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

Optimal. Leaf size=652 \[ \frac {b^2 e^3 \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{2 c^4}+\frac {3 i b^2 d (c d-e) (c d+e) \text {Li}_2\left (1-\frac {2}{i c x+1}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^3}+\frac {3 a b^2 d e^2 x}{c^2}+\frac {b^2 e^3 x^2 \left (a+b \tan ^{-1}(c x)\right )}{4 c^2}-\frac {3 b^2 e \left (6 c^2 d^2-e^2\right ) \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{2 c^4}+\frac {i b e^3 \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^4}-\frac {3 b d e^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^3}+\frac {i d (c d-e) (c d+e) \left (a+b \tan ^{-1}(c x)\right )^3}{c^3}+\frac {3 b d (c d-e) (c d+e) \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{c^3}-\frac {3 i b e \left (6 c^2 d^2-e^2\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^4}-\frac {\left (c^4 d^4-6 c^2 d^2 e^2+e^4\right ) \left (a+b \tan ^{-1}(c x)\right )^3}{4 c^4 e}-\frac {3 b e x \left (6 c^2 d^2-e^2\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^3}-\frac {3 b d e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 c}+\frac {(d+e x)^4 \left (a+b \tan ^{-1}(c x)\right )^3}{4 e}-\frac {b e^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{4 c}+\frac {i b^3 e^3 \text {Li}_2\left (1-\frac {2}{i c x+1}\right )}{4 c^4}+\frac {b^3 e^3 \tan ^{-1}(c x)}{4 c^4}+\frac {3 b^3 d (c d-e) (c d+e) \text {Li}_3\left (1-\frac {2}{i c x+1}\right )}{2 c^3}-\frac {b^3 e^3 x}{4 c^3}+\frac {3 b^3 d e^2 x \tan ^{-1}(c x)}{c^2}-\frac {3 i b^3 e \left (6 c^2 d^2-e^2\right ) \text {Li}_2\left (1-\frac {2}{i c x+1}\right )}{4 c^4}-\frac {3 b^3 d e^2 \log \left (c^2 x^2+1\right )}{2 c^3} \]

[Out]

3*a*b^2*d*e^2*x/c^2-1/4*b^3*e^3*x/c^3+1/4*b^3*e^3*arctan(c*x)/c^4+3*b^3*d*e^2*x*arctan(c*x)/c^2+1/4*b^2*e^3*x^
2*(a+b*arctan(c*x))/c^2-3/2*b*d*e^2*(a+b*arctan(c*x))^2/c^3+3*I*b^2*d*(c*d-e)*(c*d+e)*(a+b*arctan(c*x))*polylo
g(2,1-2/(1+I*c*x))/c^3-3/4*I*b^3*e*(6*c^2*d^2-e^2)*polylog(2,1-2/(1+I*c*x))/c^4-3/4*b*e*(6*c^2*d^2-e^2)*x*(a+b
*arctan(c*x))^2/c^3-3/2*b*d*e^2*x^2*(a+b*arctan(c*x))^2/c-1/4*b*e^3*x^3*(a+b*arctan(c*x))^2/c-3/4*I*b*e*(6*c^2
*d^2-e^2)*(a+b*arctan(c*x))^2/c^4-1/4*(c^4*d^4-6*c^2*d^2*e^2+e^4)*(a+b*arctan(c*x))^3/c^4/e+1/4*(e*x+d)^4*(a+b
*arctan(c*x))^3/e+1/2*b^2*e^3*(a+b*arctan(c*x))*ln(2/(1+I*c*x))/c^4-3/2*b^2*e*(6*c^2*d^2-e^2)*(a+b*arctan(c*x)
)*ln(2/(1+I*c*x))/c^4+3*b*d*(c*d-e)*(c*d+e)*(a+b*arctan(c*x))^2*ln(2/(1+I*c*x))/c^3-3/2*b^3*d*e^2*ln(c^2*x^2+1
)/c^3+1/4*I*b^3*e^3*polylog(2,1-2/(1+I*c*x))/c^4+1/4*I*b*e^3*(a+b*arctan(c*x))^2/c^4+I*d*(c*d-e)*(c*d+e)*(a+b*
arctan(c*x))^3/c^3+3/2*b^3*d*(c*d-e)*(c*d+e)*polylog(3,1-2/(1+I*c*x))/c^3

