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

Optimal. Leaf size=313 \[ \frac {2 i c^2 \text {Li}_2\left (1-\frac {2}{i a x+1}\right )}{21 a^4}+\frac {1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)^3-\frac {c^2 \tan ^{-1}(a x)^3}{24 a^4}+\frac {2 i c^2 \tan ^{-1}(a x)^2}{21 a^4}-\frac {c^2 \tan ^{-1}(a x)}{21 a^4}+\frac {4 c^2 \log \left (\frac {2}{1+i a x}\right ) \tan ^{-1}(a x)}{21 a^4}-\frac {3}{56} a^3 c^2 x^7 \tan ^{-1}(a x)^2+\frac {c^2 x}{21 a^3}+\frac {c^2 x \tan ^{-1}(a x)^2}{8 a^3}+\frac {1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)^3+\frac {1}{56} a^2 c^2 x^6 \tan ^{-1}(a x)-\frac {5 c^2 x^2 \tan ^{-1}(a x)}{168 a^2}-\frac {1}{280} a c^2 x^5-\frac {1}{8} a c^2 x^5 \tan ^{-1}(a x)^2+\frac {1}{4} c^2 x^4 \tan ^{-1}(a x)^3+\frac {1}{28} c^2 x^4 \tan ^{-1}(a x)-\frac {c^2 x^3}{168 a}-\frac {c^2 x^3 \tan ^{-1}(a x)^2}{24 a} \]

[Out]

1/21*c^2*x/a^3-1/168*c^2*x^3/a-1/280*a*c^2*x^5-1/21*c^2*arctan(a*x)/a^4-5/168*c^2*x^2*arctan(a*x)/a^2+1/28*c^2
*x^4*arctan(a*x)+1/56*a^2*c^2*x^6*arctan(a*x)+2/21*I*c^2*arctan(a*x)^2/a^4+1/8*c^2*x*arctan(a*x)^2/a^3-1/24*c^
2*x^3*arctan(a*x)^2/a-1/8*a*c^2*x^5*arctan(a*x)^2-3/56*a^3*c^2*x^7*arctan(a*x)^2-1/24*c^2*arctan(a*x)^3/a^4+1/
4*c^2*x^4*arctan(a*x)^3+1/3*a^2*c^2*x^6*arctan(a*x)^3+1/8*a^4*c^2*x^8*arctan(a*x)^3+4/21*c^2*arctan(a*x)*ln(2/
(1+I*a*x))/a^4+2/21*I*c^2*polylog(2,1-2/(1+I*a*x))/a^4

________________________________________________________________________________________

Rubi [A]  time = 2.28, antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps used = 106, number of rules used = 12, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.546, Rules used = {4948, 4852, 4916, 321, 203, 4920, 4854, 2402, 2315, 4846, 4884, 302} \[ \frac {2 i c^2 \text {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{21 a^4}+\frac {1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)^3-\frac {3}{56} a^3 c^2 x^7 \tan ^{-1}(a x)^2+\frac {1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)^3+\frac {1}{56} a^2 c^2 x^6 \tan ^{-1}(a x)-\frac {5 c^2 x^2 \tan ^{-1}(a x)}{168 a^2}+\frac {c^2 x}{21 a^3}+\frac {c^2 x \tan ^{-1}(a x)^2}{8 a^3}-\frac {c^2 \tan ^{-1}(a x)^3}{24 a^4}+\frac {2 i c^2 \tan ^{-1}(a x)^2}{21 a^4}-\frac {c^2 \tan ^{-1}(a x)}{21 a^4}+\frac {4 c^2 \log \left (\frac {2}{1+i a x}\right ) \tan ^{-1}(a x)}{21 a^4}-\frac {1}{280} a c^2 x^5-\frac {c^2 x^3}{168 a}-\frac {1}{8} a c^2 x^5 \tan ^{-1}(a x)^2+\frac {1}{4} c^2 x^4 \tan ^{-1}(a x)^3+\frac {1}{28} c^2 x^4 \tan ^{-1}(a x)-\frac {c^2 x^3 \tan ^{-1}(a x)^2}{24 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c^2*x)/(21*a^3) - (c^2*x^3)/(168*a) - (a*c^2*x^5)/280 - (c^2*ArcTan[a*x])/(21*a^4) - (5*c^2*x^2*ArcTan[a*x])/
(168*a^2) + (c^2*x^4*ArcTan[a*x])/28 + (a^2*c^2*x^6*ArcTan[a*x])/56 + (((2*I)/21)*c^2*ArcTan[a*x]^2)/a^4 + (c^
2*x*ArcTan[a*x]^2)/(8*a^3) - (c^2*x^3*ArcTan[a*x]^2)/(24*a) - (a*c^2*x^5*ArcTan[a*x]^2)/8 - (3*a^3*c^2*x^7*Arc
Tan[a*x]^2)/56 - (c^2*ArcTan[a*x]^3)/(24*a^4) + (c^2*x^4*ArcTan[a*x]^3)/4 + (a^2*c^2*x^6*ArcTan[a*x]^3)/3 + (a
^4*c^2*x^8*ArcTan[a*x]^3)/8 + (4*c^2*ArcTan[a*x]*Log[2/(1 + I*a*x)])/(21*a^4) + (((2*I)/21)*c^2*PolyLog[2, 1 -
2/(1 + I*a*x)])/a^4

