\(\int x^2 (c+a^2 c x^2)^3 \arctan (a x)^2 \, dx\) [275]

Optimal result

Integrand size = 22, antiderivative size = 274 \[ \int x^2 \left (c+a^2 c x^2\right )^3 \arctan (a x)^2 \, dx=-\frac {47 c^3 x}{3780 a^2}+\frac {239 c^3 x^3}{11340}+\frac {59 a^2 c^3 x^5}{3780}+\frac {1}{252} a^4 c^3 x^7+\frac {47 c^3 \arctan (a x)}{3780 a^3}-\frac {16 c^3 x^2 \arctan (a x)}{315 a}-\frac {89}{630} a c^3 x^4 \arctan (a x)-\frac {20}{189} a^3 c^3 x^6 \arctan (a x)-\frac {1}{36} a^5 c^3 x^8 \arctan (a x)-\frac {16 i c^3 \arctan (a x)^2}{315 a^3}+\frac {1}{3} c^3 x^3 \arctan (a x)^2+\frac {3}{5} a^2 c^3 x^5 \arctan (a x)^2+\frac {3}{7} a^4 c^3 x^7 \arctan (a x)^2+\frac {1}{9} a^6 c^3 x^9 \arctan (a x)^2-\frac {32 c^3 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{315 a^3}-\frac {16 i c^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{315 a^3} \]

[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*polylog(2,1-2/(1+I*a*x))/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*arcta
n(a*x)^2/a^3

Rubi [A] (verified)

Time = 0.84 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.00, number of steps used = 68, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {5068, 4946, 5036, 327, 209, 5040, 4964, 2449, 2352, 308} \[ \int x^2 \left (c+a^2 c x^2\right )^3 \arctan (a x)^2 \, dx=\frac {1}{9} a^6 c^3 x^9 \arctan (a x)^2-\frac {1}{36} a^5 c^3 x^8 \arctan (a x)+\frac {3}{7} a^4 c^3 x^7 \arctan (a x)^2+\frac {1}{252} a^4 c^3 x^7-\frac {20}{189} a^3 c^3 x^6 \arctan (a x)-\frac {16 i c^3 \arctan (a x)^2}{315 a^3}+\frac {47 c^3 \arctan (a x)}{3780 a^3}-\frac {32 c^3 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{315 a^3}-\frac {16 i c^3 \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{315 a^3}+\frac {3}{5} a^2 c^3 x^5 \arctan (a x)^2+\frac {59 a^2 c^3 x^5}{3780}-\frac {47 c^3 x}{3780 a^2}-\frac {89}{630} a c^3 x^4 \arctan (a x)+\frac {1}{3} c^3 x^3 \arctan (a x)^2-\frac {16 c^3 x^2 \arctan (a x)}{315 a}+\frac {239 c^3 x^3}{11340} \]

[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 209

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

Rule 308

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 327

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 2352

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

Rule 2449

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 4946

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

Rule 4964

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTan[c*x])^p)*(
Log[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 5036

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 5040

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-I)*((a + b*ArcT
an[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 5068

