\(\int \frac {\arctan (a x)^2}{x^3 (c+a^2 c x^2)^{3/2}} \, dx\) [345]

Optimal result

Integrand size = 24, antiderivative size = 422 \[ \int \frac {\arctan (a x)^2}{x^3 \left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {2 a^2}{c \sqrt {c+a^2 c x^2}}+\frac {2 a^3 x \arctan (a x)}{c \sqrt {c+a^2 c x^2}}-\frac {a \sqrt {c+a^2 c x^2} \arctan (a x)}{c^2 x}-\frac {a^2 \arctan (a x)^2}{c \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 c^2 x^2}+\frac {3 a^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}}-\frac {a^2 \text {arctanh}\left (\frac {\sqrt {c+a^2 c x^2}}{\sqrt {c}}\right )}{c^{3/2}}-\frac {3 i a^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}}+\frac {3 i a^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}}+\frac {3 a^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}}-\frac {3 a^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.80 (sec) , antiderivative size = 422, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {5086, 5082, 5064, 272, 65, 214, 5078, 5076, 4268, 2611, 2320, 6724, 5050, 5014} \[ \int \frac {\arctan (a x)^2}{x^3 \left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {3 a^2 \sqrt {a^2 x^2+1} \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{c \sqrt {a^2 c x^2+c}}-\frac {a \arctan (a x) \sqrt {a^2 c x^2+c}}{c^2 x}-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 c^2 x^2}-\frac {3 i a^2 \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{c \sqrt {a^2 c x^2+c}}+\frac {3 i a^2 \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{c \sqrt {a^2 c x^2+c}}+\frac {3 a^2 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{c \sqrt {a^2 c x^2+c}}-\frac {3 a^2 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{c \sqrt {a^2 c x^2+c}}-\frac {a^2 \arctan (a x)^2}{c \sqrt {a^2 c x^2+c}}-\frac {a^2 \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )}{c^{3/2}}+\frac {2 a^2}{c \sqrt {a^2 c x^2+c}}+\frac {2 a^3 x \arctan (a x)}{c \sqrt {a^2 c x^2+c}} \]

[In]

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

[Out]

(2*a^2)/(c*Sqrt[c + a^2*c*x^2]) + (2*a^3*x*ArcTan[a*x])/(c*Sqrt[c + a^2*c*x^2]) - (a*Sqrt[c + a^2*c*x^2]*ArcTa
n[a*x])/(c^2*x) - (a^2*ArcTan[a*x]^2)/(c*Sqrt[c + a^2*c*x^2]) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(2*c^2*x^2
) + (3*a^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a*x])])/(c*Sqrt[c + a^2*c*x^2]) - (a^2*ArcTanh[
Sqrt[c + a^2*c*x^2]/Sqrt[c]])/c^(3/2) - ((3*I)*a^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])
])/(c*Sqrt[c + a^2*c*x^2]) + ((3*I)*a^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])])/(c*Sqrt[c
+ a^2*c*x^2]) + (3*a^2*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])])/(c*Sqrt[c + a^2*c*x^2]) - (3*a^2*Sqr
t[1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])])/(c*Sqrt[c + a^2*c*x^2])

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

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

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 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 4268

Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^(I*(e + f*
x))]/f), x] + (-Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Dist[d*(m/f), Int[(c +
d*x)^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IGtQ[m, 0]

Rule 5014

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[b/(c*d*Sqrt[d + e*x^2]),
x] + Simp[x*((a + b*ArcTan[c*x])/(d*Sqrt[d + e*x^2])), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d]

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 5064

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> Simp
[(f*x)^(m + 1)*(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^p/(d*f*(m + 1))), x] - Dist[b*c*(p/(f*(m + 1))), Int[(
f*x)^(m + 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[e,
c^2*d] && EqQ[m + 2*q + 3, 0] && GtQ[p, 0] && NeQ[m, -1]

Rule 5076

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

Rule 5078

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

Rule 5082

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

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]

