3.5.24 \(\int \frac {(c+a^2 c x^2)^{3/2} \text {ArcTan}(a x)^3}{x} \, dx\) [424]

Optimal. Leaf size=726 \[ c \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)-\frac {1}{2} a c x \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^2+\frac {7 i c^2 \sqrt {1+a^2 x^2} \text {ArcTan}\left (e^{i \text {ArcTan}(a x)}\right ) \text {ArcTan}(a x)^2}{\sqrt {c+a^2 c x^2}}+c \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^3+\frac {1}{3} \left (c+a^2 c x^2\right )^{3/2} \text {ArcTan}(a x)^3-\frac {2 c^2 \sqrt {1+a^2 x^2} \text {ArcTan}(a x)^3 \tanh ^{-1}\left (e^{i \text {ArcTan}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-c^{3/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )+\frac {3 i c^2 \sqrt {1+a^2 x^2} \text {ArcTan}(a x)^2 \text {PolyLog}\left (2,-e^{i \text {ArcTan}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {7 i c^2 \sqrt {1+a^2 x^2} \text {ArcTan}(a x) \text {PolyLog}\left (2,-i e^{i \text {ArcTan}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {7 i c^2 \sqrt {1+a^2 x^2} \text {ArcTan}(a x) \text {PolyLog}\left (2,i e^{i \text {ArcTan}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i c^2 \sqrt {1+a^2 x^2} \text {ArcTan}(a x)^2 \text {PolyLog}\left (2,e^{i \text {ArcTan}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 c^2 \sqrt {1+a^2 x^2} \text {ArcTan}(a x) \text {PolyLog}\left (3,-e^{i \text {ArcTan}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {7 c^2 \sqrt {1+a^2 x^2} \text {PolyLog}\left (3,-i e^{i \text {ArcTan}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {7 c^2 \sqrt {1+a^2 x^2} \text {PolyLog}\left (3,i e^{i \text {ArcTan}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 c^2 \sqrt {1+a^2 x^2} \text {ArcTan}(a x) \text {PolyLog}\left (3,e^{i \text {ArcTan}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 i c^2 \sqrt {1+a^2 x^2} \text {PolyLog}\left (4,-e^{i \text {ArcTan}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 i c^2 \sqrt {1+a^2 x^2} \text {PolyLog}\left (4,e^{i \text {ArcTan}(a x)}\right )}{\sqrt {c+a^2 c x^2}} \]

[Out]

1/3*(a^2*c*x^2+c)^(3/2)*arctan(a*x)^3-c^(3/2)*arctanh(a*x*c^(1/2)/(a^2*c*x^2+c)^(1/2))+3*I*c^2*arctan(a*x)^2*p
olylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)-2*c^2*arctan(a*x)^3*arctanh((1+I*
a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)+7*I*c^2*arctan((1+I*a*x)/(a^2*x^2+1)^(1/2))*arct
an(a*x)^2*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)+7*I*c^2*arctan(a*x)*polylog(2,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(
a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)+6*I*c^2*polylog(4,(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x
^2+c)^(1/2)-6*I*c^2*polylog(4,-(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)-6*c^2*arctan
(a*x)*polylog(3,-(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)+7*c^2*polylog(3,-I*(1+I*a*
x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)-7*c^2*polylog(3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^
2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)+6*c^2*arctan(a*x)*polylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/
(a^2*c*x^2+c)^(1/2)-3*I*c^2*arctan(a*x)^2*polylog(2,(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+
c)^(1/2)-7*I*c^2*arctan(a*x)*polylog(2,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)+c
*arctan(a*x)*(a^2*c*x^2+c)^(1/2)-1/2*a*c*x*arctan(a*x)^2*(a^2*c*x^2+c)^(1/2)+c*arctan(a*x)^3*(a^2*c*x^2+c)^(1/
2)

