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

Optimal. Leaf size=274 \[ \frac {1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)^2-\frac {1}{36} a^5 c^3 x^8 \tan ^{-1}(a x)+\frac {1}{252} a^4 c^3 x^7+\frac {3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)^2-\frac {16 i c^3 \text {Li}_2\left (1-\frac {2}{i a x+1}\right )}{315 a^3}-\frac {20}{189} a^3 c^3 x^6 \tan ^{-1}(a x)-\frac {16 i c^3 \tan ^{-1}(a x)^2}{315 a^3}+\frac {47 c^3 \tan ^{-1}(a x)}{3780 a^3}-\frac {32 c^3 \log \left (\frac {2}{1+i a x}\right ) \tan ^{-1}(a x)}{315 a^3}+\frac {59 a^2 c^3 x^5}{3780}+\frac {3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)^2-\frac {47 c^3 x}{3780 a^2}-\frac {89}{630} a c^3 x^4 \tan ^{-1}(a x)+\frac {1}{3} c^3 x^3 \tan ^{-1}(a x)^2-\frac {16 c^3 x^2 \tan ^{-1}(a x)}{315 a}+\frac {239 c^3 x^3}{11340} \]

[Out]

-47/3780*c^3*x/a^2+239/11340*c^3*x^3+59/3780*a^2*c^3*x^5+1/252*a^4*c^3*x^7+47/3780*c^3*arctan(a*x)/a^3-16/315*
c^3*x^2*arctan(a*x)/a-89/630*a*c^3*x^4*arctan(a*x)-20/189*a^3*c^3*x^6*arctan(a*x)-1/36*a^5*c^3*x^8*arctan(a*x)
-16/315*I*c^3*arctan(a*x)^2/a^3+1/3*c^3*x^3*arctan(a*x)^2+3/5*a^2*c^3*x^5*arctan(a*x)^2+3/7*a^4*c^3*x^7*arctan
(a*x)^2+1/9*a^6*c^3*x^9*arctan(a*x)^2-32/315*c^3*arctan(a*x)*ln(2/(1+I*a*x))/a^3-16/315*I*c^3*polylog(2,1-2/(1
+I*a*x))/a^3

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Rubi [A]  time = 1.15, antiderivative size = 274, normalized size of antiderivative = 1.00, number of steps used = 68, number of rules used = 10, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {4948, 4852, 4916, 321, 203, 4920, 4854, 2402, 2315, 302} \[ -\frac {16 i c^3 \text {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{315 a^3}+\frac {1}{252} a^4 c^3 x^7+\frac {59 a^2 c^3 x^5}{3780}+\frac {1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)^2-\frac {1}{36} a^5 c^3 x^8 \tan ^{-1}(a x)+\frac {3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)^2-\frac {20}{189} a^3 c^3 x^6 \tan ^{-1}(a x)+\frac {3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)^2-\frac {47 c^3 x}{3780 a^2}-\frac {16 i c^3 \tan ^{-1}(a x)^2}{315 a^3}+\frac {47 c^3 \tan ^{-1}(a x)}{3780 a^3}-\frac {32 c^3 \log \left (\frac {2}{1+i a x}\right ) \tan ^{-1}(a x)}{315 a^3}-\frac {89}{630} a c^3 x^4 \tan ^{-1}(a x)+\frac {1}{3} c^3 x^3 \tan ^{-1}(a x)^2-\frac {16 c^3 x^2 \tan ^{-1}(a x)}{315 a}+\frac {239 c^3 x^3}{11340} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-47*c^3*x)/(3780*a^2) + (239*c^3*x^3)/11340 + (59*a^2*c^3*x^5)/3780 + (a^4*c^3*x^7)/252 + (47*c^3*ArcTan[a*x]
)/(3780*a^3) - (16*c^3*x^2*ArcTan[a*x])/(315*a) - (89*a*c^3*x^4*ArcTan[a*x])/630 - (20*a^3*c^3*x^6*ArcTan[a*x]
)/189 - (a^5*c^3*x^8*ArcTan[a*x])/36 - (((16*I)/315)*c^3*ArcTan[a*x]^2)/a^3 + (c^3*x^3*ArcTan[a*x]^2)/3 + (3*a
^2*c^3*x^5*ArcTan[a*x]^2)/5 + (3*a^4*c^3*x^7*ArcTan[a*x]^2)/7 + (a^6*c^3*x^9*ArcTan[a*x]^2)/9 - (32*c^3*ArcTan
