3.385 \(\int \frac {(c+a^2 c x^2)^3 \tan ^{-1}(a x)^3}{x^3} \, dx\)

Optimal. Leaf size=503 \[ \frac {1}{4} a^6 c^3 x^4 \tan ^{-1}(a x)^3-\frac {1}{4} a^5 c^3 x^3 \tan ^{-1}(a x)^2+\frac {3}{2} a^4 c^3 x^2 \tan ^{-1}(a x)^3+\frac {1}{4} a^4 c^3 x^2 \tan ^{-1}(a x)-\frac {1}{4} a^3 c^3 x-\frac {15}{4} a^3 c^3 x \tan ^{-1}(a x)^2-\frac {3}{2} i a^2 c^3 \text {Li}_2\left (\frac {2}{1-i a x}-1\right )-\frac {7}{2} i a^2 c^3 \text {Li}_2\left (1-\frac {2}{i a x+1}\right )+\frac {9}{4} i a^2 c^3 \text {Li}_4\left (1-\frac {2}{i a x+1}\right )-\frac {9}{4} i a^2 c^3 \text {Li}_4\left (\frac {2}{i a x+1}-1\right )-\frac {9}{2} i a^2 c^3 \text {Li}_2\left (1-\frac {2}{i a x+1}\right ) \tan ^{-1}(a x)^2+\frac {9}{2} i a^2 c^3 \text {Li}_2\left (\frac {2}{i a x+1}-1\right ) \tan ^{-1}(a x)^2-\frac {9}{2} a^2 c^3 \text {Li}_3\left (1-\frac {2}{i a x+1}\right ) \tan ^{-1}(a x)+\frac {9}{2} a^2 c^3 \text {Li}_3\left (\frac {2}{i a x+1}-1\right ) \tan ^{-1}(a x)+\frac {3}{4} a^2 c^3 \tan ^{-1}(a x)^3-5 i a^2 c^3 \tan ^{-1}(a x)^2+\frac {1}{4} a^2 c^3 \tan ^{-1}(a x)-7 a^2 c^3 \log \left (\frac {2}{1+i a x}\right ) \tan ^{-1}(a x)+3 a^2 c^3 \log \left (2-\frac {2}{1-i a x}\right ) \tan ^{-1}(a x)+6 a^2 c^3 \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )-\frac {c^3 \tan ^{-1}(a x)^3}{2 x^2}-\frac {3 a c^3 \tan ^{-1}(a x)^2}{2 x} \]

[Out]

-1/4*a^3*c^3*x+1/4*a^2*c^3*arctan(a*x)+1/4*a^4*c^3*x^2*arctan(a*x)-3/2*I*a^2*c^3*polylog(2,-1+2/(1-I*a*x))-3/2
*a*c^3*arctan(a*x)^2/x-15/4*a^3*c^3*x*arctan(a*x)^2-1/4*a^5*c^3*x^3*arctan(a*x)^2+3/4*a^2*c^3*arctan(a*x)^3-1/
2*c^3*arctan(a*x)^3/x^2+3/2*a^4*c^3*x^2*arctan(a*x)^3+1/4*a^6*c^3*x^4*arctan(a*x)^3-6*a^2*c^3*arctan(a*x)^3*ar
ctanh(-1+2/(1+I*a*x))-7*a^2*c^3*arctan(a*x)*ln(2/(1+I*a*x))+3*a^2*c^3*arctan(a*x)*ln(2-2/(1-I*a*x))-7/2*I*a^2*
c^3*polylog(2,1-2/(1+I*a*x))-5*I*a^2*c^3*arctan(a*x)^2-9/2*I*a^2*c^3*arctan(a*x)^2*polylog(2,1-2/(1+I*a*x))+9/
4*I*a^2*c^3*polylog(4,1-2/(1+I*a*x))-9/2*a^2*c^3*arctan(a*x)*polylog(3,1-2/(1+I*a*x))+9/2*a^2*c^3*arctan(a*x)*
polylog(3,-1+2/(1+I*a*x))-9/4*I*a^2*c^3*polylog(4,-1+2/(1+I*a*x))+9/2*I*a^2*c^3*arctan(a*x)^2*polylog(2,-1+2/(
1+I*a*x))