________________________________________________________________________________________

Rubi [A]
time = 0.84, antiderivative size = 726, normalized size of antiderivative = 1.00, number of steps used = 36, number of rules used = 15, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5070, 5078, 5076, 4268, 2611, 6744, 2320, 6724, 5050, 5010, 5008, 4266, 5000, 223, 212} \begin {gather*} \frac {3 i c^2 \sqrt {a^2 x^2+1} \text {ArcTan}(a x)^2 \text {Li}_2\left (-e^{i \text {ArcTan}(a x)}\right )}{\sqrt {a^2 c x^2+c}}-\frac {3 i c^2 \sqrt {a^2 x^2+1} \text {ArcTan}(a x)^2 \text {Li}_2\left (e^{i \text {ArcTan}(a x)}\right )}{\sqrt {a^2 c x^2+c}}-\frac {7 i c^2 \sqrt {a^2 x^2+1} \text {ArcTan}(a x) \text {Li}_2\left (-i e^{i \text {ArcTan}(a x)}\right )}{\sqrt {a^2 c x^2+c}}+\frac {7 i c^2 \sqrt {a^2 x^2+1} \text {ArcTan}(a x) \text {Li}_2\left (i e^{i \text {ArcTan}(a x)}\right )}{\sqrt {a^2 c x^2+c}}-\frac {6 c^2 \sqrt {a^2 x^2+1} \text {ArcTan}(a x) \text {Li}_3\left (-e^{i \text {ArcTan}(a x)}\right )}{\sqrt {a^2 c x^2+c}}+\frac {6 c^2 \sqrt {a^2 x^2+1} \text {ArcTan}(a x) \text {Li}_3\left (e^{i \text {ArcTan}(a x)}\right )}{\sqrt {a^2 c x^2+c}}+\frac {7 c^2 \sqrt {a^2 x^2+1} \text {Li}_3\left (-i e^{i \text {ArcTan}(a x)}\right )}{\sqrt {a^2 c x^2+c}}-\frac {7 c^2 \sqrt {a^2 x^2+1} \text {Li}_3\left (i e^{i \text {ArcTan}(a x)}\right )}{\sqrt {a^2 c x^2+c}}-\frac {6 i c^2 \sqrt {a^2 x^2+1} \text {Li}_4\left (-e^{i \text {ArcTan}(a x)}\right )}{\sqrt {a^2 c x^2+c}}+\frac {6 i c^2 \sqrt {a^2 x^2+1} \text {Li}_4\left (e^{i \text {ArcTan}(a x)}\right )}{\sqrt {a^2 c x^2+c}}+\frac {7 i c^2 \sqrt {a^2 x^2+1} \text {ArcTan}\left (e^{i \text {ArcTan}(a x)}\right ) \text {ArcTan}(a x)^2}{\sqrt {a^2 c x^2+c}}-\frac {2 c^2 \sqrt {a^2 x^2+1} \text {ArcTan}(a x)^3 \tanh ^{-1}\left (e^{i \text {ArcTan}(a x)}\right )}{\sqrt {a^2 c x^2+c}}+\frac {1}{3} \text {ArcTan}(a x)^3 \left (a^2 c x^2+c\right )^{3/2}+c \text {ArcTan}(a x)^3 \sqrt {a^2 c x^2+c}-\frac {1}{2} a c x \text {ArcTan}(a x)^2 \sqrt {a^2 c x^2+c}+c \text {ArcTan}(a x) \sqrt {a^2 c x^2+c}-c^{3/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x] - (a*c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/2 + ((7*I)*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] + c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3 + ((c + a^
2*c*x^2)^(3/2)*ArcTan[a*x]^3)/3 - (2*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c +
a^2*c*x^2] - c^(3/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]] + ((3*I)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*P
olyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((7*I)*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)*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] - ((3*I)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[c +
a^2*c*x^2] - (6*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (7*c^
2*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (7*c^2*Sqrt[1 + a^2*x^2]*PolyLog
[3, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (6*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, E^(I*ArcTan[a*
x])])/Sqrt[c + a^2*c*x^2] - ((6*I)*c^2*Sqrt[1 + a^2*x^2]*PolyLog[4, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] +
((6*I)*c^2*Sqrt[1 + a^2*x^2]*PolyLog[4, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]

Rule 212

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

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 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 4266

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 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 5000

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

Rule 5008

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 5010

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 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 5070