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 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 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 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 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^3 \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^3 \, dx &=\int \left (c^2 x^3 \tan ^{-1}(a x)^3+2 a^2 c^2 x^5 \tan ^{-1}(a x)^3+a^4 c^2 x^7 \tan ^{-1}(a x)^3\right ) \, dx\\ &=c^2 \int x^3 \tan ^{-1}(a x)^3 \, dx+\left (2 a^2 c^2\right ) \int x^5 \tan ^{-1}(a x)^3 \, dx+\left (a^4 c^2\right ) \int x^7 \tan ^{-1}(a x)^3 \, dx\\ &=\frac {1}{4} c^2 x^4 \tan ^{-1}(a x)^3+\frac {1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)^3+\frac {1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)^3-\frac {1}{4} \left (3 a c^2\right ) \int \frac {x^4 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx-\left (a^3 c^2\right ) \int \frac {x^6 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx-\frac {1}{8} \left (3 a^5 c^2\right ) \int \frac {x^8 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx\\ &=\frac {1}{4} c^2 x^4 \tan ^{-1}(a x)^3+\frac {1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)^3+\frac {1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)^3-\frac {\left (3 c^2\right ) \int x^2 \tan ^{-1}(a x)^2 \, dx}{4 a}+\frac {\left (3 c^2\right ) \int \frac {x^2 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{4 a}-\left (a c^2\right ) \int x^4 \tan ^{-1}(a x)^2 \, dx+\left (a c^2\right ) \int \frac {x^4 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx-\frac {1}{8} \left (3 a^3 c^2\right ) \int x^6 \tan ^{-1}(a x)^2 \, dx+\frac {1}{8} \left (3 a^3 c^2\right ) \int \frac {x^6 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx\\ &=-\frac {c^2 x^3 \tan ^{-1}(a x)^2}{4 a}-\frac {1}{5} a c^2 x^5 \tan ^{-1}(a x)^2-\frac {3}{56} a^3 c^2 x^7 \tan ^{-1}(a x)^2+\frac {1}{4} c^2 x^4 \tan ^{-1}(a x)^3+\frac {1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)^3+\frac {1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)^3+\frac {1}{2} c^2 \int \frac {x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac {\left (3 c^2\right ) \int \tan ^{-1}(a x)^2 \, dx}{4 a^3}-\frac {\left (3 c^2\right ) \int \frac {\tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{4 a^3}+\frac {c^2 \int x^2 \tan ^{-1}(a x)^2 \, dx}{a}-\frac {c^2 \int \frac {x^2 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{a}+\frac {1}{8} \left (3 a c^2\right ) \int x^4 \tan ^{-1}(a x)^2 \, dx-\frac {1}{8} \left (3 a c^2\right ) \int \frac {x^4 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx+\frac {1}{5} \left (2 a^2 c^2\right ) \int \frac {x^5 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac {1}{28} \left (3 a^4 c^2\right ) \int \frac {x^7 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=\frac {3 c^2 x \tan ^{-1}(a x)^2}{4 a^3}+\frac {c^2 x^3 \tan ^{-1}(a x)^2}{12 a}-\frac {1}{8} a c^2 x^5 \tan ^{-1}(a x)^2-\frac {3}{56} a^3 c^2 x^7 \tan ^{-1}(a x)^2-\frac {c^2 \tan ^{-1}(a x)^3}{4 a^4}+\frac {1}{4} c^2 x^4 \tan ^{-1}(a x)^3+\frac {1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)^3+\frac {1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)^3+\frac {1}{5} \left (2 c^2\right ) \int x^3 \tan ^{-1}(a x) \, dx-\frac {1}{5} \left (2 c^2\right ) \int \frac {x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac {1}{3} \left (2 c^2\right ) \int \frac {x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac {c^2 \int \tan ^{-1}(a x)^2 \, dx}{a^3}+\frac {c^2 \int \frac {\tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{a^3}+\frac {c^2 \int x \tan ^{-1}(a x) \, dx}{2 a^2}-\frac {c^2 \int \frac {x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{2 a^2}-\frac {\left (3 c^2\right ) \int \frac {x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{2 a^2}-\frac {\left (3 c^2\right ) \int x^2 \tan ^{-1}(a x)^2 \, dx}{8 a}+\frac {\left (3 c^2\right ) \int \frac {x^2 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{8 a}+\frac {1}{28} \left (3 a^2 c^2\right ) \int x^5 \tan ^{-1}(a x) \, dx-\frac {1}{28} \left (3 a^2 c^2\right ) \int \frac {x^5 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac {1}{20} \left (3 a^2 c^2\right ) \int \frac {x^5 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=\frac {c^2 x^2 \tan ^{-1}(a x)}{4 a^2}+\frac {1}{10} c^2 x^4 \tan ^{-1}(a x)+\frac {1}{56} a^2 c^2 x^6 \tan ^{-1}(a x)+\frac {i c^2 \tan ^{-1}(a x)^2}{a^4}-\frac {c^2 x \tan ^{-1}(a x)^2}{4 a^3}-\frac {c^2 x^3 \tan ^{-1}(a x)^2}{24 a}-\frac {1}{8} a c^2 x^5 \tan ^{-1}(a x)^2-\frac {3}{56} a^3 c^2 x^7 \tan ^{-1}(a x)^2+\frac {c^2 \tan ^{-1}(a x)^3}{12 a^4}+\frac {1}{4} c^2 x^4 \tan ^{-1}(a x)^3+\frac {1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)^3+\frac {1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)^3-\frac {1}{28} \left (3 c^2\right ) \int x^3 \tan ^{-1}(a x) \, dx+\frac {1}{28} \left (3 c^2\right ) \int \frac {x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac {1}{20} \left (3 c^2\right ) \int x^3 \tan ^{-1}(a x) \, dx+\frac {1}{20} \left (3 c^2\right ) \int \frac {x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac {1}{4} c^2 \int \frac {x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac {\left (3 c^2\right ) \int \tan ^{-1}(a x)^2 \, dx}{8 a^3}-\frac {\left (3 c^2\right ) \int \frac {\tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{8 a^3}+\frac {c^2 \int \frac {\tan ^{-1}(a x)}{i-a x} \, dx}{2 a^3}+\frac {\left (3 c^2\right ) \int \frac {\tan ^{-1}(a x)}{i-a x} \, dx}{2 a^3}-\frac {\left (2 c^2\right ) \int x \tan ^{-1}(a x) \, dx}{5 a^2}+\frac {\left (2 c^2\right ) \int \frac {x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{5 a^2}-\frac {\left (2 c^2\right ) \int x \tan ^{-1}(a x) \, dx}{3 a^2}+\frac {\left (2 c^2\right ) \int \frac {x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{3 a^2}+\frac {\left (2 c^2\right ) \int \frac {x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{a^2}-\frac {c^2 \int \frac {x^2}{1+a^2 x^2} \, dx}{4 a}-\frac {1}{10} \left (a c^2\right ) \int \frac {x^4}{1+a^2 x^2} \, dx-\frac {1}{56} \left (a^3 c^2\right ) \int \frac {x^6}{1+a^2 x^2} \, dx\\ &=-\frac {c^2 x}{4 a^3}-\frac {17 c^2 x^2 \tan ^{-1}(a x)}{60 a^2}+\frac {1}{28} c^2 x^4 \tan ^{-1}(a x)+\frac {1}{56} a^2 c^2 x^6 \tan ^{-1}(a x)-\frac {8 i c^2 \tan ^{-1}(a x)^2}{15 a^4}+\frac {c^2 x \tan ^{-1}(a x)^2}{8 a^3}-\frac {c^2 x^3 \tan ^{-1}(a x)^2}{24 a}-\frac {1}{8} a c^2 x^5 \tan ^{-1}(a x)^2-\frac {3}{56} a^3 c^2 x^7 \tan ^{-1}(a x)^2-\frac {c^2 \tan ^{-1}(a x)^3}{24 a^4}+\frac {1}{4} c^2 x^4 \tan ^{-1}(a x)^3+\frac {1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)^3+\frac {1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)^3+\frac {2 c^2 \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{a^4}+\frac {c^2 \int \frac {1}{1+a^2 x^2} \, dx}{4 a^3}-\frac {\left (2 c^2\right ) \int \frac {\tan ^{-1}(a x)}{i-a x} \, dx}{5 