\(\int \frac {\arctan (a x)^3}{x (c+a^2 c x^2)^{3/2}} \, dx\) [448]

Optimal result

Integrand size = 24, antiderivative size = 443 \[ \int \frac {\arctan (a x)^3}{x \left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {6 a x}{c \sqrt {c+a^2 c x^2}}-\frac {6 \arctan (a x)}{c \sqrt {c+a^2 c x^2}}-\frac {3 a x \arctan (a x)^2}{c \sqrt {c+a^2 c x^2}}+\frac {\arctan (a x)^3}{c \sqrt {c+a^2 c x^2}}-\frac {2 \sqrt {1+a^2 x^2} \arctan (a x)^3 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}}+\frac {3 i \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}}-\frac {3 i \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}}-\frac {6 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}}+\frac {6 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}}-\frac {6 i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,-e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}}+\frac {6 i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 443, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {5086, 5078, 5076, 4268, 2611, 6744, 2320, 6724, 5050, 5018, 197} \[ \int \frac {\arctan (a x)^3}{x \left (c+a^2 c x^2\right )^{3/2}} \, dx=-\frac {2 \sqrt {a^2 x^2+1} \arctan (a x)^3 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{c \sqrt {a^2 c x^2+c}}+\frac {3 i \sqrt {a^2 x^2+1} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{c \sqrt {a^2 c x^2+c}}-\frac {3 i \sqrt {a^2 x^2+1} \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{c \sqrt {a^2 c x^2+c}}-\frac {6 \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{c \sqrt {a^2 c x^2+c}}+\frac {6 \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{c \sqrt {a^2 c x^2+c}}-\frac {6 i \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (4,-e^{i \arctan (a x)}\right )}{c \sqrt {a^2 c x^2+c}}+\frac {6 i \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (4,e^{i \arctan (a x)}\right )}{c \sqrt {a^2 c x^2+c}}+\frac {\arctan (a x)^3}{c \sqrt {a^2 c x^2+c}}-\frac {3 a x \arctan (a x)^2}{c \sqrt {a^2 c x^2+c}}-\frac {6 \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {6 a x}{c \sqrt {a^2 c x^2+c}} \]

[In]

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

[Out]

(6*a*x)/(c*Sqrt[c + a^2*c*x^2]) - (6*ArcTan[a*x])/(c*Sqrt[c + a^2*c*x^2]) - (3*a*x*ArcTan[a*x]^2)/(c*Sqrt[c +
a^2*c*x^2]) + ArcTan[a*x]^3/(c*Sqrt[c + a^2*c*x^2]) - (2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3*ArcTanh[E^(I*ArcTan[a
*x])])/(c*Sqrt[c + a^2*c*x^2]) + ((3*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, -E^(I*ArcTan[a*x])])/(c*Sqr
t[c + a^2*c*x^2]) - ((3*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, E^(I*ArcTan[a*x])])/(c*Sqrt[c + a^2*c*x^
2]) - (6*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, -E^(I*ArcTan[a*x])])/(c*Sqrt[c + a^2*c*x^2]) + (6*Sqrt[1 + a
^2*x^2]*ArcTan[a*x]*PolyLog[3, E^(I*ArcTan[a*x])])/(c*Sqrt[c + a^2*c*x^2]) - ((6*I)*Sqrt[1 + a^2*x^2]*PolyLog[
4, -E^(I*ArcTan[a*x])])/(c*Sqrt[c + a^2*c*x^2]) + ((6*I)*Sqrt[1 + a^2*x^2]*PolyLog[4, E^(I*ArcTan[a*x])])/(c*S
qrt[c + a^2*c*x^2])

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

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 5018

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

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 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 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]

