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

Optimal. Leaf size=251 \[ -i a c^3 \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )+\frac{11}{5} i a c^3 \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )+\frac{1}{30} a^4 c^3 x^3+\frac{1}{5} a^6 c^3 x^5 \tan ^{-1}(a x)^2-\frac{1}{10} a^5 c^3 x^4 \tan ^{-1}(a x)+a^4 c^3 x^3 \tan ^{-1}(a x)^2-\frac{4}{5} a^3 c^3 x^2 \tan ^{-1}(a x)+\frac{7}{10} a^2 c^3 x+3 a^2 c^3 x \tan ^{-1}(a x)^2+\frac{6}{5} i a c^3 \tan ^{-1}(a x)^2-\frac{7}{10} a c^3 \tan ^{-1}(a x)-\frac{c^3 \tan ^{-1}(a x)^2}{x}+\frac{22}{5} a c^3 \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)+2 a c^3 \log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x) \]

[Out]

(7*a^2*c^3*x)/10 + (a^4*c^3*x^3)/30 - (7*a*c^3*ArcTan[a*x])/10 - (4*a^3*c^3*x^2*ArcTan[a*x])/5 - (a^5*c^3*x^4*
ArcTan[a*x])/10 + ((6*I)/5)*a*c^3*ArcTan[a*x]^2 - (c^3*ArcTan[a*x]^2)/x + 3*a^2*c^3*x*ArcTan[a*x]^2 + a^4*c^3*
x^3*ArcTan[a*x]^2 + (a^6*c^3*x^5*ArcTan[a*x]^2)/5 + (22*a*c^3*ArcTan[a*x]*Log[2/(1 + I*a*x)])/5 + 2*a*c^3*ArcT
an[a*x]*Log[2 - 2/(1 - I*a*x)] - I*a*c^3*PolyLog[2, -1 + 2/(1 - I*a*x)] + ((11*I)/5)*a*c^3*PolyLog[2, 1 - 2/(1
+ I*a*x)]

________________________________________________________________________________________

Rubi [A]  time = 0.64533, antiderivative size = 251, normalized size of antiderivative = 1., number of steps used = 34, number of rules used = 14, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.636, Rules used = {4948, 4846, 4920, 4854, 2402, 2315, 4852, 4924, 4868, 2447, 4916, 321, 203, 302} \[ -i a c^3 \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )+\frac{11}{5} i a c^3 \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )+\frac{1}{30} a^4 c^3 x^3+\frac{1}{5} a^6 c^3 x^5 \tan ^{-1}(a x)^2-\frac{1}{10} a^5 c^3 x^4 \tan ^{-1}(a x)+a^4 c^3 x^3 \tan ^{-1}(a x)^2-\frac{4}{5} a^3 c^3 x^2 \tan ^{-1}(a x)+\frac{7}{10} a^2 c^3 x+3 a^2 c^3 x \tan ^{-1}(a x)^2+\frac{6}{5} i a c^3 \tan ^{-1}(a x)^2-\frac{7}{10} a c^3 \tan ^{-1}(a x)-\frac{c^3 \tan ^{-1}(a x)^2}{x}+\frac{22}{5} a c^3 \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)+2 a c^3 \log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(7*a^2*c^3*x)/10 + (a^4*c^3*x^3)/30 - (7*a*c^3*ArcTan[a*x])/10 - (4*a^3*c^3*x^2*ArcTan[a*x])/5 - (a^5*c^3*x^4*
ArcTan[a*x])/10 + ((6*I)/5)*a*c^3*ArcTan[a*x]^2 - (c^3*ArcTan[a*x]^2)/x + 3*a^2*c^3*x*ArcTan[a*x]^2 + a^4*c^3*
x^3*ArcTan[a*x]^2 + (a^6*c^3*x^5*ArcTan[a*x]^2)/5 + (22*a*c^3*ArcTan[a*x]*Log[2/(1 + I*a*x)])/5 + 2*a*c^3*ArcT
an[a*x]*Log[2 - 2/(1 - I*a*x)] - I*a*c^3*PolyLog[2, -1 + 2/(1 - I*a*x)] + ((11*I)/5)*a*c^3*PolyLog[2, 1 - 2/(1
+ I*a*x)]

