\(\int \frac {(c+a^2 c x^2)^2 \arctan (a x)^3}{x^3} \, dx\) [377]

Optimal result

Integrand size = 22, antiderivative size = 399 \[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^3}{x^3} \, dx=-3 i a^2 c^2 \arctan (a x)^2-\frac {3 a c^2 \arctan (a x)^2}{2 x}-\frac {3}{2} a^3 c^2 x \arctan (a x)^2-\frac {c^2 \arctan (a x)^3}{2 x^2}+\frac {1}{2} a^4 c^2 x^2 \arctan (a x)^3+4 a^2 c^2 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-3 a^2 c^2 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )+3 a^2 c^2 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {3}{2} i a^2 c^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )-\frac {3}{2} i a^2 c^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )-3 i a^2 c^2 \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+3 i a^2 c^2 \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-3 a^2 c^2 \arctan (a x) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+3 a^2 c^2 \arctan (a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right )+\frac {3}{2} i a^2 c^2 \operatorname {PolyLog}\left (4,1-\frac {2}{1+i a x}\right )-\frac {3}{2} i a^2 c^2 \operatorname {PolyLog}\left (4,-1+\frac {2}{1+i a x}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.818, Rules used = {5068, 4946, 5038, 5044, 4988, 2497, 5004, 4942, 5108, 5114, 5118, 6745, 5036, 4930, 5040, 4964, 2449, 2352} \[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^3}{x^3} \, dx=\frac {1}{2} a^4 c^2 x^2 \arctan (a x)^3-\frac {3}{2} a^3 c^2 x \arctan (a x)^2+4 a^2 c^2 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-3 i a^2 c^2 \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )+3 i a^2 c^2 \arctan (a x)^2 \operatorname {PolyLog}\left (2,\frac {2}{i a x+1}-1\right )-3 a^2 c^2 \arctan (a x) \operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right )+3 a^2 c^2 \arctan (a x) \operatorname {PolyLog}\left (3,\frac {2}{i a x+1}-1\right )-3 i a^2 c^2 \arctan (a x)^2-3 a^2 c^2 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )+3 a^2 c^2 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {3}{2} i a^2 c^2 \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )-\frac {3}{2} i a^2 c^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )+\frac {3}{2} i a^2 c^2 \operatorname {PolyLog}\left (4,1-\frac {2}{i a x+1}\right )-\frac {3}{2} i a^2 c^2 \operatorname {PolyLog}\left (4,\frac {2}{i a x+1}-1\right )-\frac {c^2 \arctan (a x)^3}{2 x^2}-\frac {3 a c^2 \arctan (a x)^2}{2 x} \]

[In]

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

[Out]

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

Rule 2352

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

Rule 2449

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 2497

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 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 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 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 5118

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

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 302, normalized size of antiderivative = 0.76 \[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^3}{x^3} \, dx=\frac {1}{32} a^2 c^2 \left (-i \pi ^4-\frac {48 \arctan (a x)^2}{a x}-48 a x \arctan (a x)^2-\frac {16 \arctan (a x)^3}{a^2 x^2}+16 a^2 x^2 \arctan (a x)^3+32 i \arctan (a x)^4+64 \arctan (a x)^3 \log \left (1-e^{-2 i \arctan (a x)}\right )+96 \arctan (a x) \log \left (1-e^{2 i \arctan (a x)}\right )-96 \arctan (a x) \log \left (1+e^{2 i \arctan (a x)}\right )-64 \arctan (a x)^3 \log \left (1+e^{2 i \arctan (a x)}\right )+96 i \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )+48 i \left (1+2 \arctan (a x)^2\right ) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )-48 i \operatorname {PolyLog}\left (2,e^{2 i \arctan (a x)}\right )+96 \arctan (a x) \operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )-96 \arctan (a x) \operatorname {PolyLog}\left (3,-e^{2 i \arctan (a x)}\right )-48 i \operatorname {PolyLog}\left (4,e^{-2 i \arctan (a x)}\right )-48 i \operatorname {PolyLog}\left (4,-e^{2 i \arctan (a x)}\right )\right ) \]

