3.326 \(\int (c+a^2 c x^2)^{5/2} \tan ^{-1}(a x)^2 \, dx\)

Optimal. Leaf size=516 \[ \frac{5 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt{a^2 c x^2+c}}-\frac{5 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt{a^2 c x^2+c}}-\frac{5 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt{a^2 c x^2+c}}+\frac{5 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt{a^2 c x^2+c}}+\frac{17}{180} c^2 x \sqrt{a^2 c x^2+c}-\frac{5 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{8 a \sqrt{a^2 c x^2+c}}+\frac{5}{16} c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{5 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{8 a}+\frac{259 c^{5/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right )}{360 a}+\frac{1}{60} c x \left (a^2 c x^2+c\right )^{3/2}+\frac{5}{24} c x \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^2-\frac{5 c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)}{36 a}+\frac{1}{6} x \left (a^2 c x^2+c\right )^{5/2} \tan ^{-1}(a x)^2-\frac{\left (a^2 c x^2+c\right )^{5/2} \tan ^{-1}(a x)}{15 a} \]

[Out]

(17*c^2*x*Sqrt[c + a^2*c*x^2])/180 + (c*x*(c + a^2*c*x^2)^(3/2))/60 - (5*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/
(8*a) - (5*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/(36*a) - ((c + a^2*c*x^2)^(5/2)*ArcTan[a*x])/(15*a) + (5*c^2*x
*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/16 + (5*c*x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/24 + (x*(c + a^2*c*x^2)^(
5/2)*ArcTan[a*x]^2)/6 - (((5*I)/8)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/(a*Sqrt[c +
a^2*c*x^2]) + (259*c^(5/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/(360*a) + (((5*I)/8)*c^3*Sqrt[1 + a^2*x
^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) - (((5*I)/8)*c^3*Sqrt[1 + a^2*x^2]
*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) - (5*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, (-
I)*E^(I*ArcTan[a*x])])/(8*a*Sqrt[c + a^2*c*x^2]) + (5*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(
8*a*Sqrt[c + a^2*c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.39048, antiderivative size = 516, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 10, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.476, Rules used = {4880, 4890, 4888, 4181, 2531, 2282, 6589, 217, 206, 195} \[ \frac{5 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt{a^2 c x^2+c}}-\frac{5 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt{a^2 c x^2+c}}-\frac{5 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt{a^2 c x^2+c}}+\frac{5 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt{a^2 c x^2+c}}+\frac{17}{180} c^2 x \sqrt{a^2 c x^2+c}-\frac{5 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{8 a \sqrt{a^2 c x^2+c}}+\frac{5}{16} c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{5 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{8 a}+\frac{259 c^{5/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right )}{360 a}+\frac{1}{60} c x \left (a^2 c x^2+c\right )^{3/2}+\frac{5}{24} c x \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^2-\frac{5 c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)}{36 a}+\frac{1}{6} x \left (a^2 c x^2+c\right )^{5/2} \tan ^{-1}(a x)^2-\frac{\left (a^2 c x^2+c\right )^{5/2} \tan ^{-1}(a x)}{15 a} \]

Antiderivative was successfully verified.

[In]

Int[(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2,x]

[Out]

(17*c^2*x*Sqrt[c + a^2*c*x^2])/180 + (c*x*(c + a^2*c*x^2)^(3/2))/60 - (5*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/
(8*a) - (5*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/(36*a) - ((c + a^2*c*x^2)^(5/2)*ArcTan[a*x])/(15*a) + (5*c^2*x
*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/16 + (5*c*x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/24 + (x*(c + a^2*c*x^2)^(
5/2)*ArcTan[a*x]^2)/6 - (((5*I)/8)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/(a*Sqrt[c +
a^2*c*x^2]) + (259*c^(5/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/(360*a) + (((5*I)/8)*c^3*Sqrt[1 + a^2*x
^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) - (((5*I)/8)*c^3*Sqrt[1 + a^2*x^2]
*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) - (5*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, (-
I)*E^(I*ArcTan[a*x])])/(8*a*Sqrt[c + a^2*c*x^2]) + (5*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(
8*a*Sqrt[c + a^2*c*x^2])