Rule 4948

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

Rule 4846

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

Rule 4920

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

Rule 4854

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

Rule 2402

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> -Dist[e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rule 4852

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcTa
n[c*x])^p)/(d*(m + 1)), x] - Dist[(b*c*p)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcTan[c*x])^(p - 1))/(1 + c^
2*x^2), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && (EqQ[p, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 4924

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

Rule 4868

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

Rule 2447

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[(Pq^m*(1 - u))/D[u, x]]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

Rule 4916

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

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 203

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

Rule 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rubi steps

\begin{align*} \int \frac{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2}{x^2} \, dx &=\int \left (3 a^2 c^3 \tan ^{-1}(a x)^2+\frac{c^3 \tan ^{-1}(a x)^2}{x^2}+3 a^4 c^3 x^2 \tan ^{-1}(a x)^2+a^6 c^3 x^4 \tan ^{-1}(a x)^2\right ) \, dx\\ &=c^3 \int \frac{\tan ^{-1}(a x)^2}{x^2} \, dx+\left (3 a^2 c^3\right ) \int \tan ^{-1}(a x)^2 \, dx+\left (3 a^4 c^3\right ) \int x^2 \tan ^{-1}(a x)^2 \, dx+\left (a^6 c^3\right ) \int x^4 \tan ^{-1}(a x)^2 \, dx\\ &=-\frac{c^3 \tan ^{-1}(a x)^2}{x}+3 a^2 c^3 x \tan ^{-1}(a x)^2+a^4 c^3 x^3 \tan ^{-1}(a x)^2+\frac{1}{5} a^6 c^3 x^5 \tan ^{-1}(a x)^2+\left (2 a c^3\right ) \int \frac{\tan ^{-1}(a x)}{x \left (1+a^2 x^2\right )} \, dx-\left (6 a^3 c^3\right ) \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\left (2 a^5 c^3\right ) \int \frac{x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{5} \left (2 a^7 c^3\right ) \int \frac{x^5 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=2 i a c^3 \tan ^{-1}(a x)^2-\frac{c^3 \tan ^{-1}(a x)^2}{x}+3 a^2 c^3 x \tan ^{-1}(a x)^2+a^4 c^3 x^3 \tan ^{-1}(a x)^2+\frac{1}{5} a^6 c^3 x^5 \tan ^{-1}(a x)^2+\left (2 i a c^3\right ) \int \frac{\tan ^{-1}(a x)}{x (i+a x)} \, dx+\left (6 a^2 c^3\right ) \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx-\left (2 a^3 c^3\right ) \int x \tan ^{-1}(a x) \, dx+\left (2 a^3 c^3\right ) \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{5} \left (2 a^5 c^3\right ) \int x^3 \tan ^{-1}(a x) \, dx+\frac{1}{5} \left (2 a^5 c^3\right ) \int \frac{x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=-a^3 c^3 x^2 \tan ^{-1}(a x)-\frac{1}{10} a^5 c^3 x^4 \tan ^{-1}(a x)+i a c^3 \tan ^{-1}(a x)^2-\frac{c^3 \tan ^{-1}(a x)^2}{x}+3 a^2 c^3 x \tan ^{-1}(a x)^2+a^4 c^3 x^3 \tan ^{-1}(a x)^2+\frac{1}{5} a^6 c^3 x^5 \tan ^{-1}(a x)^2+6 a c^3 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )+2 a c^3 \tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )-\left (2 a^2 c^3\right ) \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx-\left (2 a^2 c^3\right ) \int \frac{\log \left (2-\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx-\left (6 a^2 c^3\right ) \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx+\frac{1}{5} \left (2 a^3 c^3\right ) \int