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

Optimal. Leaf size=191 \[ \frac {1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)^2-\frac {c^2 \tan ^{-1}(a x)^2}{24 a^4}-\frac {1}{28} a^3 c^2 x^7 \tan ^{-1}(a x)+\frac {c^2 x \tan ^{-1}(a x)}{12 a^3}+\frac {1}{168} a^2 c^2 x^6+\frac {1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)^2-\frac {5 c^2 x^2}{504 a^2}-\frac {2 c^2 \log \left (a^2 x^2+1\right )}{63 a^4}-\frac {1}{12} a c^2 x^5 \tan ^{-1}(a x)+\frac {1}{4} c^2 x^4 \tan ^{-1}(a x)^2-\frac {c^2 x^3 \tan ^{-1}(a x)}{36 a}+\frac {c^2 x^4}{84} \]

[Out]

-5/504*c^2*x^2/a^2+1/84*c^2*x^4+1/168*a^2*c^2*x^6+1/12*c^2*x*arctan(a*x)/a^3-1/36*c^2*x^3*arctan(a*x)/a-1/12*a
*c^2*x^5*arctan(a*x)-1/28*a^3*c^2*x^7*arctan(a*x)-1/24*c^2*arctan(a*x)^2/a^4+1/4*c^2*x^4*arctan(a*x)^2+1/3*a^2
*c^2*x^6*arctan(a*x)^2+1/8*a^4*c^2*x^8*arctan(a*x)^2-2/63*c^2*ln(a^2*x^2+1)/a^4

________________________________________________________________________________________

Rubi [A]  time = 0.79, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 47, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {4948, 4852, 4916, 266, 43, 4846, 260, 4884} \[ \frac {1}{168} a^2 c^2 x^6-\frac {5 c^2 x^2}{504 a^2}-\frac {2 c^2 \log \left (a^2 x^2+1\right )}{63 a^4}+\frac {1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)^2-\frac {1}{28} a^3 c^2 x^7 \tan ^{-1}(a x)+\frac {1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)^2+\frac {c^2 x \tan ^{-1}(a x)}{12 a^3}-\frac {c^2 \tan ^{-1}(a x)^2}{24 a^4}-\frac {1}{12} a c^2 x^5 \tan ^{-1}(a x)+\frac {1}{4} c^2 x^4 \tan ^{-1}(a x)^2-\frac {c^2 x^3 \tan ^{-1}(a x)}{36 a}+\frac {c^2 x^4}{84} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*c^2*x^2)/(504*a^2) + (c^2*x^4)/84 + (a^2*c^2*x^6)/168 + (c^2*x*ArcTan[a*x])/(12*a^3) - (c^2*x^3*ArcTan[a*x
])/(36*a) - (a*c^2*x^5*ArcTan[a*x])/12 - (a^3*c^2*x^7*ArcTan[a*x])/28 - (c^2*ArcTan[a*x]^2)/(24*a^4) + (c^2*x^
4*ArcTan[a*x]^2)/4 + (a^2*c^2*x^6*ArcTan[a*x]^2)/3 + (a^4*c^2*x^8*ArcTan[a*x]^2)/8 - (2*c^2*Log[1 + a^2*x^2])/
(63*a^4)

Rule 43

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 260

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 266

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 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 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 4884