Rule 4880

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] && E
qQ[e, c^2*d] && GtQ[q, 0] && GtQ[p, 1]

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]

Rule 217

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 206

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

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rubi steps

\begin{align*} \int \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2 \, dx &=-\frac{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)}{15 a}+\frac{1}{6} x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2+\frac{1}{15} c \int \left (c+a^2 c x^2\right )^{3/2} \, dx+\frac{1}{6} (5 c) \int \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2 \, dx\\ &=\frac{1}{60} c x \left (c+a^2 c x^2\right )^{3/2}-\frac{5 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{36 a}-\frac{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)}{15 a}+\frac{5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+\frac{1}{6} x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2+\frac{1}{20} c^2 \int \sqrt{c+a^2 c x^2} \, dx+\frac{1}{36} \left (5 c^2\right ) \int \sqrt{c+a^2 c x^2} \, dx+\frac{1}{8} \left (5 c^2\right ) \int \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2 \, dx\\ &=\frac{17}{180} c^2 x \sqrt{c+a^2 c x^2}+\frac{1}{60} c x \left (c+a^2 c x^2\right )^{3/2}-\frac{5 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{8 a}-\frac{5 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{36 a}-\frac{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)}{15 a}+\frac{5}{16} c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+\frac{1}{6} x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2+\frac{1}{40} c^3 \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx+\frac{1}{72} \left (5 c^3\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx+\frac{1}{16} \left (5 c^3\right ) \int \frac{\tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx+\frac{1}{8} \left (5 c^3\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx\\ &=\frac{17}{180} c^2 x \sqrt{c+a^2 c x^2}+\frac{1}{60} c x \left (c+a^2 c x^2\right )^{3/2}-\frac{5 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{8 a}-\frac{5 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{36 a}-\frac{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)}{15 a}+\frac{5}{16} c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+\frac{1}{6} x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2+\frac{1}{40} c^3 \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )+\frac{1}{72} \left (5 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )+\frac{1}{8} \left (5 c^3\right ) \operatorname{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 (5 c^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)^2}{\sqrt{1+a^2 x^2}} \, dx}{16 \sqrt{c+a^2 c x^2}}\\ &=\frac{17}{180} c^2 x \sqrt{c+a^2 c x^2}+\frac{1}{60} c x \left (c+a^2 c x^2\right )^{3/2}-\frac{5 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{8 a}-\frac{5 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{36 a}-\frac{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)}{15 a}+\frac{5}{16} c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+\frac{1}{6} x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2+\frac{259 c^{5/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{360 a}+\frac{\left (5 c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{16 a \sqrt{c+a^2 c x^2}}\\ &=\frac{17}{180} c^2 x \sqrt{c+a^2 c x^2}+\frac{1}{60} c x \left (c+a^2 c x^2\right )^{3/2}-\frac{5 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{8 a}-\frac{5 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{36 a}-\frac{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)}{15 a}+\frac{5}{16} c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+\frac{1}{6} x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2-\frac{5 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{8 a \sqrt{c+a^2 c x^2}}+\frac{259 c^{5/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{360 a}-\frac{\left (5 c^3 \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 )}{8 a \sqrt{c+a^2 c x^2}}+\frac{\left (5 c^3 \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 )}{8 a \sqrt{c+a^2 c x^2}}\\ &=\frac{17}{180} c^2 x \sqrt{c+a^2 c x^2}+\frac{1}{60} c x \left (c+a^2 c x^2\right )^{3/2}-\frac{5 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{8 a}-\frac{5 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{36 a}-\frac{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)}{15 a}+\frac{5}{16} c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+\frac{1}{6} x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2-\frac{5 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{8 a \sqrt{c+a^2 c x^2}}+\frac{259 c^{5/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{360 a}+\frac{5 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt{c+a^2 c x^2}}-\frac{5 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt{c+a^2 c x^2}}-\frac{\left (5 i c^3 \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 )}{8 a \sqrt{c+a^2 c x^2}}+\frac{\left (5 i c^3 \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 )}{8 a \sqrt{c+a^2 c x^2}}\\ &=\frac{17}{180} c^2 x \sqrt{c+a^2 c x^2}+\frac{1}{60} c x \left (c+a^2 c x^2\right )^{3/2}-\frac{5 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{8 a}-\frac{5 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{36 a}-\frac{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)}{15 a}+\frac{5}{16} c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+\frac{1}{6} x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2-\frac{5 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{8 a \sqrt{c+a^2 c x^2}}+\frac{259 c^{5/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{360 a}+\frac{5 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt{c+a^2 c x^2}}-\frac{5 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt{c+a^2 c x^2}}-\frac{\left (5 c^3 \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 )}{8 a \sqrt{c+a^2 c x^2}}+\frac{\left (5 c^3 \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 )}{8 a \sqrt{c+a^2 c x^2}}\\ &=\frac{17}{180} c^2 x \sqrt{c+a^2 c x^2}+\frac{1}{60} c x \left (c+a^2 c x^2\right )^{3/2}-\frac{5 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{8 a}-\frac{5 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{36 a}-\frac{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)}{15 a}+\frac{5}{16} c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+\frac{1}{6} x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2-\frac{5 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{8 a \sqrt{c+a^2 c x^2}}+\frac{259 c^{5/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{360 a}+\frac{5 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt{c+a^2 c x^2}}-\frac{5 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt{c+a^2 c x^2}}-\frac{5 c^3 \sqrt{1+a^2 x^2} \text{Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt{c+a^2 c x^2}}+\frac{5 c^3 \sqrt{1+a^2 x^2} \text{Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt{c+a^2 c x^2}}\\ \end{align*}