________________________________________________________________________________________

Rubi [A]  time = 1.20, antiderivative size = 652, normalized size of antiderivative = 1.00, number of steps used = 29, number of rules used = 15, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {4864, 4846, 4920, 4854, 2402, 2315, 4852, 4916, 260, 4884, 321, 203, 4984, 4994, 6610} \[ \frac {3 i b^2 d (c d-e) (c d+e) \text {PolyLog}\left (2,1-\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^3}-\frac {3 i b^3 e \left (6 c^2 d^2-e^2\right ) \text {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{4 c^4}+\frac {3 b^3 d (c d-e) (c d+e) \text {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 c^3}+\frac {i b^3 e^3 \text {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{4 c^4}-\frac {3 b^2 e \left (6 c^2 d^2-e^2\right ) \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{2 c^4}+\frac {3 a b^2 d e^2 x}{c^2}+\frac {b^2 e^3 x^2 \left (a+b \tan ^{-1}(c x)\right )}{4 c^2}+\frac {b^2 e^3 \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{2 c^4}-\frac {\left (-6 c^2 d^2 e^2+c^4 d^4+e^4\right ) \left (a+b \tan ^{-1}(c x)\right )^3}{4 c^4 e}-\frac {3 i b e \left (6 c^2 d^2-e^2\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^4}-\frac {3 b e x \left (6 c^2 d^2-e^2\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^3}-\frac {3 b d e^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^3}+\frac {i d (c d-e) (c d+e) \left (a+b \tan ^{-1}(c x)\right )^3}{c^3}+\frac {3 b d (c d-e) (c d+e) \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{c^3}+\frac {i b e^3 \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^4}-\frac {3 b d e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 c}+\frac {(d+e x)^4 \left (a+b \tan ^{-1}(c x)\right )^3}{4 e}-\frac {b e^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{4 c}-\frac {3 b^3 d e^2 \log \left (c^2 x^2+1\right )}{2 c^3}+\frac {3 b^3 d e^2 x \tan ^{-1}(c x)}{c^2}-\frac {b^3 e^3 x}{4 c^3}+\frac {b^3 e^3 \tan ^{-1}(c x)}{4 c^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*a*b^2*d*e^2*x)/c^2 - (b^3*e^3*x)/(4*c^3) + (b^3*e^3*ArcTan[c*x])/(4*c^4) + (3*b^3*d*e^2*x*ArcTan[c*x])/c^2
+ (b^2*e^3*x^2*(a + b*ArcTan[c*x]))/(4*c^2) - (3*b*d*e^2*(a + b*ArcTan[c*x])^2)/(2*c^3) + ((I/4)*b*e^3*(a + b*
ArcTan[c*x])^2)/c^4 - (((3*I)/4)*b*e*(6*c^2*d^2 - e^2)*(a + b*ArcTan[c*x])^2)/c^4 - (3*b*e*(6*c^2*d^2 - e^2)*x
*(a + b*ArcTan[c*x])^2)/(4*c^3) - (3*b*d*e^2*x^2*(a + b*ArcTan[c*x])^2)/(2*c) - (b*e^3*x^3*(a + b*ArcTan[c*x])
^2)/(4*c) + (I*d*(c*d - e)*(c*d + e)*(a + b*ArcTan[c*x])^3)/c^3 - ((c^4*d^4 - 6*c^2*d^2*e^2 + e^4)*(a + b*ArcT
an[c*x])^3)/(4*c^4*e) + ((d + e*x)^4*(a + b*ArcTan[c*x])^3)/(4*e) + (b^2*e^3*(a + b*ArcTan[c*x])*Log[2/(1 + I*
c*x)])/(2*c^4) - (3*b^2*e*(6*c^2*d^2 - e^2)*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(2*c^4) + (3*b*d*(c*d - e)
*(c*d + e)*(a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/c^3 - (3*b^3*d*e^2*Log[1 + c^2*x^2])/(2*c^3) + ((I/4)*b^3
*e^3*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^4 - (((3*I)/4)*b^3*e*(6*c^2*d^2 - e^2)*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^
4 + ((3*I)*b^2*d*(c*d - e)*(c*d + e)*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^3 + (3*b^3*d*(c*d -
e)*(c*d + e)*PolyLog[3, 1 - 2/(1 + I*c*x)])/(2*c^3)