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 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 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^3 \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2 \, dx &=\int \left (c^2 x^3 \tan ^{-1}(a x)^2+2 a^2 c^2 x^5 \tan ^{-1}(a x)^2+a^4 c^2 x^7 \tan ^{-1}(a x)^2\right ) \, dx\\ &=c^2 \int x^3 \tan ^{-1}(a x)^2 \, dx+\left (2 a^2 c^2\right ) \int x^5 \tan ^{-1}(a x)^2 \, dx+\left (a^4 c^2\right ) \int x^7 \tan ^{-1}(a x)^2 \, dx\\ &=\frac {1}{4} c^2 x^4 \tan ^{-1}(a x)^2+\frac {1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)^2+\frac {1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)^2-\frac {1}{2} \left (a c^2\right ) \int \frac {x^4 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac {1}{3} \left (2 a^3 c^2\right ) \int \frac {x^6 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac {1}{4} \left (a^5 c^2\right ) \int \frac {x^8 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=\frac {1}{4} c^2 x^4 \tan ^{-1}(a x)^2+\frac {1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)^2+\frac {1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)^2-\frac {c^2 \int x^2 \tan ^{-1}(a x) \, dx}{2 a}+\frac {c^2 \int \frac {x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{2 a}-\frac {1}{3} \left (2 a c^2\right ) \int x^4 \tan ^{-1}(a x) \, dx+\frac {1}{3} \left (2 a c^2\right ) \int \frac {x^4 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac {1}{4} \left (a^3 c^2\right ) \int x^6 \tan ^{-1}(a x) \, dx+\frac {1}{4} \left (a^3 c^2\right ) \int \frac {x^6 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=-\frac {c^2 x^3 \tan ^{-1}(a x)}{6 a}-\frac {2}{15} a c^2 x^5 \tan ^{-1}(a x)-\frac {1}{28} a^3 c^2 x^7 \tan ^{-1}(a x)+\frac {1}{4} c^2 x^4 \tan ^{-1}(a x)^2+\frac {1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)^2+\frac {1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)^2+\frac {1}{6} c^2 \int \frac {x^3}{1+a^2 x^2} \, dx+\frac {c^2 \int \tan ^{-1}(a x) \, dx}{2 a^3}-\frac {c^2 \int \frac {\tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{2 a^3}+\frac {\left (2 c^2\right ) \int x^2 \tan ^{-1}(a x) \, dx}{3 a}-\frac {\left (2 c^2\right ) \int \frac {x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{3 a}+\frac {1}{4} \left (a c^2\right ) \int x^4 \tan ^{-1}(a x) \, dx-\frac {1}{4} \left (a c^2\right ) \int \frac {x^4 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac {1}{15} \left (2 a^2 c^2\right ) \int \frac {x^5}{1+a^2 x^2} \, dx+\frac {1}{28} \left (a^4 c^2\right ) \int \frac {x^7}{1+a^2 x^2} \, dx\\ &=\frac {c^2 x \tan ^{-1}(a x)}{2 a^3}+\frac {c^2 x^3 \tan ^{-1}(a x)}{18 a}-\frac {1}{12} a c^2 x^5 \tan ^{-1}(a x)-\frac {1}{28} a^3 c^2 x^7 \tan ^{-1}(a x)-\frac {c^2 \tan ^{-1}(a x)^2}{4 a^4}+\frac {1}{4} c^2 x^4 \tan ^{-1}(a x)^2+\frac {1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)^2+\frac {1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)^2+\frac {1}{12} c^2 \operatorname {Subst}\left (\int \frac {x}{1+a^2 x} \, dx,x,x^2\right )-\frac {1}{9} \left (2 c^2\right ) \int \frac {x^3}{1+a^2 x^2} \, dx-\frac {\left (2 c^2\right ) \int \tan ^{-1}(a x) \, dx}{3 a^3}+\frac {\left (2 c^2\right ) \int \frac {\tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{3 a^3}-\frac {c^2 \int \frac {x}{1+a^2 x^2} \, dx}{2 a^2}-\frac {c^2 \int x^2 \tan ^{-1}(a x) \, dx}{4 a}+\frac {c^2 \int \frac {x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{4 a}-\frac {1}{20} \left (a^2 c^2\right ) \int \frac {x^5}{1+a^2 x^2} \, dx+\frac {1}{15} \left (a^2 c^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{1+a^2 x} \, dx,x,x^2\right )+\frac {1}{56} \left (a^4 c^2\right ) \operatorname {Subst}\left (\int \frac {x^3}{1+a^2 x} \, dx,x,x^2\right )\\ &=-\frac {c^2 x \tan ^{-1}(a x)}{6 a^3}-\frac {c^2 x^3 \tan ^{-1}(a x)}{36 a}-\frac {1}{12} a c^2 x^5 \tan ^{-1}(a x)-\frac {1}{28} a^3 c^2 x^7 \tan ^{-1}(a x)+\frac {c^2 \tan ^{-1}(a x)^2}{12 a^4}+\frac {1}{4} c^2 x^4 \tan ^{-1}(a x)^2+\frac {1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)^2+\frac {1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)^2-\frac {c^2 \log \left (1+a^2 x^2\right )}{4 a^4}+\frac {1}{12} c^2 \int \frac {x^3}{1+a^2 x^2} \, dx+\frac {1}{12} c^2 \operatorname {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}{9} c^2 \operatorname {Subst}\left (\int \frac {x}{1+a^2 x} \, dx,x,x^2\right )+\frac {c^2 \int \tan ^{-1}(a x) \, dx}{4 a^3}-\frac {c^2 \int \frac {\tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{4 a^3}+\frac {\left (2 c^2\right ) \int \frac {x}{1+a^2 x^2} \, dx}{3 a^2}-\frac {1}{40} \left (a^2 c^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{1+a^2 x} \, dx,x,x^2\right )+\frac {1}{15} \left (a^2 c^2\right ) \operatorname {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}{56} \left (a^4 c^2\right ) \operatorname {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 {29 c^2 x^2}{840 a^2}+\frac {41 c^2 x^4}{1680}+\frac {1}{168} a^2 c^2 x^6+\frac {c^2 x \tan ^{-1}(a x)}{12 a^3}-\frac {c^2 x^3 \tan ^{-1}(a x)}{36 a}-\frac {1}{12} a c^2 x^5 \tan ^{-1}(a x)-\frac {1}{28} a^3 c^2 x^7 \tan ^{-1}(a x)-\frac {c^2 \tan ^{-1}(a x)^2}{24 a^4}+\frac {1}{4} c^2 x^4 \tan ^{-1}(a x)^2+\frac {1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)^2+\frac {1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)^2+\frac {41 c^2 \log \left (1+a^2 x^2\right )}{840 a^4}+\frac {1}{24} c^2 \operatorname {Subst}\left (\int \frac {x}{1+a^2 x} \, dx,x,x^2\right )-\frac {1}{9} c^2 \operatorname {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^2 \int \frac {x}{1+a^2 x^2} \, dx}{4 a^2}-\frac {1}{40} \left (a^2 c^2\right ) \operatorname {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 {13 c^2 x^2}{252 a^2}+\frac {c^2 x^4}{84}+\frac {1}{168} a^2 c^2 x^6+\frac {c^2 x \tan ^{-1}(a x)}{12 a^3}-\frac {c^2 x^3 \tan ^{-1}(a x)}{36 a}-\frac {1}{12} a c^2 x^5 \tan ^{-1}(a x)-\frac {1}{28} a^3 c^2 x^7 \tan ^{-1}(a x)-\frac {c^2 \tan ^{-1}(a x)^2}{24 a^4}+\frac {1}{4} c^2 x^4 \tan ^{-1}(a x)^2+\frac {1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)^2+\frac {1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)^2+\frac {5 c^2 \log \left (1+a^2 x^2\right )}{504 a^4}+\frac {1}{24} c^2 \operatorname {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 {5 c^2 x^2}{504 a^2}+\frac {c^2 x^4}{84}+\frac {1}{168} a^2 c^2 x^6+\frac {c^2 x \tan ^{-1}(a x)}{12 a^3}-\frac {c^2 x^3 \tan ^{-1}(a x)}{36 a}-\frac {1}{12} a c^2 x^5 \tan ^{-1}(a x)-\frac {1}{28} a^3 c^2 x^7 \tan ^{-1}(a x)-\frac {c^2 \tan ^{-1}(a x)^2}{24 a^4}+\frac {1}{4} c^2 x^4 \tan ^{-1}(a x)^2+\frac {1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)^2+\frac {1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)^2-\frac {2 c^2 \log \left (1+a^2 x^2\right )}{63 a^4}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.08, size = 110, normalized size = 0.58 \[ \frac {c^2 \left (3 a^6 x^6+6 a^4 x^4-5 a^2 x^2-16 \log \left (a^2 x^2+1\right )+21 \left (a^2 x^2+1\right )^3 \left (3 a^2 x^2-1\right ) \tan ^{-1}(a x)^2-2 a x \left (9 a^6 x^6+21 a^4 x^4+7 a^2 x^2-21\right ) \tan ^{-1}(a x)\right )}{504 a^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c^2*(-5*a^2*x^2 + 6*a^4*x^4 + 3*a^6*x^6 - 2*a*x*(-21 + 7*a^2*x^2 + 21*a^4*x^4 + 9*a^6*x^6)*ArcTan[a*x] + 21*(
1 + a^2*x^2)^3*(-1 + 3*a^2*x^2)*ArcTan[a*x]^2 - 16*Log[1 + a^2*x^2]))/(504*a^4)