x \tan ^{-1}(a x) \, dx-\frac{1}{5} \left (2 a^3 c^3\right ) \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\left (a^4 c^3\right ) \int \frac{x^2}{1+a^2 x^2} \, dx+\frac{1}{10} \left (a^6 c^3\right ) \int \frac{x^4}{1+a^2 x^2} \, dx\\ &=a^2 c^3 x-\frac{4}{5} a^3 c^3 x^2 \tan ^{-1}(a x)-\frac{1}{10} a^5 c^3 x^4 \tan ^{-1}(a x)+\frac{6}{5} i a c^3 \tan ^{-1}(a x)^2-\frac{c^3 \tan ^{-1}(a x)^2}{x}+3 a^2 c^3 x \tan ^{-1}(a x)^2+a^4 c^3 x^3 \tan ^{-1}(a x)^2+\frac{1}{5} a^6 c^3 x^5 \tan ^{-1}(a x)^2+4 a c^3 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )+2 a c^3 \tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )-i a c^3 \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )+\left (6 i a c^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i a x}\right )+\frac{1}{5} \left (2 a^2 c^3\right ) \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx-\left (a^2 c^3\right ) \int \frac{1}{1+a^2 x^2} \, dx+\left (2 a^2 c^3\right ) \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\frac{1}{5} \left (a^4 c^3\right ) \int \frac{x^2}{1+a^2 x^2} \, dx+\frac{1}{10} \left (a^6 c^3\right ) \int \left (-\frac{1}{a^4}+\frac{x^2}{a^2}+\frac{1}{a^4 \left (1+a^2 x^2\right )}\right ) \, dx\\ &=\frac{7}{10} a^2 c^3 x+\frac{1}{30} a^4 c^3 x^3-a c^3 \tan ^{-1}(a x)-\frac{4}{5} a^3 c^3 x^2 \tan ^{-1}(a x)-\frac{1}{10} a^5 c^3 x^4 \tan ^{-1}(a x)+\frac{6}{5} i a c^3 \tan ^{-1}(a x)^2-\frac{c^3 \tan ^{-1}(a x)^2}{x}+3 a^2 c^3 x \tan ^{-1}(a x)^2+a^4 c^3 x^3 \tan ^{-1}(a x)^2+\frac{1}{5} a^6 c^3 x^5 \tan ^{-1}(a x)^2+\frac{22}{5} a c^3 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )+2 a c^3 \tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )-i a c^3 \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )+3 i a c^3 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )-\left (2 i a c^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i a x}\right )+\frac{1}{10} \left (a^2 c^3\right ) \int \frac{1}{1+a^2 x^2} \, dx+\frac{1}{5} \left (a^2 c^3\right ) \int \frac{1}{1+a^2 x^2} \, dx-\frac{1}{5} \left (2 a^2 c^3\right ) \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx\\ &=\frac{7}{10} a^2 c^3 x+\frac{1}{30} a^4 c^3 x^3-\frac{7}{10} a c^3 \tan ^{-1}(a x)-\frac{4}{5} a^3 c^3 x^2 \tan ^{-1}(a x)-\frac{1}{10} a^5 c^3 x^4 \tan ^{-1}(a x)+\frac{6}{5} i a c^3 \tan ^{-1}(a x)^2-\frac{c^3 \tan ^{-1}(a x)^2}{x}+3 a^2 c^3 x \tan ^{-1}(a x)^2+a^4 c^3 x^3 \tan ^{-1}(a x)^2+\frac{1}{5} a^6 c^3 x^5 \tan ^{-1}(a x)^2+\frac{22}{5} a c^3 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )+2 a c^3 \tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )-i a c^3 \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )+2 i a c^3 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )+\frac{1}{5} \left (2 i a c^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i a x}\right )\\ &=\frac{7}{10} a^2 c^3 x+\frac{1}{30} a^4 c^3 x^3-\frac{7}{10} a c^3 \tan ^{-1}(a x)-\frac{4}{5} a^3 c^3 x^2 \tan ^{-1}(a x)-\frac{1}{10} a^5 c^3 x^4 \tan ^{-1}(a x)+\frac{6}{5} i a c^3 \tan ^{-1}(a x)^2-\frac{c^3 \tan ^{-1}(a x)^2}{x}+3 a^2 c^3 x \tan ^{-1}(a x)^2+a^4 c^3 x^3 \tan ^{-1}(a x)^2+\frac{1}{5} a^6 c^3 x^5 \tan ^{-1}(a x)^2+\frac{22}{5} a c^3 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )+2 a c^3 \tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )-i a c^3 \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )+\frac{11}{5} i a c^3 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )\\ \end{align*}