[a*x]*Log[2/(1 + I*a*x)])/(315*a^3) - (((16*I)/315)*c^3*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^3

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]

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 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 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 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 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 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 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 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])

Rubi steps

\begin {align*} \int x^2 \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2 \, dx &=\int \left (c^3 x^2 \tan ^{-1}(a x)^2+3 a^2 c^3 x^4 \tan ^{-1}(a x)^2+3 a^4 c^3 x^6 \tan ^{-1}(a x)^2+a^6 c^3 x^8 \tan ^{-1}(a x)^2\right ) \, dx\\ &=c^3 \int x^2 \tan ^{-1}(a x)^2 \, dx+\left (3 a^2 c^3\right ) \int x^4 \tan ^{-1}(a x)^2 \, dx+\left (3 a^4 c^3\right ) \int x^6 \tan ^{-1}(a x)^2 \, dx+\left (a^6 c^3\right ) \int x^8 \tan ^{-1}(a x)^2 \, dx\\ &=\frac {1}{3} c^3 x^3 \tan ^{-1}(a x)^2+\frac {3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)^2+\frac {3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)^2+\frac {1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)^2-\frac {1}{3} \left (2 a c^3\right ) \int \frac {x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac {1}{5} \left (6 a^3 c^3\right ) \int \frac {x^5 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac {1}{7} \left (6 a^5 c^3\right ) \int \frac {x^7 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac {1}{9} \left (2 a^7 c^3\right ) \int \frac {x^9 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=\frac {1}{3} c^3 x^3 \tan ^{-1}(a x)^2+\frac {3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)^2+\frac {3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)^2+\frac {1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)^2-\frac {\left (2 c^3\right ) \int x \tan ^{-1}(a x) \, dx}{3 a}+\frac {\left (2 c^3\right ) \int \frac {x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{3 a}-\frac {1}{5} \left (6 a c^3\right ) \int x^3 \tan ^{-1}(a x) \, dx+\frac {1}{5} \left (6 a c^3\right ) \int \frac {x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac {1}{7} \left (6 a^3 c^3\right ) \int x^5 \tan ^{-1}(a x) \, dx+\frac {1}{7} \left (6 a^3 c^3\right ) \int \frac {x^5 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac {1}{9} \left (2 a^5 c^3\right ) \int x^7 \tan ^{-1}(a x) \, dx+\frac {1}{9} \left (2 a^5 c^3\right ) \int \frac {x^7 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=-\frac {c^3 x^2 \tan ^{-1}(a x)}{3 a}-\frac {3}{10} a c^3 x^4 \tan ^{-1}(a x)-\frac {1}{7} a^3 c^3 x^6 \tan ^{-1}(a x)-\frac {1}{36} a^5 c^3 x^8 \tan ^{-1}(a x)-\frac {i c^3 \tan ^{-1}(a x)^2}{3 a^3}+\frac {1}{3} c^3 x^3 \tan ^{-1}(a