Rule 6744

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

Rubi steps \begin{align*} \text {integral}& = -\left (a^2 \int \frac {x \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx\right )+\frac {\int \frac {\arctan (a x)^3}{x \sqrt {c+a^2 c x^2}} \, dx}{c} \\ & = \frac {\arctan (a x)^3}{c \sqrt {c+a^2 c x^2}}-(3 a) \int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx+\frac {\sqrt {1+a^2 x^2} \int \frac {\arctan (a x)^3}{x \sqrt {1+a^2 x^2}} \, dx}{c \sqrt {c+a^2 c x^2}} \\ & = -\frac {6 \arctan (a x)}{c \sqrt {c+a^2 c x^2}}-\frac {3 a x \arctan (a x)^2}{c \sqrt {c+a^2 c x^2}}+\frac {\arctan (a x)^3}{c \sqrt {c+a^2 c x^2}}+(6 a) \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx+\frac {\sqrt {1+a^2 x^2} \text {Subst}\left (\int x^3 \csc (x) \, dx,x,\arctan (a x)\right )}{c \sqrt {c+a^2 c x^2}} \\ & = \frac {6 a x}{c \sqrt {c+a^2 c x^2}}-\frac {6 \arctan (a x)}{c \sqrt {c+a^2 c x^2}}-\frac {3 a x \arctan (a x)^2}{c \sqrt {c+a^2 c x^2}}+\frac {\arctan (a x)^3}{c \sqrt {c+a^2 c x^2}}-\frac {2 \sqrt {1+a^2 x^2} \arctan (a x)^3 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \log \left (1-e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{c \sqrt {c+a^2 c x^2}}+\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \log \left (1+e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{c \sqrt {c+a^2 c x^2}} \\ & = \frac {6 a x}{c \sqrt {c+a^2 c x^2}}-\frac {6 \arctan (a x)}{c \sqrt {c+a^2 c x^2}}-\frac {3 a x \arctan (a x)^2}{c \sqrt {c+a^2 c x^2}}+\frac {\arctan (a x)^3}{c \sqrt {c+a^2 c x^2}}-\frac {2 \sqrt {1+a^2 x^2} \arctan (a x)^3 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}}+\frac {3 i \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}}-\frac {3 i \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}}-\frac {\left (6 i \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,-e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{c \sqrt {c+a^2 c x^2}}+\frac {\left (6 i \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{c \sqrt {c+a^2 c x^2}} \\ & = \frac {6 a x}{c \sqrt {c+a^2 c x^2}}-\frac {6 \arctan (a x)}{c \sqrt {c+a^2 c x^2}}-\frac {3 a x \arctan (a x)^2}{c \sqrt {c+a^2 c x^2}}+\frac {\arctan (a x)^3}{c \sqrt {c+a^2 c x^2}}-\frac {2 \sqrt {1+a^2 x^2} \arctan (a x)^3 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}}+\frac {3 i \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}}-\frac {3 i \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}}-\frac {6 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}}+\frac {6 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}}+\frac {\left (6 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,-e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{c \sqrt {c+a^2 c x^2}}-\frac {\left (6 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{c \sqrt {c+a^2 c x^2}} \\ & = \frac {6 a x}{c \sqrt {c+a^2 c x^2}}-\frac {6 \arctan (a x)}{c \sqrt {c+a^2 c x^2}}-\frac {3 a x \arctan (a x)^2}{c \sqrt {c+a^2 c x^2}}+\frac {\arctan (a x)^3}{c \sqrt {c+a^2 c x^2}}-\frac {2 \sqrt {1+a^2 x^2} \arctan (a x)^3 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}}+\frac {3 i \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}}-\frac {3 i \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}}-\frac {6 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}}+\frac {6 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}}-\frac {\left (6 i \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-x)}{x} \, dx,x,e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}}+\frac {\left (6 i \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,x)}{x} \, dx,x,e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}} \\ & = \frac {6 a x}{c \sqrt {c+a^2 c x^2}}-\frac {6 \arctan (a x)}{c \sqrt {c+a^2 c x^2}}-\frac {3 a x \arctan (a x)^2}{c \sqrt {c+a^2 c x^2}}+\frac {\arctan (a x)^3}{c \sqrt {c+a^2 c x^2}}-\frac {2 \sqrt {1+a^2 x^2} \arctan (a x)^3 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}}+\frac {3 i \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}}-\frac {3 i \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}}-\frac {6 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}}+\frac {6 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}}-\frac {6 i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,-e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}}+\frac {6 i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 295, normalized size of antiderivative = 0.67 \[ \int \frac {\arctan (a x)^3}{x \left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {\sqrt {1+a^2 x^2} \left (-i \pi ^4+\frac {48 a x}{\sqrt {1+a^2 x^2}}-\frac {48 \arctan (a x)}{\sqrt {1+a^2 x^2}}-\frac {24 a x \arctan (a x)^2}{\sqrt {1+a^2 x^2}}+\frac {8 \arctan (a x)^3}{\sqrt {1+a^2 x^2}}+2 i \arctan (a x)^4+8 \arctan (a x)^3 \log \left (1-e^{-i \arctan (a x)}\right )-8 \arctan (a x)^3 \log \left (1+e^{i \arctan (a x)}\right )+24 i \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{-i \arctan (a x)}\right )+24 i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )+48 \arctan (a x) \operatorname {PolyLog}\left (3,e^{-i \arctan (a x)}\right )-48 \arctan (a x) \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )-48 i \operatorname {PolyLog}\left (4,e^{-i \arctan (a x)}\right )-48 i \operatorname {PolyLog}\left (4,-e^{i \arctan (a x)}\right )\right )}{8 c \sqrt {c \left (1+a^2 x^2\right )}} \]

