\(\int x^3 (c+a^2 c x^2)^3 \arctan (a x)^2 \, dx\) [274]

Optimal result

Integrand size = 22, antiderivative size = 240 \[ \int x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^2 \, dx=-\frac {107 c^3 x^2}{12600 a^2}+\frac {53 c^3 x^4}{6300}+\frac {71 a^2 c^3 x^6}{7560}+\frac {1}{360} a^4 c^3 x^8+\frac {c^3 x \arctan (a x)}{20 a^3}-\frac {c^3 x^3 \arctan (a x)}{60 a}-\frac {9}{100} a c^3 x^5 \arctan (a x)-\frac {11}{140} a^3 c^3 x^7 \arctan (a x)-\frac {1}{45} a^5 c^3 x^9 \arctan (a x)-\frac {c^3 \arctan (a x)^2}{40 a^4}+\frac {1}{4} c^3 x^4 \arctan (a x)^2+\frac {1}{2} a^2 c^3 x^6 \arctan (a x)^2+\frac {3}{8} a^4 c^3 x^8 \arctan (a x)^2+\frac {1}{10} a^6 c^3 x^{10} \arctan (a x)^2-\frac {26 c^3 \log \left (1+a^2 x^2\right )}{1575 a^4} \]

[Out]

-107/12600*c^3*x^2/a^2+53/6300*c^3*x^4+71/7560*a^2*c^3*x^6+1/360*a^4*c^3*x^8+1/20*c^3*x*arctan(a*x)/a^3-1/60*c
^3*x^3*arctan(a*x)/a-9/100*a*c^3*x^5*arctan(a*x)-11/140*a^3*c^3*x^7*arctan(a*x)-1/45*a^5*c^3*x^9*arctan(a*x)-1
/40*c^3*arctan(a*x)^2/a^4+1/4*c^3*x^4*arctan(a*x)^2+1/2*a^2*c^3*x^6*arctan(a*x)^2+3/8*a^4*c^3*x^8*arctan(a*x)^
2+1/10*a^6*c^3*x^10*arctan(a*x)^2-26/1575*c^3*ln(a^2*x^2+1)/a^4

Rubi [A] (verified)

Time = 0.87 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps used = 72, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {5068, 4946, 5036, 272, 45, 4930, 266, 5004} \[ \int x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^2 \, dx=\frac {1}{10} a^6 c^3 x^{10} \arctan (a x)^2-\frac {1}{45} a^5 c^3 x^9 \arctan (a x)+\frac {3}{8} a^4 c^3 x^8 \arctan (a x)^2-\frac {c^3 \arctan (a x)^2}{40 a^4}+\frac {1}{360} a^4 c^3 x^8-\frac {11}{140} a^3 c^3 x^7 \arctan (a x)+\frac {c^3 x \arctan (a x)}{20 a^3}+\frac {1}{2} a^2 c^3 x^6 \arctan (a x)^2+\frac {71 a^2 c^3 x^6}{7560}-\frac {107 c^3 x^2}{12600 a^2}-\frac {26 c^3 \log \left (a^2 x^2+1\right )}{1575 a^4}-\frac {9}{100} a c^3 x^5 \arctan (a x)+\frac {1}{4} c^3 x^4 \arctan (a x)^2-\frac {c^3 x^3 \arctan (a x)}{60 a}+\frac {53 c^3 x^4}{6300} \]

[In]

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

[Out]

(-107*c^3*x^2)/(12600*a^2) + (53*c^3*x^4)/6300 + (71*a^2*c^3*x^6)/7560 + (a^4*c^3*x^8)/360 + (c^3*x*ArcTan[a*x
])/(20*a^3) - (c^3*x^3*ArcTan[a*x])/(60*a) - (9*a*c^3*x^5*ArcTan[a*x])/100 - (11*a^3*c^3*x^7*ArcTan[a*x])/140
- (a^5*c^3*x^9*ArcTan[a*x])/45 - (c^3*ArcTan[a*x]^2)/(40*a^4) + (c^3*x^4*ArcTan[a*x]^2)/4 + (a^2*c^3*x^6*ArcTa
n[a*x]^2)/2 + (3*a^4*c^3*x^8*ArcTan[a*x]^2)/8 + (a^6*c^3*x^10*ArcTan[a*x]^2)/10 - (26*c^3*Log[1 + a^2*x^2])/(1
575*a^4)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 4930

