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

Optimal. Leaf size=219 \[ \frac{7 i c \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{30 a^4}+\frac{1}{6} a^2 c x^6 \tan ^{-1}(a x)^3-\frac{c x^2 \tan ^{-1}(a x)}{60 a^2}+\frac{c x}{15 a^3}+\frac{c x \tan ^{-1}(a x)^2}{4 a^3}-\frac{c \tan ^{-1}(a x)^3}{12 a^4}+\frac{7 i c \tan ^{-1}(a x)^2}{30 a^4}-\frac{c \tan ^{-1}(a x)}{15 a^4}+\frac{7 c \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)}{15 a^4}-\frac{c x^3}{60 a}-\frac{1}{10} a c x^5 \tan ^{-1}(a x)^2+\frac{1}{4} c x^4 \tan ^{-1}(a x)^3+\frac{1}{20} c x^4 \tan ^{-1}(a x)-\frac{c x^3 \tan ^{-1}(a x)^2}{12 a} \]

[Out]

(c*x)/(15*a^3) - (c*x^3)/(60*a) - (c*ArcTan[a*x])/(15*a^4) - (c*x^2*ArcTan[a*x])/(60*a^2) + (c*x^4*ArcTan[a*x]
)/20 + (((7*I)/30)*c*ArcTan[a*x]^2)/a^4 + (c*x*ArcTan[a*x]^2)/(4*a^3) - (c*x^3*ArcTan[a*x]^2)/(12*a) - (a*c*x^
5*ArcTan[a*x]^2)/10 - (c*ArcTan[a*x]^3)/(12*a^4) + (c*x^4*ArcTan[a*x]^3)/4 + (a^2*c*x^6*ArcTan[a*x]^3)/6 + (7*
c*ArcTan[a*x]*Log[2/(1 + I*a*x)])/(15*a^4) + (((7*I)/30)*c*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^4

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Rubi [A]  time = 1.11374, antiderivative size = 219, normalized size of antiderivative = 1., number of steps used = 52, number of rules used = 12, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {4950, 4852, 4916, 321, 203, 4920, 4854, 2402, 2315, 4846, 4884, 302} \[ \frac{7 i c \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{30 a^4}+\frac{1}{6} a^2 c x^6 \tan ^{-1}(a x)^3-\frac{c x^2 \tan ^{-1}(a x)}{60 a^2}+\frac{c x}{15 a^3}+\frac{c x \tan ^{-1}(a x)^2}{4 a^3}-\frac{c \tan ^{-1}(a x)^3}{12 a^4}+\frac{7 i c \tan ^{-1}(a x)^2}{30 a^4}-\frac{c \tan ^{-1}(a x)}{15 a^4}+\frac{7 c \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)}{15 a^4}-\frac{c x^3}{60 a}-\frac{1}{10} a c x^5 \tan ^{-1}(a x)^2+\frac{1}{4} c x^4 \tan ^{-1}(a x)^3+\frac{1}{20} c x^4 \tan ^{-1}(a x)-\frac{c x^3 \tan ^{-1}(a x)^2}{12 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*x)/(15*a^3) - (c*x^3)/(60*a) - (c*ArcTan[a*x])/(15*a^4) - (c*x^2*ArcTan[a*x])/(60*a^2) + (c*x^4*ArcTan[a*x]
)/20 + (((7*I)/30)*c*ArcTan[a*x]^2)/a^4 + (c*x*ArcTan[a*x]^2)/(4*a^3) - (c*x^3*ArcTan[a*x]^2)/(12*a) - (a*c*x^
5*ArcTan[a*x]^2)/10 - (c*ArcTan[a*x]^3)/(12*a^4) + (c*x^4*ArcTan[a*x]^3)/4 + (a^2*c*x^6*ArcTan[a*x]^3)/6 + (7*
c*ArcTan[a*x]*Log[2/(1 + I*a*x)])/(15*a^4) + (((7*I)/30)*c*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^4

Rule 4950

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> Dist[
d, Int[(f*x)^m*(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x])^p, x], x] + Dist[(c^2*d)/f^2, Int[(f*x)^(m + 2)*(d + e*
x^2)^(q - 1)*(a + b*ArcTan[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[q, 0] &&
IGtQ[p, 0] && (RationalQ[m] || (EqQ[p, 1] && IntegerQ[q]))

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 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 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 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 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 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 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 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 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 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 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]

