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

Optimal. Leaf size=382 \[ -\frac {6 i b^2 d^3 \text {Li}_2\left (1-\frac {2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c}-\frac {1}{4} i b^2 c d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )-\frac {11 i b^2 d^3 \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c}-3 a b^2 d^3 x+\frac {1}{4} i b c^2 d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {3}{2} b c d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {i d^3 (1+i c x)^4 \left (a+b \tan ^{-1}(c x)\right )^3}{4 c}+\frac {7 b d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{c}-\frac {21}{4} i b d^3 x \left (a+b \tan ^{-1}(c x)\right )^2+\frac {6 b d^3 \log \left (\frac {2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{c}+\frac {3 b^3 d^3 \log \left (c^2 x^2+1\right )}{2 c}+\frac {11 b^3 d^3 \text {Li}_2\left (1-\frac {2}{i c x+1}\right )}{2 c}+\frac {3 b^3 d^3 \text {Li}_3\left (1-\frac {2}{1-i c x}\right )}{c}-\frac {i b^3 d^3 \tan ^{-1}(c x)}{4 c}-3 b^3 d^3 x \tan ^{-1}(c x)+\frac {1}{4} i b^3 d^3 x \]

[Out]

-3*a*b^2*d^3*x+1/4*I*b^3*d^3*x-1/4*I*b^3*d^3*arctan(c*x)/c-3*b^3*d^3*x*arctan(c*x)-1/4*I*b^2*c*d^3*x^2*(a+b*ar
ctan(c*x))+7*b*d^3*(a+b*arctan(c*x))^2/c-21/4*I*b*d^3*x*(a+b*arctan(c*x))^2+3/2*b*c*d^3*x^2*(a+b*arctan(c*x))^
2-6*I*b^2*d^3*(a+b*arctan(c*x))*polylog(2,1-2/(1-I*c*x))/c+1/4*I*b*c^2*d^3*x^3*(a+b*arctan(c*x))^2+6*b*d^3*(a+
b*arctan(c*x))^2*ln(2/(1-I*c*x))/c-11*I*b^2*d^3*(a+b*arctan(c*x))*ln(2/(1+I*c*x))/c+3/2*b^3*d^3*ln(c^2*x^2+1)/
c-1/4*I*d^3*(1+I*c*x)^4*(a+b*arctan(c*x))^3/c+11/2*b^3*d^3*polylog(2,1-2/(1+I*c*x))/c+3*b^3*d^3*polylog(3,1-2/
(1-I*c*x))/c

________________________________________________________________________________________

Rubi [A]  time = 0.71, antiderivative size = 382, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 15, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.682, Rules used = {4864, 4846, 4920, 4854, 2402, 2315, 4852, 4916, 260, 4884, 321, 203, 1586, 4992, 6610} \[ -\frac {6 i b^2 d^3 \text {PolyLog}\left (2,1-\frac {2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c}+\frac {11 b^3 d^3 \text {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 c}+\frac {3 b^3 d^3 \text {PolyLog}\left (3,1-\frac {2}{1-i c x}\right )}{c}-\frac {1}{4} i b^2 c d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )-\frac {11 i b^2 d^3 \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c}-3 a b^2 d^3 x+\frac {1}{4} i b c^2 d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {3}{2} b c d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {i d^3 (1+i c x)^4 \left (a+b \tan ^{-1}(c x)\right )^3}{4 c}+\frac {7 b d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{c}-\frac {21}{4} i b d^3 x \left (a+b \tan ^{-1}(c x)\right )^2+\frac {6 b d^3 \log \left (\frac {2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{c}+\frac {3 b^3 d^3 \log \left (c^2 x^2+1\right )}{2 c}-\frac {i b^3 d^3 \tan ^{-1}(c x)}{4 c}-3 b^3 d^3 x \tan ^{-1}(c x)+\frac {1}{4} i b^3 d^3 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

-3*a*b^2*d^3*x + (I/4)*b^3*d^3*x - ((I/4)*b^3*d^3*ArcTan[c*x])/c - 3*b^3*d^3*x*ArcTan[c*x] - (I/4)*b^2*c*d^3*x
^2*(a + b*ArcTan[c*x]) + (7*b*d^3*(a + b*ArcTan[c*x])^2)/c - ((21*I)/4)*b*d^3*x*(a + b*ArcTan[c*x])^2 + (3*b*c
*d^3*x^2*(a + b*ArcTan[c*x])^2)/2 + (I/4)*b*c^2*d^3*x^3*(a + b*ArcTan[c*x])^2 - ((I/4)*d^3*(1 + I*c*x)^4*(a +
b*ArcTan[c*x])^3)/c + (6*b*d^3*(a + b*ArcTan[c*x])^2*Log[2/(1 - I*c*x)])/c - ((11*I)*b^2*d^3*(a + b*ArcTan[c*x
])*Log[2/(1 + I*c*x)])/c + (3*b^3*d^3*Log[1 + c^2*x^2])/(2*c) - ((6*I)*b^2*d^3*(a + b*ArcTan[c*x])*PolyLog[2,
1 - 2/(1 - I*c*x)])/c + (11*b^3*d^3*PolyLog[2, 1 - 2/(1 + I*c*x)])/(2*c) + (3*b^3*d^3*PolyLog[3, 1 - 2/(1 - I*
c*x)])/c