[In]

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

[Out]

(a^2*c^2*((-I)*Pi^4 - (48*ArcTan[a*x]^2)/(a*x) - 48*a*x*ArcTan[a*x]^2 - (16*ArcTan[a*x]^3)/(a^2*x^2) + 16*a^2*
x^2*ArcTan[a*x]^3 + (32*I)*ArcTan[a*x]^4 + 64*ArcTan[a*x]^3*Log[1 - E^((-2*I)*ArcTan[a*x])] + 96*ArcTan[a*x]*L
og[1 - E^((2*I)*ArcTan[a*x])] - 96*ArcTan[a*x]*Log[1 + E^((2*I)*ArcTan[a*x])] - 64*ArcTan[a*x]^3*Log[1 + E^((2
*I)*ArcTan[a*x])] + (96*I)*ArcTan[a*x]^2*PolyLog[2, E^((-2*I)*ArcTan[a*x])] + (48*I)*(1 + 2*ArcTan[a*x]^2)*Pol
yLog[2, -E^((2*I)*ArcTan[a*x])] - (48*I)*PolyLog[2, E^((2*I)*ArcTan[a*x])] + 96*ArcTan[a*x]*PolyLog[3, E^((-2*
I)*ArcTan[a*x])] - 96*ArcTan[a*x]*PolyLog[3, -E^((2*I)*ArcTan[a*x])] - (48*I)*PolyLog[4, E^((-2*I)*ArcTan[a*x]
)] - (48*I)*PolyLog[4, -E^((2*I)*ArcTan[a*x])]))/32

Maple [A] (verified)

Time = 32.79 (sec) , antiderivative size = 622, normalized size of antiderivative = 1.56

method	result	size
derivativedivides	\(a^{2} \left (\frac {c^{2} \arctan \left (a x \right )^{2} \left (a^{2} \arctan \left (a x \right ) x^{2}-\arctan \left (a x \right )-3 a x \right ) \left (a x -i\right ) \left (a x +i\right )}{2 a^{2} x^{2}}+2 c^{2} \arctan \left (a x \right )^{3} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+12 i c^{2} \operatorname {polylog}\left (4, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+3 c^{2} \arctan \left (a x \right ) \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )+3 i c^{2} \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )+12 c^{2} \arctan \left (a x \right ) \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-\frac {3 i c^{2} \operatorname {polylog}\left (4, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}-3 c^{2} \arctan \left (a x \right ) \ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )-6 i c^{2} \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-3 c^{2} \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )+12 i c^{2} \operatorname {polylog}\left (4, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+2 c^{2} \arctan \left (a x \right )^{3} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )-3 i c^{2} \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+12 c^{2} \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-6 i c^{2} \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+3 c^{2} \arctan \left (a x \right ) \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-3 i c^{2} \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-2 c^{2} \arctan \left (a x \right )^{3} \ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )+\frac {3 i c^{2} \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}\right )\)	\(622\)
default	\(a^{2} \left (\frac {c^{2} \arctan \left (a x \right )^{2} \left (a^{2} \arctan \left (a x \right ) x^{2}-\arctan \left (a x \right )-3 a x \right ) \left (a x -i\right ) \left (a x +i\right )}{2 a^{2} x^{2}}+2 c^{2} \arctan \left (a x \right )^{3} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+12 i c^{2} \operatorname {polylog}\left (4, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+3 c^{2} \arctan \left (a x \right ) \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )+3 i c^{2} \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )+12 c^{2} \arctan \left (a x \right ) \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-\frac {3 i c^{2} \operatorname {polylog}\left (4, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}-3 c^{2} \arctan \left (a x \right ) \ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )-6 i c^{2} \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-3 c^{2} \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )+12 i c^{2} \operatorname {polylog}\left (4, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+2 c^{2} \arctan \left (a x \right )^{3} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )-3 i c^{2} \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+12 c^{2} \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-6 i c^{2} \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+3 c^{2} \arctan \left (a x \right ) \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-3 i c^{2} \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-2 c^{2} \arctan \left (a x \right )^{3} \ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )+\frac {3 i c^{2} \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}\right )\)	\(622\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