Rubi steps

\begin{align*} \int x^3 \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^3 \, dx &=c \int x^3 \tan ^{-1}(a x)^3 \, dx+\left (a^2 c\right ) \int x^5 \tan ^{-1}(a x)^3 \, dx\\ &=\frac{1}{4} c x^4 \tan ^{-1}(a x)^3+\frac{1}{6} a^2 c x^6 \tan ^{-1}(a x)^3-\frac{1}{4} (3 a c) \int \frac{x^4 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx-\frac{1}{2} \left (a^3 c\right ) \int \frac{x^6 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx\\ &=\frac{1}{4} c x^4 \tan ^{-1}(a x)^3+\frac{1}{6} a^2 c x^6 \tan ^{-1}(a x)^3-\frac{(3 c) \int x^2 \tan ^{-1}(a x)^2 \, dx}{4 a}+\frac{(3 c) \int \frac{x^2 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{4 a}-\frac{1}{2} (a c) \int x^4 \tan ^{-1}(a x)^2 \, dx+\frac{1}{2} (a c) \int \frac{x^4 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx\\ &=-\frac{c x^3 \tan ^{-1}(a x)^2}{4 a}-\frac{1}{10} a c x^5 \tan ^{-1}(a x)^2+\frac{1}{4} c x^4 \tan ^{-1}(a x)^3+\frac{1}{6} a^2 c x^6 \tan ^{-1}(a x)^3+\frac{1}{2} c \int \frac{x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac{(3 c) \int \tan ^{-1}(a x)^2 \, dx}{4 a^3}-\frac{(3 c) \int \frac{\tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{4 a^3}+\frac{c \int x^2 \tan ^{-1}(a x)^2 \, dx}{2 a}-\frac{c \int \frac{x^2 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{2 a}+\frac{1}{5} \left (a^2 c\right ) \int \frac{x^5 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=\frac{3 c x \tan ^{-1}(a x)^2}{4 a^3}-\frac{c x^3 \tan ^{-1}(a x)^2}{12 a}-\frac{1}{10} a c x^5 \tan ^{-1}(a x)^2-\frac{c \tan ^{-1}(a x)^3}{4 a^4}+\frac{1}{4} c x^4 \tan ^{-1}(a x)^3+\frac{1}{6} a^2 c x^6 \tan ^{-1}(a x)^3+\frac{1}{5} c \int x^3 \tan ^{-1}(a x) \, dx-\frac{1}{5} c \int \frac{x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{3} c \int \frac{x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{c \int \tan ^{-1}(a x)^2 \, dx}{2 a^3}+\frac{c \int \frac{\tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{2 a^3}+\frac{c \int x \tan ^{-1}(a x) \, dx}{2 a^2}-\frac{c \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{2 a^2}-\frac{(3 c) \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{2 a^2}\\ &=\frac{c x^2 \tan ^{-1}(a x)}{4 a^2}+\frac{1}{20} c x^4 \tan ^{-1}(a x)+\frac{i c \tan ^{-1}(a x)^2}{a^4}+\frac{c x \tan ^{-1}(a x)^2}{4 a^3}-\frac{c x^3 \tan ^{-1}(a x)^2}{12 a}-\frac{1}{10} a c x^5 \tan ^{-1}(a x)^2-\frac{c \tan ^{-1}(a x)^3}{12 a^4}+\frac{1}{4} c x^4 \tan ^{-1}(a x)^3+\frac{1}{6} a^2 c x^6 \tan ^{-1}(a x)^3+\frac{c \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx}{2 a^3}+\frac{(3 c) \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx}{2 a^3}-\frac{c \int x \tan ^{-1}(a x) \, dx}{5 a^2}+\frac{c \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{5 a^2}-\frac{c \int x \tan ^{-1}(a x) \, dx}{3 a^2}+\frac{c \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{3 a^2}+\frac{c \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{a^2}-\frac{c \int \frac{x^2}{1+a^2 x^2} \, dx}{4 a}-\frac{1}{20} (a c) \int \frac{x^4}{1+a^2 x^2} \, dx\\ &=-\frac{c x}{4 a^3}-\frac{c x^2 \tan ^{-1}(a x)}{60 a^2}+\frac{1}{20} c x^4 \tan ^{-1}(a x)+\frac{7 i c \tan ^{-1}(a x)^2}{30 a^4}+\frac{c x \tan ^{-1}(a x)^2}{4 