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 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 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 4864

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[((d + e*x)^(q + 1)*(a
+ b*ArcTan[c*x])^p)/(e*(q + 1)), x] - Dist[(b*c*p)/(e*(q + 1)), Int[ExpandIntegrand[(a + b*ArcTan[c*x])^(p -
1), (d + e*x)^(q + 1)/(1 + c^2*x^2), x], x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 1] && IntegerQ[q] && N
eQ[q, -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 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 4984

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(m_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> I
nt[ExpandIntegrand[(a + b*ArcTan[c*x])^p/(d + e*x^2), (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& IGtQ[p, 0] && EqQ[e, c^2*d] && IGtQ[m, 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]

Rubi steps

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

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Mathematica [A]  time = 2.23, size = 855, normalized size = 1.31 \[ \frac {a^3 e^3 x^4 c^4+3 a^2 b x \left (4 d^3+6 e x d^2+4 e^2 x^2 d+e^3 x^3\right ) \tan ^{-1}(c x) c^4+a^2 e^2 (4 a c d-b e) x^3 c^3+6 a^2 d e (a c d-b e) x^2 c^3+12 a b^2 d^3 \left (\tan ^{-1}(c x) \left ((c x-i) \tan ^{-1}(c x)+2 \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )\right )-i \text {Li}_2\left (-e^{2 i \tan ^{-1}(c x)}\right )\right ) c^3+2 b^3 d^3 \left (2 \left ((c x-i) \tan ^{-1}(c x)+3 \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )\right ) \tan ^{-1}(c x)^2-6 i \text {Li}_2\left (-e^{2 i \tan ^{-1}(c x)}\right ) \tan ^{-1}(c x)+3 \text {Li}_3\left (-e^{2 i \tan ^{-1}(c x)}\right )\right ) c^3+18 a b^2 d^2 e \left (\left (c^2 x^2+1\right ) \tan ^{-1}(c x)^2-2 c x \tan ^{-1}(c x)+\log \left (c^2 x^2+1\right )\right ) c^2+6 b^3 d^2 e \left (\tan ^{-1}(c x) \left (\left (c^2 x^2+1\right ) \tan ^{-1}(c x)^2+(3 i-3 c x) \tan ^{-1}(c x)-6 \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )\right )+3 i \text {Li}_2\left (-e^{2 i \tan ^{-1}(c x)}\right )\right ) c^2+a^2 \left (4 a c^3 d^3+3 b e \left (e^2-6 c^2 d^2\right )\right ) x c-6 a^2 b d \left (c^2 d^2-e^2\right ) \log \left (c^2 x^2+1\right ) c+12 a b^2 d e^2 \left (\left (c^3 x^3+i\right ) \tan ^{-1}(c x)^2-\left (c^2 x^2+2 \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )+1\right ) \tan ^{-1}(c x)+c x+i \text {Li}_2\left (-e^{2 i \tan ^{-1}(c x)}\right )\right ) c+2 b^3 d e^2 \left (2 c^3 x^3 \tan ^{-1}(c x)^3+2 i \tan ^{-1}(c x)^3-3 c^2 x^2 \tan ^{-1}(c x)^2-6 \log \left (1+e^{2 i \tan ^{-1}(c x)}\right ) \tan ^{-1}(c x)^2-3 \tan ^{-1}(c x)^2+6 c x \tan ^{-1}(c x)+6 i \text {Li}_2\left (-e^{2 i \tan ^{-1}(c x)}\right ) \tan ^{-1}(c x)-3 \log \left (c^2 x^2+1\right )-3 \text {Li}_3\left (-e^{2 i \tan ^{-1}(c x)}\right )\right ) c+3 a^2 b \left (6 c^2 d^2 e-e^3\right ) \tan ^{-1}(c x)+a b^2 e^3 \left (c^2 x^2+3 \left (c^4 x^4-1\right ) \tan ^{-1}(c x)^2+\left (6 c x-2 c^3 x^3\right ) \tan ^{-1}(c x)-4 \log \left (c^2 x^2+1\right )+1\right )+b^3 e^3 \left (\left (c^4 x^4-1\right ) \tan ^{-1}(c x)^3-\left (c^3 x^3-3 c x+4 i\right ) \tan ^{-1}(c x)^2+\left (c^2 x^2+8 \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )+1\right ) \tan ^{-1}(c x)-c x-4 i \text {Li}_2\left (-e^{2 i \tan ^{-1}(c x)}\right )\right )}{4 c^4} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a^2*c*(4*a*c^3*d^3 + 3*b*e*(-6*c^2*d^2 + e^2))*x + 6*a^2*c^3*d*e*(a*c*d - b*e)*x^2 + a^2*c^3*e^2*(4*a*c*d - b
*e)*x^3 + a^3*c^4*e^3*x^4 + 3*a^2*b*(6*c^2*d^2*e - e^3)*ArcTan[c*x] + 3*a^2*b*c^4*x*(4*d^3 + 6*d^2*e*x + 4*d*e
^2*x^2 + e^3*x^3)*ArcTan[c*x] + a*b^2*e^3*(1 + c^2*x^2 + (6*c*x - 2*c^3*x^3)*ArcTan[c*x] + 3*(-1 + c^4*x^4)*Ar
cTan[c*x]^2 - 4*Log[1 + c^2*x^2]) - 6*a^2*b*c*d*(c^2*d^2 - e^2)*Log[1 + c^2*x^2] + 18*a*b^2*c^2*d^2*e*(-2*c*x*
ArcTan[c*x] + (1 + c^2*x^2)*ArcTan[c*x]^2 + Log[1 + c^2*x^2]) + 12*a*b^2*c^3*d^3*(ArcTan[c*x]*((-I + c*x)*ArcT
an[c*x] + 2*Log[1 + E^((2*I)*ArcTan[c*x])]) - I*PolyLog[2, -E^((2*I)*ArcTan[c*x])]) + 12*a*b^2*c*d*e^2*(c*x +
(I + c^3*x^3)*ArcTan[c*x]^2 - ArcTan[c*x]*(1 + c^2*x^2 + 2*Log[1 + E^((2*I)*ArcTan[c*x])]) + I*PolyLog[2, -E^(
(2*I)*ArcTan[c*x])]) + 6*b^3*c^2*d^2*e*(ArcTan[c*x]*((3*I - 3*c*x)*ArcTan[c*x] + (1 + c^2*x^2)*ArcTan[c*x]^2 -
6*Log[1 + E^((2*I)*ArcTan[c*x])]) + (3*I)*PolyLog[2, -E^((2*I)*ArcTan[c*x])]) + b^3*e^3*(-(c*x) - (4*I - 3*c*
x + c^3*x^3)*ArcTan[c*x]^2 + (-1 + c^4*x^4)*ArcTan[c*x]^3 + ArcTan[c*x]*(1 + c^2*x^2 + 8*Log[1 + E^((2*I)*ArcT
an[c*x])]) - (4*I)*PolyLog[2, -E^((2*I)*ArcTan[c*x])]) + 2*b^3*c*d*e^2*(6*c*x*ArcTan[c*x] - 3*ArcTan[c*x]^2 -
3*c^2*x^2*ArcTan[c*x]^2 + (2*I)*ArcTan[c*x]^3 + 2*c^3*x^3*ArcTan[c*x]^3 - 6*ArcTan[c*x]^2*Log[1 + E^((2*I)*Arc
Tan[c*x])] - 3*Log[1 + c^2*x^2] + (6*I)*ArcTan[c*x]*PolyLog[2, -E^((2*I)*ArcTan[c*x])] - 3*PolyLog[3, -E^((2*I
)*ArcTan[c*x])]) + 2*b^3*c^3*d^3*(2*ArcTan[c*x]^2*((-I + c*x)*ArcTan[c*x] + 3*Log[1 + E^((2*I)*ArcTan[c*x])])
- (6*I)*ArcTan[c*x]*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + 3*PolyLog[3, -E^((2*I)*ArcTan[c*x])]))/(4*c^4)