x)^2+\frac {3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)^2+\frac {3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)^2+\frac {1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)^2+\frac {1}{3} c^3 \int \frac {x^2}{1+a^2 x^2} \, dx-\frac {\left (2 c^3\right ) \int \frac {\tan ^{-1}(a x)}{i-a x} \, dx}{3 a^2}+\frac {\left (6 c^3\right ) \int x \tan ^{-1}(a x) \, dx}{5 a}-\frac {\left (6 c^3\right ) \int \frac {x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{5 a}+\frac {1}{7} \left (6 a c^3\right ) \int x^3 \tan ^{-1}(a x) \, dx-\frac {1}{7} \left (6 a c^3\right ) \int \frac {x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac {1}{10} \left (3 a^2 c^3\right ) \int \frac {x^4}{1+a^2 x^2} \, dx+\frac {1}{9} \left (2 a^3 c^3\right ) \int x^5 \tan ^{-1}(a x) \, dx-\frac {1}{9} \left (2 a^3 c^3\right ) \int \frac {x^5 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac {1}{7} \left (a^4 c^3\right ) \int \frac {x^6}{1+a^2 x^2} \, dx+\frac {1}{36} \left (a^6 c^3\right ) \int \frac {x^8}{1+a^2 x^2} \, dx\\ &=\frac {c^3 x}{3 a^2}+\frac {4 c^3 x^2 \tan ^{-1}(a x)}{15 a}-\frac {3}{35} a c^3 x^4 \tan ^{-1}(a x)-\frac {20}{189} a^3 c^3 x^6 \tan ^{-1}(a x)-\frac {1}{36} a^5 c^3 x^8 \tan ^{-1}(a x)+\frac {4 i c^3 \tan ^{-1}(a x)^2}{15 a^3}+\frac {1}{3} c^3 x^3 \tan ^{-1}(a x)^2+\frac {3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)^2+\frac {3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)^2+\frac {1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)^2-\frac {2 c^3 \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{3 a^3}-\frac {1}{5} \left (3 c^3\right ) \int \frac {x^2}{1+a^2 x^2} \, dx-\frac {c^3 \int \frac {1}{1+a^2 x^2} \, dx}{3 a^2}+\frac {\left (2 c^3\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{3 a^2}+\frac {\left (6 c^3\right ) \int \frac {\tan ^{-1}(a x)}{i-a x} \, dx}{5 a^2}-\frac {\left (6 c^3\right ) \int x \tan ^{-1}(a x) \, dx}{7 a}+\frac {\left (6 c^3\right ) \int \frac {x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{7 a}-\frac {1}{9} \left (2 a c^3\right ) \int x^3 \tan ^{-1}(a x) \, dx+\frac {1}{9} \left (2 a c^3\right ) \int \frac {x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac {1}{14} \left (3 a^2 c^3\right ) \int \frac {x^4}{1+a^2 x^2} \, dx+\frac {1}{10} \left (3 a^2 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 {1}{27} \left (a^4 c^3\right ) \int \frac {x^6}{1+a^2 x^2} \, dx+\frac {1}{7} \left (a^4 c^3\right ) \int \left (\frac {1}{a^6}-\frac {x^2}{a^4}+\frac {x^4}{a^2}-\frac {1}{a^6 \left (1+a^2 x^2\right )}\right ) \, dx+\frac {1}{36} \left (a^6 c^3\right ) \int \left (-\frac {1}{a^8}+\frac {x^2}{a^6}-\frac {x^4}{a^4}+\frac {x^6}{a^2}+\frac {1}{a^8 \left (1+a^2 x^2\right )}\right ) \, dx\\ &=-\frac {569 c^3 x}{1260 a^2}+\frac {233 c^3 x^3}{3780}+\frac {29 a^2 c^3 x^5}{1260}+\frac {1}{252} a^4 c^3 x^7-\frac {c^3 \tan ^{-1}(a