________________________________________________________________________________________

fricas [A]  time = 0.63, size = 148, normalized size = 0.77 \[ \frac {3 \, a^{6} c^{2} x^{6} + 6 \, a^{4} c^{2} x^{4} - 5 \, a^{2} c^{2} x^{2} + 21 \, {\left (3 \, a^{8} c^{2} x^{8} + 8 \, a^{6} c^{2} x^{6} + 6 \, a^{4} c^{2} x^{4} - c^{2}\right )} \arctan \left (a x\right )^{2} - 16 \, c^{2} \log \left (a^{2} x^{2} + 1\right ) - 2 \, {\left (9 \, a^{7} c^{2} x^{7} + 21 \, a^{5} c^{2} x^{5} + 7 \, a^{3} c^{2} x^{3} - 21 \, a c^{2} x\right )} \arctan \left (a x\right )}{504 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/504*(3*a^6*c^2*x^6 + 6*a^4*c^2*x^4 - 5*a^2*c^2*x^2 + 21*(3*a^8*c^2*x^8 + 8*a^6*c^2*x^6 + 6*a^4*c^2*x^4 - c^2
)*arctan(a*x)^2 - 16*c^2*log(a^2*x^2 + 1) - 2*(9*a^7*c^2*x^7 + 21*a^5*c^2*x^5 + 7*a^3*c^2*x^3 - 21*a*c^2*x)*ar
ctan(a*x))/a^4

________________________________________________________________________________________

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

[Out]

sage0*x

________________________________________________________________________________________

maple [A]  time = 0.04, size = 168, normalized size = 0.88 \[ -\frac {5 c^{2} x^{2}}{504 a^{2}}+\frac {c^{2} x^{4}}{84}+\frac {a^{2} c^{2} x^{6}}{168}+\frac {c^{2} x \arctan \left (a x \right )}{12 a^{3}}-\frac {c^{2} x^{3} \arctan \left (a x \right )}{36 a}-\frac {a \,c^{2} x^{5} \arctan \left (a x \right )}{12}-\frac {a^{3} c^{2} x^{7} \arctan \left (a x \right )}{28}-\frac {c^{2} \arctan \left (a x \right )^{2}}{24 a^{4}}+\frac {c^{2} x^{4} \arctan \left (a x \right )^{2}}{4}+\frac {a^{2} c^{2} x^{6} \arctan \left (a x \right )^{2}}{3}+\frac {a^{4} c^{2} x^{8} \arctan \left (a x \right )^{2}}{8}-\frac {2 c^{2} \ln \left (a^{2} x^{2}+1\right )}{63 a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/504*c^2*x^2/a^2+1/84*c^2*x^4+1/168*a^2*c^2*x^6+1/12*c^2*x*arctan(a*x)/a^3-1/36*c^2*x^3*arctan(a*x)/a-1/12*a
*c^2*x^5*arctan(a*x)-1/28*a^3*c^2*x^7*arctan(a*x)-1/24*c^2*arctan(a*x)^2/a^4+1/4*c^2*x^4*arctan(a*x)^2+1/3*a^2
*c^2*x^6*arctan(a*x)^2+1/8*a^4*c^2*x^8*arctan(a*x)^2-2/63*c^2*ln(a^2*x^2+1)/a^4

________________________________________________________________________________________

maxima [A]  time = 0.43, size = 169, normalized size = 0.88 \[ -\frac {1}{252} \, a {\left (\frac {21 \, c^{2} \arctan \left (a x\right )}{a^{5}} + \frac {9 \, a^{6} c^{2} x^{7} + 21 \, a^{4} c^{2} x^{5} + 7 \, a^{2} c^{2} x^{3} - 21 \, c^{2} x}{a^{4}}\right )} \arctan \left (a x\right ) + \frac {1}{24} \, {\left (3 \, a^{4} c^{2} x^{8} + 8 \, a^{2} c^{2} x^{6} + 6 \, c^{2} x^{4}\right )} \arctan \left (a x\right )^{2} + \frac {3 \, a^{6} c^{2} x^{6} + 6 \, a^{4} c^{2} x^{4} - 5 \, a^{2} c^{2} x^{2} + 21 \, c^{2} \arctan \left (a x\right )^{2} - 16 \, c^{2} \log \left (a^{2} x^{2} + 1\right )}{504 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/252*a*(21*c^2*arctan(a*x)/a^5 + (9*a^6*c^2*x^7 + 21*a^4*c^2*x^5 + 7*a^2*c^2*x^3 - 21*c^2*x)/a^4)*arctan(a*x
) + 1/24*(3*a^4*c^2*x^8 + 8*a^2*c^2*x^6 + 6*c^2*x^4)*arctan(a*x)^2 + 1/504*(3*a^6*c^2*x^6 + 6*a^4*c^2*x^4 - 5*
a^2*c^2*x^2 + 21*c^2*arctan(a*x)^2 - 16*c^2*log(a^2*x^2 + 1))/a^4

