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

Optimal. Leaf size=211 \[ -\frac{c \text{PolyLog}\left (3,1-\frac{2}{1+i a x}\right )}{5 a^3}-\frac{2 i c \tan ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{5 a^3}+\frac{1}{5} a^2 c x^5 \tan ^{-1}(a x)^3+\frac{c x \tan ^{-1}(a x)}{10 a^2}-\frac{2 i c \tan ^{-1}(a x)^3}{15 a^3}-\frac{c \tan ^{-1}(a x)^2}{20 a^3}-\frac{2 c \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)^2}{5 a^3}-\frac{c x^2}{20 a}-\frac{3}{20} a c x^4 \tan ^{-1}(a x)^2+\frac{1}{3} c x^3 \tan ^{-1}(a x)^3+\frac{1}{10} c x^3 \tan ^{-1}(a x)-\frac{c x^2 \tan ^{-1}(a x)^2}{5 a} \]

[Out]

-(c*x^2)/(20*a) + (c*x*ArcTan[a*x])/(10*a^2) + (c*x^3*ArcTan[a*x])/10 - (c*ArcTan[a*x]^2)/(20*a^3) - (c*x^2*Ar
cTan[a*x]^2)/(5*a) - (3*a*c*x^4*ArcTan[a*x]^2)/20 - (((2*I)/15)*c*ArcTan[a*x]^3)/a^3 + (c*x^3*ArcTan[a*x]^3)/3
+ (a^2*c*x^5*ArcTan[a*x]^3)/5 - (2*c*ArcTan[a*x]^2*Log[2/(1 + I*a*x)])/(5*a^3) - (((2*I)/5)*c*ArcTan[a*x]*Pol
yLog[2, 1 - 2/(1 + I*a*x)])/a^3 - (c*PolyLog[3, 1 - 2/(1 + I*a*x)])/(5*a^3)

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Rubi [A]  time = 0.881975, antiderivative size = 211, normalized size of antiderivative = 1., number of steps used = 34, number of rules used = 12, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {4950, 4852, 4916, 4846, 260, 4884, 4920, 4854, 4994, 6610, 266, 43} \[ -\frac{c \text{PolyLog}\left (3,1-\frac{2}{1+i a x}\right )}{5 a^3}-\frac{2 i c \tan ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{5 a^3}+\frac{1}{5} a^2 c x^5 \tan ^{-1}(a x)^3+\frac{c x \tan ^{-1}(a x)}{10 a^2}-\frac{2 i c \tan ^{-1}(a x)^3}{15 a^3}-\frac{c \tan ^{-1}(a x)^2}{20 a^3}-\frac{2 c \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)^2}{5 a^3}-\frac{c x^2}{20 a}-\frac{3}{20} a c x^4 \tan ^{-1}(a x)^2+\frac{1}{3} c x^3 \tan ^{-1}(a x)^3+\frac{1}{10} c x^3 \tan ^{-1}(a x)-\frac{c x^2 \tan ^{-1}(a x)^2}{5 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c*x^2)/(20*a) + (c*x*ArcTan[a*x])/(10*a^2) + (c*x^3*ArcTan[a*x])/10 - (c*ArcTan[a*x]^2)/(20*a^3) - (c*x^2*Ar
cTan[a*x]^2)/(5*a) - (3*a*c*x^4*ArcTan[a*x]^2)/20 - (((2*I)/15)*c*ArcTan[a*x]^3)/a^3 + (c*x^3*ArcTan[a*x]^3)/3
+ (a^2*c*x^5*ArcTan[a*x]^3)/5 - (2*c*ArcTan[a*x]^2*Log[2/(1 + I*a*x)])/(5*a^3) - (((2*I)/5)*c*ArcTan[a*x]*Pol
yLog[2, 1 - 2/(1 + I*a*x)])/a^3 - (c*PolyLog[3, 1 - 2/(1 + I*a*x)])/(5*a^3)

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 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 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 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 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 4994

Int[(Log[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Simp[(I*(a + b*Arc
Tan[c*x])^p*PolyLog[2, 1 - u])/(2*c*d), x] + Dist[(b*p*I)/2, Int[((a + b*ArcTan[c*x])^(p - 1)*PolyLog[2, 1 - u
])/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[(1 - u)^2 - (1 - (2*
I)/(I - c*x))^2, 0]

Rule 6610

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /
;  !FalseQ[w]] /; FreeQ[n, x]

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 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])