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 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 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*} \int \frac {\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3}{x} \, dx &=c \int \frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{x} \, dx+\left (a^2 c\right ) \int x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3 \, dx\\ &=\frac {1}{3} \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3-(a c) \int \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2 \, dx+c^2 \int \frac {\tan ^{-1}(a x)^3}{x \sqrt {c+a^2 c x^2}} \, dx+\left (a^2 c^2\right ) \int \frac {x \tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx\\ &=c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {1}{2} a c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{3} \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3-\frac {1}{2} \left (a c^2\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx-\left (a c^2\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx-\left (3 a c^2\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {\left (c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)^3}{x \sqrt {1+a^2 x^2}} \, dx}{\sqrt {c+a^2 c x^2}}\\ &=c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {1}{2} a c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{3} \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3-\left (a c^2\right ) \text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )+\frac {\left (c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^3 \csc (x) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (a c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{2 \sqrt {c+a^2 c x^2}}-\frac {\left (3 a 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}}\\ &=c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {1}{2} a c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{3} \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3-\frac {2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-c^{3/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )-\frac {\left (c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {\left (3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \log \left (1-e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \log \left (1+e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}\\ &=c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {1}{2} a c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {7 i 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}}+c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{3} \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3-\frac {2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-c^{3/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )+\frac {3 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (6 i c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \text {Li}_2\left (-e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (6 i c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \text {Li}_2\left (e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (c^2 \sqrt {1+a^2 x^2}\right ) \text {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 (c^2 \sqrt {1+a^2 x^2}\right ) \text {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 (6 c^2 \sqrt {1+a^2 x^2}\right ) \text {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 (6 c^2 \sqrt {1+a^2 x^2}\right ) \text {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}}\\ &=c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {1}{2} a c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {7 i 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}}+c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{3} \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3-\frac {2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-c^{3/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )+\frac {3 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {7 i 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 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 {3 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (i c^2 \sqrt {1+a^2 x^2}\right ) \text {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 (i c^2 \sqrt {1+a^2 x^2}\right ) \text {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 (6 i c^2 \sqrt {1+a^2 x^2}\right ) \text {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 (6 i c^2 \sqrt {1+a^2 x^2}\right ) \text {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 (6 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \text {Li}_3\left (-e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (6 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \text {Li}_3\left (e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}\\ &=c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {1}{2} a c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {7 i 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}}+c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{3} \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3-\frac {2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-c^{3/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )+\frac {3 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {7 i 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 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 {3 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (6 i c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (6 i c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (c^2 \sqrt {1+a^2 x^2}\right ) \text {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 (c^2 \sqrt {1+a^2 x^2}\right ) \text {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 (6 c^2 \sqrt {1+a^2 x^2}\right ) \text {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 (6 c^2 \sqrt {1+a^2 x^2}\right ) \text {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}}\\ &=c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {1}{2} a c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {7 i 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}}+c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{3} \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3-\frac {2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-c^{3/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )+\frac {3 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {7 i 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 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 {3 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {7 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 {7 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 {6 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 i c^2 \sqrt {1+a^2 x^2} \text {Li}_4\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 i c^2 \sqrt {1+a^2 x^2} \text {Li}_4\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}\\ \end {align*}