a^3}-\frac {c^2 \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{2 a^3}-\frac {\left (2 c^2\right ) \int \frac {\tan ^{-1}(a x)}{i-a x} \, dx}{3 a^3}-\frac {\left (3 c^2\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{2 a^3}-\frac {\left (2 c^2\right ) \int \frac {\tan ^{-1}(a x)}{i-a x} \, dx}{a^3}+\frac {\left (3 c^2\right ) \int x \tan ^{-1}(a x) \, dx}{28 a^2}-\frac {\left (3 c^2\right ) \int \frac {x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{28 a^2}+\frac {\left (3 c^2\right ) \int x \tan ^{-1}(a x) \, dx}{20 a^2}-\frac {\left (3 c^2\right ) \int \frac {x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{20 a^2}+\frac {c^2 \int x \tan ^{-1}(a x) \, dx}{4 a^2}-\frac {c^2 \int \frac {x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{4 a^2}-\frac {\left (3 c^2\right ) \int \frac {x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{4 a^2}+\frac {c^2 \int \frac {x^2}{1+a^2 x^2} \, dx}{5 a}+\frac {c^2 \int \frac {x^2}{1+a^2 x^2} \, dx}{3 a}+\frac {1}{112} \left (3 a c^2\right ) \int \frac {x^4}{1+a^2 x^2} \, dx+\frac {1}{80} \left (3 a c^2\right ) \int \frac {x^4}{1+a^2 x^2} \, dx-\frac {1}{10} \left (a c^2\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}{56} \left (a^3 c^2\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 {307 c^2 x}{840 a^3}-\frac {23 c^2 x^3}{840 a}-\frac {1}{280} a c^2 x^5+\frac {c^2 \tan ^{-1}(a x)}{4 a^4}-\frac {5 c^2 x^2 \tan ^{-1}(a x)}{168 a^2}+\frac {1}{28} c^2 x^4 \tan ^{-1}(a x)+\frac {1}{56} a^2 c^2 x^6 \tan ^{-1}(a x)+\frac {2 i c^2 \tan ^{-1}(a x)^2}{21 a^4}+\frac {c^2 x \tan ^{-1}(a x)^2}{8 a^3}-\frac {c^2 x^3 \tan ^{-1}(a x)^2}{24 a}-\frac {1}{8} a c^2 x^5 \tan ^{-1}(a x)^2-\frac {3}{56} a^3 c^2 x^7 \tan ^{-1}(a x)^2-\frac {c^2 \tan ^{-1}(a x)^3}{24 a^4}+\frac {1}{4} c^2 x^4 \tan ^{-1}(a x)^3+\frac {1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)^3+\frac {1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)^3-\frac {16 c^2 \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{15 a^4}+\frac {\left (i c^2\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{2 a^4}+\frac {\left (3 i c^2\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{2 a^4}+\frac {c^2 \int \frac {1}{1+a^2 x^2} \, dx}{56 a^3}-\frac {c^2 \int \frac {1}{1+a^2 x^2} \, dx}{10 a^3}+\frac {\left (3 c^2\right ) \int \frac {\tan ^{-1}(a x)}{i-a x} \, dx}{28 a^3}+\frac {\left (3 c^2\right ) \int \frac {\tan ^{-1}(a x)}{i-a x} \, dx}{20 a^3}-\frac {c^2 \int \frac {1}{1+a^2 x^2} \, dx}{5 a^3}+\frac {c^2 \int \frac {\tan ^{-1}(a x)}{i-a x} \, dx}{4 a^3}-\frac {c^2 \int \frac {1}{1+a^2 x^2} \, dx}{3 a^3}+\frac {\left (2 c^2\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{5 a^3}+\frac {\left (2 c^2\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{3 a^3}+\frac {\left (3 c^2\right ) \int \frac {\tan ^{-1}(a x)}{i-a x} \, dx}{4 a^3}+\frac {\left (2 c^2\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a^3}-\frac {\left (3 c^2\right ) \int \frac {x^2}{1+a^2 x^2} \, dx}{56 a}-\frac {\left (3 c^2\right ) \int \frac {x^2}{1+a^2 x^2} \, dx}{40 a}-\frac {c^2 \int \frac {x^2}{1+a^2 x^2} \, dx}{8 a}+\frac {1}{112} \left (3 a c^2\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}{80} \left (3 a c^2\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 {c^2 x}{21 a^3}-\frac {c^2 x^3}{168 a}-\frac {1}{280} a c^2 x^5-\frac {307 c^2 \tan ^{-1}(a x)}{840 a^4}-\frac {5 c^2 x^2 \tan ^{-1}(a x)}{168 a^2}+\frac {1}{28} c^2 x^4 \tan ^{-1}(a x)+\frac {1}{56} a^2 c^2 x^6 \tan ^{-1}(a x)+\frac {2 i c^2 \tan ^{-1}(a x)^2}{21 a^4}+\frac {c^2 x \tan ^{-1}(a x)^2}{8 a^3}-\frac {c^2 x^3 \tan ^{-1}(a x)^2}{24 a}-\frac {1}{8} a c^2 x^5 \tan ^{-1}(a x)^2-\frac {3}{56} a^3 c^2 x^7 \tan ^{-1}(a x)^2-\frac {c^2 \tan ^{-1}(a x)^3}{24 a^4}+\frac {1}{4} c^2 x^4 \tan ^{-1}(a x)^3+\frac {1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)^3+\frac {1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)^3+\frac {4 c^2 \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{21 a^4}+\frac {i c^2 \text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{a^4}-\frac {\left (2 i c^2\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{5 a^4}-\frac {\left (2 i c^2\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{3 a^4}-\frac {\left (2 i c^2\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{a^4}+\frac {\left (3 c^2\right ) \int \frac {1}{1+a^2 x^2} \, dx}{112 a^3}+\frac {\left (3 c^2\right ) \int \frac {1}{1+a^2 x^2} \, dx}{80 a^3}+\frac {\left (3 c^2\right ) \int \frac {1}{1+a^2 x^2} \, dx}{56 a^3}+\frac {\left (3 c^2\right ) \int \frac {1}{1+a^2 x^2} \, dx}{40 a^3}-\frac {\left (3 c^2\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{28 a^3}+\frac {c^2 \int \frac {1}{1+a^2 x^2} \, dx}{8 a^3}-\frac {\left (3 c^2\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{20 a^3}-\frac {c^2 \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{4 a^3}-\frac {\left (3 c^2\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{4 a^3}\\ &=\frac {c^2 x}{21 a^3}-\frac {c^2 x^3}{168 a}-\frac {1}{280} a c^2 x^5-\frac {c^2 \tan ^{-1}(a x)}{21 a^4}-\frac {5 c^2 x^2 \tan ^{-1}(a x)}{168 a^2}+\frac {1}{28} c^2 x^4 \tan ^{-1}(a x)+\frac {1}{56} a^2 c^2 x^6 \tan ^{-1}(a x)+\frac {2 i c^2 \tan ^{-1}(a x)^2}{21 a^4}+\frac {c^2 x \tan ^{-1}(a x)^2}{8 a^3}-\frac {c^2 x^3 \tan ^{-1}(a x)^2}{24 a}-\frac {1}{8} a c^2 x^5 \tan ^{-1}(a x)^2-\frac {3}{56} a^3 c^2 x^7 \tan ^{-1}(a x)^2-\frac {c^2 \tan ^{-1}(a x)^3}{24 a^4}+\frac {1}{4} c^2 x^4 \tan ^{-1}(a x)^3+\frac {1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)^3+\frac {1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)^3+\frac {4 c^2 \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{21 a^4}-\frac {8 i c^2 \text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{15 a^4}+\frac {\left (3 i c^2\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{28 a^4}+\frac {\left (3 i c^2\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{20 a^4}+\frac {\left (i c^2\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{4 a^4}+\frac {\left (3 i c^2\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{4 a^4}\\ &=\frac {c^2 x}{21 a^3}-\frac {c^2 x^3}{168 a}-\frac {1}{280} a c^2 x^5-\frac {c^2 \tan ^{-1}(a x)}{21 a^4}-\frac {5 c^2 x^2 \tan ^{-1}(a x)}{168 a^2}+\frac {1}{28} c^2 x^4 \tan ^{-1}(a x)+\frac {1}{56} a^2 c^2 x^6 \tan ^{-1}(a x)+\frac {2 i c^2 \tan ^{-1}(a x)^2}{21 a^4}+\frac {c^2 x \tan ^{-1}(a x)^2}{8 a^3}-\frac {c^2 x^3 \tan ^{-1}(a x)^2}{24 a}-\frac {1}{8} a c^2 x^5 \tan ^{-1}(a x)^2-\frac {3}{56} a^3 c^2 x^7 \tan ^{-1}(a x)^2-\frac {c^2 \tan ^{-1}(a x)^3}{24 a^4}+\frac {1}{4} c^2 x^4 \tan ^{-1}(a x)^3+\frac {1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)^3+\frac {1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)^3+\frac {4 c^2 \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{21 a^4}+\frac {2 i c^2 \text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{21 a^4}\\ \end {align*}