________________________________________________________________________________________

Rubi [A]  time = 1.19, antiderivative size = 503, normalized size of antiderivative = 1.00, number of steps used = 43, number of rules used = 20, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.909, Rules used = {4948, 4852, 4918, 4924, 4868, 2447, 4884, 4850, 4988, 4994, 4998, 6610, 4916, 4846, 4920, 4854, 2402, 2315, 321, 203} \[ -\frac {3}{2} i a^2 c^3 \text {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )-\frac {7}{2} i a^2 c^3 \text {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\frac {9}{4} i a^2 c^3 \text {PolyLog}\left (4,1-\frac {2}{1+i a x}\right )-\frac {9}{4} i a^2 c^3 \text {PolyLog}\left (4,-1+\frac {2}{1+i a x}\right )-\frac {9}{2} i a^2 c^3 \tan ^{-1}(a x)^2 \text {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\frac {9}{2} i a^2 c^3 \tan ^{-1}(a x)^2 \text {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-\frac {9}{2} a^2 c^3 \tan ^{-1}(a x) \text {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\frac {9}{2} a^2 c^3 \tan ^{-1}(a x) \text {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right )+\frac {1}{4} a^6 c^3 x^4 \tan ^{-1}(a x)^3-\frac {1}{4} a^5 c^3 x^3 \tan ^{-1}(a x)^2+\frac {3}{2} a^4 c^3 x^2 \tan ^{-1}(a x)^3+\frac {1}{4} a^4 c^3 x^2 \tan ^{-1}(a x)-\frac {1}{4} a^3 c^3 x-\frac {15}{4} a^3 c^3 x \tan ^{-1}(a x)^2+\frac {3}{4} a^2 c^3 \tan ^{-1}(a x)^3-5 i a^2 c^3 \tan ^{-1}(a x)^2+\frac {1}{4} a^2 c^3 \tan ^{-1}(a x)-7 a^2 c^3 \log \left (\frac {2}{1+i a x}\right ) \tan ^{-1}(a x)+3 a^2 c^3 \log \left (2-\frac {2}{1-i a x}\right ) \tan ^{-1}(a x)+6 a^2 c^3 \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )-\frac {c^3 \tan ^{-1}(a x)^3}{2 x^2}-\frac {3 a c^3 \tan ^{-1}(a x)^2}{2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^3*c^3*x)/4 + (a^2*c^3*ArcTan[a*x])/4 + (a^4*c^3*x^2*ArcTan[a*x])/4 - (5*I)*a^2*c^3*ArcTan[a*x]^2 - (3*a*c^
3*ArcTan[a*x]^2)/(2*x) - (15*a^3*c^3*x*ArcTan[a*x]^2)/4 - (a^5*c^3*x^3*ArcTan[a*x]^2)/4 + (3*a^2*c^3*ArcTan[a*
x]^3)/4 - (c^3*ArcTan[a*x]^3)/(2*x^2) + (3*a^4*c^3*x^2*ArcTan[a*x]^3)/2 + (a^6*c^3*x^4*ArcTan[a*x]^3)/4 + 6*a^
2*c^3*ArcTan[a*x]^3*ArcTanh[1 - 2/(1 + I*a*x)] - 7*a^2*c^3*ArcTan[a*x]*Log[2/(1 + I*a*x)] + 3*a^2*c^3*ArcTan[a
*x]*Log[2 - 2/(1 - I*a*x)] - ((3*I)/2)*a^2*c^3*PolyLog[2, -1 + 2/(1 - I*a*x)] - ((7*I)/2)*a^2*c^3*PolyLog[2, 1
- 2/(1 + I*a*x)] - ((9*I)/2)*a^2*c^3*ArcTan[a*x]^2*PolyLog[2, 1 - 2/(1 + I*a*x)] + ((9*I)/2)*a^2*c^3*ArcTan[a
*x]^2*PolyLog[2, -1 + 2/(1 + I*a*x)] - (9*a^2*c^3*ArcTan[a*x]*PolyLog[3, 1 - 2/(1 + I*a*x)])/2 + (9*a^2*c^3*Ar
cTan[a*x]*PolyLog[3, -1 + 2/(1 + I*a*x)])/2 + ((9*I)/4)*a^2*c^3*PolyLog[4, 1 - 2/(1 + I*a*x)] - ((9*I)/4)*a^2*
c^3*PolyLog[4, -1 + 2/(1 + I*a*x)]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rule 2402

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> -Dist[e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 2447

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[(Pq^m*(1 - u))/D[u, x]]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

Rule 4846

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcTan[c*x])^p, x] - Dist[b*c*p, Int[
(x*(a + b*ArcTan[c*x])^(p - 1))/(1 + c^2*x^2), x], x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 0]

Rule 4850

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 4852

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcTa
n[c*x])^p)/(d*(m + 1)), x] - Dist[(b*c*p)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcTan[c*x])^(p - 1))/(1 + c^
2*x^2), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && (EqQ[p, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 4854

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcTan[c*x])^p*Lo
g[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 4868

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 4884

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 4916

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 4918

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 4920

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] - 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 4924