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 1586

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 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 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 4992

Int[(Log[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(I*(a + b*ArcT
an[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+i c d x)^3 \left (a+b \tan ^{-1}(c x)\right )^3 \, dx &=-\frac {i d^3 (1+i c x)^4 \left (a+b \tan ^{-1}(c x)\right )^3}{4 c}+\frac {(3 i b) \int \left (-7 d^4 \left (a+b \tan ^{-1}(c x)\right )^2-4 i c d^4 x \left (a+b \tan ^{-1}(c x)\right )^2+c^2 d^4 x^2 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {8 i \left (i d^4-c d^4 x\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2}\right ) \, dx}{4 d}\\ &=-\frac {i d^3 (1+i c x)^4 \left (a+b \tan ^{-1}(c x)\right )^3}{4 c}+\frac {(6 b) \int \frac {\left (i d^4-c d^4 x\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2} \, dx}{d}-\frac {1}{4} \left (21 i b d^3\right ) \int \left (a+b \tan ^{-1}(c x)\right )^2 \, dx+\left (3 b c d^3\right ) \int x \left (a+b \tan ^{-1}(c x)\right )^2 \, dx+\frac {1}{4} \left (3 i b c^2 d^3\right ) \int x^2 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx\\ &=-\frac {21}{4} i b d^3 x \left (a+b \tan ^{-1}(c x)\right )^2+\frac {3}{2} b c d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{4} i b c^2 d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {i d^3 (1+i c x)^4 \left (a+b \tan ^{-1}(c x)\right )^3}{4 c}+\frac {(6 b) \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{-\frac {i}{d^4}-\frac {c x}{d^4}} \, dx}{d}+\frac {1}{2} \left (21 i b^2 c d^3\right ) \int \frac {x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx-\left (3 b^2 c^2 d^3\right ) \int \frac {x^2 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx-\frac {1}{2} \left (i b^2 c^3 d^3\right ) \int \frac {x^3 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx\\ &=\frac {21 b d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{4 c}-\frac {21}{4} i b d^3 x \left (a+b \tan ^{-1}(c x)\right )^2+\frac {3}{2} b c d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{4} i b c^2 d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {i d^3 (1+i c x)^4 \left (a+b \tan ^{-1}(c x)\right )^3}{4 c}+\frac {6 b d^3 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-i c x}\right )}{c}-\frac {1}{2} \left (21 i b^2 d^3\right ) \int \frac {a+b \tan ^{-1}(c x)}{i-c x} \, dx-\left (3 b^2 d^3\right ) \int \left (a+b \tan ^{-1}(c x)\right ) \, dx+\left (3 b^2 d^3\right ) \int \frac {a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx-\left (12 b^2 d^3\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-\frac {1}{2} \left (i b^2 c d^3\right ) \int x \left (a+b \tan ^{-1}(c x)\right ) \, dx+\frac {1}{2} \left (i b^2 c d^3\right ) \int \frac {x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx\\ &=-3 a b^2 d^3 x-\frac {1}{4} i b^2 c d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac {7 b d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{c}-\frac {21}{4} i b d^3 x \left (a+b \tan ^{-1}(c x)\right )^2+\frac {3}{2} b c d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{4} i b c^2 d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {i d^3 (1+i c x)^4 \left (a+b \tan ^{-1}(c x)\right )^3}{4 c}+\frac {6 b d^3 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-i c x}\right )}{c}-\frac {21 i b^2 d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{2 c}-\frac {6 i b^2 d^3 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{c}-\frac {1}{2} \left (i b^2 d^3\right ) \int \frac {a+b \tan ^{-1}(c x)}{i-c x} \, dx+\left (6 i b^3 d^3\right ) \int \frac {\text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{1+c^2 x^2} \, dx+\frac {1}{2} \left (21 i b^3 d^3\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx-\left (3 b^3 d^3\right ) \int \tan ^{-1}(c x) \, dx+\frac {1}{4} \left (i b^3 c^2 d^3\right ) \int \frac {x^2}{1+c^2 x^2} \, dx\\ &=-3 a b^2 d^3 x+\frac {1}{4} i b^3 d^3 x-3 b^3 d^3 x \tan ^{-1}(c x)-\frac {1}{4} i b^2 c d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac {7 b d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{c}-\frac {21}{4} i b d^3 x \left (a+b \tan ^{-1}(c x)\right )^2+\frac {3}{2} b c d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{4} i b c^2 d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {i d^3 (1+i c x)^4 \left (a+b \tan ^{-1}(c x)\right )^3}{4 c}+\frac {6 b d^3 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-i c x}\right )}{c}-\frac {11 i b^2 d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c}-\frac {6 i b^2 d^3 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{c}+\frac {3 b^3 d^3 \text {Li}_3\left (1-\frac {2}{1-i c x}\right )}{c}-\frac {1}{4} \left (i b^3 d^3\right ) \int \frac {1}{1+c^2 x^2} \, dx+\frac {1}{2} \left (i b^3 d^3\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx+\frac {\left (21 b^3 d^3\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{2 c}+\left (3 b^3 c d^3\right ) \int \frac {x}{1+c^2 x^2} \, dx\\ &=-3 a b^2 d^3 x+\frac {1}{4} i b^3 d^3 x-\frac {i b^3 d^3 \tan ^{-1}(c x)}{4 c}-3 b^3 d^3 x \tan ^{-1}(c x)-\frac {1}{4} i b^2 c d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac {7 b d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{c}-\frac {21}{4} i b d^3 x \left (a+b \tan ^{-1}(c x)\right )^2+\frac {3}{2} b c d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{4} i b c^2 d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {i d^3 (1+i c x)^4 \left (a+b \tan ^{-1}(c x)\right )^3}{4 c}+\frac {6 b d^3 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-i c x}\right )}{c}-\frac {11 i b^2 d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c}+\frac {3 b^3 d^3 \log \left (1+c^2 x^2\right )}{2 c}-\frac {6 i b^2 d^3 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{c}+\frac {21 b^3 d^3 \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{4 c}+\frac {3 b^3 d^3 \text {Li}_3\left (1-\frac {2}{1-i c x}\right )}{c}+\frac {\left (b^3 d^3\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{2 c}\\ &=-3 a b^2 d^3 x+\frac {1}{4} i b^3 d^3 x-\frac {i b^3 d^3 \tan ^{-1}(c x)}{4 c}-3 b^3 d^3 x \tan ^{-1}(c x)-\frac {1}{4} i b^2 c d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac {7 b d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{c}-\frac {21}{4} i b d^3 x \left (a+b \tan ^{-1}(c x)\right )^2+\frac {3}{2} b c d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{4} i b c^2 d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {i d^3 (1+i c x)^4 \left (a+b \tan ^{-1}(c x)\right )^3}{4 c}+\frac {6 b d^3 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-i c x}\right )}{c}-\frac {11 i b^2 d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c}+\frac {3 b^3 d^3 \log \left (1+c^2 x^2\right )}{2 c}-\frac {6 i b^2 d^3 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{c}+\frac {11 b^3 d^3 \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{2 c}+\frac {3 b^3 d^3 \text {Li}_3\left (1-\frac {2}{1-i c x}\right )}{c}\\ \end {align*}