x)}{3 a^3}-\frac {17 c^3 x^2 \tan ^{-1}(a x)}{105 a}-\frac {89}{630} a c^3 x^4 \tan ^{-1}(a x)-\frac {20}{189} a^3 c^3 x^6 \tan ^{-1}(a x)-\frac {1}{36} a^5 c^3 x^8 \tan ^{-1}(a x)-\frac {17 i c^3 \tan ^{-1}(a x)^2}{105 a^3}+\frac {1}{3} c^3 x^3 \tan ^{-1}(a x)^2+\frac {3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)^2+\frac {3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)^2+\frac {1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)^2+\frac {8 c^3 \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{15 a^3}+\frac {1}{7} \left (3 c^3\right ) \int \frac {x^2}{1+a^2 x^2} \, dx-\frac {\left (2 i c^3\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{3 a^3}+\frac {c^3 \int \frac {1}{1+a^2 x^2} \, dx}{36 a^2}-\frac {c^3 \int \frac {1}{1+a^2 x^2} \, dx}{7 a^2}+\frac {\left (3 c^3\right ) \int \frac {1}{1+a^2 x^2} \, dx}{10 a^2}+\frac {\left (3 c^3\right ) \int \frac {1}{1+a^2 x^2} \, dx}{5 a^2}-\frac {\left (6 c^3\right ) \int \frac {\tan ^{-1}(a x)}{i-a x} \, dx}{7 a^2}-\frac {\left (6 c^3\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{5 a^2}+\frac {\left (2 c^3\right ) \int x \tan ^{-1}(a x) \, dx}{9 a}-\frac {\left (2 c^3\right ) \int \frac {x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{9 a}+\frac {1}{18} \left (a^2 c^3\right ) \int \frac {x^4}{1+a^2 x^2} \, dx-\frac {1}{14} \left (3 a^2 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 {1}{27} \left (a^4 c^3\right ) \int \left (\frac {1}{a^6}-\frac {x^2}{a^4}+\frac {x^4}{a^2}-\frac {1}{a^6 \left (1+a^2 x^2\right )}\right ) \, dx\\ &=\frac {583 c^3 x}{3780 a^2}+\frac {29 c^3 x^3}{11340}+\frac {59 a^2 c^3 x^5}{3780}+\frac {1}{252} a^4 c^3 x^7+\frac {569 c^3 \tan ^{-1}(a x)}{1260 a^3}-\frac {16 c^3 x^2 \tan ^{-1}(a x)}{315 a}-\frac {89}{630} a c^3 x^4 \tan ^{-1}(a x)-\frac {20}{189} a^3 c^3 x^6 \tan ^{-1}(a x)-\frac {1}{36} a^5 c^3 x^8 \tan ^{-1}(a x)-\frac {16 i c^3 \tan ^{-1}(a x)^2}{315 a^3}+\frac {1}{3} c^3 x^3 \tan ^{-1}(a x)^2+\frac {3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)^2+\frac {3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)^2+\frac {1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)^2-\frac {34 c^3 \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{105 a^3}-\frac {i c^3 \text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{3 a^3}-\frac {1}{9} c^3 \int \frac {x^2}{1+a^2 x^2} \, dx+\frac {\left (6 i c^3\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{5 a^3}+\frac {c^3 \int \frac {1}{1+a^2 x^2} \, dx}{27 a^2}-\frac {\left (3 c^3\right ) \int \frac {1}{1+a^2 x^2} \, dx}{14 a^2}+\frac {\left (2 c^3\right ) \int \frac {\tan ^{-1}(a x)}{i-a x} \, dx}{9 a^2}-\frac {\left (3 c^3\right ) \int \frac {1}{1+a^2 x^2} \, dx}{7 a^2}+\frac {\left (6 c^3\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{7 a^2}+\frac {1}{18} \left (a^2 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 {47 