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fricas [F]  time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (a^{3} e^{3} x^{3} + 3 \, a^{3} d e^{2} x^{2} + 3 \, a^{3} d^{2} e x + a^{3} d^{3} + {\left (b^{3} e^{3} x^{3} + 3 \, b^{3} d e^{2} x^{2} + 3 \, b^{3} d^{2} e x + b^{3} d^{3}\right )} \arctan \left (c x\right )^{3} + 3 \, {\left (a b^{2} e^{3} x^{3} + 3 \, a b^{2} d e^{2} x^{2} + 3 \, a b^{2} d^{2} e x + a b^{2} d^{3}\right )} \arctan \left (c x\right )^{2} + 3 \, {\left (a^{2} b e^{3} x^{3} + 3 \, a^{2} b d e^{2} x^{2} + 3 \, a^{2} b d^{2} e x + a^{2} b d^{3}\right )} \arctan \left (c x\right ), x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

sage0*x

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maple [C]  time = 18.82, size = 3577, normalized size = 5.49 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*b^3*e^3*x/c^3+1/4*b^3*e^3*arctan(c*x)/c^4+3*a*b^2*d*e^2*x/c^2+3*b^3*d*e^2*x*arctan(c*x)/c^2+2/c^4*b^3*e^3
*arctan(c*x)*ln(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+2/c^4*b^3*e^3*arctan(c*x)*ln(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))
+3*a*b^2*arctan(c*x)^2*x*d^3-I/c*b^3*d^3*arctan(c*x)^3-2*I/c^4*b^3*e^3*dilog(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+
3/4*a^2*b/c^3*e^3*x-1/4/c*a^2*b*x^3*e^3+1/4/c^2*a*b^2*x^2*e^3+a^3*x*d^3+1/4*a^3*e^3*x^4-3/4*I/c*b^3*Pi*d^3*csg
n(I/((1+I*c*x)^2/(c^2*x^2+1)+1)^2)*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))*csgn(I*(1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2
/(c^2*x^2+1)+1)^2)*arctan(c*x)^2-3/4*I/c^3*b^3*e^2*Pi*d*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1)^2)*csgn(I*(1+I*c*x)
^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1)^2)^2*arctan(c*x)^2+3/2*I/c^3*b^3*e^2*Pi*d*csgn(I*((1+I*c*x)^2/(c^2*
x^2+1)+1))*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)+1)^2)^2*arctan(c*x)^2-3/4*I/c^3*b^3*e^2*Pi*d*csgn(I*((1+I*c*x)^2/(c
^2*x^2+1)+1))^2*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)+1)^2)*arctan(c*x)^2-3/4*I/c^3*b^3*e^2*Pi*d*csgn(I*(1+I*c*x)^2/
(c^2*x^2+1))*csgn(I*(1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1)^2)^2*arctan(c*x)^2+3/4*I/c^3*b^3*e^2*P
i*d*csgn(I*(1+I*c*x)/(c^2*x^2+1)^(1/2))^2*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))*arctan(c*x)^2-3/2*I/c^3*b^3*e^2*Pi*d