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*} \text {integral}& = \int \left (c^3 x^2 \arctan (a x)^2+3 a^2 c^3 x^4 \arctan (a x)^2+3 a^4 c^3 x^6 \arctan (a x)^2+a^6 c^3 x^8 \arctan (a x)^2\right ) \, dx \\ & = c^3 \int x^2 \arctan (a x)^2 \, dx+\left (3 a^2 c^3\right ) \int x^4 \arctan (a x)^2 \, dx+\left (3 a^4 c^3\right ) \int x^6 \arctan (a x)^2 \, dx+\left (a^6 c^3\right ) \int x^8 \arctan (a x)^2 \, dx \\ & = \frac {1}{3} c^3 x^3 \arctan (a x)^2+\frac {3}{5} a^2 c^3 x^5 \arctan (a x)^2+\frac {3}{7} a^4 c^3 x^7 \arctan (a x)^2+\frac {1}{9} a^6 c^3 x^9 \arctan (a x)^2-\frac {1}{3} \left (2 a c^3\right ) \int \frac {x^3 \arctan (a x)}{1+a^2 x^2} \, dx-\frac {1}{5} \left (6 a^3 c^3\right ) \int \frac {x^5 \arctan (a x)}{1+a^2 x^2} \, dx-\frac {1}{7} \left (6 a^5 c^3\right ) \int \frac {x^7 \arctan (a x)}{1+a^2 x^2} \, dx-\frac {1}{9} \left (2 a^7 c^3\right ) \int \frac {x^9 \arctan (a x)}{1+a^2 x^2} \, dx \\ & = \frac {1}{3} c^3 x^3 \arctan (a x)^2+\frac {3}{5} a^2 c^3 x^5 \arctan (a x)^2+\frac {3}{7} a^4 c^3 x^7 \arctan (a x)^2+\frac {1}{9} a^6 c^3 x^9 \arctan (a x)^2-\frac {\left (2 c^3\right ) \int x \arctan (a x) \, dx}{3 a}+\frac {\left (2 c^3\right ) \int \frac {x \arctan (a x)}{1+a^2 x^2} \, dx}{3 a}-\frac {1}{5} \left (6 a c^3\right ) \int x^3 \arctan (a x) \, dx+\frac {1}{5} \left (6 a c^3\right ) \int \frac {x^3 \arctan (a x)}{1+a^2 x^2} \, dx-\frac {1}{7} \left (6 a^3 c^3\right ) \int x^5 \arctan (a x) \, dx+\frac {1}{7} \left (6 a^3 c^3\right ) \int \frac {x^5 \arctan (a x)}{1+a^2 x^2} \, dx-\frac {1}{9} \left (2 a^5 c^3\right ) \int x^7 \arctan (a x) \, dx+\frac {1}{9} \left (2 a^5 c^3\right ) \int \frac {x^7 \arctan (a x)}{1+a^2 x^2} \, dx \\ & = -\frac {c^3 x^2 \arctan (a x)}{3 a}-\frac {3}{10} a c^3 x^4 \arctan (a x)-\frac {1}{7} a^3 c^3 x^6 \arctan (a x)-\frac {1}{36} a^5 c^3 x^8 \arctan (a x)-\frac {i c^3 \arctan (a x)^2}{3 a^3}+\frac {1}{3} c^3 x^3 \arctan (a x)^2+\frac {3}{5} a^2 c^3 x^5 \arctan (a x)^2+\frac {3}{7} a^4 c^3 x^7 \arctan (a x)^2+\frac {1}{9} a^6 c^3 x^9 \arctan (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 {\arctan (a x)}{i-a x} \, dx}{3 a^2}+\frac {\left (6 c^3\right ) \int x \arctan (a x) \, dx}{5 a}-\frac {\left (6 c^3\right ) \int \frac {x \arctan (a x)}{1+a^2 x^2} \, dx}{5 a}+\frac {1}{7} \left (6 a c^3\right ) \int x^3 \arctan (a x) \, dx-\frac {1}{7} \left (6 a c^3\right ) \int \frac {x^3 \arctan (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 \arctan (a x) \, dx-\frac {1}{9} \left (2 a^3 c^3\right ) \int \frac {x^5 \arctan (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 \arctan (a x)}{15 a}-\frac {3}{35} a c^3 x^4 \arctan (a x)-\frac {20}{189} a^3 c^3 x^6 \arctan (a x)-\frac {1}{36} a^5 c^3 x^8 \arctan (a x)+\frac {4 i c^3 \arctan (a x)^2}{15 a^3}+\frac {1}{3} c^3 x^3 \arctan (a x)^2+\frac {3}{5} a^2 c^3 x^5 \arctan (a x)^2+\frac {3}{7} a^4 c^3 x^7 \arctan (a x)^2+\frac {1}{9} a^6 c^3 x^9 \arctan (a x)^2-\frac {2 c^3 \arctan (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 {\arctan (a x)}{i-a x} \, dx}{5 a^2}-\frac {\left (6 c^3\right ) \int x \arctan (a x) \, dx}{7 a}+\frac {\left (6 c^3\right ) \int \frac {x \arctan (a x)}{1+a^2 x^2} \, dx}{7 a}-\frac {1}{9} \left (2 a c^3\right ) \int x^3 \arctan (a x) \, dx+\frac {1}{9} \left (2 a c^3\right ) \int \frac {x^3 \arctan (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 \arctan (a x)}{3 a^3}-\frac {17 c^3 x^2 \arctan (a x)}{105 a}-\frac {89}{630} a c^3 x^4 \arctan (a x)-\frac {20}{189} a^3 c^3 x^6 \arctan (a x)-\frac {1}{36} a^5 c^3 x^8 \arctan (a x)-\frac {17 i c^3 \arctan (a x)^2}{105 a^3}+\frac {1}{3} c^3 x^3 \arctan (a x)^2+\frac {3}{5} a^2 c^3 x^5 \arctan (a x)^2+\frac {3}{7} a^4 c^3 x^7 \arctan (a x)^2+\frac {1}{9} a^6 c^3 x^9 \arctan (a x)^2+\frac {8 c^3 \arctan (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 ) \text {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 {\arctan (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 \arctan (a x) \, dx}{9 a}-\frac {\left (2 c^3\right ) \int \frac {x \arctan (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 \arctan (a x)}{1260 a^3}-\frac {16 c^3 x^2 \arctan (a x)}{315 a}-\frac {89}{630} a c^3 x^4 \arctan (a x)-\frac {20}{189} a^3 c^3 x^6 \arctan (a x)-\frac {1}{36} a^5 c^3 x^8 \arctan (a x)-\frac {16 i c^3 \arctan (a x)^2}{315 a^3}+\frac {1}{3} c^3 x^3 \arctan (a x)^2+\frac {3}{5} a^2 c^3 x^5 \arctan (a x)^2+\frac {3}{7} a^4 c^3 x^7 \arctan (a x)^2+\frac {1}{9} a^6 c^3 x^9 \arctan (a x)^2-\frac {34 c^3 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{105 a^3}-\frac {i c^3 \operatorname {PolyLog}\left (2,1-\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 ) \text {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 {\arctan (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 \arctan (a x)}{3780 a^3}-\frac {16 c^3 x^2 \arctan (a x)}{315 a}-\frac {89}{630} a c^3 x^4 \arctan (a x)-\frac {20}{189} a^3 c^3 x^6 \arctan (a x)-\frac {1}{36} a^5 c^3 x^8 \arctan (a x)-\frac {16 i c^3 \arctan (a x)^2}{315 a^3}+\frac {1}{3} c^3 x^3 \arctan (a x)^2+\frac {3}{5} a^2 c^3 x^5 \arctan (a x)^2+\frac {3}{7} a^4 c^3 x^7 \arctan (a x)^2+\frac {1}{9} a^6 c^3 x^9 \arctan (a x)^2-\frac {32 c^3 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{315 a^3}+\frac {4 i c^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{15 a^3}-\frac {\left (6 i c^3\right ) \text {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 \arctan (a x)}{3780 a^3}-\frac {16 c^3 x^2 \arctan (a x)}{315 a}-\frac {89}{630} a c^3 x^4 \arctan (a x)-\frac {20}{189} a^3 c^3 x^6 \arctan (a x)-\frac {1}{36} a^5 c^3 x^8 \arctan (a x)-\frac {16 i c^3 \arctan (a x)^2}{315 a^3}+\frac {1}{3} c^3 x^3 \arctan (a x)^2+\frac {3}{5} a^2 c^3 x^5 \arctan (a x)^2+\frac {3}{7} a^4 c^3 x^7 \arctan (a x)^2+\frac {1}{9} a^6 c^3 x^9 \arctan (a x)^2-\frac {32 c^3 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{315 a^3}-\frac {17 i c^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{105 a^3}+\frac {\left (2 i c^3\right ) \text {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 \arctan (a x)}{3780 a^3}-\frac {16 c^3 x^2 \arctan (a x)}{315 a}-\frac {89}{630} a c^3 x^4 \arctan (a x)-\frac {20}{189} a^3 c^3 x^6 \arctan (a x)-\frac {1}{36} a^5 c^3 x^8 \arctan (a x)-\frac {16 i c^3 \arctan (a x)^2}{315 a^3}+\frac {1}{3} c^3 x^3 \arctan (a x)^2+\frac {3}{5} a^2 c^3 x^5 \arctan (a x)^2+\frac {3}{7} a^4 c^3 x^7 \arctan (a x)^2+\frac {1}{9} a^6 c^3 x^9 \arctan (a x)^2-\frac {32 c^3 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{315 a^3}-\frac {16 i c^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{315 a^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.93 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.57 \[ \int x^2 \left (c+a^2 c x^2\right )^3 \arctan (a x)^2 \, dx=\frac {c^3 \left (a x \left (-141+239 a^2 x^2+177 a^4 x^4+45 a^6 x^6\right )+36 \left (16 i+105 a^3 x^3+189 a^5 x^5+135 a^7 x^7+35 a^9 x^9\right ) \arctan (a x)^2-3 \arctan (a x) \left (-47+192 a^2 x^2+534 a^4 x^4+400 a^6 x^6+105 a^8 x^8+384 \log \left (1+e^{2 i \arctan (a x)}\right )\right )+576 i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )\right )}{11340 a^3} \]