Mathematica [A]  time = 1.57735, size = 771, normalized size = 1.49 \[ \frac{c^2 \sqrt{a^2 c x^2+c} \left (7200 i \tan ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )-7200 i \tan ^{-1}(a x) \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )-7200 \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )+7200 \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )-56 a^5 x^5 \sqrt{a^2 x^2+1}+368 a^3 x^3 \sqrt{a^2 x^2+1}+424 a x \sqrt{a^2 x^2+1}+1170 a^5 x^5 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2+12 a^4 x^4 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)+7380 a^3 x^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2+504 a^2 x^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)+11970 a x \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2-11028 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)+8288 \tanh ^{-1}\left (\frac{a x}{\sqrt{a^2 x^2+1}}\right )-108 a^6 x^6 \sin \left (3 \tan ^{-1}(a x)\right )-705 a^6 x^6 \tan ^{-1}(a x)^2 \sin \left (3 \tan ^{-1}(a x)\right )-52 a^6 x^6 \sin \left (5 \tan ^{-1}(a x)\right )+45 a^6 x^6 \tan ^{-1}(a x)^2 \sin \left (5 \tan ^{-1}(a x)\right )+156 a^4 x^4 \sin \left (3 \tan ^{-1}(a x)\right )-2835 a^4 x^4 \tan ^{-1}(a x)^2 \sin \left (3 \tan ^{-1}(a x)\right )-156 a^4 x^4 \sin \left (5 \tan ^{-1}(a x)\right )+135 a^4 x^4 \tan ^{-1}(a x)^2 \sin \left (5 \tan ^{-1}(a x)\right )+636 a^2 x^2 \sin \left (3 \tan ^{-1}(a x)\right )-3555 a^2 x^2 \tan ^{-1}(a x)^2 \sin \left (3 \tan ^{-1}(a x)\right )-156 a^2 x^2 \sin \left (5 \tan ^{-1}(a x)\right )+135 a^2 x^2 \tan ^{-1}(a x)^2 \sin \left (5 \tan ^{-1}(a x)\right )+110 a^6 x^6 \tan ^{-1}(a x) \cos \left (3 \tan ^{-1}(a x)\right )-90 a^6 x^6 \tan ^{-1}(a x) \cos \left (5 \tan ^{-1}(a x)\right )+1770 a^4 x^4 \tan ^{-1}(a x) \cos \left (3 \tan ^{-1}(a x)\right )-270 a^4 x^4 \tan ^{-1}(a x) \cos \left (5 \tan ^{-1}(a x)\right )+3210 a^2 x^2 \tan ^{-1}(a x) \cos \left (3 \tan ^{-1}(a x)\right )-270 a^2 x^2 \tan ^{-1}(a x) \cos \left (5 \tan ^{-1}(a x)\right )-7200 i \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2-1425 \tan ^{-1}(a x)^2 \sin \left (3 \tan ^{-1}(a x)\right )+372 \sin \left (3 \tan ^{-1}(a x)\right )+45 \tan ^{-1}(a x)^2 \sin \left (5 \tan ^{-1}(a x)\right )-52 \sin \left (5 \tan ^{-1}(a x)\right )+1550 \tan ^{-1}(a x) \cos \left (3 \tan ^{-1}(a x)\right )-90 \tan ^{-1}(a x) \cos \left (5 \tan ^{-1}(a x)\right )\right )}{11520 a \sqrt{a^2 x^2+1}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2,x]

