\(\int \frac {\arctan (a x)^2}{x^4 (c+a^2 c x^2)^3} \, dx\) [306]

Optimal result

Integrand size = 22, antiderivative size = 317 \[ \int \frac {\arctan (a x)^2}{x^4 \left (c+a^2 c x^2\right )^3} \, dx=-\frac {a^2}{3 c^3 x}-\frac {a^4 x}{32 c^3 \left (1+a^2 x^2\right )^2}-\frac {47 a^4 x}{64 c^3 \left (1+a^2 x^2\right )}-\frac {205 a^3 \arctan (a x)}{192 c^3}-\frac {a \arctan (a x)}{3 c^3 x^2}+\frac {a^3 \arctan (a x)}{8 c^3 \left (1+a^2 x^2\right )^2}+\frac {11 a^3 \arctan (a x)}{8 c^3 \left (1+a^2 x^2\right )}+\frac {10 i a^3 \arctan (a x)^2}{3 c^3}-\frac {\arctan (a x)^2}{3 c^3 x^3}+\frac {3 a^2 \arctan (a x)^2}{c^3 x}+\frac {a^4 x \arctan (a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {11 a^4 x \arctan (a x)^2}{8 c^3 \left (1+a^2 x^2\right )}+\frac {35 a^3 \arctan (a x)^3}{24 c^3}-\frac {20 a^3 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{3 c^3}+\frac {10 i a^3 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{3 c^3} \]

[Out]

-1/3*a^2/c^3/x-1/32*a^4*x/c^3/(a^2*x^2+1)^2-47/64*a^4*x/c^3/(a^2*x^2+1)-205/192*a^3*arctan(a*x)/c^3-1/3*a*arct
an(a*x)/c^3/x^2+1/8*a^3*arctan(a*x)/c^3/(a^2*x^2+1)^2+11/8*a^3*arctan(a*x)/c^3/(a^2*x^2+1)+10/3*I*a^3*arctan(a
*x)^2/c^3-1/3*arctan(a*x)^2/c^3/x^3+3*a^2*arctan(a*x)^2/c^3/x+1/4*a^4*x*arctan(a*x)^2/c^3/(a^2*x^2+1)^2+11/8*a
^4*x*arctan(a*x)^2/c^3/(a^2*x^2+1)+35/24*a^3*arctan(a*x)^3/c^3-20/3*a^3*arctan(a*x)*ln(2-2/(1-I*a*x))/c^3+10/3
*I*a^3*polylog(2,-1+2/(1-I*a*x))/c^3

Rubi [A] (verified)

Time = 1.15 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.00, number of steps used = 48, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {5086, 5038, 4946, 331, 209, 5044, 4988, 2497, 5004, 5012, 5050, 205, 211, 5020} \[ \int \frac {\arctan (a x)^2}{x^4 \left (c+a^2 c x^2\right )^3} \, dx=\frac {35 a^3 \arctan (a x)^3}{24 c^3}+\frac {10 i a^3 \arctan (a x)^2}{3 c^3}-\frac {205 a^3 \arctan (a x)}{192 c^3}-\frac {20 a^3 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{3 c^3}+\frac {10 i a^3 \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{3 c^3}+\frac {3 a^2 \arctan (a x)^2}{c^3 x}-\frac {a^2}{3 c^3 x}+\frac {11 a^4 x \arctan (a x)^2}{8 c^3 \left (a^2 x^2+1\right )}+\frac {a^4 x \arctan (a x)^2}{4 c^3 \left (a^2 x^2+1\right )^2}-\frac {47 a^4 x}{64 c^3 \left (a^2 x^2+1\right )}-\frac {a^4 x}{32 c^3 \left (a^2 x^2+1\right )^2}+\frac {11 a^3 \arctan (a x)}{8 c^3 \left (a^2 x^2+1\right )}+\frac {a^3 \arctan (a x)}{8 c^3 \left (a^2 x^2+1\right )^2}-\frac {\arctan (a x)^2}{3 c^3 x^3}-\frac {a \arctan (a x)}{3 c^3 x^2} \]

[In]

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

[Out]