a^3}-\frac{c x^3 \tan ^{-1}(a x)^2}{12 a}-\frac{1}{10} a c x^5 \tan ^{-1}(a x)^2-\frac{c \tan ^{-1}(a x)^3}{12 a^4}+\frac{1}{4} c x^4 \tan ^{-1}(a x)^3+\frac{1}{6} a^2 c x^6 \tan ^{-1}(a x)^3+\frac{2 c \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{a^4}-\frac{c \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx}{5 a^3}+\frac{c \int \frac{1}{1+a^2 x^2} \, dx}{4 a^3}-\frac{c \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx}{3 a^3}-\frac{c \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{2 a^3}-\frac{c \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx}{a^3}-\frac{(3 c) \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{2 a^3}+\frac{c \int \frac{x^2}{1+a^2 x^2} \, dx}{10 a}+\frac{c \int \frac{x^2}{1+a^2 x^2} \, dx}{6 a}-\frac{1}{20} (a c) \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 x}{15 a^3}-\frac{c x^3}{60 a}+\frac{c \tan ^{-1}(a x)}{4 a^4}-\frac{c x^2 \tan ^{-1}(a x)}{60 a^2}+\frac{1}{20} c x^4 \tan ^{-1}(a x)+\frac{7 i c \tan ^{-1}(a x)^2}{30 a^4}+\frac{c x \tan ^{-1}(a x)^2}{4 a^3}-\frac{c x^3 \tan ^{-1}(a x)^2}{12 a}-\frac{1}{10} a c x^5 \tan ^{-1}(a x)^2-\frac{c \tan ^{-1}(a x)^3}{12 a^4}+\frac{1}{4} c x^4 \tan ^{-1}(a x)^3+\frac{1}{6} a^2 c x^6 \tan ^{-1}(a x)^3+\frac{7 c \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{15 a^4}+\frac{(i c) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i a x}\right )}{2 a^4}+\frac{(3 i c) \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 \int \frac{1}{1+a^2 x^2} \, dx}{20 a^3}-\frac{c \int \frac{1}{1+a^2 x^2} \, dx}{10 a^3}-\frac{c \int \frac{1}{1+a^2 x^2} \, dx}{6 a^3}+\frac{c \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{5 a^3}+\frac{c \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{3 a^3}+\frac{c \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a^3}\\ &=\frac{c x}{15 a^3}-\frac{c x^3}{60 a}-\frac{c \tan ^{-1}(a x)}{15 a^4}-\frac{c x^2 \tan ^{-1}(a x)}{60 a^2}+\frac{1}{20} c x^4 \tan ^{-1}(a x)+\frac{7 i c \tan ^{-1}(a x)^2}{30 a^4}+\frac{c x \tan ^{-1}(a x)^2}{4 a^3}-\frac{c x^3 \tan ^{-1}(a x)^2}{12 a}-\frac{1}{10} a c x^5 \tan ^{-1}(a x)^2-\frac{c \tan ^{-1}(a x)^3}{12 a^4}+\frac{1}{4} c x^4 \tan ^{-1}(a x)^3+\frac{1}{6} a^2 c x^6 \tan ^{-1}(a x)^3+\frac{7 c \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{15 a^4}+\frac{i c \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{a^4}-\frac{(i c) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i a x}\right )}{5 a^4}-\frac{(i c) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i a x}\right )}{3 a^4}-\frac{(i c) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i a x}\right )}{a^4}\\ &=\frac{c x}{15 a^3}-\frac{c x^3}{60 a}-\frac{c \tan ^{-1}(a x)}{15 a^4}-\frac{c x^2 \tan ^{-1}(a x)}{60 a^2}+\frac{1}{20} c x^4 \tan ^{-1}(a x)+\frac{7 i c \tan ^{-1}(a x)^2}{30 a^4}+\frac{c x \tan ^{-1}(a x)^2}{4 a^3}-\frac{c x^3 \tan ^{-1}(a x)^2}{12 a}-\frac{1}{10} a c x^5 \tan ^{-1}(a x)^2-\frac{c \tan ^{-1}(a x)^3}{12 a^4}+\frac{1}{4} c x^4 \tan ^{-1}(a x)^3+\frac{1}{6} a^2 c x^6 \tan ^{-1}(a x)^3+\frac{7 c \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{15 a^4}+\frac{7 i c \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{30 a^4}\\ \end{align*}

