3.419 \(\int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{x^4} \, dx\)

Optimal. Leaf size=361 \[ \frac{i a^3 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}-\frac{i a^3 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}-\frac{a^3 c \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}+\frac{a^3 c \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}-\frac{a^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{x}-\frac{a \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{2 x^2}-\frac{\left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^3}{3 c x^3}-a^3 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right )-\frac{a^3 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.01622, antiderivative size = 361, normalized size of antiderivative = 1., number of steps used = 25, number of rules used = 12, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4944, 4950, 4962, 266, 63, 208, 4958, 4956, 4183, 2531, 2282, 6589} \[ \frac{i a^3 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}-\frac{i a^3 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}-\frac{a^3 c \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}+\frac{a^3 c \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}-\frac{a^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{x}-\frac{a \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{2 x^2}-\frac{\left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^3}{3 c x^3}-a^3 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right )-\frac{a^3 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4944

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 4950

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

Rule 4962

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 266

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 63

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 208

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

Rule 4958

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 4956

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 4183

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 2531

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 2282

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 6589

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*} \int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{x^4} \, dx &=-\frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3}{3 c x^3}+a \int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{x^3} \, dx\\ &=-\frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3}{3 c x^3}+(a c) \int \frac{\tan ^{-1}(a x)^2}{x^3 \sqrt{c+a^2 c x^2}} \, dx+\left (a^3 c\right ) \int \frac{\tan ^{-1}(a x)^2}{x \sqrt{c+a^2 c x^2}} \, dx\\ &=-\frac{a \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 x^2}-\frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3}{3 c x^3}+\left (a^2 c\right ) \int \frac{\tan ^{-1}(a x)}{x^2 \sqrt{c+a^2 c x^2}} \, dx-\frac{1}{2} \left (a^3 c\right ) \int \frac{\tan ^{-1}(a x)^2}{x \sqrt{c+a^2 c x^2}} \, dx+\frac{\left (a^3 c \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)^2}{x \sqrt{1+a^2 x^2}} \, dx}{\sqrt{c+a^2 c x^2}}\\ &=-\frac{a^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{x}-\frac{a \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 x^2}-\frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3}{3 c x^3}+\left (a^3 c\right ) \int \frac{1}{x \sqrt{c+a^2 c x^2}} \, dx-\frac{\left (a^3 c \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)^2}{x \sqrt{1+a^2 x^2}} \, dx}{2 \sqrt{c+a^2 c x^2}}+\frac{\left (a^3 c \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \csc (x) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}\\ &=-\frac{a^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{x}-\frac{a \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 x^2}-\frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3}{3 c x^3}-\frac{2 a^3 c \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{1}{2} \left (a^3 c\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c+a^2 c x}} \, dx,x,x^2\right )-\frac{\left (a^3 c \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \csc (x) \, dx,x,\tan ^{-1}(a x)\right )}{2 \sqrt{c+a^2 c x^2}}-\frac{\left (2 a^3 c \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \log \left (1-e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}+\frac{\left (2 a^3 c \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \log \left (1+e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}\\ &=-\frac{a^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{x}-\frac{a \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 x^2}-\frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3}{3 c x^3}-\frac{a^3 c \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{2 i a^3 c \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{2 i a^3 c \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+a \operatorname{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 )-\frac{\left (2 i a^3 c \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}+\frac{\left (2 i a^3 c \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}+\frac{\left (a^3 c \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \log \left (1-e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}-\frac{\left (a^3 c \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \log \left (1+e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}\\ &=-\frac{a^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{x}-\frac{a \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 x^2}-\frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3}{3 c x^3}-\frac{a^3 c \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-a^3 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+a^2 c x^2}}{\sqrt{c}}\right )+\frac{i a^3 c \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{i a^3 c \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{\left (i a^3 c \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}-\frac{\left (i a^3 c \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}-\frac{\left (2 a^3 c \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{\left (2 a^3 c \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}\\ &=-\frac{a^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{x}-\frac{a \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 x^2}-\frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3}{3 c x^3}-\frac{a^3 c \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-a^3 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+a^2 c x^2}}{\sqrt{c}}\right )+\frac{i a^3 c \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{i a^3 c \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{2 a^3 c \sqrt{1+a^2 x^2} \text{Li}_3\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{2 a^3 c \sqrt{1+a^2 x^2} \text{Li}_3\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{\left (a^3 c \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{\left (a^3 c \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}\\ &=-\frac{a^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{x}-\frac{a \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 x^2}-\frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3}{3 c x^3}-\frac{a^3 c \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-a^3 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+a^2 c x^2}}{\sqrt{c}}\right )+\frac{i a^3 c \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{i a^3 c \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{a^3 c \sqrt{1+a^2 x^2} \text{Li}_3\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{a^3 c \sqrt{1+a^2 x^2} \text{Li}_3\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}\\ \end{align*}

