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

Optimal. Leaf size=579 \[ \frac{7 i a^3 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 \sqrt{a^2 c x^2+c}}-\frac{7 i a^3 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 \sqrt{a^2 c x^2+c}}+\frac{2 i a^3 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}-\frac{2 i a^3 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}-\frac{2 a^3 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}+\frac{2 a^3 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}-\frac{2 i a^3 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{\sqrt{a^2 c x^2+c}}-\frac{14 a^3 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 \sqrt{a^2 c x^2+c}}-\frac{a^2 c \sqrt{a^2 c x^2+c}}{3 x}-\frac{a^2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{x}-\frac{a c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{3 x^2}-\frac{\left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^2}{3 x^3} \]

[Out]

-(a^2*c*Sqrt[c + a^2*c*x^2])/(3*x) - (a*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(3*x^2) - (a^2*c*Sqrt[c + a^2*c*x^2
]*ArcTan[a*x]^2)/x - ((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/(3*x^3) - ((2*I)*a^3*c^2*Sqrt[1 + a^2*x^2]*ArcTan[E
^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/Sqrt[c + a^2*c*x^2] - (14*a^3*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[
1 + I*a*x]/Sqrt[1 - I*a*x]])/(3*Sqrt[c + a^2*c*x^2]) + ((2*I)*a^3*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2,
(-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((2*I)*a^3*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I
*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (((7*I)/3)*a^3*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1
- I*a*x])])/Sqrt[c + a^2*c*x^2] - (((7*I)/3)*a^3*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*
a*x]])/Sqrt[c + a^2*c*x^2] - (2*a^3*c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x
^2] + (2*a^3*c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]

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Rubi [A]  time = 1.14642, antiderivative size = 579, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 13, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.542, Rules used = {4950, 4944, 4946, 4962, 264, 4958, 4954, 4890, 4888, 4181, 2531, 2282, 6589} \[ \frac{7 i a^3 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 \sqrt{a^2 c x^2+c}}-\frac{7 i a^3 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 \sqrt{a^2 c x^2+c}}+\frac{2 i a^3 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}-\frac{2 i a^3 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}-\frac{2 a^3 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}+\frac{2 a^3 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}-\frac{2 i a^3 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{\sqrt{a^2 c x^2+c}}-\frac{14 a^3 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 \sqrt{a^2 c x^2+c}}-\frac{a^2 c \sqrt{a^2 c x^2+c}}{3 x}-\frac{a^2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{x}-\frac{a c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{3 x^2}-\frac{\left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^2}{3 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^2*c*Sqrt[c + a^2*c*x^2])/(3*x) - (a*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(3*x^2) - (a^2*c*Sqrt[c + a^2*c*x^2
]*ArcTan[a*x]^2)/x - ((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/(3*x^3) - ((2*I)*a^3*c^2*Sqrt[1 + a^2*x^2]*ArcTan[E
^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/Sqrt[c + a^2*c*x^2] - (14*a^3*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[
1 + I*a*x]/Sqrt[1 - I*a*x]])/(3*Sqrt[c + a^2*c*x^2]) + ((2*I)*a^3*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2,
(-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((2*I)*a^3*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I
*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (((7*I)/3)*a^3*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1
- I*a*x])])/Sqrt[c + a^2*c*x^2] - (((7*I)/3)*a^3*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*
a*x]])/Sqrt[c + a^2*c*x^2] - (2*a^3*c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x
^2] + (2*a^3*c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]

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 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 4946

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((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]))/(f*(m + 2)), x] + (Dist[d/(m + 2), Int[((f*x)^m*(a + b*ArcTan[c*x]
))/Sqrt[d + e*x^2], x], x] - Dist[(b*c*d)/(f*(m + 2)), Int[(f*x)^(m + 1)/Sqrt[d + e*x^2], x], x]) /; FreeQ[{a,
b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && NeQ[m, -2]