*csgn(I*(1+I*c*x)/(c^2*x^2+1)^(1/2))*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))^2*arctan(c*x)^2-1/4*I/c^4*b^3*e^3+1/4*a^3
/e*d^4+a^3*e^2*x^3*d+3/2*a^3*e*x^2*d^2+b^3*arctan(c*x)^3*x*d^3-1/4/c^4*b^3*e^3*arctan(c*x)^3+3/2/c*b^3*d^3*pol
ylog(3,-(1+I*c*x)^2/(c^2*x^2+1))+1/4*b^3*e^3*arctan(c*x)^3*x^4+3/4*a*b^2*e^3*arctan(c*x)^2*x^4+3/4*a^2*b*e^3*a
rctan(c*x)*x^4+b^3*e^2*arctan(c*x)^3*x^3*d+3/2*b^3*e*arctan(c*x)^3*x^2*d^2+3*a^2*b*arctan(c*x)*x*d^3-3/4/c^4*a
^2*b*e^3*arctan(c*x)+3/4/c^3*b^3*e^3*arctan(c*x)^2*x+1/4/c^2*b^3*e^3*arctan(c*x)*x^2+3/c*b^3*d^3*ln((1+I*c*x)/
(c^2*x^2+1)^(1/2))*arctan(c*x)^2-3/2/c*b^3*arctan(c*x)^2*ln(c^2*x^2+1)*d^3-1/4/c*b^3*e^3*arctan(c*x)^2*x^3-3/2
/c*a^2*b*ln(c^2*x^2+1)*d^3+3/c*b^3*ln(2)*d^3*arctan(c*x)^2-1/c^4*a*b^2*e^3*ln(c^2*x^2+1)-3/4/c^4*a*b^2*e^3*arc
tan(c*x)^2-3/2/c^3*b^3*e^2*d*polylog(3,-(1+I*c*x)^2/(c^2*x^2+1))-3/2/c^3*b^3*e^2*d*arctan(c*x)^2+3/c^3*b^3*e^2
*d*ln((1+I*c*x)^2/(c^2*x^2+1)+1)+3/2/c^2*b^3*e*arctan(c*x)^3*d^2-I/c^4*b^3*e^3*arctan(c*x)^2-2*I/c^4*b^3*e^3*d
ilog(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-9/2*a^2*b/c*x*d^2*e-3/2/c*a^2*b*x^2*d*e^2+9/2*a*b^2*e*arctan(c*x)^2*d^2*
x^2+3*a^2*b*e^2*arctan(c*x)*d*x^3+9/2*a^2*b*e*arctan(c*x)*d^2*x^2+3*a*b^2*e^2*arctan(c*x)^2*d*x^3-9/c^2*b^3*e*
d^2*arctan(c*x)*ln(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+3/2/c^3*b^3*e^2*arctan(c*x)^2*ln(c^2*x^2+1)*d-3/c^3*b^3*e^
2*d*ln((1+I*c*x)/(c^2*x^2+1)^(1/2))*arctan(c*x)^2-9/c^2*b^3*e*d^2*arctan(c*x)*ln(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/
2))+3/2/c^3*a*b^2*e^3*arctan(c*x)*x-9/2/c*b^3*e*arctan(c*x)^2*d^2*x-1/2/c*a*b^2*e^3*arctan(c*x)*x^3-3/c*a*b^2*
arctan(c*x)*ln(c^2*x^2+1)*d^3+I/c^3*b^3*e^2*d*arctan(c*x)^3-3/c^3*b^3*e^2*ln(2)*d*arctan(c*x)^2+3/2/c^3*a^2*b*
e^2*ln(c^2*x^2+1)*d+9/2/c^2*a^2*b*e*arctan(c*x)*d^2+9/2/c^2*a*b^2*e*ln(c^2*x^2+1)*d^2-3/c^3*a*b^2*e^2*d*arctan
(c*x)+9/2/c^2*a*b^2*e*arctan(c*x)^2*d^2-3/2/c*b^3*e^2*arctan(c*x)^2*d*x^2+9/2*I/c^2*b^3*e*d^2*arctan(c*x)^2+9*
I/c^2*b^3*e*d^2*dilog(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+9*I/c^2*b^3*e*d^2*dilog(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2)
)+3/4*I/c*a*b^2*d^3*ln(c*x-I)^2-9/c*a*b^2*e*arctan(c*x)*d^2*x-3/c*a*b^2*e^2*arctan(c*x)*d*x^2+3/c^3*a*b^2*e^2*
arctan(c*x)*ln(c^2*x^2+1)*d+3/4*I/c^3*a*b^2*e^2*d*ln(I+c*x)^2+3/2*I/c^3*a*b^2*e^2*d*dilog(1/2*I*(c*x-I))-3/2*I