Mathematica [A]  time = 0.612491, size = 135, normalized size = 0.62 \[ \frac{c \left (-14 i \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(a x)}\right )-a^3 x^3+5 \left (2 a^6 x^6+3 a^4 x^4-1\right ) \tan ^{-1}(a x)^3-\left (6 a^5 x^5+5 a^3 x^3-15 a x+14 i\right ) \tan ^{-1}(a x)^2+\tan ^{-1}(a x) \left (3 a^4 x^4-a^2 x^2+28 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )-4\right )+4 a x\right )}{60 a^4} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c*(4*a*x - a^3*x^3 - (14*I - 15*a*x + 5*a^3*x^3 + 6*a^5*x^5)*ArcTan[a*x]^2 + 5*(-1 + 3*a^4*x^4 + 2*a^6*x^6)*A
rcTan[a*x]^3 + ArcTan[a*x]*(-4 - a^2*x^2 + 3*a^4*x^4 + 28*Log[1 + E^((2*I)*ArcTan[a*x])]) - (14*I)*PolyLog[2,
-E^((2*I)*ArcTan[a*x])]))/(60*a^4)

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Maple [A]  time = 0.097, size = 313, normalized size = 1.4 \begin{align*}{\frac{{a}^{2}c{x}^{6} \left ( \arctan \left ( ax \right ) \right ) ^{3}}{6}}+{\frac{c{x}^{4} \left ( \arctan \left ( ax \right ) \right ) ^{3}}{4}}-{\frac{ac{x}^{5} \left ( \arctan \left ( ax \right ) \right ) ^{2}}{10}}-{\frac{c{x}^{3} \left ( \arctan \left ( ax \right ) \right ) ^{2}}{12\,a}}+{\frac{cx \left ( \arctan \left ( ax \right ) \right ) ^{2}}{4\,{a}^{3}}}-{\frac{c \left ( \arctan \left ( ax \right ) \right ) ^{3}}{12\,{a}^{4}}}+{\frac{c{x}^{4}\arctan \left ( ax \right ) }{20}}-{\frac{c{x}^{2}\arctan \left ( ax \right ) }{60\,{a}^{2}}}-{\frac{7\,c\arctan \left ( ax \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) }{30\,{a}^{4}}}-{\frac{c{x}^{3}}{60\,a}}+{\frac{cx}{15\,{a}^{3}}}-{\frac{c\arctan \left ( ax \right ) }{15\,{a}^{4}}}-{\frac{{\frac{7\,i}{60}}c\ln \left ( ax-i \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) }{{a}^{4}}}+{\frac{{\frac{7\,i}{60}}c\ln \left ( ax-i \right ) \ln \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) }{{a}^{4}}}+{\frac{{\frac{7\,i}{120}}c \left ( \ln \left ( ax-i \right ) \right ) ^{2}}{{a}^{4}}}-{\frac{{\frac{7\,i}{60}}c\ln \left ( ax+i \right ) \ln \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) }{{a}^{4}}}+{\frac{{\frac{7\,i}{60}}c\ln \left ( ax+i \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) }{{a}^{4}}}-{\frac{{\frac{7\,i}{120}}c \left ( \ln \left ( ax+i \right ) \right ) ^{2}}{{a}^{4}}}+{\frac{{\frac{7\,i}{60}}c{\it dilog} \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) }{{a}^{4}}}-{\frac{{\frac{7\,i}{60}}c{\it dilog} \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) }{{a}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*a^2*c*x^6*arctan(a*x)^3+1/4*c*x^4*arctan(a*x)^3-1/10*a*c*x^5*arctan(a*x)^2-1/12*c*x^3*arctan(a*x)^2/a+1/4*
c*x*arctan(a*x)^2/a^3-1/12*c*arctan(a*x)^3/a^4+1/20*c*x^4*arctan(a*x)-1/60*c*x^2*arctan(a*x)/a^2-7/30/a^4*c*ar
ctan(a*x)*ln(a^2*x^2+1)-1/60*c*x^3/a+1/15*c*x/a^3-1/15*c*arctan(a*x)/a^4-7/60*I/a^4*c*ln(a*x-I)*ln(a^2*x^2+1)+
7/60*I/a^4*c*ln(a*x-I)*ln(-1/2*I*(a*x+I))+7/120*I/a^4*c*ln(a*x-I)^2-7/60*I/a^4*c*ln(a*x+I)*ln(1/2*I*(a*x-I))+7
/60*I/a^4*c*ln(a*x+I)*ln(a^2*x^2+1)-7/120*I/a^4*c*ln(a*x+I)^2+7/60*I/a^4*c*dilog(-1/2*I*(a*x+I))-7/60*I/a^4*c*
dilog(1/2*I*(a*x-I))

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a^{2} c x^{5} + c x^{3}\right )} \arctan \left (a x\right )^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((a^2*c*x^5 + c*x^3)*arctan(a*x)^3, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} c \left (\int x^{3} \operatorname{atan}^{3}{\left (a x \right )}\, dx + \int a^{2} x^{5} \operatorname{atan}^{3}{\left (a x \right )}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a^{2} c x^{2} + c\right )} x^{3} \arctan \left (a x\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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