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

Optimal. Leaf size=250 \[ -\frac{8}{3} i a^3 c^3 \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )+\frac{8}{3} i a^3 c^3 \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )+\frac{1}{3} a^6 c^3 x^3 \tan ^{-1}(a x)^2-\frac{1}{3} a^5 c^3 x^2 \tan ^{-1}(a x)+\frac{1}{3} a^4 c^3 x-\frac{a^2 c^3}{3 x}+3 a^4 c^3 x \tan ^{-1}(a x)^2-\frac{2}{3} a^3 c^3 \tan ^{-1}(a x)-\frac{3 a^2 c^3 \tan ^{-1}(a x)^2}{x}+\frac{16}{3} a^3 c^3 \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)+\frac{16}{3} a^3 c^3 \log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)-\frac{a c^3 \tan ^{-1}(a x)}{3 x^2}-\frac{c^3 \tan ^{-1}(a x)^2}{3 x^3} \]

[Out]

-(a^2*c^3)/(3*x) + (a^4*c^3*x)/3 - (2*a^3*c^3*ArcTan[a*x])/3 - (a*c^3*ArcTan[a*x])/(3*x^2) - (a^5*c^3*x^2*ArcT
an[a*x])/3 - (c^3*ArcTan[a*x]^2)/(3*x^3) - (3*a^2*c^3*ArcTan[a*x]^2)/x + 3*a^4*c^3*x*ArcTan[a*x]^2 + (a^6*c^3*
x^3*ArcTan[a*x]^2)/3 + (16*a^3*c^3*ArcTan[a*x]*Log[2/(1 + I*a*x)])/3 + (16*a^3*c^3*ArcTan[a*x]*Log[2 - 2/(1 -
I*a*x)])/3 - ((8*I)/3)*a^3*c^3*PolyLog[2, -1 + 2/(1 - I*a*x)] + ((8*I)/3)*a^3*c^3*PolyLog[2, 1 - 2/(1 + I*a*x)
]

________________________________________________________________________________________

Rubi [A]  time = 0.608375, antiderivative size = 250, normalized size of antiderivative = 1., number of steps used = 28, number of rules used = 15, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.682, Rules used = {4948, 4846, 4920, 4854, 2402, 2315, 4852, 4918, 325, 203, 4924, 4868, 2447, 4916, 321} \[ -\frac{8}{3} i a^3 c^3 \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )+\frac{8}{3} i a^3 c^3 \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )+\frac{1}{3} a^6 c^3 x^3 \tan ^{-1}(a x)^2-\frac{1}{3} a^5 c^3 x^2 \tan ^{-1}(a x)+\frac{1}{3} a^4 c^3 x-\frac{a^2 c^3}{3 x}+3 a^4 c^3 x \tan ^{-1}(a x)^2-\frac{2}{3} a^3 c^3 \tan ^{-1}(a x)-\frac{3 a^2 c^3 \tan ^{-1}(a x)^2}{x}+\frac{16}{3} a^3 c^3 \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)+\frac{16}{3} a^3 c^3 \log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)-\frac{a c^3 \tan ^{-1}(a x)}{3 x^2}-\frac{c^3 \tan ^{-1}(a x)^2}{3 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^2*c^3)/(3*x) + (a^4*c^3*x)/3 - (2*a^3*c^3*ArcTan[a*x])/3 - (a*c^3*ArcTan[a*x])/(3*x^2) - (a^5*c^3*x^2*ArcT
an[a*x])/3 - (c^3*ArcTan[a*x]^2)/(3*x^3) - (3*a^2*c^3*ArcTan[a*x]^2)/x + 3*a^4*c^3*x*ArcTan[a*x]^2 + (a^6*c^3*
x^3*ArcTan[a*x]^2)/3 + (16*a^3*c^3*ArcTan[a*x]*Log[2/(1 + I*a*x)])/3 + (16*a^3*c^3*ArcTan[a*x]*Log[2 - 2/(1 -
I*a*x)])/3 - ((8*I)/3)*a^3*c^3*PolyLog[2, -1 + 2/(1 - I*a*x)] + ((8*I)/3)*a^3*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 4918

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

Rule 325

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] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && 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 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]