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Mathematica [A]  time = 1.33, size = 165, normalized size = 0.53 \[ \frac {c^2 \left (-3 a^5 x^5-5 a^3 x^3+35 \left (a^2 x^2+1\right )^3 \left (3 a^2 x^2-1\right ) \tan ^{-1}(a x)^3-5 \left (9 a^7 x^7+21 a^5 x^5+7 a^3 x^3-21 a x+16 i\right ) \tan ^{-1}(a x)^2+5 \tan ^{-1}(a x) \left (3 a^6 x^6+6 a^4 x^4-5 a^2 x^2+32 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )-8\right )-80 i \text {Li}_2\left (-e^{2 i \tan ^{-1}(a x)}\right )+40 a x\right )}{840 a^4} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c^2*(40*a*x - 5*a^3*x^3 - 3*a^5*x^5 - 5*(16*I - 21*a*x + 7*a^3*x^3 + 21*a^5*x^5 + 9*a^7*x^7)*ArcTan[a*x]^2 +
35*(1 + a^2*x^2)^3*(-1 + 3*a^2*x^2)*ArcTan[a*x]^3 + 5*ArcTan[a*x]*(-8 - 5*a^2*x^2 + 6*a^4*x^4 + 3*a^6*x^6 + 32
*Log[1 + E^((2*I)*ArcTan[a*x])]) - (80*I)*PolyLog[2, -E^((2*I)*ArcTan[a*x])]))/(840*a^4)