-1/3*a^2/(c^3*x) - (a^4*x)/(32*c^3*(1 + a^2*x^2)^2) - (47*a^4*x)/(64*c^3*(1 + a^2*x^2)) - (205*a^3*ArcTan[a*x]
)/(192*c^3) - (a*ArcTan[a*x])/(3*c^3*x^2) + (a^3*ArcTan[a*x])/(8*c^3*(1 + a^2*x^2)^2) + (11*a^3*ArcTan[a*x])/(
8*c^3*(1 + a^2*x^2)) + (((10*I)/3)*a^3*ArcTan[a*x]^2)/c^3 - ArcTan[a*x]^2/(3*c^3*x^3) + (3*a^2*ArcTan[a*x]^2)/
(c^3*x) + (a^4*x*ArcTan[a*x]^2)/(4*c^3*(1 + a^2*x^2)^2) + (11*a^4*x*ArcTan[a*x]^2)/(8*c^3*(1 + a^2*x^2)) + (35
*a^3*ArcTan[a*x]^3)/(24*c^3) - (20*a^3*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)])/(3*c^3) + (((10*I)/3)*a^3*PolyLog[2
, -1 + 2/(1 - I*a*x)])/c^3

Rule 205

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

Rule 209

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

Rule 211

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

Rule 331

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

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 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 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 5012

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

Rule 5020

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Simp[b*p*(d + e*x^2)^(q +
1)*((a + b*ArcTan[c*x])^(p - 1)/(4*c*d*(q + 1)^2)), x] + (Dist[(2*q + 3)/(2*d*(q + 1)), Int[(d + e*x^2)^(q +
1)*(a + b*ArcTan[c*x])^p, x], x] - Dist[b^2*p*((p - 1)/(4*(q + 1)^2)), Int[(d + e*x^2)^q*(a + b*ArcTan[c*x])^(
p - 2), x], x] - Simp[x*(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^p/(2*d*(q + 1))), x]) /; FreeQ[{a, b, c, d, e
}, x] && EqQ[e, c^2*d] && LtQ[q, -1] && GtQ[p, 1] && NeQ[q, -3/2]

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 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 5050

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[(d + e*x^2)^(
q + 1)*((a + b*ArcTan[c*x])^p/(2*e*(q + 1))), x] - Dist[b*(p/(2*c*(q + 1))), Int[(d + e*x^2)^q*(a + b*ArcTan[c
*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && NeQ[q, -1]