Rubi steps

\begin{align*} \int x^2 \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^3 \, dx &=c \int x^2 \tan ^{-1}(a x)^3 \, dx+\left (a^2 c\right ) \int x^4 \tan ^{-1}(a x)^3 \, dx\\ &=\frac{1}{3} c x^3 \tan ^{-1}(a x)^3+\frac{1}{5} a^2 c x^5 \tan ^{-1}(a x)^3-(a c) \int \frac{x^3 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx-\frac{1}{5} \left (3 a^3 c\right ) \int \frac{x^5 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx\\ &=\frac{1}{3} c x^3 \tan ^{-1}(a x)^3+\frac{1}{5} a^2 c x^5 \tan ^{-1}(a x)^3-\frac{c \int x \tan ^{-1}(a x)^2 \, dx}{a}+\frac{c \int \frac{x \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{a}-\frac{1}{5} (3 a c) \int x^3 \tan ^{-1}(a x)^2 \, dx+\frac{1}{5} (3 a c) \int \frac{x^3 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx\\ &=-\frac{c x^2 \tan ^{-1}(a x)^2}{2 a}-\frac{3}{20} a c x^4 \tan ^{-1}(a x)^2-\frac{i c \tan ^{-1}(a x)^3}{3 a^3}+\frac{1}{3} c x^3 \tan ^{-1}(a x)^3+\frac{1}{5} a^2 c x^5 \tan ^{-1}(a x)^3+c \int \frac{x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{c \int \frac{\tan ^{-1}(a x)^2}{i-a x} \, dx}{a^2}+\frac{(3 c) \int x \tan ^{-1}(a x)^2 \, dx}{5 a}-\frac{(3 c) \int \frac{x \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{5 a}+\frac{1}{10} \left (3 a^2 c\right ) \int \frac{x^4 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=-\frac{c x^2 \tan ^{-1}(a x)^2}{5 a}-\frac{3}{20} a c x^4 \tan ^{-1}(a x)^2-\frac{2 i c \tan ^{-1}(a x)^3}{15 a^3}+\frac{1}{3} c x^3 \tan ^{-1}(a x)^3+\frac{1}{5} a^2 c x^5 \tan ^{-1}(a x)^3-\frac{c \tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )}{a^3}+\frac{1}{10} (3 c) \int x^2 \tan ^{-1}(a x) \, dx-\frac{1}{10} (3 c) \int \frac{x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{5} (3 c) \int \frac{x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac{(3 c) \int \frac{\tan ^{-1}(a x)^2}{i-a x} \, dx}{5 a^2}+\frac{c \int \tan ^{-1}(a x) \, dx}{a^2}-\frac{c \int \frac{\tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{a^2}+\frac{(2 c) \int \frac{\tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a^2}\\ &=\frac{c x \tan ^{-1}(a x)}{a^2}+\frac{1}{10} c x^3 \tan ^{-1}(a x)-\frac{c \tan ^{-1}(a x)^2}{2 a^3}-\frac{c x^2 \tan ^{-1}(a x)^2}{5 a}-\frac{3}{20} a c x^4 \tan ^{-1}(a x)^2-\frac{2 i c \tan ^{-1}(a x)^3}{15 a^3}+\frac{1}{3} c x^3 \tan ^{-1}(a x)^3+\frac{1}{5} a^2 c x^5 \tan ^{-1}(a x)^3-\frac{2 c \tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )}{5 a^3}-\frac{i c \tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{a^3}+\frac{(i c) \int \frac{\text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a^2}-\frac{(3 c) \int \tan ^{-1}(a x) \, dx}{10 a^2}+\frac{(3 c) \int \frac{\tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{10 a^2}-\frac{(3 c) \int \tan ^{-1}(a x) \, dx}{5 a^2}+\frac{(3 c) \int \frac{\tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{5 a^2}-\frac{(6 c) \int \frac{\tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{5 a^2}-\frac{c \int \frac{x}{1+a^2 x^2} \, dx}{a}-\frac{1}{10} (a c) \int \frac{x^3}{1+a^2 x^2} \, dx\\ &=\frac{c x \tan ^{-1}(a x)}{10 a^2}+\frac{1}{10} c x^3 \tan ^{-1}(a x)-\frac{c \tan ^{-1}(a x)^2}{20 a^3}-\frac{c x^2 \tan ^{-1}(a x)^2}{5 a}-\frac{3}{20} a c x^4 \tan ^{-1}(a x)^2-\frac{2 i c \tan ^{-1}(a x)^3}{15 a^3}+\frac{1}{3} c x^3 \tan ^{-1}(a x)^3+\frac{1}{5} a^2 c x^5 \tan ^{-1}(a x)^3-\frac{2 c \tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )}{5 a^3}-\frac{c \log \left (1+a^2 x^2\right )}{2 a^3}-\frac{2 i c \tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{5 a^3}-\frac{c \text{Li}_3\left (1-\frac{2}{1+i a x}\right )}{2 a^3}-\frac{(3 i c) \int \frac{\text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{5 a^2}+\frac{(3 c) \int \frac{x}{1+a^2 x^2} \, dx}{10 a}+\frac{(3 c) \int \frac{x}{1+a^2 x^2} \, dx}{5 a}-\frac{1}{20} (a c) \operatorname{Subst}\left (\int \frac{x}{1+a^2 x} \, dx,x,x^2\right )\\ &=\frac{c x \tan ^{-1}(a x)}{10 a^2}+\frac{1}{10} c x^3 \tan ^{-1}(a x)-\frac{c \tan ^{-1}(a x)^2}{20 a^3}-\frac{c x^2 \tan ^{-1}(a x)^2}{5 a}-\frac{3}{20} a c x^4 \tan ^{-1}(a x)^2-\frac{2 i c \tan ^{-1}(a x)^3}{15 a^3}+\frac{1}{3} c x^3 \tan ^{-1}(a x)^3+\frac{1}{5} a^2 c x^5 \tan ^{-1}(a x)^3-\frac{2 c \tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )}{5 a^3}-\frac{c \log \left (1+a^2 x^2\right )}{20 a^3}-\frac{2 i c \tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{5 a^3}-\frac{c \text{Li}_3\left (1-\frac{2}{1+i a x}\right )}{5 a^3}-\frac{1}{20} (a c) \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 x^2}{20 a}+\frac{c x \tan ^{-1}(a x)}{10 a^2}+\frac{1}{10} c x^3 \tan ^{-1}(a x)-\frac{c \tan ^{-1}(a x)^2}{20 a^3}-\frac{c x^2 \tan ^{-1}(a x)^2}{5 a}-\frac{3}{20} a c x^4 \tan ^{-1}(a x)^2-\frac{2 i c \tan ^{-1}(a x)^3}{15 a^3}+\frac{1}{3} c x^3 \tan ^{-1}(a x)^3+\frac{1}{5} a^2 c x^5 \tan ^{-1}(a x)^3-\frac{2 c \tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )}{5 a^3}-\frac{2 i c \tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{5 a^3}-\frac{c \text{Li}_3\left (1-\frac{2}{1+i a x}\right )}{5 a^3}\\ \end{align*}

