3.141 \(\int \frac{x^3 (a+b \tan ^{-1}(c x))^2}{d+e x} \, dx\)

Optimal. Leaf size=598 \[ -\frac{i b d^3 \text{PolyLog}\left (2,1-\frac{2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{e^4}+\frac{i b d^3 \left (a+b \tan ^{-1}(c x)\right ) \text{PolyLog}\left (2,1-\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{e^4}-\frac{i b^2 \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )}{3 c^3 e}+\frac{b^2 d^3 \text{PolyLog}\left (3,1-\frac{2}{1-i c x}\right )}{2 e^4}-\frac{b^2 d^3 \text{PolyLog}\left (3,1-\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{2 e^4}+\frac{i b^2 d^2 \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )}{c e^3}-\frac{d \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^2 e^2}-\frac{i \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^3 e}-\frac{2 b \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{3 c^3 e}+\frac{d^2 x \left (a+b \tan ^{-1}(c x)\right )^2}{e^3}+\frac{i d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{c e^3}+\frac{d^3 \log \left (\frac{2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{e^4}-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{e^4}+\frac{2 b d^2 \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c e^3}-\frac{d x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 e^2}+\frac{a b d x}{c e^2}+\frac{x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 e}-\frac{b x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c e}-\frac{b^2 d \log \left (c^2 x^2+1\right )}{2 c^2 e^2}+\frac{b^2 x}{3 c^2 e}-\frac{b^2 \tan ^{-1}(c x)}{3 c^3 e}+\frac{b^2 d x \tan ^{-1}(c x)}{c e^2} \]

[Out]

(a*b*d*x)/(c*e^2) + (b^2*x)/(3*c^2*e) - (b^2*ArcTan[c*x])/(3*c^3*e) + (b^2*d*x*ArcTan[c*x])/(c*e^2) - (b*x^2*(
a + b*ArcTan[c*x]))/(3*c*e) + (I*d^2*(a + b*ArcTan[c*x])^2)/(c*e^3) - (d*(a + b*ArcTan[c*x])^2)/(2*c^2*e^2) -
((I/3)*(a + b*ArcTan[c*x])^2)/(c^3*e) + (d^2*x*(a + b*ArcTan[c*x])^2)/e^3 - (d*x^2*(a + b*ArcTan[c*x])^2)/(2*e
^2) + (x^3*(a + b*ArcTan[c*x])^2)/(3*e) + (d^3*(a + b*ArcTan[c*x])^2*Log[2/(1 - I*c*x)])/e^4 + (2*b*d^2*(a + b
*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(c*e^3) - (2*b*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(3*c^3*e) - (d^3*(a +
b*ArcTan[c*x])^2*Log[(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e^4 - (b^2*d*Log[1 + c^2*x^2])/(2*c^2*e^2) -
(I*b*d^3*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 - I*c*x)])/e^4 + (I*b^2*d^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/(
c*e^3) - ((I/3)*b^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^3*e) + (I*b*d^3*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*
(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e^4 + (b^2*d^3*PolyLog[3, 1 - 2/(1 - I*c*x)])/(2*e^4) - (b^2*d^3*PolyLo
g[3, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(2*e^4)

________________________________________________________________________________________

Rubi [A]  time = 0.671122, antiderivative size = 598, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 13, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.619, Rules used = {4876, 4846, 4920, 4854, 2402, 2315, 4852, 4916, 260, 4884, 321, 203, 4858} \[ -\frac{i b d^3 \text{PolyLog}\left (2,1-\frac{2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{e^4}+\frac{i b d^3 \left (a+b \tan ^{-1}(c x)\right ) \text{PolyLog}\left (2,1-\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{e^4}-\frac{i b^2 \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )}{3 c^3 e}+\frac{b^2 d^3 \text{PolyLog}\left (3,1-\frac{2}{1-i c x}\right )}{2 e^4}-\frac{b^2 d^3 \text{PolyLog}\left (3,1-\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{2 e^4}+\frac{i b^2 d^2 \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )}{c e^3}-\frac{d \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^2 e^2}-\frac{i \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^3 e}-\frac{2 b \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{3 c^3 e}+\frac{d^2 x \left (a+b \tan ^{-1}(c x)\right )^2}{e^3}+\frac{i d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{c e^3}+\frac{d^3 \log \left (\frac{2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{e^4}-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{e^4}+\frac{2 b d^2 \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c e^3}-\frac{d x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 e^2}+\frac{a b d x}{c e^2}+\frac{x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 e}-\frac{b x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c e}-\frac{b^2 d \log \left (c^2 x^2+1\right )}{2 c^2 e^2}+\frac{b^2 x}{3 c^2 e}-\frac{b^2 \tan ^{-1}(c x)}{3 c^3 e}+\frac{b^2 d x \tan ^{-1}(c x)}{c e^2} \]

Antiderivative was successfully verified.