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

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 5004

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

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 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^3 \arctan (a x)^2+3 a^2 c^3 x^5 \arctan (a x)^2+3 a^4 c^3 x^7 \arctan (a x)^2+a^6 c^3 x^9 \arctan (a x)^2\right ) \, dx \\ & = c^3 \int x^3 \arctan (a x)^2 \, dx+\left (3 a^2 c^3\right ) \int x^5 \arctan (a x)^2 \, dx+\left (3 a^4 c^3\right ) \int x^7 \arctan (a x)^2 \, dx+\left (a^6 c^3\right ) \int x^9 \arctan (a x)^2 \, dx \\ & = \frac {1}{4} c^3 x^4 \arctan (a x)^2+\frac {1}{2} a^2 c^3 x^6 \arctan (a x)^2+\frac {3}{8} a^4 c^3 x^8 \arctan (a x)^2+\frac {1}{10} a^6 c^3 x^{10} \arctan (a x)^2-\frac {1}{2} \left (a c^3\right ) \int \frac {x^4 \arctan (a x)}{1+a^2 x^2} \, dx-\left (a^3 c^3\right ) \int \frac {x^6 \arctan (a x)}{1+a^2 x^2} \, dx-\frac {1}{4} \left (3 a^5 c^3\right ) \int \frac {x^8 \arctan (a x)}{1+a^2 x^2} \, dx-\frac {1}{5} \left (a^7 c^3\right ) \int \frac {x^{10} \arctan (a x)}{1+a^2 x^2} \, dx \\ & = \frac {1}{4} c^3 x^4 \arctan (a x)^2+\frac {1}{2} a^2 c^3 x^6 \arctan (a x)^2+\frac {3}{8} a^4 c^3 x^8 \arctan (a x)^2+\frac {1}{10} a^6 c^3 x^{10} \arctan (a x)^2-\frac {c^3 \int x^2 \arctan (a x) \, dx}{2 a}+\frac {c^3 \int \frac {x^2 \arctan (a x)}{1+a^2 x^2} \, dx}{2 a}-\left (a c^3\right ) \int x^4 \arctan (a x) \, dx+\left (a c^3\right ) \int \frac {x^4 \arctan (a x)}{1+a^2 x^2} \, dx-\frac {1}{4} \left (3 a^3 c^3\right ) \int x^6 \arctan (a x) \, dx+\frac {1}{4} \left (3 a^3 c^3\right ) \int \frac {x^6 \arctan (a x)}{1+a^2 x^2} \, dx-\frac {1}{5} \left (a^5 c^3\right ) \int x^8 \arctan (a x) \, dx+\frac {1}{5} \left (a^5 c^3\right ) \int \frac {x^8 \arctan (a x)}{1+a^2 x^2} \, dx \\ & = -\frac {c^3 x^3 \arctan (a x)}{6 a}-\frac {1}{5} a c^3 x^5 \arctan (a x)-\frac {3}{28} a^3 c^3 x^7 \arctan (a x)-\frac {1}{45} a^5 c^3 x^9 \arctan (a x)+\frac {1}{4} c^3 x^4 \arctan (a x)^2+\frac {1}{2} a^2 c^3 x^6 \arctan (a x)^2+\frac {3}{8} a^4 c^3 x^8 \arctan (a x)^2+\frac {1}{10} a^6 c^3 x^{10} \arctan (a x)^2+\frac {1}{6} c^3 \int \frac {x^3}{1+a^2 x^2} \, dx+\frac {c^3 \int \arctan (a x) \, dx}{2 a^3}-\frac {c^3 \int \frac {\arctan (a x)}{1+a^2 x^2} \, dx}{2 a^3}+\frac {c^3 \int x^2 \arctan (a x) \, dx}{a}-\frac {c^3 \int \frac {x^2 \arctan (a x)}{1+a^2 x^2} \, dx}{a}+\frac {1}{4} \left (3 a c^3\right ) \int x^4 \arctan (a x) \, dx-\frac {1}{4} \left (3 a c^3\right ) \int \frac {x^4 \arctan (a x)}{1+a^2 x^2} \, dx+\frac {1}{5} \left (a^2 c^3\right ) \int \frac {x^5}{1+a^2 x^2} \, dx+\frac {1}{5} \left (a^3 c^3\right ) \int x^6 \arctan (a x) \, dx-\frac {1}{5} \left (a^3 c^3\right ) \int \frac {x^6 \arctan (a x)}{1+a^2 x^2} \, dx+\frac {1}{28} \left (3 a^4 c^3\right ) \int \frac {x^7}{1+a^2 x^2} \, dx+\frac {1}{45} \left (a^6 c^3\right ) \int \frac {x^9}{1+a^2 x^2} \, dx \\ & = \frac {c^3 x \arctan (a x)}{2 a^3}+\frac {c^3 x^3 \arctan (a x)}{6 a}-\frac {1}{20} a c^3 x^5 \arctan (a x)-\frac {11}{140} a^3 c^3 x^7 \arctan (a x)-\frac {1}{45} a^5 c^3 x^9 \arctan (a x)-\frac {c^3 \arctan (a x)^2}{4 a^4}+\frac {1}{4} c^3 x^4 \arctan (a x)^2+\frac {1}{2} a^2 c^3 x^6 \arctan (a x)^2+\frac {3}{8} a^4 c^3 x^8 \arctan (a x)^2+\frac {1}{10} a^6 c^3 x^{10} \arctan (a x)^2+\frac {1}{12} c^3 \text {Subst}\left (\int \frac {x}{1+a^2 x} \, dx,x,x^2\right )-\frac {1}{3} c^3 \int \frac {x^3}{1+a^2 x^2} \, dx-\frac {c^3 \int \arctan (a x) \, dx}{a^3}+\frac {c^3 \int \frac {\arctan (a x)}{1+a^2 x^2} \, dx}{a^3}-\frac {c^3 \int \frac {x}{1+a^2 x^2} \, dx}{2 a^2}-\frac {\left (3 c^3\right ) \int x^2 \arctan (a x) \, dx}{4 a}+\frac {\left (3 c^3\right ) \int \frac {x^2 \arctan (a x)}{1+a^2 x^2} \, dx}{4 a}-\frac {1}{5} \left (a c^3\right ) \int x^4 \arctan (a x) \, dx+\frac {1}{5} \left (a c^3\right ) \int \frac {x^4 \arctan (a x)}{1+a^2 x^2} \, dx+\frac {1}{10} \left (a^2 c^3\right ) \text {Subst}\left (\int \frac {x^2}{1+a^2 x} \, dx,x,x^2\right )-\frac {1}{20} \left (3 a^2 c^3\right ) \int \frac {x^5}{1+a^2 x^2} \, dx-\frac {1}{35} \left (a^4 c^3\right ) \int \frac {x^7}{1+a^2 x^2} \, dx+\frac {1}{56} \left (3 a^4 c^3\right ) \text {Subst}\left (\int \frac {x^3}{1+a^2 x} \, dx,x,x^2\right )+\frac {1}{90} \left (a^6 c^3\right ) \text {Subst}\left (\int \frac {x^4}{1+a^2 x} \, dx,x,x^2\right ) \\ & = -\frac {c^3 x \arctan (a x)}{2 a^3}-\frac {c^3 x^3 \arctan (a x)}{12 a}-\frac {9}{100} a c^3 x^5 \arctan (a x)-\frac {11}{140} a^3 c^3 x^7 \arctan (a x)-\frac {1}{45} a^5 c^3 x^9 \arctan (a x)+\frac {c^3 \arctan (a x)^2}{4 a^4}+\frac {1}{4} c^3 x^4 \arctan (a x)^2+\frac {1}{2} a^2 c^3 x^6 \arctan (a x)^2+\frac {3}{8} a^4 c^3 x^8 \arctan (a x)^2+\frac {1}{10} a^6 c^3 x^{10} \arctan (a x)^2-\frac {c^3 \log \left (1+a^2 x^2\right )}{4 a^4}+\frac {1}{12} c^3 \text {Subst}\left (\int \left (\frac {1}{a^2}-\frac {1}{a^2 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )-\frac {1}{6} c^3 \text {Subst}\left (\int \frac {x}{1+a^2 x} \, dx,x,x^2\right )+\frac {1}{4} c^3 \int \frac {x^3}{1+a^2 x^2} \, dx+\frac {\left (3 c^3\right ) \int \arctan (a x) \, dx}{4 a^3}-\frac {\left (3 c^3\right ) \int \frac {\arctan (a x)}{1+a^2 x^2} \, dx}{4 a^3}+\frac {c^3 \int \frac {x}{1+a^2 x^2} \, dx}{a^2}+\frac {c^3 \int x^2 \arctan (a x) \, dx}{5 a}-\frac {c^3 \int \frac {x^2 \arctan (a x)}{1+a^2 x^2} \, dx}{5 a}+\frac {1}{25} \left (a^2 c^3\right ) \int \frac {x^5}{1+a^2 x^2} \, dx-\frac {1}{40} \left (3 a^2 c^3\right ) \text {Subst}\left (\int \frac {x^2}{1+a^2 x} \, dx,x,x^2\right )+\frac {1}{10} \left (a^2 c^3\right ) \text {Subst}\left (\int \left (-\frac {1}{a^4}+\frac {x}{a^2}+\frac {1}{a^4 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )-\frac {1}{70} \left (a^4 c^3\right ) \text {Subst}\left (\int \frac {x^3}{1+a^2 x} \, dx,x,x^2\right )+\frac {1}{56} \left (3 a^4 c^3\right ) \text {Subst}\left (\int \left (\frac {1}{a^6}-\frac {x}{a^4}+\frac {x^2}{a^2}-\frac {1}{a^6 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )+\frac {1}{90} \left (a^6 c^3\right ) \text {Subst}\left (\int \left (-\frac {1}{a^8}+\frac {x}{a^6}-\frac {x^2}{a^4}+\frac {x^3}{a^2}+\frac {1}{a^8 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right ) \\ & = \frac {13 c^3 x^2}{504 a^2}+\frac {29 c^3 x^4}{1008}+\frac {107 a^2 c^3 x^6}{7560}+\frac {1}{360} a^4 c^3 x^8+\frac {c^3 x \arctan (a x)}{4 a^3}-\frac {c^3 x^3 \arctan (a x)}{60 a}-\frac {9}{100} a c^3 x^5 \arctan (a x)-\frac {11}{140} a^3 c^3 x^7 \arctan (a x)-\frac {1}{45} a^5 c^3 x^9 \arctan (a x)-\frac {c^3 \arctan (a x)^2}{8 a^4}+\frac {1}{4} c^3 x^4 \arctan (a x)^2+\frac {1}{2} a^2 c^3 x^6 \arctan (a x)^2+\frac {3}{8} a^4 c^3 x^8 \arctan (a x)^2+\frac {1}{10} a^6 c^3 x^{10} \arctan (a x)^2+\frac {113 c^3 \log \left (1+a^2 x^2\right )}{504 a^4}-\frac {1}{15} c^3 \int \frac {x^3}{1+a^2 x^2} \, dx+\frac {1}{8} c^3 \text {Subst}\left (\int \frac {x}{1+a^2 x} \, dx,x,x^2\right )-\frac {1}{6} c^3 \text {Subst}\left (\int \left (\frac {1}{a^2}-\frac {1}{a^2 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )-\frac {c^3 \int \arctan (a x) \, dx}{5 a^3}+\frac {c^3 \int \frac {\arctan (a x)}{1+a^2 x^2} \, dx}{5 a^3}-\frac {\left (3 c^3\right ) \int \frac {x}{1+a^2 x^2} \, dx}{4 a^2}+\frac {1}{50} \left (a^2 c^3\right ) \text {Subst}\left (\int \frac {x^2}{1+a^2 x} \, dx,x,x^2\right )-\frac {1}{40} \left (3 a^2 c^3\right ) \text {Subst}\left (\int \left (-\frac {1}{a^4}+\frac {x}{a^2}+\frac {1}{a^4 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )-\frac {1}{70} \left (a^4 c^3\right ) \text {Subst}\left (\int \left (\frac {1}{a^6}-\frac {x}{a^4}+\frac {x^2}{a^2}-\frac {1}{a^6 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right ) \\ & = -\frac {101 c^3 x^2}{1260 a^2}-\frac {c^3 x^4}{630}+\frac {71 a^2 c^3 x^6}{7560}+\frac {1}{360} a^4 c^3 x^8+\frac {c^3 x \arctan (a x)}{20 a^3}-\frac {c^3 x^3 \arctan (a x)}{60 a}-\frac {9}{100} a c^3 x^5 \arctan (a x)-\frac {11}{140} a^3 c^3 x^7 \arctan (a x)-\frac {1}{45} a^5 c^3 x^9 \arctan (a x)-\frac {c^3 \arctan (a x)^2}{40 a^4}+\frac {1}{4} c^3 x^4 \arctan (a x)^2+\frac {1}{2} a^2 c^3 x^6 \arctan (a x)^2+\frac {3}{8} a^4 c^3 x^8 \arctan (a x)^2+\frac {1}{10} a^6 c^3 x^{10} \arctan (a x)^2-\frac {113 c^3 \log \left (1+a^2 x^2\right )}{2520 a^4}-\frac {1}{30} c^3 \text {Subst}\left (\int \frac {x}{1+a^2 x} \, dx,x,x^2\right )+\frac {1}{8} c^3 \text {Subst}\left (\int \left (\frac {1}{a^2}-\frac {1}{a^2 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )+\frac {c^3 \int \frac {x}{1+a^2 x^2} \, dx}{5 a^2}+\frac {1}{50} \left (a^2 c^3\right ) \text {Subst}\left (\int \left (-\frac {1}{a^4}+\frac {x}{a^2}+\frac {1}{a^4 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right ) \\ & = \frac {313 c^3 x^2}{12600 a^2}+\frac {53 c^3 x^4}{6300}+\frac {71 a^2 c^3 x^6}{7560}+\frac {1}{360} a^4 c^3 x^8+\frac {c^3 x \arctan (a x)}{20 a^3}-\frac {c^3 x^3 \arctan (a x)}{60 a}-\frac {9}{100} a c^3 x^5 \arctan (a x)-\frac {11}{140} a^3 c^3 x^7 \arctan (a x)-\frac {1}{45} a^5 c^3 x^9 \arctan (a x)-\frac {c^3 \arctan (a x)^2}{40 a^4}+\frac {1}{4} c^3 x^4 \arctan (a x)^2+\frac {1}{2} a^2 c^3 x^6 \arctan (a x)^2+\frac {3}{8} a^4 c^3 x^8 \arctan (a x)^2+\frac {1}{10} a^6 c^3 x^{10} \arctan (a x)^2-\frac {157 c^3 \log \left (1+a^2 x^2\right )}{3150 a^4}-\frac {1}{30} c^3 \text {Subst}\left (\int \left (\frac {1}{a^2}-\frac {1}{a^2 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right ) \\ & = -\frac {107 c^3 x^2}{12600 a^2}+\frac {53 c^3 x^4}{6300}+\frac {71 a^2 c^3 x^6}{7560}+\frac {1}{360} a^4 c^3 x^8+\frac {c^3 x \arctan (a x)}{20 a^3}-\frac {c^3 x^3 \arctan (a x)}{60 a}-\frac {9}{100} a c^3 x^5 \arctan (a x)-\frac {11}{140} a^3 c^3 x^7 \arctan (a x)-\frac {1}{45} a^5 c^3 x^9 \arctan (a x)-\frac {c^3 \arctan (a x)^2}{40 a^4}+\frac {1}{4} c^3 x^4 \arctan (a x)^2+\frac {1}{2} a^2 c^3 x^6 \arctan (a x)^2+\frac {3}{8} a^4 c^3 x^8 \arctan (a x)^2+\frac {1}{10} a^6 c^3 x^{10} \arctan (a x)^2-\frac {26 c^3 \log \left (1+a^2 x^2\right )}{1575 a^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.52 \[ \int x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^2 \, dx=\frac {c^3 \left (-321 a^2 x^2+318 a^4 x^4+355 a^6 x^6+105 a^8 x^8-6 a x \left (-315+105 a^2 x^2+567 a^4 x^4+495 a^6 x^6+140 a^8 x^8\right ) \arctan (a x)+945 \left (1+a^2 x^2\right )^4 \left (-1+4 a^2 x^2\right ) \arctan (a x)^2-624 \log \left (1+a^2 x^2\right )\right )}{37800 a^4} \]