[Out]

(c^2*Sqrt[c + a^2*c*x^2]*(424*a*x*Sqrt[1 + a^2*x^2] + 368*a^3*x^3*Sqrt[1 + a^2*x^2] - 56*a^5*x^5*Sqrt[1 + a^2*
x^2] - 11028*Sqrt[1 + a^2*x^2]*ArcTan[a*x] + 504*a^2*x^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x] + 12*a^4*x^4*Sqrt[1 + a
^2*x^2]*ArcTan[a*x] + 11970*a*x*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2 + 7380*a^3*x^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2
+ 1170*a^5*x^5*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2 - (7200*I)*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 + 8288*ArcT
anh[(a*x)/Sqrt[1 + a^2*x^2]] + 1550*ArcTan[a*x]*Cos[3*ArcTan[a*x]] + 3210*a^2*x^2*ArcTan[a*x]*Cos[3*ArcTan[a*x
]] + 1770*a^4*x^4*ArcTan[a*x]*Cos[3*ArcTan[a*x]] + 110*a^6*x^6*ArcTan[a*x]*Cos[3*ArcTan[a*x]] - 90*ArcTan[a*x]
*Cos[5*ArcTan[a*x]] - 270*a^2*x^2*ArcTan[a*x]*Cos[5*ArcTan[a*x]] - 270*a^4*x^4*ArcTan[a*x]*Cos[5*ArcTan[a*x]]
- 90*a^6*x^6*ArcTan[a*x]*Cos[5*ArcTan[a*x]] + (7200*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - (7200*
I)*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] - 7200*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] + 7200*PolyLog[3, I*E
^(I*ArcTan[a*x])] + 372*Sin[3*ArcTan[a*x]] + 636*a^2*x^2*Sin[3*ArcTan[a*x]] + 156*a^4*x^4*Sin[3*ArcTan[a*x]] -
108*a^6*x^6*Sin[3*ArcTan[a*x]] - 1425*ArcTan[a*x]^2*Sin[3*ArcTan[a*x]] - 3555*a^2*x^2*ArcTan[a*x]^2*Sin[3*Arc
Tan[a*x]] - 2835*a^4*x^4*ArcTan[a*x]^2*Sin[3*ArcTan[a*x]] - 705*a^6*x^6*ArcTan[a*x]^2*Sin[3*ArcTan[a*x]] - 52*
Sin[5*ArcTan[a*x]] - 156*a^2*x^2*Sin[5*ArcTan[a*x]] - 156*a^4*x^4*Sin[5*ArcTan[a*x]] - 52*a^6*x^6*Sin[5*ArcTan
[a*x]] + 45*ArcTan[a*x]^2*Sin[5*ArcTan[a*x]] + 135*a^2*x^2*ArcTan[a*x]^2*Sin[5*ArcTan[a*x]] + 135*a^4*x^4*ArcT
an[a*x]^2*Sin[5*ArcTan[a*x]] + 45*a^6*x^6*ArcTan[a*x]^2*Sin[5*ArcTan[a*x]]))/(11520*a*Sqrt[1 + a^2*x^2])