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 264

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] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

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 4954

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

Rule 4890

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[1 + c^2*x^2]/Sq
rt[d + e*x^2], Int[(a + b*ArcTan[c*x])^p/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 4888

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Dist[1/(c*Sqrt[d]), Subst
[Int[(a + b*x)^p*Sec[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 4181

Int[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c + d*x)^m*ArcTanh[E
^(I*k*Pi)*E^(I*(e + f*x))])/f, x] + (-Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))],
x], x] + Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e,
f}, x] && IntegerQ[2*k] && 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{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{x^4} \, dx &=c \int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{x^4} \, dx+\left (a^2 c\right ) \int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{x^2} \, dx\\ &=-\frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{3 x^3}+\frac{1}{3} (2 a c) \int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{x^3} \, dx+\left (a^2 c^2\right ) \int \frac{\tan ^{-1}(a x)^2}{x^2 \sqrt{c+a^2 c x^2}} \, dx+\left (a^4 c^2\right ) \int \frac{\tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx\\ &=-\frac{2 a c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{3 x^2}-\frac{a^2 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{x}-\frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{3 x^3}-\frac{1}{3} \left (2 a c^2\right ) \int \frac{\tan ^{-1}(a x)}{x^3 \sqrt{c+a^2 c x^2}} \, dx+\frac{1}{3} \left (2 a^2 c^2\right ) \int \frac{1}{x^2 \sqrt{c+a^2 c x^2}} \, dx+\left (2 a^3 c^2\right ) \int \frac{\tan ^{-1}(a x)}{x \sqrt{c+a^2 c x^2}} \, dx+\frac{\left (a^4 c^2 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)^2}{\sqrt{1+a^2 x^2}} \, dx}{\sqrt{c+a^2 c x^2}}\\ &=-\frac{2 a^2 c \sqrt{c+a^2 c x^2}}{3 x}-\frac{a c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{3 x^2}-\frac{a^2 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{x}-\frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{3 x^3}-\frac{1}{3} \left (a^2 c^2\right ) \int \frac{1}{x^2 \sqrt{c+a^2 c x^2}} \, dx+\frac{1}{3} \left (a^3 c^2\right ) \int \frac{\tan ^{-1}(a x)}{x \sqrt{c+a^2 c x^2}} \, dx+\frac{\left (a^3 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}+\frac{\left (2 a^3 c^2 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{x \sqrt{1+a^2 x^2}} \, dx}{\sqrt{c+a^2 c x^2}}\\ &=-\frac{a^2 c \sqrt{c+a^2 c x^2}}{3 x}-\frac{a c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{3 x^2}-\frac{a^2 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{x}-\frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{3 x^3}-\frac{2 i a^3 c^2 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}}-\frac{4 a^3 c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}+\frac{2 i a^3 c^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}-\frac{2 i a^3 c^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}+\frac{\left (a^3 c^2 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{x \sqrt{1+a^2 x^2}} \, dx}{3 \sqrt{c+a^2 c x^2}}-\frac{\left (2 a^3 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}+\frac{\left (2 a^3 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}\\ &=-\frac{a^2 c \sqrt{c+a^2 c x^2}}{3 x}-\frac{a c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{3 x^2}-\frac{a^2 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{x}-\frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{3 x^3}-\frac{2 i a^3 c^2 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}}-\frac{14 a^3 c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 \sqrt{c+a^2 c x^2}}+\frac{2 i a^3 c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{2 i a^3 c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{7 i a^3 c^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 \sqrt{c+a^2 c x^2}}-\frac{7 i a^3 c^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 \sqrt{c+a^2 c x^2}}-\frac{\left (2 i a^3 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-i 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^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}\\ &=-\frac{a^2 c \sqrt{c+a^2 c x^2}}{3 x}-\frac{a c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{3 x^2}-\frac{a^2 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{x}-\frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{3 x^3}-\frac{2 i a^3 c^2 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}}-\frac{14 a^3 c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 \sqrt{c+a^2 c x^2}}+\frac{2 i a^3 c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{2 i a^3 c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{7 i a^3 c^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 \sqrt{c+a^2 c x^2}}-\frac{7 i a^3 c^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 \sqrt{c+a^2 c x^2}}-\frac{\left (2 a^3 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{\left (2 a^3 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}\\ &=-\frac{a^2 c \sqrt{c+a^2 c x^2}}{3 x}-\frac{a c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{3 x^2}-\frac{a^2 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{x}-\frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{3 x^3}-\frac{2 i a^3 c^2 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}}-\frac{14 a^3 c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 \sqrt{c+a^2 c x^2}}+\frac{2 i a^3 c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{2 i a^3 c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{7 i a^3 c^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 \sqrt{c+a^2 c x^2}}-\frac{7 i a^3 c^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 \sqrt{c+a^2 c x^2}}-\frac{2 a^3 c^2 \sqrt{1+a^2 x^2} \text{Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{2 a^3 c^2 \sqrt{1+a^2 x^2} \text{Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}\\ \end{align*}