Rubi steps

\begin{align*} \int \frac{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2}{x^4} \, dx &=\int \left (3 a^4 c^3 \tan ^{-1}(a x)^2+\frac{c^3 \tan ^{-1}(a x)^2}{x^4}+\frac{3 a^2 c^3 \tan ^{-1}(a x)^2}{x^2}+a^6 c^3 x^2 \tan ^{-1}(a x)^2\right ) \, dx\\ &=c^3 \int \frac{\tan ^{-1}(a x)^2}{x^4} \, dx+\left (3 a^2 c^3\right ) \int \frac{\tan ^{-1}(a x)^2}{x^2} \, dx+\left (3 a^4 c^3\right ) \int \tan ^{-1}(a x)^2 \, dx+\left (a^6 c^3\right ) \int x^2 \tan ^{-1}(a x)^2 \, dx\\ &=-\frac{c^3 \tan ^{-1}(a x)^2}{3 x^3}-\frac{3 a^2 c^3 \tan ^{-1}(a x)^2}{x}+3 a^4 c^3 x \tan ^{-1}(a x)^2+\frac{1}{3} a^6 c^3 x^3 \tan ^{-1}(a x)^2+\frac{1}{3} \left (2 a c^3\right ) \int \frac{\tan ^{-1}(a x)}{x^3 \left (1+a^2 x^2\right )} \, dx+\left (6 a^3 c^3\right ) \int \frac{\tan ^{-1}(a x)}{x \left (1+a^2 x^2\right )} \, dx-\left (6 a^5 c^3\right ) \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{3} \left (2 a^7 c^3\right ) \int \frac{x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=-\frac{c^3 \tan ^{-1}(a x)^2}{3 x^3}-\frac{3 a^2 c^3 \tan ^{-1}(a x)^2}{x}+3 a^4 c^3 x \tan ^{-1}(a x)^2+\frac{1}{3} a^6 c^3 x^3 \tan ^{-1}(a x)^2+\frac{1}{3} \left (2 a c^3\right ) \int \frac{\tan ^{-1}(a x)}{x^3} \, dx+\left (6 i a^3 c^3\right ) \int \frac{\tan ^{-1}(a x)}{x (i+a x)} \, dx-\frac{1}{3} \left (2 a^3 c^3\right ) \int \frac{\tan ^{-1}(a x)}{x \left (1+a^2 x^2\right )} \, dx+\left (6 a^4 c^3\right ) \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx-\frac{1}{3} \left (2 a^5 c^3\right ) \int x \tan ^{-1}(a x) \, dx+\frac{1}{3} \left (2 a^5 c^3\right ) \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=-\frac{a c^3 \tan ^{-1}(a x)}{3 x^2}-\frac{1}{3} a^5 c^3 x^2 \tan ^{-1}(a x)-\frac{c^3 \tan ^{-1}(a x)^2}{3 x^3}-\frac{3 a^2 c^3 \tan ^{-1}(a x)^2}{x}+3 a^4 c^3 x \tan ^{-1}(a x)^2+\frac{1}{3} a^6 c^3 x^3 \tan ^{-1}(a x)^2+6 a^3 c^3 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )+6 a^3 c^3 \tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )+\frac{1}{3} \left (a^2 c^3\right ) \int \frac{1}{x^2 \left (1+a^2 x^2\right )} \, dx-\frac{1}{3} \left (2 i a^3 c^3\right ) \int \frac{\tan ^{-1}(a x)}{x (i+a x)} \, dx-\frac{1}{3} \left (2 a^4 c^3\right ) \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx-\left (6 a^4 c^3\right ) \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (6 a^4 c^3\right ) \int \frac{\log \left (2-\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx+\frac{1}{3} \left (a^6 c^3\right ) \int \frac{x^2}{1+a^2 x^2} \, dx\\ &=-\frac{a^2 c^3}{3 x}+\frac{1}{3} a^4 c^3 x-\frac{a c^3 \tan ^{-1}(a x)}{3 x^2}-\frac{1}{3} a^5 c^3 x^2 \tan ^{-1}(a x)-\frac{c^3 \tan ^{-1}(a x)^2}{3 x^3}-\frac{3 a^2 c^3 \tan ^{-1}(a x)^2}{x}+3 a^4 c^3 x \tan ^{-1}(a x)^2+\frac{1}{3} a^6 c^3 x^3 \tan ^{-1}(a x)^2+\frac{16}{3} a^3 c^3 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )+\frac{16}{3} a^3 c^3 \tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )-3 i a^3 c^3 \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )+\left (6 i a^3 c^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i a x}\right )-2 \left (\frac{1}{3} \left (a^4 c^3\right ) \int \frac{1}{1+a^2 x^2} \, dx\right )+\frac{1}{3} \left (2 a^4 c^3\right ) \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx+\frac{1}{3} \left (2 a^4 c^3\right ) \int \frac{\log \left (2-\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx\\ &=-\frac{a^2 c^3}{3 x}+\frac{1}{3} a^4 c^3 x-\frac{2}{3} a^3 c^3 \tan ^{-1}(a x)-\frac{a c^3 \tan ^{-1}(a x)}{3 x^2}-\frac{1}{3} a^5 c^3 x^2 \tan ^{-1}(a x)-\frac{c^3 \tan ^{-1}(a x)^2}{3 x^3}-\frac{3 a^2 c^3 \tan ^{-1}(a x)^2}{x}+3 a^4 c^3 x \tan ^{-1}(a x)^2+\frac{1}{3} a^6 c^3 x^3 \tan ^{-1}(a x)^2+\frac{16}{3} a^3 c^3 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )+\frac{16}{3} a^3 c^3 \tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )-\frac{8}{3} i a^3 c^3 \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )+3 i a^3 c^3 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )-\frac{1}{3} \left (2 i a^3 c^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i a x}\right )\\ &=-\frac{a^2 c^3}{3 x}+\frac{1}{3} a^4 c^3 x-\frac{2}{3} a^3 c^3 \tan ^{-1}(a x)-\frac{a c^3 \tan ^{-1}(a x)}{3 x^2}-\frac{1}{3} a^5 c^3 x^2 \tan ^{-1}(a x)-\frac{c^3 \tan ^{-1}(a x)^2}{3 x^3}-\frac{3 a^2 c^3 \tan ^{-1}(a x)^2}{x}+3 a^4 c^3 x \tan ^{-1}(a x)^2+\frac{1}{3} a^6 c^3 x^3 \tan ^{-1}(a x)^2+\frac{16}{3} a^3 c^3 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )+\frac{16}{3} a^3 c^3 \tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )-\frac{8}{3} i a^3 c^3 \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )+\frac{8}{3} i a^3 c^3 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )\\ \end{align*}