Rule 5086

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Dist[1/d, Int[
x^m*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p, x], x] - Dist[e/d, Int[x^(m + 2)*(d + e*x^2)^q*(a + b*ArcTan[c*
x])^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IntegersQ[p, 2*q] && LtQ[q, -1] && ILtQ[m, 0] &
& NeQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = -\left (a^2 \int \frac {\arctan (a x)^2}{x^2 \left (c+a^2 c x^2\right )^3} \, dx\right )+\frac {\int \frac {\arctan (a x)^2}{x^4 \left (c+a^2 c x^2\right )^2} \, dx}{c} \\ & = a^4 \int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^3} \, dx+\frac {\int \frac {\arctan (a x)^2}{x^4 \left (c+a^2 c x^2\right )} \, dx}{c^2}-2 \frac {a^2 \int \frac {\arctan (a x)^2}{x^2 \left (c+a^2 c x^2\right )^2} \, dx}{c} \\ & = \frac {a^3 \arctan (a x)}{8 c^3 \left (1+a^2 x^2\right )^2}+\frac {a^4 x \arctan (a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac {1}{8} a^4 \int \frac {1}{\left (c+a^2 c x^2\right )^3} \, dx+\frac {\int \frac {\arctan (a x)^2}{x^4} \, dx}{c^3}-\frac {a^2 \int \frac {\arctan (a x)^2}{x^2 \left (c+a^2 c x^2\right )} \, dx}{c^2}+\frac {\left (3 a^4\right ) \int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx}{4 c}-2 \left (\frac {a^2 \int \frac {\arctan (a x)^2}{x^2 \left (c+a^2 c x^2\right )} \, dx}{c^2}-\frac {a^4 \int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx}{c}\right ) \\ & = -\frac {a^4 x}{32 c^3 \left (1+a^2 x^2\right )^2}+\frac {a^3 \arctan (a x)}{8 c^3 \left (1+a^2 x^2\right )^2}-\frac {\arctan (a x)^2}{3 c^3 x^3}+\frac {a^4 x \arctan (a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 a^4 x \arctan (a x)^2}{8 c^3 \left (1+a^2 x^2\right )}+\frac {a^3 \arctan (a x)^3}{8 c^3}+\frac {(2 a) \int \frac {\arctan (a x)}{x^3 \left (1+a^2 x^2\right )} \, dx}{3 c^3}-\frac {a^2 \int \frac {\arctan (a x)^2}{x^2} \, dx}{c^3}+\frac {a^4 \int \frac {\arctan (a x)^2}{c+a^2 c x^2} \, dx}{c^2}-\frac {\left (3 a^4\right ) \int \frac {1}{\left (c+a^2 c x^2\right )^2} \, dx}{32 c}-\frac {\left (3 a^5\right ) \int \frac {x \arctan (a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{4 c}-2 \left (-\frac {a^4 x \arctan (a x)^2}{2 c^3 \left (1+a^2 x^2\right )}-\frac {a^3 \arctan (a x)^3}{6 c^3}+\frac {a^2 \int \frac {\arctan (a x)^2}{x^2} \, dx}{c^3}-\frac {a^4 \int \frac {\arctan (a x)^2}{c+a^2 c x^2} \, dx}{c^2}+\frac {a^5 \int \frac {x \arctan (a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{c}\right ) \\ & = -\frac {a^4 x}{32 c^3 \left (1+a^2 x^2\right )^2}-\frac {3 a^4 x}{64 c^3 \left (1+a^2 x^2\right )}+\frac {a^3 \arctan (a x)}{8 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 a^3 \arctan (a x)}{8 c^3 \left (1+a^2 x^2\right )}-\frac {\arctan (a x)^2}{3 c^3 x^3}+\frac {a^2 \arctan (a x)^2}{c^3 x}+\frac {a^4 x \arctan (a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 a^4 x \arctan (a x)^2}{8 c^3 \left (1+a^2 x^2\right )}+\frac {11 a^3 \arctan (a x)^3}{24 c^3}+\frac {(2 a) \int \frac {\arctan (a x)}{x^3} \, dx}{3 c^3}-\frac {\left (2 a^3\right ) \int \frac {\arctan (a x)}{x \left (1+a^2 x^2\right )} \, dx}{3 c^3}-\frac {\left (2 a^3\right ) \int \frac {\arctan (a x)}{x \left (1+a^2 x^2\right )} \, dx}{c^3}-\frac {\left (3 a^4\right ) \int \frac {1}{c+a^2 c x^2} \, dx}{64 c^2}-\frac {\left (3 a^4\right ) \int \frac {1}{\left (c+a^2 c x^2\right )^2} \, dx}{8 c}-2 \left (-\frac {a^3 \arctan (a x)}{2 c^3 \left (1+a^2 x^2\right )}-\frac {a^2 \arctan (a x)^2}{c^3 x}-\frac {a^4 x \arctan (a x)^2}{2 c^3 \left (1+a^2 x^2\right )}-\frac {a^3 \arctan (a x)^3}{2 c^3}+\frac {\left (2 a^3\right ) \int \frac {\arctan (a x)}{x \left (1+a^2 x^2\right )} \, dx}{c^3}+\frac {a^4 \int \frac {1}{\left (c+a^2 c x^2\right )^2} \, dx}{2 c}\right ) \\ & = -\frac {a^4 x}{32 c^3 \left (1+a^2 x^2\right )^2}-\frac {15 a^4 x}{64 c^3 \left (1+a^2 x^2\right )}-\frac {3 a^3 \arctan (a x)}{64 c^3}-\frac {a \arctan (a x)}{3 c^3 x^2}+\frac {a^3 \arctan (a x)}{8 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 a^3 \arctan (a x)}{8 c^3 \left (1+a^2 x^2\right )}+\frac {4 i a^3 \arctan (a x)^2}{3 c^3}-\frac {\arctan (a x)^2}{3 c^3 x^3}+\frac {a^2 \arctan (a x)^2}{c^3 x}+\frac {a^4 x \arctan (a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 a^4 x \arctan (a x)^2}{8 c^3 \left (1+a^2 x^2\right )}+\frac {11 a^3 \arctan (a x)^3}{24 c^3}+\frac {a^2 \int \frac {1}{x^2 \left (1+a^2 x^2\right )} \, dx}{3 c^3}-\frac {\left (2 i a^3\right ) \int \frac {\arctan (a x)}{x (i+a x)} \, dx}{3 c^3}-\frac {\left (2 i a^3\right ) \int \frac {\arctan (a x)}{x (i+a x)} \, dx}{c^3}-\frac {\left (3 a^4\right ) \int \frac {1}{c+a^2 c x^2} \, dx}{16 c^2}-2 \left (\frac {a^4 x}{4 c^3 \left (1+a^2 x^2\right )}-\frac {a^3 \arctan (a x)}{2 c^3 \left (1+a^2 x^2\right )}-\frac {i a^3 \arctan (a x)^2}{c^3}-\frac {a^2 \arctan (a x)^2}{c^3 x}-\frac {a^4 x \arctan (a x)^2}{2 c^3 \left (1+a^2 x^2\right )}-\frac {a^3 \arctan (a x)^3}{2 c^3}+\frac {\left (2 i a^3\right ) \int \frac {\arctan (a x)}{x (i+a x)} \, dx}{c^3}+\frac {a^4 \int \frac {1}{c+a^2 c x^2} \, dx}{4 c^2}\right ) \\ & = -\frac {a^2}{3 c^3 x}-\frac {a^4 x}{32 c^3 \left (1+a^2 x^2\right )^2}-\frac {15 a^4 x}{64 c^3 \left (1+a^2 x^2\right )}-\frac {15 a^3 \arctan (a x)}{64 c^3}-\frac {a \arctan (a x)}{3 c^3 x^2}+\frac {a^3 \arctan (a x)}{8 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 a^3 \arctan (a x)}{8 c^3 \left (1+a^2 x^2\right )}+\frac {4 i a^3 \arctan (a x)^2}{3 c^3}-\frac {\arctan (a x)^2}{3 c^3 x^3}+\frac {a^2 \arctan (a x)^2}{c^3 x}+\frac {a^4 x \arctan (a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 a^4 x \arctan (a x)^2}{8 c^3 \left (1+a^2 x^2\right )}+\frac {11 a^3 \arctan (a x)^3}{24 c^3}-\frac {8 a^3 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{3 c^3}-\frac {a^4 \int \frac {1}{1+a^2 x^2} \, dx}{3 c^3}+\frac {\left (2 a^4\right ) \int \frac {\log \left (2-\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{3 c^3}-2 \left (\frac {a^4 x}{4 c^3 \left (1+a^2 x^2\right )}+\frac {a^3 \arctan (a x)}{4 c^3}-\frac {a^3 \arctan (a x)}{2 c^3 \left (1+a^2 x^2\right )}-\frac {i a^3 \arctan (a x)^2}{c^3}-\frac {a^2 \arctan (a x)^2}{c^3 x}-\frac {a^4 x \arctan (a x)^2}{2 c^3 \left (1+a^2 x^2\right )}-\frac {a^3 \arctan (a x)^3}{2 c^3}+\frac {2 a^3 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{c^3}-\frac {\left (2 a^4\right ) \int \frac {\log \left (2-\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c^3}\right )+\frac {\left (2 a^4\right ) \int \frac {\log \left (2-\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c^3} \\ & = -\frac {a^2}{3 c^3 x}-\frac {a^4 x}{32 c^3 \left (1+a^2 x^2\right )^2}-\frac {15 a^4 x}{64 c^3 \left (1+a^2 x^2\right )}-\frac {109 a^3 \arctan (a x)}{192 c^3}-\frac {a \arctan (a x)}{3 c^3 x^2}+\frac {a^3 \arctan (a x)}{8 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 a^3 \arctan (a x)}{8 c^3 \left (1+a^2 x^2\right )}+\frac {4 i a^3 \arctan (a x)^2}{3 c^3}-\frac {\arctan (a x)^2}{3 c^3 x^3}+\frac {a^2 \arctan (a x)^2}{c^3 x}+\frac {a^4 x \arctan (a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 a^4 x \arctan (a x)^2}{8 c^3 \left (1+a^2 x^2\right )}+\frac {11 a^3 \arctan (a x)^3}{24 c^3}-\frac {8 a^3 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{3 c^3}+\frac {4 i a^3 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{3 c^3}-2 \left (\frac {a^4 x}{4 c^3 \left (1+a^2 x^2\right )}+\frac {a^3 \arctan (a x)}{4 c^3}-\frac {a^3 \arctan (a x)}{2 c^3 \left (1+a^2 x^2\right )}-\frac {i a^3 \arctan (a x)^2}{c^3}-\frac {a^2 \arctan (a x)^2}{c^3 x}-\frac {a^4 x \arctan (a x)^2}{2 c^3 \left (1+a^2 x^2\right )}-\frac {a^3 \arctan (a x)^3}{2 c^3}+\frac {2 a^3 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{c^3}-\frac {i a^3 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{c^3}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.71 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.60 \[ \int \frac {\arctan (a x)^2}{x^4 \left (c+a^2 c x^2\right )^3} \, dx=\frac {a^3 \left (-\frac {256 \left (1+a^2 x^2\right ) \arctan (a x)}{a^2 x^2}-\frac {256 \left (1+a^2 x^2\right ) \arctan (a x)^2}{a^3 x^3}+1120 \arctan (a x)^3+\frac {256 \left (-1+10 \arctan (a x)^2\right )}{a x}+576 \arctan (a x) \cos (2 \arctan (a x))+12 \arctan (a x) \cos (4 \arctan (a x))-5120 \arctan (a x) \log \left (1-e^{2 i \arctan (a x)}\right )+2560 i \left (\arctan (a x)^2+\operatorname {PolyLog}\left (2,e^{2 i \arctan (a x)}\right )\right )+288 \left (-1+2 \arctan (a x)^2\right ) \sin (2 \arctan (a x))+3 \left (-1+8 \arctan (a x)^2\right ) \sin (4 \arctan (a x))\right )}{768 c^3} \]