Mathematica [A]  time = 7.29432, size = 537, normalized size = 0.93 \[ \frac{a^3 c^2 \sqrt{a^2 x^2+1} \left (8 i \text{PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right )-\frac{2 \left (a^2 x^2+1\right )^{3/2} \left (\frac{4 i a^3 x^3 \text{PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right )}{\left (a^2 x^2+1\right )^{3/2}}-\frac{3 a x \tan ^{-1}(a x) \log \left (1-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 x^2+1}}+\frac{3 a x \tan ^{-1}(a x) \log \left (1+e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 x^2+1}}+4 \tan ^{-1}(a x)^2+2 \tan ^{-1}(a x) \sin \left (2 \tan ^{-1}(a x)\right )-2 \cos \left (2 \tan ^{-1}(a x)\right )+\tan ^{-1}(a x) \log \left (1-e^{i \tan ^{-1}(a x)}\right ) \sin \left (3 \tan ^{-1}(a x)\right )-\tan ^{-1}(a x) \log \left (1+e^{i \tan ^{-1}(a x)}\right ) \sin \left (3 \tan ^{-1}(a x)\right )+2\right )}{a^3 x^3}\right )}{24 \sqrt{c \left (a^2 x^2+1\right )}}-\frac{a^3 c \sqrt{c \left (a^2 x^2+1\right )} \left (-2 i \tan ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )+2 i \tan ^{-1}(a x) \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )-2 i \text{PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right )+2 i \text{PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right )+2 \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )-2 \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )+\frac{\sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2}{a x}+\tan ^{-1}(a x)^2 \left (-\log \left (1-i e^{i \tan ^{-1}(a x)}\right )\right )+\tan ^{-1}(a x)^2 \log \left (1+i e^{i \tan ^{-1}(a x)}\right )-2 \tan ^{-1}(a x) \log \left (1-e^{i \tan ^{-1}(a x)}\right )+2 \tan ^{-1}(a x) \log \left (1+e^{i \tan ^{-1}(a x)}\right )\right )}{\sqrt{a^2 x^2+1}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-((a^3*c*Sqrt[c*(1 + a^2*x^2)]*((Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2)/(a*x) - 2*ArcTan[a*x]*Log[1 - E^(I*ArcTan[a*
x])] - ArcTan[a*x]^2*Log[1 - I*E^(I*ArcTan[a*x])] + ArcTan[a*x]^2*Log[1 + I*E^(I*ArcTan[a*x])] + 2*ArcTan[a*x]
*Log[1 + E^(I*ArcTan[a*x])] - (2*I)*PolyLog[2, -E^(I*ArcTan[a*x])] - (2*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*Ar
cTan[a*x])] + (2*I)*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] + (2*I)*PolyLog[2, E^(I*ArcTan[a*x])] + 2*Poly
Log[3, (-I)*E^(I*ArcTan[a*x])] - 2*PolyLog[3, I*E^(I*ArcTan[a*x])]))/Sqrt[1 + a^2*x^2]) + (a^3*c^2*Sqrt[1 + a^
2*x^2]*((8*I)*PolyLog[2, -E^(I*ArcTan[a*x])] - (2*(1 + a^2*x^2)^(3/2)*(2 + 4*ArcTan[a*x]^2 - 2*Cos[2*ArcTan[a*
x]] - (3*a*x*ArcTan[a*x]*Log[1 - E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + (3*a*x*ArcTan[a*x]*Log[1 + E^(I*ArcTa
n[a*x])])/Sqrt[1 + a^2*x^2] + ((4*I)*a^3*x^3*PolyLog[2, E^(I*ArcTan[a*x])])/(1 + a^2*x^2)^(3/2) + 2*ArcTan[a*x
]*Sin[2*ArcTan[a*x]] + ArcTan[a*x]*Log[1 - E^(I*ArcTan[a*x])]*Sin[3*ArcTan[a*x]] - ArcTan[a*x]*Log[1 + E^(I*Ar
cTan[a*x])]*Sin[3*ArcTan[a*x]]))/(a^3*x^3)))/(24*Sqrt[c*(1 + a^2*x^2)])