/c*a*b^2*d^3*ln(c^2*x^2+1)*ln(c*x-I)+3/2*I/c*a*b^2*d^3*ln(c*x-I)*ln(-1/2*I*(I+c*x))+3/2*I/c*a*b^2*d^3*ln(I+c*x
)*ln(c^2*x^2+1)-3/2*I/c*a*b^2*d^3*ln(I+c*x)*ln(1/2*I*(c*x-I))-3/4*I/c*b^3*Pi*d^3*csgn(I*(1+I*c*x)^2/(c^2*x^2+1
))^3*arctan(c*x)^2-3/4*I/c*b^3*Pi*d^3*csgn(I*(1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1)^2)^3*arctan(c
*x)^2+3/4*I/c*b^3*Pi*d^3*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)+1)^2)^3*arctan(c*x)^2+3*I/c^3*b^3*e^2*d*arctan(c*x)*p
olylog(2,-(1+I*c*x)^2/(c^2*x^2+1))-3/4*I/c^3*a*b^2*e^2*d*ln(c*x-I)^2-3/2*I/c^3*a*b^2*e^2*d*dilog(-1/2*I*(I+c*x
))+3/2*I/c*a*b^2*d^3*dilog(-1/2*I*(I+c*x))-3/4*I/c*a*b^2*d^3*ln(I+c*x)^2-3/2*I/c*a*b^2*d^3*dilog(1/2*I*(c*x-I)
)-3*I/c^3*b^3*e^2*arctan(c*x)*d-3*I/c*b^3*d^3*arctan(c*x)*polylog(2,-(1+I*c*x)^2/(c^2*x^2+1))+3/4*I/c^3*b^3*e^
2*Pi*d*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1)^2)*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))*csgn(I*(1+I*c*x)^2/(c^2*x^2+1)/((
1+I*c*x)^2/(c^2*x^2+1)+1)^2)*arctan(c*x)^2+3/4*I/c*b^3*Pi*d^3*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))*csgn(I*(1+I*c*x)
^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1)^2)^2*arctan(c*x)^2+3/4*I/c*b^3*Pi*d^3*csgn(I/((1+I*c*x)^2/(c^2*x^2+
1)+1)^2)*csgn(I*(1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1)^2)^2*arctan(c*x)^2+3/4*I/c*b^3*Pi*d^3*csgn
(I*((1+I*c*x)^2/(c^2*x^2+1)+1))^2*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)+1)^2)*arctan(c*x)^2-3/2*I/c*b^3*Pi*d^3*csgn(
I*((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)+1)^2)^2*arctan(c*x)^2+3/4*I/c^3*b^3*e^2*Pi*d*cs
gn(I*(1+I*c*x)^2/(c^2*x^2+1))^3*arctan(c*x)^2-3/4*I/c^3*b^3*e^2*Pi*d*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)+1)^2)^3*a
rctan(c*x)^2+3/4*I/c^3*b^3*e^2*Pi*d*csgn(I*(1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1)^2)^3*arctan(c*x
)^2+3/2*I/c^3*a*b^2*e^2*d*ln(c^2*x^2+1)*ln(c*x-I)-3/2*I/c^3*a*b^2*e^2*d*ln(c*x-I)*ln(-1/2*I*(I+c*x))-3/2*I/c^3
*a*b^2*e^2*d*ln(I+c*x)*ln(c^2*x^2+1)+3/2*I/c^3*a*b^2*e^2*d*ln(I+c*x)*ln(1/2*I*(c*x-I))-3/4*I/c*b^3*Pi*d^3*csgn
(I*(1+I*c*x)/(c^2*x^2+1)^(1/2))^2*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))*arctan(c*x)^2+3/2*I/c*b^3*Pi*d^3*csgn(I*(1+I
*c*x)/(c^2*x^2+1)^(1/2))*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))^2*arctan(c*x)^2