[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)

Maple [A] (verified)

Time = 3.81 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.12

method	result	size
derivativedivides	\(\frac {\frac {c^{3} \arctan \left (a x \right )^{2} a^{9} x^{9}}{9}+\frac {3 c^{3} \arctan \left (a x \right )^{2} a^{7} x^{7}}{7}+\frac {3 a^{5} c^{3} x^{5} \arctan \left (a x \right )^{2}}{5}+\frac {a^{3} c^{3} x^{3} \arctan \left (a x \right )^{2}}{3}-\frac {2 c^{3} \left (\frac {35 \arctan \left (a x \right ) a^{8} x^{8}}{8}+\frac {50 a^{6} \arctan \left (a x \right ) x^{6}}{3}+\frac {89 \arctan \left (a x \right ) a^{4} x^{4}}{4}+8 a^{2} \arctan \left (a x \right ) x^{2}-8 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )-\frac {5 a^{7} x^{7}}{8}-\frac {59 a^{5} x^{5}}{24}-\frac {239 a^{3} x^{3}}{72}+\frac {47 a x}{24}-\frac {47 \arctan \left (a x \right )}{24}-4 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )+4 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )\right )}{315}}{a^{3}}\)	\(306\)
default	\(\frac {\frac {c^{3} \arctan \left (a x \right )^{2} a^{9} x^{9}}{9}+\frac {3 c^{3} \arctan \left (a x \right )^{2} a^{7} x^{7}}{7}+\frac {3 a^{5} c^{3} x^{5} \arctan \left (a x \right )^{2}}{5}+\frac {a^{3} c^{3} x^{3} \arctan \left (a x \right )^{2}}{3}-\frac {2 c^{3} \left (\frac {35 \arctan \left (a x \right ) a^{8} x^{8}}{8}+\frac {50 a^{6} \arctan \left (a x \right ) x^{6}}{3}+\frac {89 \arctan \left (a x \right ) a^{4} x^{4}}{4}+8 a^{2} \arctan \left (a x \right ) x^{2}-8 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )-\frac {5 a^{7} x^{7}}{8}-\frac {59 a^{5} x^{5}}{24}-\frac {239 a^{3} x^{3}}{72}+\frac {47 a x}{24}-\frac {47 \arctan \left (a x \right )}{24}-4 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )+4 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )\right )}{315}}{a^{3}}\)	\(306\)
parts	\(\frac {a^{6} c^{3} x^{9} \arctan \left (a x \right )^{2}}{9}+\frac {3 a^{4} c^{3} x^{7} \arctan \left (a x \right )^{2}}{7}+\frac {3 a^{2} c^{3} x^{5} \arctan \left (a x \right )^{2}}{5}+\frac {c^{3} x^{3} \arctan \left (a x \right )^{2}}{3}-\frac {2 c^{3} \left (\frac {35 a^{5} \arctan \left (a x \right ) x^{8}}{8}+\frac {50 a^{3} \arctan \left (a x \right ) x^{6}}{3}+\frac {89 a \arctan \left (a x \right ) x^{4}}{4}+\frac {8 \arctan \left (a x \right ) x^{2}}{a}-\frac {8 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{a^{3}}-\frac {15 a^{7} x^{7}+59 a^{5} x^{5}+\frac {239 a^{3} x^{3}}{3}-47 a x +47 \arctan \left (a x \right )+96 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )-96 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )}{24 a^{3}}\right )}{315}\)	\(306\)
risch	\(-\frac {8 i c^{3} \ln \left (-i a x +1\right ) x^{2}}{315 a}-\frac {16 i c^{3} \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (\frac {1}{2}-\frac {i a x}{2}\right )}{315 a^{3}}+\frac {16 i c^{3} \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (-i a x +1\right )}{315 a^{3}}-\frac {8 i c^{3} \ln \left (i a x +1\right ) \ln \left (-i a x +1\right )}{315 a^{3}}+\frac {3 c^{3} a^{2} \ln \left (i a x +1\right ) \ln \left (-i a x +1\right ) x^{5}}{10}+\frac {c^{3} a^{6} \ln \left (i a x +1\right ) \ln \left (-i a x +1\right ) x^{9}}{18}+\frac {3 c^{3} a^{4} \ln \left (i a x +1\right ) \ln \left (-i a x +1\right ) x^{7}}{14}-\frac {47 c^{3} x}{3780 a^{2}}+\frac {59 a^{2} c^{3} x^{5}}{3780}+\frac {a^{4} c^{3} x^{7}}{252}+\frac {47 c^{3} \arctan \left (a x \right )}{3780 a^{3}}+\frac {239 c^{3} x^{3}}{11340}+\frac {i c^{3} a^{5} \ln \left (i a x +1\right ) x^{8}}{72}+\frac {10 i c^{3} a^{3} \ln \left (i a x +1\right ) x^{6}}{189}+\frac {8 i c^{3} \ln \left (i a x +1\right ) x^{2}}{315 a}-\frac {i c^{3} a^{5} \ln \left (-i a x +1\right ) x^{8}}{72}-\frac {10 i c^{3} a^{3} \ln \left (-i a x +1\right ) x^{6}}{189}-\frac {89 i c^{3} a \ln \left (-i a x +1\right ) x^{4}}{1260}+\frac {89 i c^{3} a \ln \left (i a x +1\right ) x^{4}}{1260}-\frac {c^{3} \ln \left (i a x +1\right )^{2} x^{3}}{12}-\frac {c^{3} \ln \left (-i a x +1\right )^{2} x^{3}}{12}-\frac {1613119 i c^{3}}{31255875 a^{3}}-\frac {c^{3} a^{6} \ln \left (-i a x +1\right )^{2} x^{9}}{36}-\frac {3 c^{3} a^{4} \ln \left (-i a x +1\right )^{2} x^{7}}{28}+\frac {c^{3} \ln \left (i a x +1\right ) \ln \left (-i a x +1\right ) x^{3}}{6}-\frac {4 i c^{3} \ln \left (i a x +1\right )^{2}}{315 a^{3}}+\frac {4 i c^{3} \ln \left (-i a x +1\right )^{2}}{315 a^{3}}-\frac {16 i c^{3} \operatorname {dilog}\left (\frac {1}{2}-\frac {i a x}{2}\right )}{315 a^{3}}-\frac {3 c^{3} a^{4} \ln \left (i a x +1\right )^{2} x^{7}}{28}-\frac {3 c^{3} a^{2} \ln \left (i a x +1\right )^{2} x^{5}}{20}-\frac {c^{3} a^{6} \ln \left (i a x +1\right )^{2} x^{9}}{36}-\frac {3 c^{3} a^{2} \ln \left (-i a x +1\right )^{2} x^{5}}{20}\)	\(615\)