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Mathematica [A]
time = 1.66, size = 854, normalized size = 1.18 \begin {gather*} \frac {c \sqrt {c+a^2 c x^2} \left (-3 i \pi ^4+\frac {12 \text {ArcTan}(a x)}{\sqrt {1+a^2 x^2}}-\frac {12 a^4 x^4 \text {ArcTan}(a x)}{\sqrt {1+a^2 x^2}}+12 \sqrt {1+a^2 x^2} \text {ArcTan}(a x)+12 a^2 x^2 \sqrt {1+a^2 x^2} \text {ArcTan}(a x)-\frac {12 a x \text {ArcTan}(a x)^2}{\sqrt {1+a^2 x^2}}-\frac {12 a^3 x^3 \text {ArcTan}(a x)^2}{\sqrt {1+a^2 x^2}}+32 \sqrt {1+a^2 x^2} \text {ArcTan}(a x)^3+8 a^2 x^2 \sqrt {1+a^2 x^2} \text {ArcTan}(a x)^3+6 i \text {ArcTan}(a x)^4+24 \text {ArcTan}(a x)^3 \log \left (1-e^{-i \text {ArcTan}(a x)}\right )-84 \text {ArcTan}(a x)^2 \log \left (1-i e^{i \text {ArcTan}(a x)}\right )-12 \pi \text {ArcTan}(a x) \log \left (\frac {1}{2} \sqrt [4]{-1} e^{-\frac {1}{2} i \text {ArcTan}(a x)} \left (1-i e^{i \text {ArcTan}(a x)}\right )\right )+84 \text {ArcTan}(a x)^2 \log \left (1+i e^{i \text {ArcTan}(a x)}\right )+12 \text {ArcTan}(a x)^2 \log \left (\left (\frac {1}{2}+\frac {i}{2}\right ) e^{-\frac {1}{2} i \text {ArcTan}(a x)} \left (-i+e^{i \text {ArcTan}(a x)}\right )\right )-12 \pi \text {ArcTan}(a x) \log \left (-\frac {1}{2} \sqrt [4]{-1} e^{-\frac {1}{2} i \text {ArcTan}(a x)} \left (-i+e^{i \text {ArcTan}(a x)}\right )\right )-24 \text {ArcTan}(a x)^3 \log \left (1+e^{i \text {ArcTan}(a x)}\right )-12 \text {ArcTan}(a x)^2 \log \left (\frac {1}{2} e^{-\frac {1}{2} i \text {ArcTan}(a x)} \left ((1+i)+(1-i) e^{i \text {ArcTan}(a x)}\right )\right )+12 \pi \text {ArcTan}(a x) \log \left (-\cos \left (\frac {1}{4} (\pi +2 \text {ArcTan}(a x))\right )\right )+24 \log \left (\cos \left (\frac {1}{2} \text {ArcTan}(a x)\right )-\sin \left (\frac {1}{2} \text {ArcTan}(a x)\right )\right )-12 \text {ArcTan}(a x)^2 \log \left (\cos \left (\frac {1}{2} \text {ArcTan}(a x)\right )-\sin \left (\frac {1}{2} \text {ArcTan}(a x)\right )\right )-24 \log \left (\cos \left (\frac {1}{2} \text {ArcTan}(a x)\right )+\sin \left (\frac {1}{2} \text {ArcTan}(a x)\right )\right )+12 \text {ArcTan}(a x)^2 \log \left (\cos \left (\frac {1}{2} \text {ArcTan}(a x)\right )+\sin \left (\frac {1}{2} \text {ArcTan}(a x)\right )\right )+12 \pi \text {ArcTan}(a x) \log \left (\sin \left (\frac {1}{4} (\pi +2 \text {ArcTan}(a x))\right )\right )+72 i \text {ArcTan}(a x)^2 \text {PolyLog}\left (2,e^{-i \text {ArcTan}(a x)}\right )+72 i \text {ArcTan}(a x)^2 \text {PolyLog}\left (2,-e^{i \text {ArcTan}(a x)}\right )-168 i \text {ArcTan}(a x) \text {PolyLog}\left (2,-i e^{i \text {ArcTan}(a x)}\right )+168 i \text {ArcTan}(a x) \text {PolyLog}\left (2,i e^{i \text {ArcTan}(a x)}\right )+144 \text {ArcTan}(a x) \text {PolyLog}\left (3,e^{-i \text {ArcTan}(a x)}\right )-144 \text {ArcTan}(a x) \text {PolyLog}\left (3,-e^{i \text {ArcTan}(a x)}\right )+168 \text {PolyLog}\left (3,-i e^{i \text {ArcTan}(a x)}\right )-168 \text {PolyLog}\left (3,i e^{i \text {ArcTan}(a x)}\right )-144 i \text {PolyLog}\left (4,e^{-i \text {ArcTan}(a x)}\right )-144 i \text {PolyLog}\left (4,-e^{i \text {ArcTan}(a x)}\right )\right )}{24 \sqrt {1+a^2 x^2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c*Sqrt[c + a^2*c*x^2]*((-3*I)*Pi^4 + (12*ArcTan[a*x])/Sqrt[1 + a^2*x^2] - (12*a^4*x^4*ArcTan[a*x])/Sqrt[1 + a
^2*x^2] + 12*Sqrt[1 + a^2*x^2]*ArcTan[a*x] + 12*a^2*x^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x] - (12*a*x*ArcTan[a*x]^2)
/Sqrt[1 + a^2*x^2] - (12*a^3*x^3*ArcTan[a*x]^2)/Sqrt[1 + a^2*x^2] + 32*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3 + 8*a^2
*x^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3 + (6*I)*ArcTan[a*x]^4 + 24*ArcTan[a*x]^3*Log[1 - E^((-I)*ArcTan[a*x])] -
84*ArcTan[a*x]^2*Log[1 - I*E^(I*ArcTan[a*x])] - 12*Pi*ArcTan[a*x]*Log[((-1)^(1/4)*(1 - I*E^(I*ArcTan[a*x])))/(
2*E^((I/2)*ArcTan[a*x]))] + 84*ArcTan[a*x]^2*Log[1 + I*E^(I*ArcTan[a*x])] + 12*ArcTan[a*x]^2*Log[((1/2 + I/2)*
(-I + E^(I*ArcTan[a*x])))/E^((I/2)*ArcTan[a*x])] - 12*Pi*ArcTan[a*x]*Log[-1/2*((-1)^(1/4)*(-I + E^(I*ArcTan[a*
x])))/E^((I/2)*ArcTan[a*x])] - 24*ArcTan[a*x]^3*Log[1 + E^(I*ArcTan[a*x])] - 12*ArcTan[a*x]^2*Log[((1 + I) + (
1 - I)*E^(I*ArcTan[a*x]))/(2*E^((I/2)*ArcTan[a*x]))] + 12*Pi*ArcTan[a*x]*Log[-Cos[(Pi + 2*ArcTan[a*x])/4]] + 2
4*Log[Cos[ArcTan[a*x]/2] - Sin[ArcTan[a*x]/2]] - 12*ArcTan[a*x]^2*Log[Cos[ArcTan[a*x]/2] - Sin[ArcTan[a*x]/2]]
- 24*Log[Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2]] + 12*ArcTan[a*x]^2*Log[Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]
/2]] + 12*Pi*ArcTan[a*x]*Log[Sin[(Pi + 2*ArcTan[a*x])/4]] + (72*I)*ArcTan[a*x]^2*PolyLog[2, E^((-I)*ArcTan[a*x
])] + (72*I)*ArcTan[a*x]^2*PolyLog[2, -E^(I*ArcTan[a*x])] - (168*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*
x])] + (168*I)*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] + 144*ArcTan[a*x]*PolyLog[3, E^((-I)*ArcTan[a*x])]
- 144*ArcTan[a*x]*PolyLog[3, -E^(I*ArcTan[a*x])] + 168*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] - 168*PolyLog[3, I*E
^(I*ArcTan[a*x])] - (144*I)*PolyLog[4, E^((-I)*ArcTan[a*x])] - (144*I)*PolyLog[4, -E^(I*ArcTan[a*x])]))/(24*Sq
rt[1 + a^2*x^2])