Mathematica [A]  time = 3.49065, size = 341, normalized size = 0.94 \[ \frac{a^3 c \sqrt{a^2 x^2+1} \left (24 i \tan ^{-1}(a x) \text{PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right )-24 i \tan ^{-1}(a x) \text{PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right )-24 \text{PolyLog}\left (3,-e^{i \tan ^{-1}(a x)}\right )+24 \text{PolyLog}\left (3,e^{i \tan ^{-1}(a x)}\right )-\frac{8 \left (a^2 x^2+1\right )^{3/2} \tan ^{-1}(a x)^3 \sin ^4\left (\frac{1}{2} \tan ^{-1}(a x)\right )}{a^3 x^3}-\frac{a x \tan ^{-1}(a x)^3 \csc ^4\left (\frac{1}{2} \tan ^{-1}(a x)\right )}{2 \sqrt{a^2 x^2+1}}-2 \tan ^{-1}(a x)^3 \tan \left (\frac{1}{2} \tan ^{-1}(a x)\right )-12 \tan ^{-1}(a x) \tan \left (\frac{1}{2} \tan ^{-1}(a x)\right )+12 \tan ^{-1}(a x)^2 \log \left (1-e^{i \tan ^{-1}(a x)}\right )-12 \tan ^{-1}(a x)^2 \log \left (1+e^{i \tan ^{-1}(a x)}\right )+24 \log \left (\tan \left (\frac{1}{2} \tan ^{-1}(a x)\right )\right )-2 \tan ^{-1}(a x)^3 \cot \left (\frac{1}{2} \tan ^{-1}(a x)\right )-12 \tan ^{-1}(a x) \cot \left (\frac{1}{2} \tan ^{-1}(a x)\right )-3 \tan ^{-1}(a x)^2 \csc ^2\left (\frac{1}{2} \tan ^{-1}(a x)\right )+3 \tan ^{-1}(a x)^2 \sec ^2\left (\frac{1}{2} \tan ^{-1}(a x)\right )\right )}{24 \sqrt{c \left (a^2 x^2+1\right )}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a^3*c*Sqrt[1 + a^2*x^2]*(-12*ArcTan[a*x]*Cot[ArcTan[a*x]/2] - 2*ArcTan[a*x]^3*Cot[ArcTan[a*x]/2] - 3*ArcTan[a
*x]^2*Csc[ArcTan[a*x]/2]^2 - (a*x*ArcTan[a*x]^3*Csc[ArcTan[a*x]/2]^4)/(2*Sqrt[1 + a^2*x^2]) + 12*ArcTan[a*x]^2
*Log[1 - E^(I*ArcTan[a*x])] - 12*ArcTan[a*x]^2*Log[1 + E^(I*ArcTan[a*x])] + 24*Log[Tan[ArcTan[a*x]/2]] + (24*I
)*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])] - (24*I)*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])] - 24*PolyLog[3
, -E^(I*ArcTan[a*x])] + 24*PolyLog[3, E^(I*ArcTan[a*x])] + 3*ArcTan[a*x]^2*Sec[ArcTan[a*x]/2]^2 - (8*(1 + a^2*
x^2)^(3/2)*ArcTan[a*x]^3*Sin[ArcTan[a*x]/2]^4)/(a^3*x^3) - 12*ArcTan[a*x]*Tan[ArcTan[a*x]/2] - 2*ArcTan[a*x]^3
*Tan[ArcTan[a*x]/2]))/(24*Sqrt[c*(1 + a^2*x^2)])

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Maple [A]  time = 2.786, size = 462, normalized size = 1.3 \begin{align*} -{\frac{\arctan \left ( ax \right ) \left ( 2\, \left ( \arctan \left ( ax \right ) \right ) ^{2}{x}^{2}{a}^{2}+6\,{a}^{2}{x}^{2}+3\,\arctan \left ( ax \right ) xa+2\, \left ( \arctan \left ( ax \right ) \right ) ^{2} \right ) }{6\,{x}^{3}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{\frac{{a}^{3} \left ( \arctan \left ( ax \right ) \right ) ^{2}}{2}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }\ln \left ( 1-{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}-{i{a}^{3}\arctan \left ( ax \right ) \sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }{\it polylog} \left ( 2,{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}+{{a}^{3}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }{\it polylog} \left ( 3,{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}-{\frac{{a}^{3} \left ( \arctan \left ( ax \right ) \right ) ^{2}}{2}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }\ln \left ( 1+{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}+{i{a}^{3}\arctan \left ( ax \right ) \sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }{\it polylog} \left ( 2,-{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}-{{a}^{3}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }{\it polylog} \left ( 3,-{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}-2\,{\frac{{a}^{3}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}{\sqrt{{a}^{2}{x}^{2}+1}}{\it Artanh} \left ({\frac{1+iax}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctan(a*x)^3*(a^2*c*x^2+c)^(1/2)/x^4,x)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(a*x)^3*(a^2*c*x^2+c)^(1/2)/x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{a^{2} c x^{2} + c} \arctan \left (a x\right )^{3}}{x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(a*x)^3*(a^2*c*x^2+c)^(1/2)/x^4,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c \left (a^{2} x^{2} + 1\right )} \operatorname{atan}^{3}{\left (a x \right )}}{x^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atan(a*x)**3*(a**2*c*x**2+c)**(1/2)/x**4,x)

[Out]

Integral(sqrt(c*(a**2*x**2 + 1))*atan(a*x)**3/x**4, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a^{2} c x^{2} + c} \arctan \left (a x\right )^{3}}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(a*x)^3*(a^2*c*x^2+c)^(1/2)/x^4,x, algorithm="giac")

[Out]

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