[In]

Int[(x^3*(a + b*ArcTan[c*x])^2)/(d + e*x),x]

[Out]

(a*b*d*x)/(c*e^2) + (b^2*x)/(3*c^2*e) - (b^2*ArcTan[c*x])/(3*c^3*e) + (b^2*d*x*ArcTan[c*x])/(c*e^2) - (b*x^2*(
a + b*ArcTan[c*x]))/(3*c*e) + (I*d^2*(a + b*ArcTan[c*x])^2)/(c*e^3) - (d*(a + b*ArcTan[c*x])^2)/(2*c^2*e^2) -
((I/3)*(a + b*ArcTan[c*x])^2)/(c^3*e) + (d^2*x*(a + b*ArcTan[c*x])^2)/e^3 - (d*x^2*(a + b*ArcTan[c*x])^2)/(2*e
^2) + (x^3*(a + b*ArcTan[c*x])^2)/(3*e) + (d^3*(a + b*ArcTan[c*x])^2*Log[2/(1 - I*c*x)])/e^4 + (2*b*d^2*(a + b
*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(c*e^3) - (2*b*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(3*c^3*e) - (d^3*(a +
b*ArcTan[c*x])^2*Log[(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e^4 - (b^2*d*Log[1 + c^2*x^2])/(2*c^2*e^2) -
(I*b*d^3*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 - I*c*x)])/e^4 + (I*b^2*d^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/(
c*e^3) - ((I/3)*b^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^3*e) + (I*b*d^3*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*
(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e^4 + (b^2*d^3*PolyLog[3, 1 - 2/(1 - I*c*x)])/(2*e^4) - (b^2*d^3*PolyLo
g[3, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(2*e^4)

Rule 4876

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

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 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 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 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 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 4858

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^2/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcTan[c*x])^2*Log[2/
(1 - I*c*x)])/e, x] + (Simp[((a + b*ArcTan[c*x])^2*Log[(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e, x] + Sim
p[(I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 - I*c*x)])/e, x] - Simp[(I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 -
(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e, x] - Simp[(b^2*PolyLog[3, 1 - 2/(1 - I*c*x)])/(2*e), x] + Simp
[(b^2*PolyLog[3, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(2*e), x]) /; FreeQ[{a, b, c, d, e}, x] && Ne
Q[c^2*d^2 + e^2, 0]

Rubi steps

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

Mathematica [F]  time = 180.002, size = 0, normalized size = 0. \[ \text{\$Aborted} \]

Verification is Not applicable to the result.

[In]

Integrate[(x^3*(a + b*ArcTan[c*x])^2)/(d + e*x),x]

[Out]

$Aborted

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Maple [C]  time = 14.474, size = 2136, normalized size = 3.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a+b*arctan(c*x))^2/(e*x+d),x)

[Out]