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*a^3*e^3*x^4 + a^3*d*e^2*x^3 + 7/32*b^3*d^3*arctan(c*x)^4/c + 112*b^3*c^2*e^3*integrate(1/128*x^5*arctan(c*
x)^3/(c^2*x^2 + 1), x) + 12*b^3*c^2*e^3*integrate(1/128*x^5*arctan(c*x)*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) +
384*a*b^2*c^2*e^3*integrate(1/128*x^5*arctan(c*x)^2/(c^2*x^2 + 1), x) + 336*b^3*c^2*d*e^2*integrate(1/128*x^4
*arctan(c*x)^3/(c^2*x^2 + 1), x) + 12*b^3*c^2*e^3*integrate(1/128*x^5*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^2 +
1), x) + 36*b^3*c^2*d*e^2*integrate(1/128*x^4*arctan(c*x)*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + 1152*a*b^2*c^
2*d*e^2*integrate(1/128*x^4*arctan(c*x)^2/(c^2*x^2 + 1), x) + 336*b^3*c^2*d^2*e*integrate(1/128*x^3*arctan(c*x
)^3/(c^2*x^2 + 1), x) + 48*b^3*c^2*d*e^2*integrate(1/128*x^4*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^2 + 1), x) +
36*b^3*c^2*d^2*e*integrate(1/128*x^3*arctan(c*x)*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + 1152*a*b^2*c^2*d^2*e*i
ntegrate(1/128*x^3*arctan(c*x)^2/(c^2*x^2 + 1), x) + 112*b^3*c^2*d^3*integrate(1/128*x^2*arctan(c*x)^3/(c^2*x^
2 + 1), x) + 72*b^3*c^2*d^2*e*integrate(1/128*x^3*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^2 + 1), x) + 12*b^3*c^2*
d^3*integrate(1/128*x^2*arctan(c*x)*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + 384*a*b^2*c^2*d^3*integrate(1/128*x
^2*arctan(c*x)^2/(c^2*x^2 + 1), x) + 48*b^3*c^2*d^3*integrate(1/128*x^2*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^2
+ 1), x) + 3/2*a^3*d^2*e*x^2 + a*b^2*d^3*arctan(c*x)^3/c - 12*b^3*c*e^3*integrate(1/128*x^4*arctan(c*x)^2/(c^2
*x^2 + 1), x) + 3*b^3*c*e^3*integrate(1/128*x^4*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) - 48*b^3*c*d*e^2*integrat
e(1/128*x^3*arctan(c*x)^2/(c^2*x^2 + 1), x) + 12*b^3*c*d*e^2*integrate(1/128*x^3*log(c^2*x^2 + 1)^2/(c^2*x^2 +
1), x) - 72*b^3*c*d^2*e*integrate(1/128*x^2*arctan(c*x)^2/(c^2*x^2 + 1), x) + 18*b^3*c*d^2*e*integrate(1/128*
x^2*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) - 48*b^3*c*d^3*integrate(1/128*x*arctan(c*x)^2/(c^2*x^2 + 1), x) + 12
*b^3*c*d^3*integrate(1/128*x*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + 9/2*(x^2*arctan(c*x) - c*(x/c^2 - arctan(c
*x)/c^3))*a^2*b*d^2*e + 3/2*(2*x^3*arctan(c*x) - c*(x^2/c^2 - log(c^2*x^2 + 1)/c^4))*a^2*b*d*e^2 + 1/4*(3*x^4*
arctan(c*x) - c*((c^2*x^3 - 3*x)/c^4 + 3*arctan(c*x)/c^5))*a^2*b*e^3 + a^3*d^3*x + 112*b^3*e^3*integrate(1/128
*x^3*arctan(c*x)^3/(c^2*x^2 + 1), x) + 12*b^3*e^3*integrate(1/128*x^3*arctan(c*x)*log(c^2*x^2 + 1)^2/(c^2*x^2
+ 1), x) + 384*a*b^2*e^3*integrate(1/128*x^3*arctan(c*x)^2/(c^2*x^2 + 1), x) + 336*b^3*d*e^2*integrate(1/128*x
^2*arctan(c*x)^3/(c^2*x^2 + 1), x) + 36*b^3*d*e^2*integrate(1/128*x^2*arctan(c*x)*log(c^2*x^2 + 1)^2/(c^2*x^2
+ 1), x) + 1152*a*b^2*d*e^2*integrate(1/128*x^2*arctan(c*x)^2/(c^2*x^2 + 1), x) + 336*b^3*d^2*e*integrate(1/12
8*x*arctan(c*x)^3/(c^2*x^2 + 1), x) + 36*b^3*d^2*e*integrate(1/128*x*arctan(c*x)*log(c^2*x^2 + 1)^2/(c^2*x^2 +
1), x) + 1152*a*b^2*d^2*e*integrate(1/128*x*arctan(c*x)^2/(c^2*x^2 + 1), x) + 12*b^3*d^3*integrate(1/128*arct
an(c*x)*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + 3/2*(2*c*x*arctan(c*x) - log(c^2*x^2 + 1))*a^2*b*d^3/c + 1/32*(
b^3*e^3*x^4 + 4*b^3*d*e^2*x^3 + 6*b^3*d^2*e*x^2 + 4*b^3*d^3*x)*arctan(c*x)^3 - 3/128*(b^3*e^3*x^4 + 4*b^3*d*e^
2*x^3 + 6*b^3*d^2*e*x^2 + 4*b^3*d^3*x)*arctan(c*x)*log(c^2*x^2 + 1)^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atan(c*x))^3*(d + e*x)^3,x)

[Out]

int((a + b*atan(c*x))^3*(d + e*x)^3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**3*(a+b*atan(c*x))**3,x)

[Out]

Integral((a + b*atan(c*x))**3*(d + e*x)**3, x)

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