Mathematica [A]  time = 0.531563, size = 171, normalized size = 0.81 \[ \frac{c \left (24 i \tan ^{-1}(a x) \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(a x)}\right )-12 \text{PolyLog}\left (3,-e^{2 i \tan ^{-1}(a x)}\right )-3 a^2 x^2+12 a^5 x^5 \tan ^{-1}(a x)^3-9 a^4 x^4 \tan ^{-1}(a x)^2+20 a^3 x^3 \tan ^{-1}(a x)^3+6 a^3 x^3 \tan ^{-1}(a x)-12 a^2 x^2 \tan ^{-1}(a x)^2+6 a x \tan ^{-1}(a x)+8 i \tan ^{-1}(a x)^3-3 \tan ^{-1}(a x)^2-24 \tan ^{-1}(a x)^2 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )-3\right )}{60 a^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c*(-3 - 3*a^2*x^2 + 6*a*x*ArcTan[a*x] + 6*a^3*x^3*ArcTan[a*x] - 3*ArcTan[a*x]^2 - 12*a^2*x^2*ArcTan[a*x]^2 -
9*a^4*x^4*ArcTan[a*x]^2 + (8*I)*ArcTan[a*x]^3 + 20*a^3*x^3*ArcTan[a*x]^3 + 12*a^5*x^5*ArcTan[a*x]^3 - 24*ArcTa
n[a*x]^2*Log[1 + E^((2*I)*ArcTan[a*x])] + (24*I)*ArcTan[a*x]*PolyLog[2, -E^((2*I)*ArcTan[a*x])] - 12*PolyLog[3
, -E^((2*I)*ArcTan[a*x])]))/(60*a^3)