1/64*(12*a^4*c^2*x^2*integrate(4*x*arctan(a*x)^3 + x*arctan(a*x)*log(a^2*x^2 + 1)^2, x) + 8*a^3*c^2*x^2*integr
ate(-1/8*(24*(a^2*x^2 + 1)*a*x*arctan(a*x)^3 - 18*(a^2*x^2 + 1)*a*x*arctan(a*x)*log(a^2*x^2 + 1)^2 + 36*(a^2*x
^2 + 1)*arctan(a*x)^2*log(a^2*x^2 + 1) - 3*(a^2*x^2 + 1)*log(a^2*x^2 + 1)^3 - sqrt(a^2*x^2 + 1)*(12*sqrt(a^2*x
^2 + 1)*arctan(a*x)^2*log(a^2*x^2 + 1) - sqrt(a^2*x^2 + 1)*log(a^2*x^2 + 1)^3 - (12*(a^2*x^2 + 1)^2*arctan(a*x
)^2*log(a^2*x^2 + 1) - (a^2*x^2 + 1)^2*log(a^2*x^2 + 1)^3)*cos(3*arctan(a*x)) + 3*(12*(a^2*x^2 + 1)^(3/2)*arct
an(a*x)^2*log(a^2*x^2 + 1) - (a^2*x^2 + 1)^(3/2)*log(a^2*x^2 + 1)^3)*cos(2*arctan(a*x)) - 2*(4*(a^2*x^2 + 1)^2
*arctan(a*x)^3 - 3*(a^2*x^2 + 1)^2*arctan(a*x)*log(a^2*x^2 + 1)^2)*sin(3*arctan(a*x)) + 6*(4*(a^2*x^2 + 1)^(3/
2)*arctan(a*x)^3 - 3*(a^2*x^2 + 1)^(3/2)*arctan(a*x)*log(a^2*x^2 + 1)^2)*sin(2*arctan(a*x))))/((a^2*x^2 + 1)^4
*cos(3*arctan(a*x))^2 + (a^2*x^2 + 1)^4*sin(3*arctan(a*x))^2 - 6*(a^2*x^2 + 1)^(7/2)*sin(3*arctan(a*x))*sin(2*
arctan(a*x)) + 9*(a^2*x^2 + 1)^3*cos(2*arctan(a*x))^2 + 9*(a^2*x^2 + 1)^3*sin(2*arctan(a*x))^2 + a^2*x^2 + 6*(
a^2*x^2 + 1)^2*cos(2*arctan(a*x)) + 9*(a^2*x^2 + 1)^2 - 2*(3*(a^2*x^2 + 1)^(7/2)*cos(2*arctan(a*x)) + (a^2*x^2
+ 1)^(5/2))*cos(3*arctan(a*x)) + 6*((a^2*x^2 + 1)^2*a*x*sin(3*arctan(a*x)) - 3*(a^2*x^2 + 1)^(3/2)*a*x*sin(2*
arctan(a*x)) + (a^2*x^2 + 1)^2*cos(3*arctan(a*x)) - 3*(a^2*x^2 + 1)^(3/2)*cos(2*arctan(a*x)) - sqrt(a^2*x^2 +
1))*sqrt(a^2*x^2 + 1) + 1), x) - 12*a^3*c^2*x^2*integrate(1/4*(8*(a^2*x^2 + 1)*a*x*arctan(a*x)*log(a^2*x^2 + 1
) - 8*(a^2*x^2 + 1)*arctan(a*x)^2 + 2*(a^2*x^2 + 1)*log(a^2*x^2 + 1)^2 - (4*(a^2*x^2 + 1)^(3/2)*arctan(a*x)*lo
g(a^2*x^2 + 1)*sin(2*arctan(a*x)) - 4*sqrt(a^2*x^2 + 1)*arctan(a*x)^2 + sqrt(a^2*x^2 + 1)*log(a^2*x^2 + 1)^2 -
(4*(a^2*x^2 + 1)^(3/2)*arctan(a*x)^2 - (a^2*x^2 + 1)^(3/2)*log(a^2*x^2 + 1)^2)*cos(2*arctan(a*x)))*sqrt(a^2*x