Mathematica [A]  time = 0.573921, size = 221, normalized size = 0.88 \[ \frac{c^3 \left (-8 i a^3 x^3 \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(a x)}\right )-8 i a^3 x^3 \text{PolyLog}\left (2,e^{2 i \tan ^{-1}(a x)}\right )+a^4 x^4-a^2 x^2+a^6 x^6 \tan ^{-1}(a x)^2-a^5 x^5 \tan ^{-1}(a x)+9 a^4 x^4 \tan ^{-1}(a x)^2-16 i a^3 x^3 \tan ^{-1}(a x)^2-2 a^3 x^3 \tan ^{-1}(a x)-9 a^2 x^2 \tan ^{-1}(a x)^2+16 a^3 x^3 \tan ^{-1}(a x) \log \left (1-e^{2 i \tan ^{-1}(a x)}\right )+16 a^3 x^3 \tan ^{-1}(a x) \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )-a x \tan ^{-1}(a x)-\tan ^{-1}(a x)^2\right )}{3 x^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c^3*(-(a^2*x^2) + a^4*x^4 - a*x*ArcTan[a*x] - 2*a^3*x^3*ArcTan[a*x] - a^5*x^5*ArcTan[a*x] - ArcTan[a*x]^2 - 9
*a^2*x^2*ArcTan[a*x]^2 - (16*I)*a^3*x^3*ArcTan[a*x]^2 + 9*a^4*x^4*ArcTan[a*x]^2 + a^6*x^6*ArcTan[a*x]^2 + 16*a
^3*x^3*ArcTan[a*x]*Log[1 - E^((2*I)*ArcTan[a*x])] + 16*a^3*x^3*ArcTan[a*x]*Log[1 + E^((2*I)*ArcTan[a*x])] - (8
*I)*a^3*x^3*PolyLog[2, -E^((2*I)*ArcTan[a*x])] - (8*I)*a^3*x^3*PolyLog[2, E^((2*I)*ArcTan[a*x])]))/(3*x^3)