[In]

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

[Out]

(a^3*((-256*(1 + a^2*x^2)*ArcTan[a*x])/(a^2*x^2) - (256*(1 + a^2*x^2)*ArcTan[a*x]^2)/(a^3*x^3) + 1120*ArcTan[a
*x]^3 + (256*(-1 + 10*ArcTan[a*x]^2))/(a*x) + 576*ArcTan[a*x]*Cos[2*ArcTan[a*x]] + 12*ArcTan[a*x]*Cos[4*ArcTan
[a*x]] - 5120*ArcTan[a*x]*Log[1 - E^((2*I)*ArcTan[a*x])] + (2560*I)*(ArcTan[a*x]^2 + PolyLog[2, E^((2*I)*ArcTa
n[a*x])]) + 288*(-1 + 2*ArcTan[a*x]^2)*Sin[2*ArcTan[a*x]] + 3*(-1 + 8*ArcTan[a*x]^2)*Sin[4*ArcTan[a*x]]))/(768
*c^3)

Maple [A] (verified)

Time = 1.79 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.29

method	result	size
derivativedivides	\(a^{3} \left (-\frac {\arctan \left (a x \right )^{2}}{3 c^{3} a^{3} x^{3}}+\frac {3 \arctan \left (a x \right )^{2}}{c^{3} a x}+\frac {11 \arctan \left (a x \right )^{2} a^{3} x^{3}}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {13 a x \arctan \left (a x \right )^{2}}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {35 \arctan \left (a x \right )^{3}}{8 c^{3}}-\frac {\frac {4 \arctan \left (a x \right )}{a^{2} x^{2}}+80 \arctan \left (a x \right ) \ln \left (a x \right )-40 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )-\frac {3 \arctan \left (a x \right )}{2 \left (a^{2} x^{2}+1\right )^{2}}-\frac {33 \arctan \left (a x \right )}{2 \left (a^{2} x^{2}+1\right )}+40 i \ln \left (a x \right ) \ln \left (i a x +1\right )-40 i \ln \left (a x \right ) \ln \left (-i a x +1\right )+40 i \operatorname {dilog}\left (i a x +1\right )-40 i \operatorname {dilog}\left (-i a x +1\right )-20 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )+20 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )+\frac {4}{a x}+\frac {\frac {141}{8} a^{3} x^{3}+\frac {147}{8} a x}{2 \left (a^{2} x^{2}+1\right )^{2}}+\frac {205 \arctan \left (a x \right )}{16}+35 \arctan \left (a x \right )^{3}}{12 c^{3}}\right )\)	\(409\)
default	\(a^{3} \left (-\frac {\arctan \left (a x \right )^{2}}{3 c^{3} a^{3} x^{3}}+\frac {3 \arctan \left (a x \right )^{2}}{c^{3} a x}+\frac {11 \arctan \left (a x \right )^{2} a^{3} x^{3}}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {13 a x \arctan \left (a x \right )^{2}}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {35 \arctan \left (a x \right )^{3}}{8 c^{3}}-\frac {\frac {4 \arctan \left (a x \right )}{a^{2} x^{2}}+80 \arctan \left (a x \right ) \ln \left (a x \right )-40 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )-\frac {3 \arctan \left (a x \right )}{2 \left (a^{2} x^{2}+1\right )^{2}}-\frac {33 \arctan \left (a x \right )}{2 \left (a^{2} x^{2}+1\right )}+40 i \ln \left (a x \right ) \ln \left (i a x +1\right )-40 i \ln \left (a x \right ) \ln \left (-i a x +1\right )+40 i \operatorname {dilog}\left (i a x +1\right )-40 i \operatorname {dilog}\left (-i a x +1\right )-20 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )+20 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )+\frac {4}{a x}+\frac {\frac {141}{8} a^{3} x^{3}+\frac {147}{8} a x}{2 \left (a^{2} x^{2}+1\right )^{2}}+\frac {205 \arctan \left (a x \right )}{16}+35 \arctan \left (a x \right )^{3}}{12 c^{3}}\right )\)	\(409\)
parts	\(\frac {11 \arctan \left (a x \right )^{2} a^{6} x^{3}}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {13 a^{4} x \arctan \left (a x \right )^{2}}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {35 a^{3} \arctan \left (a x \right )^{3}}{8 c^{3}}-\frac {\arctan \left (a x \right )^{2}}{3 c^{3} x^{3}}+\frac {3 a^{2} \arctan \left (a x \right )^{2}}{c^{3} x}-\frac {2 \left (\frac {35 a^{3} \arctan \left (a x \right )^{3}}{8}+\frac {a^{3} \left (\frac {4 \arctan \left (a x \right )}{a^{2} x^{2}}+80 \arctan \left (a x \right ) \ln \left (a x \right )-40 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )-\frac {3 \arctan \left (a x \right )}{2 \left (a^{2} x^{2}+1\right )^{2}}-\frac {33 \arctan \left (a x \right )}{2 \left (a^{2} x^{2}+1\right )}+40 i \ln \left (a x \right ) \ln \left (i a x +1\right )-40 i \ln \left (a x \right ) \ln \left (-i a x +1\right )+40 i \operatorname {dilog}\left (i a x +1\right )-40 i \operatorname {dilog}\left (-i a x +1\right )-20 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )+20 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )+\frac {4}{a x}+\frac {\frac {141}{8} a^{3} x^{3}+\frac {147}{8} a x}{2 \left (a^{2} x^{2}+1\right )^{2}}+\frac {205 \arctan \left (a x \right )}{16}\right )}{8}\right )}{3 c^{3}}\)	\(416\)