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> -Simp[(I*(a + b*ArcTan
[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 4948

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 4988

Int[(ArcTanh[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/2, Int[(
Log[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 4994

Int[(Log[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Simp[(I*(a + b*Arc
Tan[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 4998

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

Rule 6610

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*} \int \frac {\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^3}{x^3} \, dx &=\int \left (\frac {c^3 \tan ^{-1}(a x)^3}{x^3}+\frac {3 a^2 c^3 \tan ^{-1}(a x)^3}{x}+3 a^4 c^3 x \tan ^{-1}(a x)^3+a^6 c^3 x^3 \tan ^{-1}(a x)^3\right ) \, dx\\ &=c^3 \int \frac {\tan ^{-1}(a x)^3}{x^3} \, dx+\left (3 a^2 c^3\right ) \int \frac {\tan ^{-1}(a x)^3}{x} \, dx+\left (3 a^4 c^3\right ) \int x \tan ^{-1}(a x)^3 \, dx+\left (a^6 c^3\right ) \int x^3 \tan ^{-1}(a x)^3 \, dx\\ &=-\frac {c^3 \tan ^{-1}(a x)^3}{2 x^2}+\frac {3}{2} a^4 c^3 x^2 \tan ^{-1}(a x)^3+\frac {1}{4} a^6 c^3 x^4 \tan ^{-1}(a x)^3+6 a^2 c^3 \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )+\frac {1}{2} \left (3 a c^3\right ) \int \frac {\tan ^{-1}(a x)^2}{x^2 \left (1+a^2 x^2\right )} \, dx-\left (18 a^3 c^3\right ) \int \frac {\tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\frac {1}{2} \left (9 a^5 c^3\right ) \int \frac {x^2 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx-\frac {1}{4} \left (3 a^7 c^3\right ) \int \frac {x^4 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx\\ &=-\frac {c^3 \tan ^{-1}(a x)^3}{2 x^2}+\frac {3}{2} a^4 c^3 x^2 \tan ^{-1}(a x)^3+\frac {1}{4} a^6 c^3 x^4 \tan ^{-1}(a x)^3+6 a^2 c^3 \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )+\frac {1}{2} \left (3 a c^3\right ) \int \frac {\tan ^{-1}(a x)^2}{x^2} \, dx-\frac {1}{2} \left (3 a^3 c^3\right ) \int \frac {\tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx-\frac {1}{2} \left (9 a^3 c^3\right ) \int \tan ^{-1}(a x)^2 \, dx+\frac {1}{2} \left (9 a^3 c^3\right ) \int \frac {\tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx+\left (9 a^3 c^3\right ) \int \frac {\tan ^{-1}(a x)^2 \log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (9 a^3 c^3\right ) \int \frac {\tan ^{-1}(a x)^2 \log \left (2-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\frac {1}{4} \left (3 a^5 c^3\right ) \int x^2 \tan ^{-1}(a x)^2 \, dx+\frac {1}{4} \left (3 a^5 c^3\right ) \int \frac {x^2 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx\\ &=-\frac {3 a c^3 \tan ^{-1}(a x)^2}{2 x}-\frac {9}{2} a^3 c^3 x \tan ^{-1}(a x)^2-\frac {1}{4} a^5 c^3 x^3 \tan ^{-1}(a x)^2+a^2 c^3 \tan ^{-1}(a x)^3-\frac {c^3 \tan ^{-1}(a x)^3}{2 x^2}+\frac {3}{2} a^4 c^3 x^2 \tan ^{-1}(a x)^3+\frac {1}{4} a^6 c^3 x^4 \tan ^{-1}(a x)^3+6 a^2 c^3 \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )-\frac {9}{2} i a^2 c^3 \tan ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2}{1+i a x}\right )+\frac {9}{2} i a^2 c^3 \tan ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1+i a x}\right )+\left (3 a^2 c^3\right ) \int \frac {\tan ^{-1}(a x)}{x \left (1+a^2 x^2\right )} \, dx+\left (9 i a^3 c^3\right ) \int \frac {\tan ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (9 i a^3 c^3\right ) \int \frac {\tan ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx+\frac {1}{4} \left (3 a^3 c^3\right ) \int \tan ^{-1}(a x)^2 \, dx-\frac {1}{4} \left (3 a^3 c^3\right ) \int \frac {\tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx+\left (9 a^4 c^3\right ) \int \frac {x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac {1}{2} \left (a^6 c^3\right ) \int \frac {x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=-6 i a^2 c^3 \tan ^{-1}(a x)^2-\frac {3 a c^3 \tan ^{-1}(a x)^2}{2 x}-\frac {15}{4} a^3 c^3 x \tan ^{-1}(a x)^2-\frac {1}{4} a^5 c^3 x^3 \tan ^{-1}(a x)^2+\frac {3}{4} a^2 c^3 \tan ^{-1}(a x)^3-\frac {c^3 \tan ^{-1}(a x)^3}{2 x^2}+\frac {3}{2} a^4 c^3 x^2 \tan ^{-1}(a x)^3+\frac {1}{4} a^6 c^3 x^4 \tan ^{-1}(a x)^3+6 a^2 c^3 \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )-\frac {9}{2} i a^2 c^3 \tan ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2}{1+i a x}\right )+\frac {9}{2} i a^2 c^3 \tan ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1+i a x}\right )-\frac {9}{2} a^2 c^3 \tan ^{-1}(a x) \text {Li}_3\left (1-\frac {2}{1+i a x}\right )+\frac {9}{2} a^2 c^3 \tan ^{-1}(a x) \text {Li}_3\left (-1+\frac {2}{1+i a x}\right )+\left (3 i a^2 c^3\right ) \int \frac {\tan ^{-1}(a x)}{x (i+a x)} \, dx+\frac {1}{2} \left (9 a^3 c^3\right ) \int \frac {\text {Li}_3\left (1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\frac {1}{2} \left (9 a^3 c^3\right ) \int \frac {\text {Li}_3\left (-1+\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (9 a^3 c^3\right ) \int \frac {\tan ^{-1}(a x)}{i-a x} \, dx+\frac {1}{2} \left (a^4 c^3\right ) \int x \tan ^{-1}(a x) \, dx-\frac {1}{2} \left (a^4 c^3\right ) \int \frac {x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac {1}{2} \left (3 a^4 c^3\right ) \int \frac {x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=\frac {1}{4} a^4 c^3 x^2 \tan ^{-1}(a x)-5 i a^2 c^3 \tan ^{-1}(a x)^2-\frac {3 a c^3 \tan ^{-1}(a x)^2}{2 x}-\frac {15}{4} a^3 c^3 x \tan ^{-1}(a x)^2-\frac {1}{4} a^5 c^3 x^3 \tan ^{-1}(a x)^2+\frac {3}{4} a^2 c^3 \tan ^{-1}(a x)^3-\frac {c^3 \tan ^{-1}(a x)^3}{2 x^2}+\frac {3}{2} a^4 c^3 x^2 \tan ^{-1}(a x)^3+\frac {1}{4} a^6 c^3 x^4 \tan ^{-1}(a x)^3+6 a^2 c^3 \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )-9 a^2 c^3 \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )+3 a^2 c^3 \tan ^{-1}(a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {9}{2} i a^2 c^3 \tan ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2}{1+i a x}\right )+\frac {9}{2} i a^2 c^3 \tan ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1+i a x}\right )-\frac {9}{2} a^2 c^3 \tan ^{-1}(a x) \text {Li}_3\left (1-\frac {2}{1+i a x}\right )+\frac {9}{2} a^2 c^3 \tan ^{-1}(a x) \text {Li}_3\left (-1+\frac {2}{1+i a x}\right )+\frac {9}{4} i a^2 c^3 \text {Li}_4\left (1-\frac {2}{1+i a x}\right )-\frac {9}{4} i a^2 c^3 \text {Li}_4\left (-1+\frac {2}{1+i a x}\right )+\frac {1}{2} \left (a^3 c^3\right ) \int \frac {\tan ^{-1}(a x)}{i-a x} \, dx+\frac {1}{2} \left (3 a^3 c^3\right ) \int \frac {\tan ^{-1}(a x)}{i-a x} \, dx-\left (3 a^3 c^3\right ) \int \frac {\log \left (2-\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx+\left (9 a^3 c^3\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\frac {1}{4} \left (a^5 c^3\right ) \int \frac {x^2}{1+a^2 x^2} \, dx\\ &=-\frac {1}{4} a^3 c^3 x+\frac {1}{4} a^4 c^3 x^2 \tan ^{-1}(a x)-5 i a^2 c^3 \tan ^{-1}(a x)^2-\frac {3 a c^3 \tan ^{-1}(a x)^2}{2 x}-\frac {15}{4} a^3 c^3 x \tan ^{-1}(a x)^2-\frac {1}{4} a^5 c^3 x^3 \tan ^{-1}(a x)^2+\frac {3}{4} a^2 c^3 \tan ^{-1}(a x)^3-\frac {c^3 \tan ^{-1}(a x)^3}{2 x^2}+\frac {3}{2} a^4 c^3 x^2 \tan ^{-1}(a x)^3+\frac {1}{4} a^6 c^3 x^4 \tan ^{-1}(a x)^3+6 a^2 c^3 \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )-7 a^2 c^3 \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )+3 a^2 c^3 \tan ^{-1}(a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {3}{2} i