Mathematica [A]  time = 0.705295, size = 202, normalized size = 0.8 \[ \frac{c^3 \left (-66 i a x \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(a x)}\right )-30 i a x \text{PolyLog}\left (2,e^{2 i \tan ^{-1}(a x)}\right )+a^4 x^4+21 a^2 x^2+6 a^6 x^6 \tan ^{-1}(a x)^2-3 a^5 x^5 \tan ^{-1}(a x)+30 a^4 x^4 \tan ^{-1}(a x)^2-24 a^3 x^3 \tan ^{-1}(a x)+90 a^2 x^2 \tan ^{-1}(a x)^2-96 i a x \tan ^{-1}(a x)^2-21 a x \tan ^{-1}(a x)-30 \tan ^{-1}(a x)^2+60 a x \tan ^{-1}(a x) \log \left (1-e^{2 i \tan ^{-1}(a x)}\right )+132 a x \tan ^{-1}(a x) \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )\right )}{30 x} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c^3*(21*a^2*x^2 + a^4*x^4 - 21*a*x*ArcTan[a*x] - 24*a^3*x^3*ArcTan[a*x] - 3*a^5*x^5*ArcTan[a*x] - 30*ArcTan[a
*x]^2 - (96*I)*a*x*ArcTan[a*x]^2 + 90*a^2*x^2*ArcTan[a*x]^2 + 30*a^4*x^4*ArcTan[a*x]^2 + 6*a^6*x^6*ArcTan[a*x]
^2 + 60*a*x*ArcTan[a*x]*Log[1 - E^((2*I)*ArcTan[a*x])] + 132*a*x*ArcTan[a*x]*Log[1 + E^((2*I)*ArcTan[a*x])] -
(66*I)*a*x*PolyLog[2, -E^((2*I)*ArcTan[a*x])] - (30*I)*a*x*PolyLog[2, E^((2*I)*ArcTan[a*x])]))/(30*x)