Rule 6724

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

Rubi steps \begin{align*} \text {integral}& = -\left (a^2 \int \frac {\arctan (a x)^2}{x \left (c+a^2 c x^2\right )^{3/2}} \, dx\right )+\frac {\int \frac {\arctan (a x)^2}{x^3 \sqrt {c+a^2 c x^2}} \, dx}{c} \\ & = -\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 c^2 x^2}+a^4 \int \frac {x \arctan (a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx+\frac {a \int \frac {\arctan (a x)}{x^2 \sqrt {c+a^2 c x^2}} \, dx}{c}-\frac {a^2 \int \frac {\arctan (a x)^2}{x \sqrt {c+a^2 c x^2}} \, dx}{2 c}-\frac {a^2 \int \frac {\arctan (a x)^2}{x \sqrt {c+a^2 c x^2}} \, dx}{c} \\ & = -\frac {a \sqrt {c+a^2 c x^2} \arctan (a x)}{c^2 x}-\frac {a^2 \arctan (a x)^2}{c \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 c^2 x^2}+\left (2 a^3\right ) \int \frac {\arctan (a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx+\frac {a^2 \int \frac {1}{x \sqrt {c+a^2 c x^2}} \, dx}{c}-\frac {\left (a^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)^2}{x \sqrt {1+a^2 x^2}} \, dx}{2 c \sqrt {c+a^2 c x^2}}-\frac {\left (a^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)^2}{x \sqrt {1+a^2 x^2}} \, dx}{c \sqrt {c+a^2 c x^2}} \\ & = \frac {2 a^2}{c \sqrt {c+a^2 c x^2}}+\frac {2 a^3 x \arctan (a x)}{c \sqrt {c+a^2 c x^2}}-\frac {a \sqrt {c+a^2 c x^2} \arctan (a x)}{c^2 x}-\frac {a^2 \arctan (a x)^2}{c \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 c^2 x^2}+\frac {a^2 \text {Subst}\left (\int \frac {1}{x \sqrt {c+a^2 c x}} \, dx,x,x^2\right )}{2 c}-\frac {\left (a^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \csc (x) \, dx,x,\arctan (a x)\right )}{2 c \sqrt {c+a^2 c x^2}}-\frac {\left (a^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \csc (x) \, dx,x,\arctan (a x)\right )}{c \sqrt {c+a^2 c x^2}} \\ & = \frac {2 a^2}{c \sqrt {c+a^2 c x^2}}+\frac {2 a^3 x \arctan (a x)}{c \sqrt {c+a^2 c x^2}}-\frac {a \sqrt {c+a^2 c x^2} \arctan (a x)}{c^2 x}-\frac {a^2 \arctan (a x)^2}{c \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 c^2 x^2}+\frac {3 a^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}}+\frac {\text {Subst}\left (\int \frac {1}{-\frac {1}{a^2}+\frac {x^2}{a^2 c}} \, dx,x,\sqrt {c+a^2 c x^2}\right )}{c^2}+\frac {\left (a^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1-e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{c \sqrt {c+a^2 c x^2}}-\frac {\left (a^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1+e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{c \sqrt {c+a^2 c x^2}}+\frac {\left (2 a^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1-e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{c \sqrt {c+a^2 c x^2}}-\frac {\left (2 a^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1+e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{c \sqrt {c+a^2 c x^2}} \\ & = \frac {2 a^2}{c \sqrt {c+a^2 c x^2}}+\frac {2 a^3 x \arctan (a x)}{c \sqrt {c+a^2 c x^2}}-\frac {a \sqrt {c+a^2 c x^2} \arctan (a x)}{c^2 x}-\frac {a^2 \arctan (a x)^2}{c \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 c^2 x^2}+\frac {3 a^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}}-\frac {a^2 \text {arctanh}\left (\frac {\sqrt {c+a^2 c x^2}}{\sqrt {c}}\right )}{c^{3/2}}-\frac {3 i a^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}}+\frac {3 i a^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}}+\frac {\left (i a^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{c \sqrt {c+a^2 c x^2}}-\frac {\left (i a^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{c \sqrt {c+a^2 c x^2}}+\frac {\left (2 i a^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{c \sqrt {c+a^2 c x^2}}-\frac {\left (2 i a^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{c \sqrt {c+a^2 c x^2}} \\ & = \frac {2 a^2}{c \sqrt {c+a^2 c x^2}}+\frac {2 a^3 x \arctan (a x)}{c \sqrt {c+a^2 c x^2}}-\frac {a \sqrt {c+a^2 c x^2} \arctan (a x)}{c^2 x}-\frac {a^2 \arctan (a x)^2}{c \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 c^2 x^2}+\frac {3 a^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}}-\frac {a^2 \text {arctanh}\left (\frac {\sqrt {c+a^2 c x^2}}{\sqrt {c}}\right )}{c^{3/2}}-\frac {3 i a^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}}+\frac {3 i a^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}}+\frac {\left (a^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}}-\frac {\left (a^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}}+\frac {\left (2 a^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}}-\frac {\left (2 a^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}} \\ & = \frac {2 a^2}{c \sqrt {c+a^2 c x^2}}+\frac {2 a^3 x \arctan (a x)}{c \sqrt {c+a^2 c x^2}}-\frac {a \sqrt {c+a^2 c x^2} \arctan (a x)}{c^2 x}-\frac {a^2 \arctan (a x)^2}{c \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 c^2 x^2}+\frac {3 a^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}}-\frac {a^2 \text {arctanh}\left (\frac {\sqrt {c+a^2 c x^2}}{\sqrt {c}}\right )}{c^{3/2}}-\frac {3 i a^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}}+\frac {3 i a^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}}+\frac {3 a^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}}-\frac {3 a^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.93 (sec) , antiderivative size = 371, normalized size of antiderivative = 0.88 \[ \int \frac {\arctan (a x)^2}{x^3 \left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {a^2 \left (16+16 a x \arctan (a x)-8 \arctan (a x)^2-2 a x \arctan (a x) \csc ^2\left (\frac {1}{2} \arctan (a x)\right )-\sqrt {1+a^2 x^2} \arctan (a x)^2 \csc ^2\left (\frac {1}{2} \arctan (a x)\right )-12 \sqrt {1+a^2 x^2} \arctan (a x)^2 \log \left (1-e^{i \arctan (a x)}\right )+12 \sqrt {1+a^2 x^2} \arctan (a x)^2 \log \left (1+e^{i \arctan (a x)}\right )+8 \sqrt {1+a^2 x^2} \log \left (\tan \left (\frac {1}{2} \arctan (a x)\right )\right )-24 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )+24 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )+24 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )-24 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )+\sqrt {1+a^2 x^2} \arctan (a x)^2 \sec ^2\left (\frac {1}{2} \arctan (a x)\right )-4 \sqrt {1+a^2 x^2} \arctan (a x) \tan \left (\frac {1}{2} \arctan (a x)\right )\right )}{8 c \sqrt {c+a^2 c x^2}} \]