a^2 c^3 \text {Li}_2\left (-1+\frac {2}{1-i a x}\right )-\frac {9}{2} i a^2 c^3 \tan ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2}{1+i a x}\right )+\frac {9}{2} i a^2 c^3 \tan ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1+i a x}\right )-\frac {9}{2} a^2 c^3 \tan ^{-1}(a x) \text {Li}_3\left (1-\frac {2}{1+i a x}\right )+\frac {9}{2} a^2 c^3 \tan ^{-1}(a x) \text {Li}_3\left (-1+\frac {2}{1+i a x}\right )+\frac {9}{4} i a^2 c^3 \text {Li}_4\left (1-\frac {2}{1+i a x}\right )-\frac {9}{4} i a^2 c^3 \text {Li}_4\left (-1+\frac {2}{1+i a x}\right )-\left (9 i a^2 c^3\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )+\frac {1}{4} \left (a^3 c^3\right ) \int \frac {1}{1+a^2 x^2} \, dx-\frac {1}{2} \left (a^3 c^3\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\frac {1}{2} \left (3 a^3 c^3\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx\\ &=-\frac {1}{4} a^3 c^3 x+\frac {1}{4} a^2 c^3 \tan ^{-1}(a x)+\frac {1}{4} a^4 c^3 x^2 \tan ^{-1}(a x)-5 i a^2 c^3 \tan ^{-1}(a x)^2-\frac {3 a c^3 \tan ^{-1}(a x)^2}{2 x}-\frac {15}{4} a^3 c^3 x \tan ^{-1}(a x)^2-\frac {1}{4} a^5 c^3 x^3 \tan ^{-1}(a x)^2+\frac {3}{4} a^2 c^3 \tan ^{-1}(a x)^3-\frac {c^3 \tan ^{-1}(a x)^3}{2 x^2}+\frac {3}{2} a^4 c^3 x^2 \tan ^{-1}(a x)^3+\frac {1}{4} a^6 c^3 x^4 \tan ^{-1}(a x)^3+6 a^2 c^3 \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )-7 a^2 c^3 \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )+3 a^2 c^3 \tan ^{-1}(a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {3}{2} i a^2 c^3 \text {Li}_2\left (-1+\frac {2}{1-i a x}\right )-\frac {9}{2} i a^2 c^3 \text {Li}_2\left (1-\frac {2}{1+i a x}\right )-\frac {9}{2} i a^2 c^3 \tan ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2}{1+i a x}\right )+\frac {9}{2} i a^2 c^3 \tan ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1+i a x}\right )-\frac {9}{2} a^2 c^3 \tan ^{-1}(a x) \text {Li}_3\left (1-\frac {2}{1+i a x}\right )+\frac {9}{2} a^2 c^3 \tan ^{-1}(a x) \text {Li}_3\left (-1+\frac {2}{1+i a x}\right )+\frac {9}{4} i a^2 c^3 \text {Li}_4\left (1-\frac {2}{1+i a x}\right )-\frac {9}{4} i a^2 c^3 \text {Li}_4\left (-1+\frac {2}{1+i a x}\right )+\frac {1}{2} \left (i a^2 c^3\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )+\frac {1}{2} \left (3 i a^2 c^3\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )\\ &=-\frac {1}{4} a^3 c^3 x+\frac {1}{4} a^2 c^3 \tan ^{-1}(a x)+\frac {1}{4} a^4 c^3 x^2 \tan ^{-1}(a x)-5 i a^2 c^3 \tan ^{-1}(a x)^2-\frac {3 a c^3 \tan ^{-1}(a x)^2}{2 x}-\frac {15}{4} a^3 c^3 x \tan ^{-1}(a x)^2-\frac {1}{4} a^5 c^3 x^3 \tan ^{-1}(a x)^2+\frac {3}{4} a^2 c^3 \tan ^{-1}(a x)^3-\frac {c^3 \tan ^{-1}(a x)^3}{2 x^2}+\frac {3}{2} a^4 c^3 x^2 \tan ^{-1}(a x)^3+\frac {1}{4} a^6 c^3 x^4 \tan ^{-1}(a x)^3+6 a^2 c^3 \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )-7 a^2 c^3 \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )+3 a^2 c^3 \tan ^{-1}(a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {3}{2} i a^2 c^3 \text {Li}_2\left (-1+\frac {2}{1-i a x}\right )-\frac {7}{2} i a^2 c^3 \text {Li}_2\left (1-\frac {2}{1+i a x}\right )-\frac {9}{2} i a^2 c^3 \tan ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2}{1+i a x}\right )+\frac {9}{2} i a^2 c^3 \tan ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1+i a x}\right )-\frac {9}{2} a^2 c^3 \tan ^{-1}(a x) \text {Li}_3\left (1-\frac {2}{1+i a x}\right )+\frac {9}{2} a^2 c^3 \tan ^{-1}(a x) \text {Li}_3\left (-1+\frac {2}{1+i a x}\right )+\frac {9}{4} i a^2 c^3 \text {Li}_4\left (1-\frac {2}{1+i a x}\right )-\frac {9}{4} i a^2 c^3 \text {Li}_4\left (-1+\frac {2}{1+i a x}\right )\\ \end {align*}