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Maple [A]  time = 0.106, size = 417, normalized size = 1.7 \begin{align*}{\frac{{a}^{6}{c}^{3}{x}^{3} \left ( \arctan \left ( ax \right ) \right ) ^{2}}{3}}+3\,{a}^{4}{c}^{3}x \left ( \arctan \left ( ax \right ) \right ) ^{2}-3\,{\frac{{a}^{2}{c}^{3} \left ( \arctan \left ( ax \right ) \right ) ^{2}}{x}}-{\frac{{c}^{3} \left ( \arctan \left ( ax \right ) \right ) ^{2}}{3\,{x}^{3}}}-{\frac{{a}^{5}{c}^{3}{x}^{2}\arctan \left ( ax \right ) }{3}}-{\frac{16\,{a}^{3}{c}^{3}\arctan \left ( ax \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) }{3}}-{\frac{a{c}^{3}\arctan \left ( ax \right ) }{3\,{x}^{2}}}+{\frac{16\,{a}^{3}{c}^{3}\arctan \left ( ax \right ) \ln \left ( ax \right ) }{3}}+{\frac{{a}^{4}{c}^{3}x}{3}}-{\frac{2\,{a}^{3}{c}^{3}\arctan \left ( ax \right ) }{3}}-{\frac{{c}^{3}{a}^{2}}{3\,x}}+{\frac{8\,i}{3}}{a}^{3}{c}^{3}{\it dilog} \left ( 1+iax \right ) +{\frac{8\,i}{3}}{a}^{3}{c}^{3}\ln \left ( ax-i \right ) \ln \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) +{\frac{8\,i}{3}}{a}^{3}{c}^{3}{\it dilog} \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) -{\frac{8\,i}{3}}{a}^{3}{c}^{3}\ln \left ( ax \right ) \ln \left ( 1-iax \right ) +{\frac{8\,i}{3}}{a}^{3}{c}^{3}\ln \left ({a}^{2}{x}^{2}+1 \right ) \ln \left ( ax+i \right ) -{\frac{8\,i}{3}}{a}^{3}{c}^{3}\ln \left ( ax+i \right ) \ln \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) +{\frac{8\,i}{3}}{a}^{3}{c}^{3}\ln \left ( ax \right ) \ln \left ( 1+iax \right ) +{\frac{4\,i}{3}}{a}^{3}{c}^{3} \left ( \ln \left ( ax-i \right ) \right ) ^{2}-{\frac{4\,i}{3}}{a}^{3}{c}^{3} \left ( \ln \left ( ax+i \right ) \right ) ^{2}-{\frac{8\,i}{3}}{a}^{3}{c}^{3}{\it dilog} \left ( 1-iax \right ) -{\frac{8\,i}{3}}{a}^{3}{c}^{3}\ln \left ({a}^{2}{x}^{2}+1 \right ) \ln \left ( ax-i \right ) -{\frac{8\,i}{3}}{a}^{3}{c}^{3}{\it dilog} \left ({\frac{i}{2}} \left ( ax-i \right ) \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^4,x)

[Out]

1/3*a^6*c^3*x^3*arctan(a*x)^2+3*a^4*c^3*x*arctan(a*x)^2-3*a^2*c^3*arctan(a*x)^2/x-1/3*c^3*arctan(a*x)^2/x^3-1/
3*a^5*c^3*x^2*arctan(a*x)-16/3*a^3*c^3*arctan(a*x)*ln(a^2*x^2+1)-1/3*a*c^3*arctan(a*x)/x^2+16/3*a^3*c^3*arctan
(a*x)*ln(a*x)+1/3*a^4*c^3*x-2/3*a^3*c^3*arctan(a*x)-1/3*a^2*c^3/x+8/3*I*a^3*c^3*dilog(1+I*a*x)+8/3*I*a^3*c^3*l
n(a*x-I)*ln(-1/2*I*(a*x+I))+8/3*I*a^3*c^3*dilog(-1/2*I*(a*x+I))-8/3*I*a^3*c^3*ln(a*x)*ln(1-I*a*x)+8/3*I*a^3*c^
3*ln(a^2*x^2+1)*ln(a*x+I)-8/3*I*a^3*c^3*ln(a*x+I)*ln(1/2*I*(a*x-I))+8/3*I*a^3*c^3*ln(a*x)*ln(1+I*a*x)+4/3*I*a^
3*c^3*ln(a*x-I)^2-4/3*I*a^3*c^3*ln(a*x+I)^2-8/3*I*a^3*c^3*dilog(1-I*a*x)-8/3*I*a^3*c^3*ln(a^2*x^2+1)*ln(a*x-I)
-8/3*I*a^3*c^3*dilog(1/2*I*(a*x-I))

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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^4,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^{4}}, 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^4,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^4, x)

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

[Out]

c**3*(Integral(3*a**4*atan(a*x)**2, x) + Integral(atan(a*x)**2/x**4, x) + Integral(3*a**2*atan(a*x)**2/x**2, x
) + Integral(a**6*x**2*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^{4}}\,{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^4,x, algorithm="giac")

[Out]

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