[In]

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

[Out]

(a^2*(16 + 16*a*x*ArcTan[a*x] - 8*ArcTan[a*x]^2 - 2*a*x*ArcTan[a*x]*Csc[ArcTan[a*x]/2]^2 - Sqrt[1 + a^2*x^2]*A
rcTan[a*x]^2*Csc[ArcTan[a*x]/2]^2 - 12*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*Log[1 - E^(I*ArcTan[a*x])] + 12*Sqrt[1
+ a^2*x^2]*ArcTan[a*x]^2*Log[1 + E^(I*ArcTan[a*x])] + 8*Sqrt[1 + a^2*x^2]*Log[Tan[ArcTan[a*x]/2]] - (24*I)*Sqr
t[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])] + (24*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^
(I*ArcTan[a*x])] + 24*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])] - 24*Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*
ArcTan[a*x])] + Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*Sec[ArcTan[a*x]/2]^2 - 4*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*Tan[Arc
Tan[a*x]/2]))/(8*c*Sqrt[c + a^2*c*x^2])

Maple [A] (verified)

Time = 1.09 (sec) , antiderivative size = 376, normalized size of antiderivative = 0.89

method	result	size
default	\(-\frac {a^{2} \left (\arctan \left (a x \right )^{2}-2+2 i \arctan \left (a x \right )\right ) \left (i a x +1\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 \left (a^{2} x^{2}+1\right ) c^{2}}+\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i a x -1\right ) \left (\arctan \left (a x \right )^{2}-2-2 i \arctan \left (a x \right )\right ) a^{2}}{2 \left (a^{2} x^{2}+1\right ) c^{2}}-\frac {\left (2 a x +\arctan \left (a x \right )\right ) \arctan \left (a x \right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 c^{2} x^{2}}+\frac {a^{2} \left (3 \arctan \left (a x \right )^{2} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )-3 \arctan \left (a x \right )^{2} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-6 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-4 \,\operatorname {arctanh}\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 \operatorname {polylog}\left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-6 \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 \sqrt {a^{2} x^{2}+1}\, c^{2}}\)	\(376\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral(sqrt(a^2*c*x^2 + c)*arctan(a*x)^2/(a^4*c^2*x^7 + 2*a^2*c^2*x^5 + c^2*x^3), x)

Sympy [F]

\[ \int \frac {\arctan (a x)^2}{x^3 \left (c+a^2 c x^2\right )^{3/2}} \, dx=\int \frac {\operatorname {atan}^{2}{\left (a x \right )}}{x^{3} \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \]

[In]

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

[Out]

Integral(atan(a*x)**2/(x**3*(c*(a**2*x**2 + 1))**(3/2)), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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