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Mathematica [A]  time = 0.78, size = 464, normalized size = 0.92 \[ \frac {c^3 \left (16 a^6 x^6 \tan ^{-1}(a x)^3-16 a^5 x^5 \tan ^{-1}(a x)^2+96 a^4 x^4 \tan ^{-1}(a x)^3+16 a^4 x^4 \tan ^{-1}(a x)-16 a^3 x^3-240 a^3 x^3 \tan ^{-1}(a x)^2+288 i a^2 x^2 \tan ^{-1}(a x)^2 \text {Li}_2\left (e^{-2 i \tan ^{-1}(a x)}\right )+32 i a^2 x^2 \left (9 \tan ^{-1}(a x)^2+7\right ) \text {Li}_2\left (-e^{2 i \tan ^{-1}(a x)}\right )-96 i a^2 x^2 \text {Li}_2\left (e^{2 i \tan ^{-1}(a x)}\right )+288 a^2 x^2 \tan ^{-1}(a x) \text {Li}_3\left (e^{-2 i \tan ^{-1}(a x)}\right )-288 a^2 x^2 \tan ^{-1}(a x) \text {Li}_3\left (-e^{2 i \tan ^{-1}(a x)}\right )-144 i a^2 x^2 \text {Li}_4\left (e^{-2 i \tan ^{-1}(a x)}\right )-144 i a^2 x^2 \text {Li}_4\left (-e^{2 i \tan ^{-1}(a x)}\right )-3 i \pi ^4 a^2 x^2+96 i a^2 x^2 \tan ^{-1}(a x)^4+48 a^2 x^2 \tan ^{-1}(a x)^3+128 i a^2 x^2 \tan ^{-1}(a x)^2+16 a^2 x^2 \tan ^{-1}(a x)+192 a^2 x^2 \tan ^{-1}(a x)^3 \log \left (1-e^{-2 i \tan ^{-1}(a x)}\right )+192 a^2 x^2 \tan ^{-1}(a x) \log \left (1-e^{2 i \tan ^{-1}(a x)}\right )-192 a^2 x^2 \tan ^{-1}(a x)^3 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )-448 a^2 x^2 \tan ^{-1}(a x) \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )-96 a x \tan ^{-1}(a x)^2-32 \tan ^{-1}(a x)^3\right )}{64 x^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c^3*((-3*I)*a^2*Pi^4*x^2 - 16*a^3*x^3 + 16*a^2*x^2*ArcTan[a*x] + 16*a^4*x^4*ArcTan[a*x] - 96*a*x*ArcTan[a*x]^
2 + (128*I)*a^2*x^2*ArcTan[a*x]^2 - 240*a^3*x^3*ArcTan[a*x]^2 - 16*a^5*x^5*ArcTan[a*x]^2 - 32*ArcTan[a*x]^3 +
48*a^2*x^2*ArcTan[a*x]^3 + 96*a^4*x^4*ArcTan[a*x]^3 + 16*a^6*x^6*ArcTan[a*x]^3 + (96*I)*a^2*x^2*ArcTan[a*x]^4
+ 192*a^2*x^2*ArcTan[a*x]^3*Log[1 - E^((-2*I)*ArcTan[a*x])] + 192*a^2*x^2*ArcTan[a*x]*Log[1 - E^((2*I)*ArcTan[
a*x])] - 448*a^2*x^2*ArcTan[a*x]*Log[1 + E^((2*I)*ArcTan[a*x])] - 192*a^2*x^2*ArcTan[a*x]^3*Log[1 + E^((2*I)*A
rcTan[a*x])] + (288*I)*a^2*x^2*ArcTan[a*x]^2*PolyLog[2, E^((-2*I)*ArcTan[a*x])] + (32*I)*a^2*x^2*(7 + 9*ArcTan
[a*x]^2)*PolyLog[2, -E^((2*I)*ArcTan[a*x])] - (96*I)*a^2*x^2*PolyLog[2, E^((2*I)*ArcTan[a*x])] + 288*a^2*x^2*A
rcTan[a*x]*PolyLog[3, E^((-2*I)*ArcTan[a*x])] - 288*a^2*x^2*ArcTan[a*x]*PolyLog[3, -E^((2*I)*ArcTan[a*x])] - (
144*I)*a^2*x^2*PolyLog[4, E^((-2*I)*ArcTan[a*x])] - (144*I)*a^2*x^2*PolyLog[4, -E^((2*I)*ArcTan[a*x])]))/(64*x
^2)