c^3 x}{3780 a^2}+\frac {239 c^3 x^3}{11340}+\frac {59 a^2 c^3 x^5}{3780}+\frac {1}{252} a^4 c^3 x^7-\frac {583 c^3 \tan ^{-1}(a x)}{3780 a^3}-\frac {16 c^3 x^2 \tan ^{-1}(a x)}{315 a}-\frac {89}{630} a c^3 x^4 \tan ^{-1}(a x)-\frac {20}{189} a^3 c^3 x^6 \tan ^{-1}(a x)-\frac {1}{36} a^5 c^3 x^8 \tan ^{-1}(a x)-\frac {16 i c^3 \tan ^{-1}(a x)^2}{315 a^3}+\frac {1}{3} c^3 x^3 \tan ^{-1}(a x)^2+\frac {3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)^2+\frac {3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)^2+\frac {1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)^2-\frac {32 c^3 \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{315 a^3}+\frac {4 i c^3 \text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{15 a^3}-\frac {\left (6 i c^3\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{7 a^3}+\frac {c^3 \int \frac {1}{1+a^2 x^2} \, dx}{18 a^2}+\frac {c^3 \int \frac {1}{1+a^2 x^2} \, dx}{9 a^2}-\frac {\left (2 c^3\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{9 a^2}\\ &=-\frac {47 c^3 x}{3780 a^2}+\frac {239 c^3 x^3}{11340}+\frac {59 a^2 c^3 x^5}{3780}+\frac {1}{252} a^4 c^3 x^7+\frac {47 c^3 \tan ^{-1}(a x)}{3780 a^3}-\frac {16 c^3 x^2 \tan ^{-1}(a x)}{315 a}-\frac {89}{630} a c^3 x^4 \tan ^{-1}(a x)-\frac {20}{189} a^3 c^3 x^6 \tan ^{-1}(a x)-\frac {1}{36} a^5 c^3 x^8 \tan ^{-1}(a x)-\frac {16 i c^3 \tan ^{-1}(a x)^2}{315 a^3}+\frac {1}{3} c^3 x^3 \tan ^{-1}(a x)^2+\frac {3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)^2+\frac {3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)^2+\frac {1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)^2-\frac {32 c^3 \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{315 a^3}-\frac {17 i c^3 \text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{105 a^3}+\frac {\left (2 i c^3\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{9 a^3}\\ &=-\frac {47 c^3 x}{3780 a^2}+\frac {239 c^3 x^3}{11340}+\frac {59 a^2 c^3 x^5}{3780}+\frac {1}{252} a^4 c^3 x^7+\frac {47 c^3 \tan ^{-1}(a x)}{3780 a^3}-\frac {16 c^3 x^2 \tan ^{-1}(a x)}{315 a}-\frac {89}{630} a c^3 x^4 \tan ^{-1}(a x)-\frac {20}{189} a^3 c^3 x^6 \tan ^{-1}(a x)-\frac {1}{36} a^5 c^3 x^8 \tan ^{-1}(a x)-\frac {16 i c^3 \tan ^{-1}(a x)^2}{315 a^3}+\frac {1}{3} c^3 x^3 \tan ^{-1}(a x)^2+\frac {3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)^2+\frac {3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)^2+\frac {1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)^2-\frac {32 c^3 \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{315 a^3}-\frac {16 i c^3 \text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{315 a^3}\\ \end {align*}