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fricas [F]  time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{4} c^{2} x^{7} + 2 \, a^{2} c^{2} x^{5} + c^{2} x^{3}\right )} \arctan \left (a x\right )^{3}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((a^4*c^2*x^7 + 2*a^2*c^2*x^5 + c^2*x^3)*arctan(a*x)^3, 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^3*(a^2*c*x^2+c)^2*arctan(a*x)^3,x, algorithm="giac")

[Out]

sage0*x

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maple [A]  time = 0.16, size = 411, normalized size = 1.31 \[ \frac {a^{4} c^{2} x^{8} \arctan \left (a x \right )^{3}}{8}+\frac {a^{2} c^{2} x^{6} \arctan \left (a x \right )^{3}}{3}+\frac {c^{2} x^{4} \arctan \left (a x \right )^{3}}{4}-\frac {3 a^{3} c^{2} x^{7} \arctan \left (a x \right )^{2}}{56}-\frac {a \,c^{2} x^{5} \arctan \left (a x \right )^{2}}{8}-\frac {c^{2} x^{3} \arctan \left (a x \right )^{2}}{24 a}+\frac {c^{2} x \arctan \left (a x \right )^{2}}{8 a^{3}}-\frac {c^{2} \arctan \left (a x \right )^{3}}{24 a^{4}}+\frac {a^{2} c^{2} x^{6} \arctan \left (a x \right )}{56}+\frac {c^{2} x^{4} \arctan \left (a x \right )}{28}-\frac {5 c^{2} x^{2} \arctan \left (a x \right )}{168 a^{2}}-\frac {2 c^{2} \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{21 a^{4}}-\frac {a \,c^{2} x^{5}}{280}-\frac {c^{2} x^{3}}{168 a}+\frac {c^{2} x}{21 a^{3}}-\frac {c^{2} \arctan \left (a x \right )}{21 a^{4}}-\frac {i c^{2} \ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )}{21 a^{4}}+\frac {i c^{2} \ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )}{21 a^{4}}-\frac {i c^{2} \dilog \left (\frac {i \left (a x -i\right )}{2}\right )}{21 a^{4}}+\frac {i c^{2} \ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )}{21 a^{4}}+\frac {i c^{2} \dilog \left (-\frac {i \left (a x +i\right )}{2}\right )}{21 a^{4}}-\frac {i c^{2} \ln \left (a x +i\right )^{2}}{42 a^{4}}+\frac {i c^{2} \ln \left (a x -i\right )^{2}}{42 a^{4}}-\frac {i c^{2} \ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )}{21 a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*a^4*c^2*x^8*arctan(a*x)^3+1/3*a^2*c^2*x^6*arctan(a*x)^3+1/4*c^2*x^4*arctan(a*x)^3-3/56*a^3*c^2*x^7*arctan(
a*x)^2-1/8*a*c^2*x^5*arctan(a*x)^2-1/24*c^2*x^3*arctan(a*x)^2/a+1/8*c^2*x*arctan(a*x)^2/a^3-1/24*c^2*arctan(a*
x)^3/a^4+1/56*a^2*c^2*x^6*arctan(a*x)+1/28*c^2*x^4*arctan(a*x)-5/168*c^2*x^2*arctan(a*x)/a^2-2/21/a^4*c^2*arct
an(a*x)*ln(a^2*x^2+1)-1/280*a*c^2*x^5-1/168*c^2*x^3/a+1/21*c^2*x/a^3-1/21*c^2*arctan(a*x)/a^4-1/21*I/a^4*c^2*l
n(a*x-I)*ln(a^2*x^2+1)+1/21*I/a^4*c^2*ln(a*x-I)*ln(-1/2*I*(I+a*x))+1/21*I/a^4*c^2*dilog(-1/2*I*(I+a*x))+1/21*I
/a^4*c^2*ln(I+a*x)*ln(a^2*x^2+1)-1/42*I/a^4*c^2*ln(I+a*x)^2-1/21*I/a^4*c^2*dilog(1/2*I*(a*x-I))+1/42*I/a^4*c^2
*ln(a*x-I)^2-1/21*I/a^4*c^2*ln(I+a*x)*ln(1/2*I*(a*x-I))

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maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ c^{2} \left (\int x^{3} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int 2 a^{2} x^{5} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int a^{4} x^{7} \operatorname {atan}^{3}{\left (a x \right )}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c**2*(Integral(x**3*atan(a*x)**3, x) + Integral(2*a**2*x**5*atan(a*x)**3, x) + Integral(a**4*x**7*atan(a*x)**3
, x))

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