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fricas [F]  time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{6} c^{3} x^{6} + 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} + c^{3}\right )} \arctan \left (a x\right )^{3}}{x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [A]  time = 21.30, size = 790, normalized size = 1.57 \[ \frac {9 i a^{2} c^{3} \arctan \left (a x \right )^{2} \polylog \left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}-9 i a^{2} c^{3} \arctan \left (a x \right )^{2} \polylog \left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-9 i a^{2} c^{3} \arctan \left (a x \right )^{2} \polylog \left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-\frac {a^{3} c^{3} x}{4}+\frac {a^{2} c^{3} \arctan \left (a x \right )}{4}+\frac {3 a^{2} c^{3} \arctan \left (a x \right )^{3}}{4}-\frac {c^{3} \arctan \left (a x \right )^{3}}{2 x^{2}}-\frac {3 a \,c^{3} \arctan \left (a x \right )^{2}}{2 x}-\frac {15 a^{3} c^{3} x \arctan \left (a x \right )^{2}}{4}-\frac {a^{5} c^{3} x^{3} \arctan \left (a x \right )^{2}}{4}+\frac {3 a^{4} c^{3} x^{2} \arctan \left (a x \right )^{3}}{2}+\frac {a^{6} c^{3} x^{4} \arctan \left (a x \right )^{3}}{4}+\frac {a^{4} c^{3} x^{2} \arctan \left (a x \right )}{4}+18 a^{2} c^{3} \arctan \left (a x \right ) \polylog \left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+18 i a^{2} c^{3} \polylog \left (4, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+3 a^{2} c^{3} \arctan \left (a x \right )^{3} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-7 a^{2} c^{3} \arctan \left (a x \right ) \ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )-\frac {9 i a^{2} c^{3} \polylog \left (4, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{4}+18 i a^{2} c^{3} \polylog \left (4, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-3 i a^{2} c^{3} \polylog \left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-3 i a^{2} c^{3} \polylog \left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+\frac {7 i a^{2} c^{3} \polylog \left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}+18 a^{2} c^{3} \arctan \left (a x \right ) \polylog \left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-\frac {9 a^{2} c^{3} \arctan \left (a x \right ) \polylog \left (3, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}+3 a^{2} c^{3} \arctan \left (a x \right ) \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-3 a^{2} c^{3} \arctan \left (a x \right )^{3} \ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )+3 a^{2} c^{3} \arctan \left (a x \right )^{3} \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+3 a^{2} c^{3} \arctan \left (a x \right ) \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+2 i a^{2} c^{3} \arctan \left (a x \right )^{2}-\frac {i a^{2} c^{3}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*a^3*c^3*x+1/4*a^2*c^3*arctan(a*x)+9/2*I*a^2*c^3*arctan(a*x)^2*polylog(2,-(1+I*a*x)^2/(a^2*x^2+1))+3/4*a^2
*c^3*arctan(a*x)^3-1/2*c^3*arctan(a*x)^3/x^2-9*I*a^2*c^3*arctan(a*x)^2*polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))
-9*I*a^2*c^3*arctan(a*x)^2*polylog(2,(1+I*a*x)/(a^2*x^2+1)^(1/2))-3/2*a*c^3*arctan(a*x)^2/x-15/4*a^3*c^3*x*arc
tan(a*x)^2-1/4*a^5*c^3*x^3*arctan(a*x)^2+3/2*a^4*c^3*x^2*arctan(a*x)^3+1/4*a^6*c^3*x^4*arctan(a*x)^3+1/4*a^4*c
^3*x^2*arctan(a*x)+3*a^2*c^3*arctan(a*x)*ln(1+(1+I*a*x)/(a^2*x^2+1)^(1/2))-3*a^2*c^3*arctan(a*x)^3*ln((1+I*a*x
)^2/(a^2*x^2+1)+1)+3*a^2*c^3*arctan(a*x)^3*ln(1+(1+I*a*x)/(a^2*x^2+1)^(1/2))+3*a^2*c^3*arctan(a*x)*ln(1-(1+I*a
*x)/(a^2*x^2+1)^(1/2))+18*a^2*c^3*arctan(a*x)*polylog(3,-(1+I*a*x)/(a^2*x^2+1)^(1/2))+18*I*a^2*c^3*polylog(4,-
(1+I*a*x)/(a^2*x^2+1)^(1/2))-9/4*I*a^2*c^3*polylog(4,-(1+I*a*x)^2/(a^2*x^2+1))+2*I*a^2*c^3*arctan(a*x)^2+18*I*
a^2*c^3*polylog(4,(1+I*a*x)/(a^2*x^2+1)^(1/2))-3*I*a^2*c^3*polylog(2,(1+I*a*x)/(a^2*x^2+1)^(1/2))-3*I*a^2*c^3*
polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))+7/2*I*a^2*c^3*polylog(2,-(1+I*a*x)^2/(a^2*x^2+1))+3*a^2*c^3*arctan(a*x
)^3*ln(1-(1+I*a*x)/(a^2*x^2+1)^(1/2))+18*a^2*c^3*arctan(a*x)*polylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))-7*a^2*c^3*
arctan(a*x)*ln((1+I*a*x)^2/(a^2*x^2+1)+1)-9/2*a^2*c^3*arctan(a*x)*polylog(3,-(1+I*a*x)^2/(a^2*x^2+1))-1/4*I*a^
2*c^3