[In]

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

[Out]

(c^3*(-321*a^2*x^2 + 318*a^4*x^4 + 355*a^6*x^6 + 105*a^8*x^8 - 6*a*x*(-315 + 105*a^2*x^2 + 567*a^4*x^4 + 495*a
^6*x^6 + 140*a^8*x^8)*ArcTan[a*x] + 945*(1 + a^2*x^2)^4*(-1 + 4*a^2*x^2)*ArcTan[a*x]^2 - 624*Log[1 + a^2*x^2])
)/(37800*a^4)

Maple [A] (verified)

Time = 1.22 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.78

method	result	size
derivativedivides	\(\frac {\frac {c^{3} \arctan \left (a x \right )^{2} a^{10} x^{10}}{10}+\frac {3 c^{3} \arctan \left (a x \right )^{2} a^{8} x^{8}}{8}+\frac {a^{6} c^{3} x^{6} \arctan \left (a x \right )^{2}}{2}+\frac {a^{4} c^{3} x^{4} \arctan \left (a x \right )^{2}}{4}-\frac {c^{3} \left (\frac {4 \arctan \left (a x \right ) a^{9} x^{9}}{9}+\frac {11 \arctan \left (a x \right ) a^{7} x^{7}}{7}+\frac {9 \arctan \left (a x \right ) a^{5} x^{5}}{5}+\frac {\arctan \left (a x \right ) x^{3} a^{3}}{3}-x \arctan \left (a x \right ) a +\frac {\arctan \left (a x \right )^{2}}{2}-\frac {a^{8} x^{8}}{18}-\frac {71 a^{6} x^{6}}{378}-\frac {53 a^{4} x^{4}}{315}+\frac {107 a^{2} x^{2}}{630}+\frac {104 \ln \left (a^{2} x^{2}+1\right )}{315}\right )}{20}}{a^{4}}\)	\(188\)
default	\(\frac {\frac {c^{3} \arctan \left (a x \right )^{2} a^{10} x^{10}}{10}+\frac {3 c^{3} \arctan \left (a x \right )^{2} a^{8} x^{8}}{8}+\frac {a^{6} c^{3} x^{6} \arctan \left (a x \right )^{2}}{2}+\frac {a^{4} c^{3} x^{4} \arctan \left (a x \right )^{2}}{4}-\frac {c^{3} \left (\frac {4 \arctan \left (a x \right ) a^{9} x^{9}}{9}+\frac {11 \arctan \left (a x \right ) a^{7} x^{7}}{7}+\frac {9 \arctan \left (a x \right ) a^{5} x^{5}}{5}+\frac {\arctan \left (a x \right ) x^{3} a^{3}}{3}-x \arctan \left (a x \right ) a +\frac {\arctan \left (a x \right )^{2}}{2}-\frac {a^{8} x^{8}}{18}-\frac {71 a^{6} x^{6}}{378}-\frac {53 a^{4} x^{4}}{315}+\frac {107 a^{2} x^{2}}{630}+\frac {104 \ln \left (a^{2} x^{2}+1\right )}{315}\right )}{20}}{a^{4}}\)	\(188\)
parts	\(\frac {a^{6} c^{3} x^{10} \arctan \left (a x \right )^{2}}{10}+\frac {3 a^{4} c^{3} x^{8} \arctan \left (a x \right )^{2}}{8}+\frac {a^{2} c^{3} x^{6} \arctan \left (a x \right )^{2}}{2}+\frac {c^{3} x^{4} \arctan \left (a x \right )^{2}}{4}-\frac {c^{3} \left (\frac {4 a^{5} \arctan \left (a x \right ) x^{9}}{9}+\frac {11 a^{3} \arctan \left (a x \right ) x^{7}}{7}+\frac {9 a \arctan \left (a x \right ) x^{5}}{5}+\frac {\arctan \left (a x \right ) x^{3}}{3 a}-\frac {\arctan \left (a x \right ) x}{a^{3}}+\frac {\arctan \left (a x \right )^{2}}{a^{4}}-\frac {\frac {35 a^{8} x^{8}}{2}+\frac {355 a^{6} x^{6}}{6}+53 a^{4} x^{4}-\frac {107 a^{2} x^{2}}{2}-104 \ln \left (a^{2} x^{2}+1\right )+\frac {315 \arctan \left (a x \right )^{2}}{2}}{315 a^{4}}\right )}{20}\)	\(197\)