^2 + 1))/((a^2*x^2 + 1)^3*cos(2*arctan(a*x))^2 + (a^2*x^2 + 1)^3*sin(2*arctan(a*x))^2 + a^2*x^2 + 2*(a^2*x^2 +
1)^2*cos(2*arctan(a*x)) + 4*(a^2*x^2 + 1)^2 - 4*((a^2*x^2 + 1)^(3/2)*a*x*sin(2*arctan(a*x)) + (a^2*x^2 + 1)^(
3/2)*cos(2*arctan(a*x)) + sqrt(a^2*x^2 + 1))*sqrt(a^2*x^2 + 1) + 1), x) - 8*a^3*c^2*x^2*integrate(1/8*((8*(a^2
*x^2 + 1)*a*x*arctan(a*x)^3 - 6*(a^2*x^2 + 1)*a*x*arctan(a*x)*log(a^2*x^2 + 1)^2 + 12*(a^2*x^2 + 1)*arctan(a*x
)^2*log(a^2*x^2 + 1) - (a^2*x^2 + 1)*log(a^2*x^2 + 1)^3)*cos(2*arctan(a*x)) + (12*(a^2*x^2 + 1)*a*x*arctan(a*x
)^2*log(a^2*x^2 + 1) - (a^2*x^2 + 1)*a*x*log(a^2*x^2 + 1)^3 - 8*(a^2*x^2 + 1)*arctan(a*x)^3 + 6*(a^2*x^2 + 1)*
arctan(a*x)*log(a^2*x^2 + 1)^2)*sin(2*arctan(a*x)) - sqrt(a^2*x^2 + 1)*(12*sqrt(a^2*x^2 + 1)*arctan(a*x)^2*log
(a^2*x^2 + 1) - sqrt(a^2*x^2 + 1)*log(a^2*x^2 + 1)^3))/(a^2*x^2 + 1), x) + 12*a^3*c^2*x^2*integrate(-1/4*((4*(
a^2*x^2 + 1)*a*x*arctan(a*x)*log(a^2*x^2 + 1) - 4*(a^2*x^2 + 1)*arctan(a*x)^2 + (a^2*x^2 + 1)*log(a^2*x^2 + 1)
^2)*cos(2*arctan(a*x)) - (4*(a^2*x^2 + 1)*a*x*arctan(a*x)^2 - (a^2*x^2 + 1)*a*x*log(a^2*x^2 + 1)^2 + 4*(a^2*x^
2 + 1)*arctan(a*x)*log(a^2*x^2 + 1))*sin(2*arctan(a*x)) + sqrt(a^2*x^2 + 1)*(4*sqrt(a^2*x^2 + 1)*arctan(a*x)^2
- sqrt(a^2*x^2 + 1)*log(a^2*x^2 + 1)^2))/(a^2*x^2 + 1), x) - 6*a^2*c^2*x^2*(I*conjugate(gamma(3, -log(-I*a*x
+ 1))) - I*gamma(3, -log(-I*a*x + 1))) - 12*a^2*c^2*x^2*integrate(arctan(a*x)*log(a^2*x^2 + 1)/x, x) + 24*a^2*
c^2*x^2*integrate((4*arctan(a*x)^3 + arctan(a*x)*log(a^2*x^2 + 1)^2)/x, x) - 4*a^2*c^2*x^2*integrate(-(4*arcta
n(a*x)^3 - 3*arctan(a*x)*log(a^2*x^2 + 1)^2)/x, x) + 12*c^2*x^2*integrate((4*arctan(a*x)^3 + arctan(a*x)*log(a
^2*x^2 + 1)^2)/x^3, x) + 4*(a^4*c^2*x^4 - c^2)*arctan(a*x)^3 - 3*(a^4*c^2*x^4 - c^2)*arctan(a*x)*log(a^2*x^2 +
1)^2)/x^2

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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