________________________________________________________________________________________

mupad [B]  time = 0.54, size = 145, normalized size = 0.76 \[ {\mathrm {atan}\left (a\,x\right )}^2\,\left (\frac {c^2\,x^4}{4}-\frac {c^2}{24\,a^4}+\frac {a^2\,c^2\,x^6}{3}+\frac {a^4\,c^2\,x^8}{8}\right )+\frac {c^2\,x^4}{84}-a^2\,\mathrm {atan}\left (a\,x\right )\,\left (\frac {a\,c^2\,x^7}{28}-\frac {c^2\,x}{12\,a^5}+\frac {c^2\,x^5}{12\,a}+\frac {c^2\,x^3}{36\,a^3}\right )-\frac {2\,c^2\,\ln \left (a^2\,x^2+1\right )}{63\,a^4}-\frac {5\,c^2\,x^2}{504\,a^2}+\frac {a^2\,c^2\,x^6}{168} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

atan(a*x)^2*((c^2*x^4)/4 - c^2/(24*a^4) + (a^2*c^2*x^6)/3 + (a^4*c^2*x^8)/8) + (c^2*x^4)/84 - a^2*atan(a*x)*((
a*c^2*x^7)/28 - (c^2*x)/(12*a^5) + (c^2*x^5)/(12*a) + (c^2*x^3)/(36*a^3)) - (2*c^2*log(a^2*x^2 + 1))/(63*a^4)
- (5*c^2*x^2)/(504*a^2) + (a^2*c^2*x^6)/168

________________________________________________________________________________________

sympy [A]  time = 3.29, size = 185, normalized size = 0.97 \[ \begin {cases} \frac {a^{4} c^{2} x^{8} \operatorname {atan}^{2}{\left (a x \right )}}{8} - \frac {a^{3} c^{2} x^{7} \operatorname {atan}{\left (a x \right )}}{28} + \frac {a^{2} c^{2} x^{6} \operatorname {atan}^{2}{\left (a x \right )}}{3} + \frac {a^{2} c^{2} x^{6}}{168} - \frac {a c^{2} x^{5} \operatorname {atan}{\left (a x \right )}}{12} + \frac {c^{2} x^{4} \operatorname {atan}^{2}{\left (a x \right )}}{4} + \frac {c^{2} x^{4}}{84} - \frac {c^{2} x^{3} \operatorname {atan}{\left (a x \right )}}{36 a} - \frac {5 c^{2} x^{2}}{504 a^{2}} + \frac {c^{2} x \operatorname {atan}{\left (a x \right )}}{12 a^{3}} - \frac {2 c^{2} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{63 a^{4}} - \frac {c^{2} \operatorname {atan}^{2}{\left (a x \right )}}{24 a^{4}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**4*c**2*x**8*atan(a*x)**2/8 - a**3*c**2*x**7*atan(a*x)/28 + a**2*c**2*x**6*atan(a*x)**2/3 + a**2*
c**2*x**6/168 - a*c**2*x**5*atan(a*x)/12 + c**2*x**4*atan(a*x)**2/4 + c**2*x**4/84 - c**2*x**3*atan(a*x)/(36*a
) - 5*c**2*x**2/(504*a**2) + c**2*x*atan(a*x)/(12*a**3) - 2*c**2*log(x**2 + a**(-2))/(63*a**4) - c**2*atan(a*x
)**2/(24*a**4), Ne(a, 0)), (0, True))

________________________________________________________________________________________