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Maple [C]  time = 2.629, size = 2555, normalized size = 12.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*c*x^3*arctan(a*x)^3-2/5/a^3*c*arctan(a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2))-2/5/a^3*c*arctan(a*x)^2*ln(2)+
1/5/a^3*c*arctan(a*x)^2*ln(a^2*x^2+1)+2/15*I/a^3*c*arctan(a*x)^3-1/20*c*x^2/a+1/10*c*x^3*arctan(a*x)-1/20*c*ar
ctan(a*x)^2/a^3-3/40*I/a*c*arctan(a*x)^2*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^2*csgn(I*((1+I*a*x)^2/(a^2*x
^2+1)+1))*x^2+3/80*I/a*c*arctan(a*x)^2*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)*csgn(I*((1+I*a*x)^2/(a^2*x^2+1
)+1))^2*x^2+3/40*I/a*c*arctan(a*x)^2*Pi*csgn(I*(1+I*a*x)^4/(a^2*x^2+1)^2+2*I*(1+I*a*x)^2/(a^2*x^2+1)+I)^2*csgn
(I*(1+I*a*x)^2/(a^2*x^2+1)+I)*x^2-3/80*I/a*c*arctan(a*x)^2*Pi*csgn(I*(1+I*a*x)^4/(a^2*x^2+1)^2+2*I*(1+I*a*x)^2
/(a^2*x^2+1)+I)*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)+I)^2*x^2+1/10*I/a^3*c*arctan(a*x)^2*Pi*csgn(I*(1+I*a*x)^2/(a^2*
x^2+1))*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)+1/
10*c*x*arctan(a*x)/a^2-1/5*c*x^2*arctan(a*x)^2/a-3/20*a*c*x^4*arctan(a*x)^2+1/5*a^2*c*x^5*arctan(a*x)^3-3/80/a
^2*c*arctan(a*x)^2*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1))^2*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)*x+3/40/a^2*c
*arctan(a*x)^2*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1))*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^2*x-1/5/a^3*c*poly
log(3,-(1+I*a*x)^2/(a^2*x^2+1))-1/80*c*arctan(a*x)^2*Pi*csgn(I*(1+I*a*x)^4/(a^2*x^2+1)^2+2*I*(1+I*a*x)^2/(a^2*
x^2+1)+I)^3*x^3+2/5*I/a^3*c*arctan(a*x)*polylog(2,-(1+I*a*x)^2/(a^2*x^2+1))-3/80/a^2*c*arctan(a*x)^2*Pi*csgn(I
*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^3*x+3/80/a^2*c*arctan(a*x)^2*Pi*csgn(I*(1+I*a*x)^4/(a^2*x^2+1)^2+2*I*(1+I*a*x)
^2/(a^2*x^2+1)+I)^3*x-1/20/a^3*c+7/40*I/a^3*c*arctan(a*x)^2*Pi*csgn(I*(1+I*a*x)^4/(a^2*x^2+1)^2+2*I*(1+I*a*x)^
2/(a^2*x^2+1)+I)^2*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)+I)-7/80*I/a^3*c*arctan(a*x)^2*Pi*csgn(I*(1+I*a*x)^4/(a^2*x^2
+1)^2+2*I*(1+I*a*x)^2/(a^2*x^2+1)+I)*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)+I)^2-1/5*I/a^3*c*arctan(a*x)^2*Pi*csgn(I*(
1+I*a*x)^2/(a^2*x^2+1))^2*csgn(I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-1/10*I/a^3*c*arctan(a*x)^2*Pi*csgn(I*(1+I*a*x)^2
/(a^2*x^2+1))*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^2+1/10*I/a^3*c*arctan(a*x)^2*Pi*cs
gn(I*(1+I*a*x)^2/(a^2*x^2+1))*csgn(I*(1+I*a*x)/(a^2*x^2+1)^(1/2))^2-1/10*I/a^3*c*arctan(a*x)^2*Pi*csgn(I/((1+I
*a*x)^2/(a^2*x^2+1)+1)^2)*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^2+3/80*I/a*c*arctan(a*
x)^2*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^3*x^2-3/80*I/a*c*arctan(a*x)^2*Pi*csgn(I*(1+I*a*x)^4/(a^2*x^2+1)
^2+2*I*(1+I*a*x)^2/(a^2*x^2+1)+I)^3*x^2+3/80/a^2*c*arctan(a*x)^2*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)+I)^2*csgn(I
*(1+I*a*x)^4/(a^2*x^2+1)^2+2*I*(1+I*a*x)^2/(a^2*x^2+1)+I)*x-3/40/a^2*c*arctan(a*x)^2*Pi*csgn(I*(1+I*a*x)^2/(a^
2*x^2+1)+I)*csgn(I*(1+I*a*x)^4/(a^2*x^2+1)^2+2*I*(1+I*a*x)^2/(a^2*x^2+1)+I)^2*x+1/40*I/a^3*c*arctan(a*x)^2*Pi*
csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^2*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1))-1/80*I/a^3*c*arctan(a*x)^2*Pi*csgn
(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1))^2+1/80*c*arctan(a*x)^2*Pi*csgn(I*((1+I*a
*x)^2/(a^2*x^2+1)+1)^2)^3*x^3+1/80*c*arctan(a*x)^2*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1))^2*csgn(I*((1+I*a*x)^
2/(a^2*x^2+1)+1)^2)*x^3-1/40*c*arctan(a*x)^2*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1))*csgn(I*((1+I*a*x)^2/(a^2*x
^2+1)+1)^2)^2*x^3-1/80*c*arctan(a*x)^2*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)+I)^2*csgn(I*(1+I*a*x)^4/(a^2*x^2+1)^2
+2*I*(1+I*a*x)^2/(a^2*x^2+1)+I)*x^3+1/40*c*arctan(a*x)^2*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)+I)*csgn(I*(1+I*a*x)
^4/(a^2*x^2+1)^2+2*I*(1+I*a*x)^2/(a^2*x^2+1)+I)^2*x^3-1/80*I/a^3*c*arctan(a*x)^2*Pi*csgn(I*((1+I*a*x)^2/(a^2*x
^2+1)+1)^2)^3-7/80*I/a^3*c*arctan(a*x)^2*Pi*csgn(I*(1+I*a*x)^4/(a^2*x^2+1)^2+2*I*(1+I*a*x)^2/(a^2*x^2+1)+I)^3+
1/10*I/a^3*c*arctan(a*x)^2*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))^3+1/10*I/a^3*c*arctan(a*x)^2*Pi*csgn(I*(1+I*a*x)
^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{120} \,{\left (3 \, a^{2} c x^{5} + 5 \, c x^{3}\right )} \arctan \left (a x\right )^{3} - \frac{1}{160} \,{\left (3 \, a^{2} c x^{5} + 5 \, c x^{3}\right )} \arctan \left (a x\right ) \log \left (a^{2} x^{2} + 1\right )^{2} + \int \frac{140 \,{\left (a^{4} c x^{6} + 2 \, a^{2} c x^{4} + c x^{2}\right )} \arctan \left (a x\right )^{3} - 4 \,{\left (3 \, a^{3} c x^{5} + 5 \, a c x^{3}\right )} \arctan \left (a x\right )^{2} + 4 \,{\left (3 \, a^{4} c x^{6} + 5 \, a^{2} c x^{4}\right )} \arctan \left (a x\right ) \log \left (a^{2} x^{2} + 1\right ) +{\left (3 \, a^{3} c x^{5} + 5 \, a c x^{3} + 15 \,{\left (a^{4} c x^{6} + 2 \, a^{2} c x^{4} + c x^{2}\right )} \arctan \left (a x\right )\right )} \log \left (a^{2} x^{2} + 1\right )^{2}}{160 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/120*(3*a^2*c*x^5 + 5*c*x^3)*arctan(a*x)^3 - 1/160*(3*a^2*c*x^5 + 5*c*x^3)*arctan(a*x)*log(a^2*x^2 + 1)^2 + i
ntegrate(1/160*(140*(a^4*c*x^6 + 2*a^2*c*x^4 + c*x^2)*arctan(a*x)^3 - 4*(3*a^3*c*x^5 + 5*a*c*x^3)*arctan(a*x)^
2 + 4*(3*a^4*c*x^6 + 5*a^2*c*x^4)*arctan(a*x)*log(a^2*x^2 + 1) + (3*a^3*c*x^5 + 5*a*c*x^3 + 15*(a^4*c*x^6 + 2*
a^2*c*x^4 + c*x^2)*arctan(a*x))*log(a^2*x^2 + 1)^2)/(a^2*x^2 + 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a^{2} c x^{4} + c x^{2}\right )} \arctan \left (a x\right )^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((a^2*c*x^4 + c*x^2)*arctan(a*x)^3, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} c \left (\int x^{2} \operatorname{atan}^{3}{\left (a x \right )}\, dx + \int a^{2} x^{4} \operatorname{atan}^{3}{\left (a x \right )}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*(Integral(x**2*atan(a*x)**3, x) + Integral(a**2*x**4*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^{2} \arctan \left (a x\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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