-c*b^2*d^4/e^4/(d*c-I*e)*arctan(c*x)^2*ln(1-(I*e-d*c)/(d*c+I*e)*(1+I*c*x)^2/(c^2*x^2+1))+I*b^2*d^3/e^3/(d*c-I*
e)*arctan(c*x)^2*ln(1-(I*e-d*c)/(d*c+I*e)*(1+I*c*x)^2/(c^2*x^2+1))+I*a*b/e^4*d^3*ln(c*e*x+c*d)*ln((I*e+e*c*x)/
(I*e-d*c))-1/2*I*b^2/e^4*d^3*Pi*csgn(I*(-I*(1+I*c*x)^2/(c^2*x^2+1)*e+c*d*(1+I*c*x)^2/(c^2*x^2+1)+I*e+d*c)/((1+
I*c*x)^2/(c^2*x^2+1)+1))^3*arctan(c*x)^2-I*a*b/e^4*d^3*ln(c*e*x+c*d)*ln((I*e-e*c*x)/(d*c+I*e))+2/c*b^2/e^3*d^2
*arctan(c*x)*ln(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-1/2*c*b^2*d^4/e^4/(d*c-I*e)*polylog(3,(I*e-d*c)/(d*c+I*e)*(1+
I*c*x)^2/(c^2*x^2+1))-1/c*a*b/e^3*ln(c^2*d^2-2*(c*e*x+c*d)*c*d+(c*e*x+c*d)^2+e^2)*d^2-1/c^2*a*b/e^2*arctan(c*x
)*d-I*b^2*d^3/e^4*arctan(c*x)*polylog(2,-(1+I*c*x)^2/(c^2*x^2+1))+1/2*I*b^2*d^3/e^3/(d*c-I*e)*polylog(3,(I*e-d
*c)/(d*c+I*e)*(1+I*c*x)^2/(c^2*x^2+1))-I*a*b/e^4*d^3*dilog((I*e-e*c*x)/(d*c+I*e))-I/c^2*b^2*d*arctan(c*x)/e^2-
2*I/c*b^2/e^3*d^2*dilog(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-I/c*b^2/e^3*d^2*arctan(c*x)^2-2*I/c*b^2/e^3*d^2*dilog
(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+b^2*d^3/e^3/(d*c-I*e)*arctan(c*x)*polylog(2,(I*e-d*c)/(d*c+I*e)*(1+I*c*x)^2/
(c^2*x^2+1))-a*b*arctan(c*x)/e^2*d*x^2+1/3*b^2*x/c^2/e-1/3*b^2*arctan(c*x)/c^3/e+I*c*b^2*d^4/e^4/(d*c-I*e)*arc
tan(c*x)*polylog(2,(I*e-d*c)/(d*c+I*e)*(1+I*c*x)^2/(c^2*x^2+1))+1/2*I*b^2/e^4*d^3*Pi*csgn(I/((1+I*c*x)^2/(c^2*
x^2+1)+1))*csgn(I*(-I*(1+I*c*x)^2/(c^2*x^2+1)*e+c*d*(1+I*c*x)^2/(c^2*x^2+1)+I*e+d*c)/((1+I*c*x)^2/(c^2*x^2+1)+
1))^2*arctan(c*x)^2+1/2*I*b^2/e^4*d^3*Pi*csgn(I*(-I*(1+I*c*x)^2/(c^2*x^2+1)*e+c*d*(1+I*c*x)^2/(c^2*x^2+1)+I*e+
d*c))*csgn(I*(-I*(1+I*c*x)^2/(c^2*x^2+1)*e+c*d*(1+I*c*x)^2/(c^2*x^2+1)+I*e+d*c)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2
*arctan(c*x)^2+a*b*d*x/c/e^2+b^2*d*x*arctan(c*x)/c/e^2-1/2*I*b^2/e^4*d^3*Pi*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1)
)*csgn(I*(-I*(1+I*c*x)^2/(c^2*x^2+1)*e+c*d*(1+I*c*x)^2/(c^2*x^2+1)+I*e+d*c))*csgn(I*(-I*(1+I*c*x)^2/(c^2*x^2+1
)*e+c*d*(1+I*c*x)^2/(c^2*x^2+1)+I*e+d*c)/((1+I*c*x)^2/(c^2*x^2+1)+1))*arctan(c*x)^2+1/3*I/c^3*b^2/e+2*a*b*arct
an(c*x)/e^3*d^2*x-2*a*b*arctan(c*x)*d^3/e^4*ln(c*e*x+c*d)+I*a*b/e^4*d^3*dilog((I*e+e*c*x)/(I*e-d*c))+2/c*b^2/e
^3*d^2*arctan(c*x)*ln(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-1/3/c*a*b*x^2/e-1/2*b^2*arctan(c*x)^2/e^2*d*x^2+b^2*arc
tan(c*x)^2/e^3*x*d^2+2/3*a*b*arctan(c*x)/e*x^3+b^2*d^3/e^4*arctan(c*x)^2*ln(-I*(1+I*c*x)^2/(c^2*x^2+1)*e+c*d*(
1+I*c*x)^2/(c^2*x^2+1)+I*e+d*c)-b^2*arctan(c*x)^2*d^3/e^4*ln(c*e*x+c*d)+1/c^2*b^2/e^2*d*ln((1+I*c*x)^2/(c^2*x^
2+1)+1)-1/2/c^2*b^2/e^2*d*arctan(c*x)^2+1/3/c^3*a*b/e*ln(c^2*d^2-2*(c*e*x+c*d)*c*d+(c*e*x+c*d)^2+e^2)-2/3/c^3*