parallelrisch	\(-\frac {-3780 c^{3} \arctan \left (a x \right )^{2} a^{10} x^{10}+840 c^{3} \arctan \left (a x \right ) a^{9} x^{9}-14175 c^{3} \arctan \left (a x \right )^{2} a^{8} x^{8}-105 a^{8} c^{3} x^{8}+2970 c^{3} \arctan \left (a x \right ) a^{7} x^{7}-18900 a^{6} c^{3} x^{6} \arctan \left (a x \right )^{2}-355 a^{6} c^{3} x^{6}+3402 a^{5} c^{3} x^{5} \arctan \left (a x \right )-9450 a^{4} c^{3} x^{4} \arctan \left (a x \right )^{2}-318 a^{4} c^{3} x^{4}+630 a^{3} c^{3} x^{3} \arctan \left (a x \right )+321 a^{2} c^{3} x^{2}-1890 a \,c^{3} x \arctan \left (a x \right )+945 c^{3} \arctan \left (a x \right )^{2}+624 c^{3} \ln \left (a^{2} x^{2}+1\right )-321 c^{3}}{37800 a^{4}}\)	\(221\)
risch	\(-\frac {c^{3} \left (4 a^{10} x^{10}+15 a^{8} x^{8}+20 a^{6} x^{6}+10 a^{4} x^{4}-1\right ) \ln \left (i a x +1\right )^{2}}{160 a^{4}}+\frac {c^{3} \left (1260 a^{10} x^{10} \ln \left (-i a x +1\right )+280 i a^{9} x^{9}+4725 a^{8} x^{8} \ln \left (-i a x +1\right )+990 i a^{7} x^{7}+6300 a^{6} x^{6} \ln \left (-i a x +1\right )+1134 i a^{5} x^{5}+3150 x^{4} \ln \left (-i a x +1\right ) a^{4}+210 i a^{3} x^{3}-630 i a x -315 \ln \left (-i a x +1\right )\right ) \ln \left (i a x +1\right )}{25200 a^{4}}-\frac {c^{3} a^{6} x^{10} \ln \left (-i a x +1\right )^{2}}{40}-\frac {11 i c^{3} a^{3} x^{7} \ln \left (-i a x +1\right )}{280}-\frac {3 c^{3} a^{4} x^{8} \ln \left (-i a x +1\right )^{2}}{32}-\frac {i c^{3} a^{5} x^{9} \ln \left (-i a x +1\right )}{90}+\frac {a^{4} c^{3} x^{8}}{360}-\frac {c^{3} a^{2} x^{6} \ln \left (-i a x +1\right )^{2}}{8}-\frac {9 i c^{3} a \,x^{5} \ln \left (-i a x +1\right )}{200}+\frac {71 a^{2} c^{3} x^{6}}{7560}-\frac {c^{3} x^{4} \ln \left (-i a x +1\right )^{2}}{16}-\frac {i c^{3} x^{3} \ln \left (-i a x +1\right )}{120 a}+\frac {53 c^{3} x^{4}}{6300}+\frac {i c^{3} x \ln \left (-i a x +1\right )}{40 a^{3}}-\frac {107 c^{3} x^{2}}{12600 a^{2}}+\frac {c^{3} \ln \left (-i a x +1\right )^{2}}{160 a^{4}}-\frac {26 c^{3} \ln \left (-a^{2} x^{2}-1\right )}{1575 a^{4}}\)	\(441\)