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Maple [A]  time = 0.5, size = 343, normalized size = 0.6 \begin{align*} -{\frac{c \left ( 4\, \left ( \arctan \left ( ax \right ) \right ) ^{2}{x}^{2}{a}^{2}+{a}^{2}{x}^{2}+\arctan \left ( ax \right ) xa+ \left ( \arctan \left ( ax \right ) \right ) ^{2} \right ) }{3\,{x}^{3}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{{a}^{3}c}{3}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) } \left ( 3\, \left ( \arctan \left ( ax \right ) \right ) ^{2}\ln \left ( 1+{\frac{i \left ( 1+iax \right ) }{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) -3\, \left ( \arctan \left ( ax \right ) \right ) ^{2}\ln \left ( 1-{\frac{i \left ( 1+iax \right ) }{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) -6\,i\arctan \left ( ax \right ){\it polylog} \left ( 2,{-i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) +6\,i\arctan \left ( ax \right ){\it polylog} \left ( 2,{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) +7\,\arctan \left ( ax \right ) \ln \left ( 1+{\frac{1+iax}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) -7\,i{\it dilog} \left ({(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -7\,i{\it dilog} \left ( 1+{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) +6\,{\it polylog} \left ( 3,{\frac{-i \left ( 1+iax \right ) }{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) -6\,{\it polylog} \left ( 3,{\frac{i \left ( 1+iax \right ) }{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*c*(c*(a*x-I)*(a*x+I))^(1/2)*(4*arctan(a*x)^2*x^2*a^2+a^2*x^2+arctan(a*x)*x*a+arctan(a*x)^2)/x^3-1/3*a^3*c
*(c*(a*x-I)*(a*x+I))^(1/2)*(3*arctan(a*x)^2*ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-3*arctan(a*x)^2*ln(1-I*(1+I*a*
x)/(a^2*x^2+1)^(1/2))-6*I*arctan(a*x)*polylog(2,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+6*I*arctan(a*x)*polylog(2,I*(1
+I*a*x)/(a^2*x^2+1)^(1/2))+7*arctan(a*x)*ln(1+(1+I*a*x)/(a^2*x^2+1)^(1/2))-7*I*dilog((1+I*a*x)/(a^2*x^2+1)^(1/
2))-7*I*dilog(1+(1+I*a*x)/(a^2*x^2+1)^(1/2))+6*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-6*polylog(3,I*(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((a^2*c*x^2+c)^(3/2)*arctan(a*x)^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{{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}} \arctan \left (a x\right )^{2}}{x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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