(1/2))-11/2*polylog(3,-(1+I*a*x)^2/(a^2*x^2+1))))

Fricas [F]

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

[In]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

1/320*(8*(a^6*c^3*x^6 + 5*a^4*c^3*x^4 + 15*a^2*c^3*x^2 - 5*c^3)*arctan(a*x)^3 - 6*(a^6*c^3*x^6 + 5*a^4*c^3*x^4
+ 15*a^2*c^3*x^2 - 5*c^3)*arctan(a*x)*log(a^2*x^2 + 1)^2 + 5*(8960*a^8*c^3*integrate(1/160*x^8*arctan(a*x)^3/
(a^2*x^4 + x^2), x) + 960*a^8*c^3*integrate(1/160*x^8*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^2*x^4 + x^2), x) + 768
*a^8*c^3*integrate(1/160*x^8*arctan(a*x)*log(a^2*x^2 + 1)/(a^2*x^4 + x^2), x) - 768*a^7*c^3*integrate(1/160*x^
7*arctan(a*x)^2/(a^2*x^4 + x^2), x) + 192*a^7*c^3*integrate(1/160*x^7*log(a^2*x^2 + 1)^2/(a^2*x^4 + x^2), x) +
35840*a^6*c^3*integrate(1/160*x^6*arctan(a*x)^3/(a^2*x^4 + x^2), x) + 3840*a^6*c^3*integrate(1/160*x^6*arctan
(a*x)*log(a^2*x^2 + 1)^2/(a^2*x^4 + x^2), x) + 3840*a^6*c^3*integrate(1/160*x^6*arctan(a*x)*log(a^2*x^2 + 1)/(
a^2*x^4 + x^2), x) - 3840*a^5*c^3*integrate(1/160*x^5*arctan(a*x)^2/(a^2*x^4 + x^2), x) + 960*a^5*c^3*integrat
e(1/160*x^5*log(a^2*x^2 + 1)^2/(a^2*x^4 + x^2), x) + 56*a*c^3*arctan(a*x)^4 + 53760*a^4*c^3*integrate(1/160*x^
4*arctan(a*x)^3/(a^2*x^4 + x^2), x) + 5760*a^4*c^3*integrate(1/160*x^4*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^2*x^4
+ x^2), x) + 11520*a^4*c^3*integrate(1/160*x^4*arctan(a*x)*log(a^2*x^2 + 1)/(a^2*x^4 + x^2), x) - 11520*a^3*c
^3*integrate(1/160*x^3*arctan(a*x)^2/(a^2*x^4 + x^2), x) + 3*a*c^3*log(a^2*x^2 + 1)^3 + 3840*a^2*c^3*integrate
(1/160*x^2*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^2*x^4 + x^2), x) - 3840*a^2*c^3*integrate(1/160*x^2*arctan(a*x)*l
og(a^2*x^2 + 1)/(a^2*x^4 + x^2), x) + 3840*a*c^3*integrate(1/160*x*arctan(a*x)^2/(a^2*x^4 + x^2), x) - 960*a*c
^3*integrate(1/160*x*log(a^2*x^2 + 1)^2/(a^2*x^4 + x^2), x) + 8960*c^3*integrate(1/160*arctan(a*x)^3/(a^2*x^4
+ x^2), x) + 960*c^3*integrate(1/160*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^2*x^4 + x^2), x))*x)/x

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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