[In]

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

[Out]

1/a^4*(1/10*c^3*arctan(a*x)^2*a^10*x^10+3/8*c^3*arctan(a*x)^2*a^8*x^8+1/2*a^6*c^3*x^6*arctan(a*x)^2+1/4*a^4*c^
3*x^4*arctan(a*x)^2-1/20*c^3*(4/9*arctan(a*x)*a^9*x^9+11/7*arctan(a*x)*a^7*x^7+9/5*arctan(a*x)*a^5*x^5+1/3*arc
tan(a*x)*x^3*a^3-x*arctan(a*x)*a+1/2*arctan(a*x)^2-1/18*a^8*x^8-71/378*a^6*x^6-53/315*a^4*x^4+107/630*a^2*x^2+
104/315*ln(a^2*x^2+1)))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.75 \[ \int x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^2 \, dx=\frac {105 \, a^{8} c^{3} x^{8} + 355 \, a^{6} c^{3} x^{6} + 318 \, a^{4} c^{3} x^{4} - 321 \, a^{2} c^{3} x^{2} - 624 \, c^{3} \log \left (a^{2} x^{2} + 1\right ) + 945 \, {\left (4 \, a^{10} c^{3} x^{10} + 15 \, a^{8} c^{3} x^{8} + 20 \, a^{6} c^{3} x^{6} + 10 \, a^{4} c^{3} x^{4} - c^{3}\right )} \arctan \left (a x\right )^{2} - 6 \, {\left (140 \, a^{9} c^{3} x^{9} + 495 \, a^{7} c^{3} x^{7} + 567 \, a^{5} c^{3} x^{5} + 105 \, a^{3} c^{3} x^{3} - 315 \, a c^{3} x\right )} \arctan \left (a x\right )}{37800 \, a^{4}} \]

[In]

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

[Out]

1/37800*(105*a^8*c^3*x^8 + 355*a^6*c^3*x^6 + 318*a^4*c^3*x^4 - 321*a^2*c^3*x^2 - 624*c^3*log(a^2*x^2 + 1) + 94
5*(4*a^10*c^3*x^10 + 15*a^8*c^3*x^8 + 20*a^6*c^3*x^6 + 10*a^4*c^3*x^4 - c^3)*arctan(a*x)^2 - 6*(140*a^9*c^3*x^
9 + 495*a^7*c^3*x^7 + 567*a^5*c^3*x^5 + 105*a^3*c^3*x^3 - 315*a*c^3*x)*arctan(a*x))/a^4

Sympy [A] (verification not implemented)

Time = 0.82 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.00 \[ \int x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^2 \, dx=\begin {cases} \frac {a^{6} c^{3} x^{10} \operatorname {atan}^{2}{\left (a x \right )}}{10} - \frac {a^{5} c^{3} x^{9} \operatorname {atan}{\left (a x \right )}}{45} + \frac {3 a^{4} c^{3} x^{8} \operatorname {atan}^{2}{\left (a x \right )}}{8} + \frac {a^{4} c^{3} x^{8}}{360} - \frac {11 a^{3} c^{3} x^{7} \operatorname {atan}{\left (a x \right )}}{140} + \frac {a^{2} c^{3} x^{6} \operatorname {atan}^{2}{\left (a x \right )}}{2} + \frac {71 a^{2} c^{3} x^{6}}{7560} - \frac {9 a c^{3} x^{5} \operatorname {atan}{\left (a x \right )}}{100} + \frac {c^{3} x^{4} \operatorname {atan}^{2}{\left (a x \right )}}{4} + \frac {53 c^{3} x^{4}}{6300} - \frac {c^{3} x^{3} \operatorname {atan}{\left (a x \right )}}{60 a} - \frac {107 c^{3} x^{2}}{12600 a^{2}} + \frac {c^{3} x \operatorname {atan}{\left (a x \right )}}{20 a^{3}} - \frac {26 c^{3} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{1575 a^{4}} - \frac {c^{3} \operatorname {atan}^{2}{\left (a x \right )}}{40 a^{4}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((a**6*c**3*x**10*atan(a*x)**2/10 - a**5*c**3*x**9*atan(a*x)/45 + 3*a**4*c**3*x**8*atan(a*x)**2/8 + a
**4*c**3*x**8/360 - 11*a**3*c**3*x**7*atan(a*x)/140 + a**2*c**3*x**6*atan(a*x)**2/2 + 71*a**2*c**3*x**6/7560 -
9*a*c**3*x**5*atan(a*x)/100 + c**3*x**4*atan(a*x)**2/4 + 53*c**3*x**4/6300 - c**3*x**3*atan(a*x)/(60*a) - 107
*c**3*x**2/(12600*a**2) + c**3*x*atan(a*x)/(20*a**3) - 26*c**3*log(x**2 + a**(-2))/(1575*a**4) - c**3*atan(a*x
)**2/(40*a**4), Ne(a, 0)), (0, True))