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Maple [A]
time = 2.73, size = 511, normalized size = 0.70

method	result	size
default	\(\frac {c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \arctan \left (a x \right ) \left (2 \arctan \left (a x \right )^{2} a^{2} x^{2}-3 \arctan \left (a x \right ) a x +8 \arctan \left (a x \right )^{2}+6\right )}{6}+\frac {c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (2 \arctan \left (a x \right )^{3} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-2 \arctan \left (a x \right )^{3} \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-6 i \arctan \left (a x \right )^{2} \polylog \left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \arctan \left (a x \right )^{2} \polylog \left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+7 \arctan \left (a x \right )^{2} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-7 \arctan \left (a x \right )^{2} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-14 i \arctan \left (a x \right ) \polylog \left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+14 i \arctan \left (a x \right ) \polylog \left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+12 \arctan \left (a x \right ) \polylog \left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-12 \arctan \left (a x \right ) \polylog \left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+12 i \polylog \left (4, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-12 i \polylog \left (4, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+4 i \arctan \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+14 \polylog \left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-14 \polylog \left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{2 \sqrt {a^{2} x^{2}+1}}\)	\(511\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*c*(c*(a*x-I)*(I+a*x))^(1/2)*arctan(a*x)*(2*arctan(a*x)^2*a^2*x^2-3*arctan(a*x)*a*x+8*arctan(a*x)^2+6)+1/2*
c*(c*(a*x-I)*(I+a*x))^(1/2)*(2*arctan(a*x)^3*ln(1-(1+I*a*x)/(a^2*x^2+1)^(1/2))-2*arctan(a*x)^3*ln(1+(1+I*a*x)/
(a^2*x^2+1)^(1/2))-6*I*arctan(a*x)^2*polylog(2,(1+I*a*x)/(a^2*x^2+1)^(1/2))+6*I*arctan(a*x)^2*polylog(2,-(1+I*
a*x)/(a^2*x^2+1)^(1/2))+7*arctan(a*x)^2*ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-7*arctan(a*x)^2*ln(1-I*(1+I*a*x)/(
a^2*x^2+1)^(1/2))-14*I*arctan(a*x)*polylog(2,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+14*I*arctan(a*x)*polylog(2,I*(1+I
*a*x)/(a^2*x^2+1)^(1/2))+12*arctan(a*x)*polylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))-12*arctan(a*x)*polylog(3,-(1+I*
a*x)/(a^2*x^2+1)^(1/2))+12*I*polylog(4,(1+I*a*x)/(a^2*x^2+1)^(1/2))-12*I*polylog(4,-(1+I*a*x)/(a^2*x^2+1)^(1/2
))+4*I*arctan((1+I*a*x)/(a^2*x^2+1)^(1/2))+14*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-14*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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \operatorname {atan}^{3}{\left (a x \right )}}{x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^{3/2}}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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