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Mathematica [A]  time = 2.26, size = 157, normalized size = 0.57 \[ \frac {c^3 \left (a x \left (45 a^6 x^6+177 a^4 x^4+239 a^2 x^2-141\right )+36 \left (35 a^9 x^9+135 a^7 x^7+189 a^5 x^5+105 a^3 x^3+16 i\right ) \tan ^{-1}(a x)^2-3 \tan ^{-1}(a x) \left (105 a^8 x^8+400 a^6 x^6+534 a^4 x^4+192 a^2 x^2+384 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )-47\right )+576 i \text {Li}_2\left (-e^{2 i \tan ^{-1}(a x)}\right )\right )}{11340 a^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c^3*(a*x*(-141 + 239*a^2*x^2 + 177*a^4*x^4 + 45*a^6*x^6) + 36*(16*I + 105*a^3*x^3 + 189*a^5*x^5 + 135*a^7*x^7
+ 35*a^9*x^9)*ArcTan[a*x]^2 - 3*ArcTan[a*x]*(-47 + 192*a^2*x^2 + 534*a^4*x^4 + 400*a^6*x^6 + 105*a^8*x^8 + 38
4*Log[1 + E^((2*I)*ArcTan[a*x])]) + (576*I)*PolyLog[2, -E^((2*I)*ArcTan[a*x])]))/(11340*a^3)

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fricas [F]  time = 0.78, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{6} c^{3} x^{8} + 3 \, a^{4} c^{3} x^{6} + 3 \, a^{2} c^{3} x^{4} + c^{3} x^{2}\right )} \arctan \left (a x\right )^{2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((a^6*c^3*x^8 + 3*a^4*c^3*x^6 + 3*a^2*c^3*x^4 + c^3*x^2)*arctan(a*x)^2, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

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maple [A]  time = 0.10, size = 376, normalized size = 1.37 \[ \frac {a^{6} c^{3} x^{9} \arctan \left (a x \right )^{2}}{9}+\frac {3 a^{4} c^{3} x^{7} \arctan \left (a x \right )^{2}}{7}+\frac {3 a^{2} c^{3} x^{5} \arctan \left (a x \right )^{2}}{5}+\frac {c^{3} x^{3} \arctan \left (a x \right )^{2}}{3}-\frac {a^{5} c^{3} x^{8} \arctan \left (a x \right )}{36}-\frac {20 a^{3} c^{3} x^{6} \arctan \left (a x \right )}{189}-\frac {89 a \,c^{3} x^{4} \arctan \left (a x \right )}{630}-\frac {16 c^{3} x^{2} \arctan \left (a x \right )}{315 a}+\frac {16 c^{3} \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{315 a^{3}}+\frac {a^{4} c^{3} x^{7}}{252}+\frac {59 a^{2} c^{3} x^{5}}{3780}+\frac {239 c^{3} x^{3}}{11340}-\frac {47 c^{3} x}{3780 a^{2}}+\frac {47 c^{3} \arctan \left (a x \right )}{3780 a^{3}}-\frac {8 i c^{3} \ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )}{315 a^{3}}-\frac {4 i c^{3} \ln \left (a x -i\right )^{2}}{315 a^{3}}+\frac {8 i c^{3} \ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )}{315 a^{3}}-\frac {8 i c^{3} \ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )}{315 a^{3}}+\frac {4 i c^{3} \ln \left (a x +i\right )^{2}}{315 a^{3}}-\frac {8 i c^{3} \dilog \left (-\frac {i \left (a x +i\right )}{2}\right )}{315 a^{3}}+\frac {8 i c^{3} \dilog \left (\frac {i \left (a x -i\right )}{2}\right )}{315 a^{3}}+\frac {8 i c^{3} \ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )}{315 a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*a^6*c^3*x^9*arctan(a*x)^2+3/7*a^4*c^3*x^7*arctan(a*x)^2+3/5*a^2*c^3*x^5*arctan(a*x)^2+1/3*c^3*x^3*arctan(a
*x)^2-1/36*a^5*c^3*x^8*arctan(a*x)-20/189*a^3*c^3*x^6*arctan(a*x)-89/630*a*c^3*x^4*arctan(a*x)-16/315*c^3*x^2*