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.84 \[ \int x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^2 \, dx=-\frac {1}{6300} \, a {\left (\frac {315 \, c^{3} \arctan \left (a x\right )}{a^{5}} + \frac {140 \, a^{8} c^{3} x^{9} + 495 \, a^{6} c^{3} x^{7} + 567 \, a^{4} c^{3} x^{5} + 105 \, a^{2} c^{3} x^{3} - 315 \, c^{3} x}{a^{4}}\right )} \arctan \left (a x\right ) + \frac {1}{40} \, {\left (4 \, a^{6} c^{3} x^{10} + 15 \, a^{4} c^{3} x^{8} + 20 \, a^{2} c^{3} x^{6} + 10 \, c^{3} x^{4}\right )} \arctan \left (a x\right )^{2} + \frac {105 \, a^{8} c^{3} x^{8} + 355 \, a^{6} c^{3} x^{6} + 318 \, a^{4} c^{3} x^{4} - 321 \, a^{2} c^{3} x^{2} + 945 \, c^{3} \arctan \left (a x\right )^{2} - 624 \, c^{3} \log \left (a^{2} x^{2} + 1\right )}{37800 \, a^{4}} \]

[In]

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

[Out]

-1/6300*a*(315*c^3*arctan(a*x)/a^5 + (140*a^8*c^3*x^9 + 495*a^6*c^3*x^7 + 567*a^4*c^3*x^5 + 105*a^2*c^3*x^3 -
315*c^3*x)/a^4)*arctan(a*x) + 1/40*(4*a^6*c^3*x^10 + 15*a^4*c^3*x^8 + 20*a^2*c^3*x^6 + 10*c^3*x^4)*arctan(a*x)
^2 + 1/37800*(105*a^8*c^3*x^8 + 355*a^6*c^3*x^6 + 318*a^4*c^3*x^4 - 321*a^2*c^3*x^2 + 945*c^3*arctan(a*x)^2 -
624*c^3*log(a^2*x^2 + 1))/a^4

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [B] (verification not implemented)

Time = 0.56 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.74 \[ \int x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^2 \, dx={\mathrm {atan}\left (a\,x\right )}^2\,\left (\frac {c^3\,x^4}{4}-\frac {c^3}{40\,a^4}+\frac {a^2\,c^3\,x^6}{2}+\frac {3\,a^4\,c^3\,x^8}{8}+\frac {a^6\,c^3\,x^{10}}{10}\right )+\frac {53\,c^3\,x^4}{6300}-\frac {26\,c^3\,\ln \left (a^2\,x^2+1\right )}{1575\,a^4}-\frac {107\,c^3\,x^2}{12600\,a^2}+\frac {71\,a^2\,c^3\,x^6}{7560}+\frac {a^4\,c^3\,x^8}{360}-a^2\,\mathrm {atan}\left (a\,x\right )\,\left (\frac {11\,a\,c^3\,x^7}{140}-\frac {c^3\,x}{20\,a^5}+\frac {9\,c^3\,x^5}{100\,a}+\frac {c^3\,x^3}{60\,a^3}+\frac {a^3\,c^3\,x^9}{45}\right ) \]

[In]

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

[Out]

atan(a*x)^2*((c^3*x^4)/4 - c^3/(40*a^4) + (a^2*c^3*x^6)/2 + (3*a^4*c^3*x^8)/8 + (a^6*c^3*x^10)/10) + (53*c^3*x
^4)/6300 - (26*c^3*log(a^2*x^2 + 1))/(1575*a^4) - (107*c^3*x^2)/(12600*a^2) + (71*a^2*c^3*x^6)/7560 + (a^4*c^3
*x^8)/360 - a^2*atan(a*x)*((11*a*c^3*x^7)/140 - (c^3*x)/(20*a^5) + (9*c^3*x^5)/(100*a) + (c^3*x^3)/(60*a^3) +
(a^3*c^3*x^9)/45)