[In]

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

[Out]

a^3*(-1/3/c^3*arctan(a*x)^2/a^3/x^3+3/c^3*arctan(a*x)^2/a/x+11/8/c^3*arctan(a*x)^2/(a^2*x^2+1)^2*a^3*x^3+13/8*
a*x*arctan(a*x)^2/c^3/(a^2*x^2+1)^2+35/8*arctan(a*x)^3/c^3-1/12/c^3*(4*arctan(a*x)/a^2/x^2+80*arctan(a*x)*ln(a
*x)-40*arctan(a*x)*ln(a^2*x^2+1)-3/2*arctan(a*x)/(a^2*x^2+1)^2-33/2*arctan(a*x)/(a^2*x^2+1)+40*I*ln(a*x)*ln(1+
I*a*x)-40*I*ln(a*x)*ln(1-I*a*x)+40*I*dilog(1+I*a*x)-40*I*dilog(1-I*a*x)-20*I*(ln(a*x-I)*ln(a^2*x^2+1)-dilog(-1
/2*I*(I+a*x))-ln(a*x-I)*ln(-1/2*I*(I+a*x))-1/2*ln(a*x-I)^2)+20*I*(ln(I+a*x)*ln(a^2*x^2+1)-dilog(1/2*I*(a*x-I))
-ln(I+a*x)*ln(1/2*I*(a*x-I))-1/2*ln(I+a*x)^2)+4/a/x+1/2*(141/8*a^3*x^3+147/8*a*x)/(a^2*x^2+1)^2+205/16*arctan(
a*x)+35*arctan(a*x)^3))

Fricas [F]

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

[In]

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

[Out]

integral(arctan(a*x)^2/(a^6*c^3*x^10 + 3*a^4*c^3*x^8 + 3*a^2*c^3*x^6 + c^3*x^4), x)

Sympy [F]

\[ \int \frac {\arctan (a x)^2}{x^4 \left (c+a^2 c x^2\right )^3} \, dx=\frac {\int \frac {\operatorname {atan}^{2}{\left (a x \right )}}{a^{6} x^{10} + 3 a^{4} x^{8} + 3 a^{2} x^{6} + x^{4}}\, dx}{c^{3}} \]

[In]

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

[Out]

Integral(atan(a*x)**2/(a**6*x**10 + 3*a**4*x**8 + 3*a**2*x**6 + x**4), x)/c**3

Maxima [F(-1)]

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

[In]

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

[Out]

Timed out

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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