[In]

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

[Out]

1/a^3*(1/9*c^3*arctan(a*x)^2*a^9*x^9+3/7*c^3*arctan(a*x)^2*a^7*x^7+3/5*a^5*c^3*x^5*arctan(a*x)^2+1/3*a^3*c^3*x
^3*arctan(a*x)^2-2/315*c^3*(35/8*arctan(a*x)*a^8*x^8+50/3*a^6*arctan(a*x)*x^6+89/4*arctan(a*x)*a^4*x^4+8*a^2*a
rctan(a*x)*x^2-8*arctan(a*x)*ln(a^2*x^2+1)-5/8*a^7*x^7-59/24*a^5*x^5-239/72*a^3*x^3+47/24*a*x-47/24*arctan(a*x
)-4*I*(ln(a*x-I)*ln(a^2*x^2+1)-dilog(-1/2*I*(I+a*x))-ln(a*x-I)*ln(-1/2*I*(I+a*x))-1/2*ln(a*x-I)^2)+4*I*(ln(I+a
*x)*ln(a^2*x^2+1)-dilog(1/2*I*(a*x-I))-ln(I+a*x)*ln(1/2*I*(a*x-I))-1/2*ln(I+a*x)^2)))

Fricas [F]

\[ \int x^2 \left (c+a^2 c x^2\right )^3 \arctan (a x)^2 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{3} x^{2} \arctan \left (a x\right )^{2} \,d x } \]

[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)

Sympy [F]

\[ \int x^2 \left (c+a^2 c x^2\right )^3 \arctan (a x)^2 \, dx=c^{3} \left (\int x^{2} \operatorname {atan}^{2}{\left (a x \right )}\, dx + \int 3 a^{2} x^{4} \operatorname {atan}^{2}{\left (a x \right )}\, dx + \int 3 a^{4} x^{6} \operatorname {atan}^{2}{\left (a x \right )}\, dx + \int a^{6} x^{8} \operatorname {atan}^{2}{\left (a x \right )}\, dx\right ) \]

[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))

Maxima [F]

\[ \int x^2 \left (c+a^2 c x^2\right )^3 \arctan (a x)^2 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{3} x^{2} \arctan \left (a x\right )^{2} \,d x } \]

[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)

Giac [F]

\[ \int x^2 \left (c+a^2 c x^2\right )^3 \arctan (a x)^2 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{3} x^{2} \arctan \left (a x\right )^{2} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int x^2 \left (c+a^2 c x^2\right )^3 \arctan (a x)^2 \, dx=\int x^2\,{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^3 \,d x \]

[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)