[In]

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

[Out]

(Sqrt[1 + a^2*x^2]*((-I)*Pi^4 + (48*a*x)/Sqrt[1 + a^2*x^2] - (48*ArcTan[a*x])/Sqrt[1 + a^2*x^2] - (24*a*x*ArcT
an[a*x]^2)/Sqrt[1 + a^2*x^2] + (8*ArcTan[a*x]^3)/Sqrt[1 + a^2*x^2] + (2*I)*ArcTan[a*x]^4 + 8*ArcTan[a*x]^3*Log
[1 - E^((-I)*ArcTan[a*x])] - 8*ArcTan[a*x]^3*Log[1 + E^(I*ArcTan[a*x])] + (24*I)*ArcTan[a*x]^2*PolyLog[2, E^((
-I)*ArcTan[a*x])] + (24*I)*ArcTan[a*x]^2*PolyLog[2, -E^(I*ArcTan[a*x])] + 48*ArcTan[a*x]*PolyLog[3, E^((-I)*Ar
cTan[a*x])] - 48*ArcTan[a*x]*PolyLog[3, -E^(I*ArcTan[a*x])] - (48*I)*PolyLog[4, E^((-I)*ArcTan[a*x])] - (48*I)
*PolyLog[4, -E^(I*ArcTan[a*x])]))/(8*c*Sqrt[c*(1 + a^2*x^2)])

Maple [A] (verified)

Time = 3.83 (sec) , antiderivative size = 388, normalized size of antiderivative = 0.88

method	result	size
default	\(\frac {\left (\arctan \left (a x \right )^{3}-6 \arctan \left (a x \right )+3 i \arctan \left (a x \right )^{2}-6 i\right ) \left (i a x +1\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 \left (a^{2} x^{2}+1\right ) c^{2}}-\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i a x -1\right ) \left (\arctan \left (a x \right )^{3}-6 \arctan \left (a x \right )-3 i \arctan \left (a x \right )^{2}+6 i\right )}{2 \left (a^{2} x^{2}+1\right ) c^{2}}+\frac {i \left (i \arctan \left (a x \right )^{3} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )-i \arctan \left (a x \right )^{3} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+3 \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-3 \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-6 i \arctan \left (a x \right ) \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-6 \operatorname {polylog}\left (4, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 \operatorname {polylog}\left (4, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{\sqrt {a^{2} x^{2}+1}\, c^{2}}\)	\(388\)

[In]

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

[Out]

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

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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