arctan(a*x)/a+16/315/a^3*c^3*arctan(a*x)*ln(a^2*x^2+1)+1/252*a^4*c^3*x^7+59/3780*a^2*c^3*x^5+239/11340*c^3*x^3
-47/3780*c^3*x/a^2+47/3780*c^3*arctan(a*x)/a^3-8/315*I/a^3*c^3*ln(I+a*x)*ln(a^2*x^2+1)+8/315*I/a^3*c^3*dilog(1
/2*I*(a*x-I))-4/315*I/a^3*c^3*ln(a*x-I)^2+8/315*I/a^3*c^3*ln(a*x-I)*ln(a^2*x^2+1)-8/315*I/a^3*c^3*ln(a*x-I)*ln
(-1/2*I*(I+a*x))+4/315*I/a^3*c^3*ln(I+a*x)^2-8/315*I/a^3*c^3*dilog(-1/2*I*(I+a*x))+8/315*I/a^3*c^3*ln(I+a*x)*l
n(1/2*I*(a*x-I))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{1260} \, {\left (35 \, a^{6} c^{3} x^{9} + 135 \, a^{4} c^{3} x^{7} + 189 \, a^{2} c^{3} x^{5} + 105 \, c^{3} x^{3}\right )} \arctan \left (a x\right )^{2} - \frac {1}{5040} \, {\left (35 \, a^{6} c^{3} x^{9} + 135 \, a^{4} c^{3} x^{7} + 189 \, a^{2} c^{3} x^{5} + 105 \, c^{3} x^{3}\right )} \log \left (a^{2} x^{2} + 1\right )^{2} + \int \frac {3780 \, {\left (a^{8} c^{3} x^{10} + 4 \, a^{6} c^{3} x^{8} + 6 \, a^{4} c^{3} x^{6} + 4 \, a^{2} c^{3} x^{4} + c^{3} x^{2}\right )} \arctan \left (a x\right )^{2} + 315 \, {\left (a^{8} c^{3} x^{10} + 4 \, a^{6} c^{3} x^{8} + 6 \, a^{4} c^{3} x^{6} + 4 \, a^{2} c^{3} x^{4} + c^{3} x^{2}\right )} \log \left (a^{2} x^{2} + 1\right )^{2} - 8 \, {\left (35 \, a^{7} c^{3} x^{9} + 135 \, a^{5} c^{3} x^{7} + 189 \, a^{3} c^{3} x^{5} + 105 \, a c^{3} x^{3}\right )} \arctan \left (a x\right ) + 4 \, {\left (35 \, a^{8} c^{3} x^{10} + 135 \, a^{6} c^{3} x^{8} + 189 \, a^{4} c^{3} x^{6} + 105 \, a^{2} c^{3} x^{4}\right )} \log \left (a^{2} x^{2} + 1\right )}{5040 \, {\left (a^{2} x^{2} + 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1260*(35*a^6*c^3*x^9 + 135*a^4*c^3*x^7 + 189*a^2*c^3*x^5 + 105*c^3*x^3)*arctan(a*x)^2 - 1/5040*(35*a^6*c^3*x
^9 + 135*a^4*c^3*x^7 + 189*a^2*c^3*x^5 + 105*c^3*x^3)*log(a^2*x^2 + 1)^2 + integrate(1/5040*(3780*(a^8*c^3*x^1
0 + 4*a^6*c^3*x^8 + 6*a^4*c^3*x^6 + 4*a^2*c^3*x^4 + c^3*x^2)*arctan(a*x)^2 + 315*(a^8*c^3*x^10 + 4*a^6*c^3*x^8
+ 6*a^4*c^3*x^6 + 4*a^2*c^3*x^4 + c^3*x^2)*log(a^2*x^2 + 1)^2 - 8*(35*a^7*c^3*x^9 + 135*a^5*c^3*x^7 + 189*a^3
*c^3*x^5 + 105*a*c^3*x^3)*arctan(a*x) + 4*(35*a^8*c^3*x^10 + 135*a^6*c^3*x^8 + 189*a^4*c^3*x^6 + 105*a^2*c^3*x
^4)*log(a^2*x^2 + 1))/(a^2*x^2 + 1), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int x^2\,{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^3 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ c^{3} \left (\int x^{2} \operatorname {atan}^{2}{\left (a x \right )}\, dx + \int 3 a^{2} x^{4} \operatorname {atan}^{2}{\left (a x \right )}\, dx + \int 3 a^{4} x^{6} \operatorname {atan}^{2}{\left (a x \right )}\, dx + \int a^{6} x^{8} \operatorname {atan}^{2}{\left (a x \right )}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c**3*(Integral(x**2*atan(a*x)**2, x) + Integral(3*a**2*x**4*atan(a*x)**2, x) + Integral(3*a**4*x**6*atan(a*x)*
*2, x) + Integral(a**6*x**8*atan(a*x)**2, x))

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