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Maple [A]  time = 0.097, size = 388, normalized size = 1.6 \begin{align*}{\frac{{a}^{6}{c}^{3}{x}^{5} \left ( \arctan \left ( ax \right ) \right ) ^{2}}{5}}+{a}^{4}{c}^{3}{x}^{3} \left ( \arctan \left ( ax \right ) \right ) ^{2}+3\,{a}^{2}{c}^{3}x \left ( \arctan \left ( ax \right ) \right ) ^{2}-{\frac{{c}^{3} \left ( \arctan \left ( ax \right ) \right ) ^{2}}{x}}-{\frac{{a}^{5}{c}^{3}{x}^{4}\arctan \left ( ax \right ) }{10}}-{\frac{4\,{a}^{3}{c}^{3}{x}^{2}\arctan \left ( ax \right ) }{5}}-{\frac{16\,a{c}^{3}\arctan \left ( ax \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) }{5}}+2\,a{c}^{3}\arctan \left ( ax \right ) \ln \left ( ax \right ) +{\frac{{a}^{4}{c}^{3}{x}^{3}}{30}}+{\frac{7\,{a}^{2}{c}^{3}x}{10}}-{\frac{7\,a{c}^{3}\arctan \left ( ax \right ) }{10}}+ia{c}^{3}{\it dilog} \left ( 1+iax \right ) +ia{c}^{3}\ln \left ( ax \right ) \ln \left ( 1+iax \right ) -{\frac{8\,i}{5}}a{c}^{3}\ln \left ({a}^{2}{x}^{2}+1 \right ) \ln \left ( ax-i \right ) -{\frac{4\,i}{5}}a{c}^{3} \left ( \ln \left ( ax+i \right ) \right ) ^{2}-{\frac{8\,i}{5}}a{c}^{3}\ln \left ( ax+i \right ) \ln \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) -ia{c}^{3}{\it dilog} \left ( 1-iax \right ) +{\frac{4\,i}{5}}a{c}^{3} \left ( \ln \left ( ax-i \right ) \right ) ^{2}-{\frac{8\,i}{5}}a{c}^{3}{\it dilog} \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) +{\frac{8\,i}{5}}a{c}^{3}{\it dilog} \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) +{\frac{8\,i}{5}}a{c}^{3}\ln \left ({a}^{2}{x}^{2}+1 \right ) \ln \left ( ax+i \right ) +{\frac{8\,i}{5}}a{c}^{3}\ln \left ( ax-i \right ) \ln \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) -ia{c}^{3}\ln \left ( ax \right ) \ln \left ( 1-iax \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*a^6*c^3*x^5*arctan(a*x)^2+a^4*c^3*x^3*arctan(a*x)^2+3*a^2*c^3*x*arctan(a*x)^2-c^3*arctan(a*x)^2/x-1/10*a^5
*c^3*x^4*arctan(a*x)-4/5*a^3*c^3*x^2*arctan(a*x)-16/5*a*c^3*arctan(a*x)*ln(a^2*x^2+1)+2*a*c^3*arctan(a*x)*ln(a
*x)+1/30*a^4*c^3*x^3+7/10*a^2*c^3*x-7/10*a*c^3*arctan(a*x)+I*a*c^3*dilog(1+I*a*x)+I*a*c^3*ln(a*x)*ln(1+I*a*x)-
8/5*I*a*c^3*ln(a^2*x^2+1)*ln(a*x-I)-4/5*I*a*c^3*ln(a*x+I)^2-8/5*I*a*c^3*ln(a*x+I)*ln(1/2*I*(a*x-I))-I*a*c^3*di
log(1-I*a*x)+4/5*I*a*c^3*ln(a*x-I)^2-8/5*I*a*c^3*dilog(1/2*I*(a*x-I))+8/5*I*a*c^3*dilog(-1/2*I*(a*x+I))+8/5*I*
a*c^3*ln(a^2*x^2+1)*ln(a*x+I)+8/5*I*a*c^3*ln(a*x-I)*ln(-1/2*I*(a*x+I))-I*a*c^3*ln(a*x)*ln(1-I*a*x)

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Maxima [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)^3*arctan(a*x)^2/x^2,x, algorithm="maxima")

[Out]

Timed out

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (a^{6} c^{3} x^{6} + 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} + c^{3}\right )} \arctan \left (a x\right )^{2}}{x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((a^6*c^3*x^6 + 3*a^4*c^3*x^4 + 3*a^2*c^3*x^2 + c^3)*arctan(a*x)^2/x^2, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} c^{3} \left (\int 3 a^{2} \operatorname{atan}^{2}{\left (a x \right )}\, dx + \int \frac{\operatorname{atan}^{2}{\left (a x \right )}}{x^{2}}\, dx + \int 3 a^{4} x^{2} \operatorname{atan}^{2}{\left (a x \right )}\, dx + \int a^{6} x^{4} \operatorname{atan}^{2}{\left (a x \right )}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c**3*(Integral(3*a**2*atan(a*x)**2, x) + Integral(atan(a*x)**2/x**2, x) + Integral(3*a**4*x**2*atan(a*x)**2, x
) + Integral(a**6*x**4*atan(a*x)**2, 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 )}^{3} \arctan \left (a x\right )^{2}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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