b^2/e*arctan(c*x)*ln(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-2/3/c^3*b^2/e*arctan(c*x)*ln(1-I*(1+I*c*x)/(c^2*x^2+1)^(
1/2))-1/3/c*b^2*arctan(c*x)/e*x^2+2/3*I/c^3*b^2/e*dilog(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+1/3*I/c^3*b^2/e*arcta
n(c*x)^2+2/3*I/c^3*b^2/e*dilog(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+4/3/c*a*b/e^3*d^2+1/3*a^2/e*x^3+1/2*b^2*d^3/e^
4*polylog(3,-(1+I*c*x)^2/(c^2*x^2+1))-a^2*d^3/e^4*ln(c*e*x+c*d)+1/3*b^2*arctan(c*x)^2/e*x^3+a^2/e^3*x*d^2-1/2*
a^2/e^2*d*x^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{6} \, a^{2}{\left (\frac{6 \, d^{3} \log \left (e x + d\right )}{e^{4}} - \frac{2 \, e^{2} x^{3} - 3 \, d e x^{2} + 6 \, d^{2} x}{e^{3}}\right )} + \frac{2 \, e^{3} \int \frac{36 \,{\left (b^{2} c^{2} e^{3} x^{5} + b^{2} e^{3} x^{3}\right )} \arctan \left (c x\right )^{2} + 3 \,{\left (b^{2} c^{2} e^{3} x^{5} + b^{2} e^{3} x^{3}\right )} \log \left (c^{2} x^{2} + 1\right )^{2} + 4 \,{\left (24 \, a b c^{2} e^{3} x^{5} - 2 \, b^{2} c e^{3} x^{4} - 3 \, b^{2} c d^{2} e x^{2} - 6 \, b^{2} c d^{3} x +{\left (b^{2} c d e^{2} + 24 \, a b e^{3}\right )} x^{3}\right )} \arctan \left (c x\right ) + 2 \,{\left (2 \, b^{2} c^{2} e^{3} x^{5} - b^{2} c^{2} d e^{2} x^{4} + 3 \, b^{2} c^{2} d^{2} e x^{3} + 6 \, b^{2} c^{2} d^{3} x^{2}\right )} \log \left (c^{2} x^{2} + 1\right )}{c^{2} e^{4} x^{3} + c^{2} d e^{3} x^{2} + e^{4} x + d e^{3}}\,{d x} + 4 \,{\left (2 \, b^{2} e^{2} x^{3} - 3 \, b^{2} d e x^{2} + 6 \, b^{2} d^{2} x\right )} \arctan \left (c x\right )^{2} -{\left (2 \, b^{2} e^{2} x^{3} - 3 \, b^{2} d e x^{2} + 6 \, b^{2} d^{2} x\right )} \log \left (c^{2} x^{2} + 1\right )^{2}}{96 \, e^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*a^2*(6*d^3*log(e*x + d)/e^4 - (2*e^2*x^3 - 3*d*e*x^2 + 6*d^2*x)/e^3) + 1/96*(96*e^3*integrate(1/48*(36*(b
^2*c^2*e^3*x^5 + b^2*e^3*x^3)*arctan(c*x)^2 + 3*(b^2*c^2*e^3*x^5 + b^2*e^3*x^3)*log(c^2*x^2 + 1)^2 + 4*(24*a*b
*c^2*e^3*x^5 - 2*b^2*c*e^3*x^4 - 3*b^2*c*d^2*e*x^2 - 6*b^2*c*d^3*x + (b^2*c*d*e^2 + 24*a*b*e^3)*x^3)*arctan(c*
x) + 2*(2*b^2*c^2*e^3*x^5 - b^2*c^2*d*e^2*x^4 + 3*b^2*c^2*d^2*e*x^3 + 6*b^2*c^2*d^3*x^2)*log(c^2*x^2 + 1))/(c^
2*e^4*x^3 + c^2*d*e^3*x^2 + e^4*x + d*e^3), x) + 4*(2*b^2*e^2*x^3 - 3*b^2*d*e*x^2 + 6*b^2*d^2*x)*arctan(c*x)^2
- (2*b^2*e^2*x^3 - 3*b^2*d*e*x^2 + 6*b^2*d^2*x)*log(c^2*x^2 + 1)^2)/e^3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b^2*x^3*arctan(c*x)^2 + 2*a*b*x^3*arctan(c*x) + a^2*x^3)/(e*x + d), x)

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Sympy [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+b*atan(c*x))**2/(e*x+d),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*arctan(c*x) + a)^2*x^3/(e*x + d), x)