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Maple [A]  time = 0.364, size = 342, normalized size = 0.7 \begin{align*}{\frac{{c}^{2} \left ( 120\, \left ( \arctan \left ( ax \right ) \right ) ^{2}{x}^{5}{a}^{5}-48\,\arctan \left ( ax \right ){x}^{4}{a}^{4}+390\, \left ( \arctan \left ( ax \right ) \right ) ^{2}{x}^{3}{a}^{3}+12\,{a}^{3}{x}^{3}-196\,\arctan \left ( ax \right ){a}^{2}{x}^{2}+495\, \left ( \arctan \left ( ax \right ) \right ) ^{2}xa+80\,ax-598\,\arctan \left ( ax \right ) \right ) }{720\,a}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{\frac{{\frac{i}{720}}{c}^{2}}{a}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) } \left ( 225\,i \left ( \arctan \left ( ax \right ) \right ) ^{2}\ln \left ( 1+{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -225\,i \left ( \arctan \left ( ax \right ) \right ) ^{2}\ln \left ( 1-{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) +450\,\arctan \left ( ax \right ){\it polylog} \left ( 2,{\frac{-i \left ( 1+iax \right ) }{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) -450\,\arctan \left ( ax \right ){\it polylog} \left ( 2,{\frac{i \left ( 1+iax \right ) }{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) +450\,i{\it polylog} \left ( 3,{-i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -450\,i{\it polylog} \left ( 3,{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -1036\,\arctan \left ({\frac{1+iax}{\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)^(5/2)*arctan(a*x)^2,x)

[Out]

1/720*c^2/a*(c*(a*x-I)*(a*x+I))^(1/2)*(120*arctan(a*x)^2*x^5*a^5-48*arctan(a*x)*x^4*a^4+390*arctan(a*x)^2*x^3*
a^3+12*a^3*x^3-196*arctan(a*x)*a^2*x^2+495*arctan(a*x)^2*x*a+80*a*x-598*arctan(a*x))+1/720*I*c^2*(c*(a*x-I)*(a
*x+I))^(1/2)*(225*I*arctan(a*x)^2*ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-225*I*arctan(a*x)^2*ln(1-I*(1+I*a*x)/(a^
2*x^2+1)^(1/2))+450*arctan(a*x)*polylog(2,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-450*arctan(a*x)*polylog(2,I*(1+I*a*x
)/(a^2*x^2+1)^(1/2))+450*I*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-450*I*polylog(3,I*(1+I*a*x)/(a^2*x^2+1)^(
1/2))-1036*arctan((1+I*a*x)/(a^2*x^2+1)^(1/2)))/a/(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)^(5/2)*arctan(a*x)^2,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 ({\left (a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} \sqrt{a^{2} c x^{2} + c} \arctan \left (a x\right )^{2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2*c*x^2+c)^(5/2)*arctan(a*x)^2,x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a**2*c*x**2+c)**(5/2)*atan(a*x)**2,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2*c*x^2+c)^(5/2)*arctan(a*x)^2,x, algorithm="giac")

[Out]

Exception raised: TypeError