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {4 \, {\left (a^{6} c^{3} x^{6} + 6 \, a^{4} c^{3} x^{4} - 2 \, c^{3}\right )} \arctan \left (a x\right )^{3} - 3 \, {\left (a^{6} c^{3} x^{6} + 6 \, a^{4} c^{3} x^{4} - 2 \, c^{3}\right )} \arctan \left (a x\right ) \log \left (a^{2} x^{2} + 1\right )^{2} + x^{2} \int \frac {112 \, {\left (a^{8} c^{3} x^{8} + 4 \, a^{6} c^{3} x^{6} + 6 \, a^{4} c^{3} x^{4} + 4 \, a^{2} c^{3} x^{2} + c^{3}\right )} \arctan \left (a x\right )^{3} - 12 \, {\left (a^{7} c^{3} x^{7} + 6 \, a^{5} c^{3} x^{5} - 2 \, a c^{3} x\right )} \arctan \left (a x\right )^{2} + 12 \, {\left (a^{8} c^{3} x^{8} + 6 \, a^{6} c^{3} x^{6} - 2 \, a^{2} c^{3} x^{2}\right )} \arctan \left (a x\right ) \log \left (a^{2} x^{2} + 1\right ) + 3 \, {\left (a^{7} c^{3} x^{7} + 6 \, a^{5} c^{3} x^{5} - 2 \, a c^{3} x + 4 \, {\left (a^{8} c^{3} x^{8} + 4 \, a^{6} c^{3} x^{6} + 6 \, a^{4} c^{3} x^{4} + 4 \, a^{2} c^{3} x^{2} + c^{3}\right )} \arctan \left (a x\right )\right )} \log \left (a^{2} x^{2} + 1\right )^{2}}{a^{2} x^{5} + x^{3}}\,{d x}}{128 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/128*(4*(a^6*c^3*x^6 + 6*a^4*c^3*x^4 - 2*c^3)*arctan(a*x)^3 - 3*(a^6*c^3*x^6 + 6*a^4*c^3*x^4 - 2*c^3)*arctan(
a*x)*log(a^2*x^2 + 1)^2 + 128*x^2*integrate(1/128*(112*(a^8*c^3*x^8 + 4*a^6*c^3*x^6 + 6*a^4*c^3*x^4 + 4*a^2*c^
3*x^2 + c^3)*arctan(a*x)^3 - 12*(a^7*c^3*x^7 + 6*a^5*c^3*x^5 - 2*a*c^3*x)*arctan(a*x)^2 + 12*(a^8*c^3*x^8 + 6*
a^6*c^3*x^6 - 2*a^2*c^3*x^2)*arctan(a*x)*log(a^2*x^2 + 1) + 3*(a^7*c^3*x^7 + 6*a^5*c^3*x^5 - 2*a*c^3*x + 4*(a^
8*c^3*x^8 + 4*a^6*c^3*x^6 + 6*a^4*c^3*x^4 + 4*a^2*c^3*x^2 + c^3)*arctan(a*x))*log(a^2*x^2 + 1)^2)/(a^2*x^5 + x
^3), x))/x^2

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^3}{x^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ c^{3} \left (\int \frac {\operatorname {atan}^{3}{\left (a x \right )}}{x^{3}}\, dx + \int \frac {3 a^{2} \operatorname {atan}^{3}{\left (a x \right )}}{x}\, dx + \int 3 a^{4} x \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int a^{6} x^{3} \operatorname {atan}^{3}{\left (a x \right )}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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