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Mathematica [A]  time = 1.80, size = 693, normalized size = 1.81 \[ -\frac {i d^3 \left (a^3 c^4 x^4-4 i a^3 c^3 x^3-6 a^3 c^2 x^2+4 i a^3 c x+3 a^2 b c^4 x^4 \tan ^{-1}(c x)-a^2 b c^3 x^3-12 i a^2 b c^3 x^3 \tan ^{-1}(c x)+6 i a^2 b c^2 x^2-12 i a^2 b \log \left (c^2 x^2+1\right )-18 a^2 b c^2 x^2 \tan ^{-1}(c x)+21 a^2 b c x+12 i a^2 b c x \tan ^{-1}(c x)-21 a^2 b \tan ^{-1}(c x)+3 a b^2 c^4 x^4 \tan ^{-1}(c x)^2-12 i a b^2 c^3 x^3 \tan ^{-1}(c x)^2-2 a b^2 c^3 x^3 \tan ^{-1}(c x)+a b^2 c^2 x^2-22 a b^2 \log \left (c^2 x^2+1\right )-18 a b^2 c^2 x^2 \tan ^{-1}(c x)^2+12 i a b^2 c^2 x^2 \tan ^{-1}(c x)+2 b^2 \text {Li}_2\left (-e^{2 i \tan ^{-1}(c x)}\right ) \left (12 a+12 b \tan ^{-1}(c x)-11 i b\right )-12 i a b^2 c x+12 i a b^2 c x \tan ^{-1}(c x)^2+42 a b^2 c x \tan ^{-1}(c x)+3 a b^2 \tan ^{-1}(c x)^2+12 i a b^2 \tan ^{-1}(c x)+48 i a b^2 \tan ^{-1}(c x) \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )+a b^2+b^3 c^4 x^4 \tan ^{-1}(c x)^3-4 i b^3 c^3 x^3 \tan ^{-1}(c x)^3-b^3 c^3 x^3 \tan ^{-1}(c x)^2+6 i b^3 \log \left (c^2 x^2+1\right )-6 b^3 c^2 x^2 \tan ^{-1}(c x)^3+6 i b^3 c^2 x^2 \tan ^{-1}(c x)^2+b^3 c^2 x^2 \tan ^{-1}(c x)+12 i b^3 \text {Li}_3\left (-e^{2 i \tan ^{-1}(c x)}\right )-b^3 c x+4 i b^3 c x \tan ^{-1}(c x)^3+21 b^3 c x \tan ^{-1}(c x)^2-12 i b^3 c x \tan ^{-1}(c x)+b^3 \tan ^{-1}(c x)^3-16 i b^3 \tan ^{-1}(c x)^2+b^3 \tan ^{-1}(c x)+24 i b^3 \tan ^{-1}(c x)^2 \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )+44 b^3 \tan ^{-1}(c x) \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )\right )}{4 c} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-1/4*I)*d^3*(a*b^2 + (4*I)*a^3*c*x + 21*a^2*b*c*x - (12*I)*a*b^2*c*x - b^3*c*x - 6*a^3*c^2*x^2 + (6*I)*a^2*b
*c^2*x^2 + a*b^2*c^2*x^2 - (4*I)*a^3*c^3*x^3 - a^2*b*c^3*x^3 + a^3*c^4*x^4 - 21*a^2*b*ArcTan[c*x] + (12*I)*a*b
^2*ArcTan[c*x] + b^3*ArcTan[c*x] + (12*I)*a^2*b*c*x*ArcTan[c*x] + 42*a*b^2*c*x*ArcTan[c*x] - (12*I)*b^3*c*x*Ar
cTan[c*x] - 18*a^2*b*c^2*x^2*ArcTan[c*x] + (12*I)*a*b^2*c^2*x^2*ArcTan[c*x] + b^3*c^2*x^2*ArcTan[c*x] - (12*I)
*a^2*b*c^3*x^3*ArcTan[c*x] - 2*a*b^2*c^3*x^3*ArcTan[c*x] + 3*a^2*b*c^4*x^4*ArcTan[c*x] + 3*a*b^2*ArcTan[c*x]^2
- (16*I)*b^3*ArcTan[c*x]^2 + (12*I)*a*b^2*c*x*ArcTan[c*x]^2 + 21*b^3*c*x*ArcTan[c*x]^2 - 18*a*b^2*c^2*x^2*Arc
Tan[c*x]^2 + (6*I)*b^3*c^2*x^2*ArcTan[c*x]^2 - (12*I)*a*b^2*c^3*x^3*ArcTan[c*x]^2 - b^3*c^3*x^3*ArcTan[c*x]^2
+ 3*a*b^2*c^4*x^4*ArcTan[c*x]^2 + b^3*ArcTan[c*x]^3 + (4*I)*b^3*c*x*ArcTan[c*x]^3 - 6*b^3*c^2*x^2*ArcTan[c*x]^
3 - (4*I)*b^3*c^3*x^3*ArcTan[c*x]^3 + b^3*c^4*x^4*ArcTan[c*x]^3 + (48*I)*a*b^2*ArcTan[c*x]*Log[1 + E^((2*I)*Ar
cTan[c*x])] + 44*b^3*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] + (24*I)*b^3*ArcTan[c*x]^2*Log[1 + E^((2*I)*Ar
cTan[c*x])] - (12*I)*a^2*b*Log[1 + c^2*x^2] - 22*a*b^2*Log[1 + c^2*x^2] + (6*I)*b^3*Log[1 + c^2*x^2] + 2*b^2*(
12*a - (11*I)*b + 12*b*ArcTan[c*x])*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + (12*I)*b^3*PolyLog[3, -E^((2*I)*ArcTa
n[c*x])]))/c

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fricas [F]  time = 0.68, size = 0, normalized size = 0.00 \[ -\frac {1}{32} \, {\left (b^{3} c^{3} d^{3} x^{4} - 4 i \, b^{3} c^{2} d^{3} x^{3} - 6 \, b^{3} c d^{3} x^{2} + 4 i \, b^{3} d^{3} x\right )} \log \left (-\frac {c x + i}{c x - i}\right )^{3} + {\rm integral}\left (\frac {-16 i \, a^{3} c^{5} d^{3} x^{5} - 48 \, a^{3} c^{4} d^{3} x^{4} + 32 i \, a^{3} c^{3} d^{3} x^{3} - 32 \, a^{3} c^{2} d^{3} x^{2} + 48 i \, a^{3} c d^{3} x + 16 \, a^{3} d^{3} + {\left (12 i \, a b^{2} c^{5} d^{3} x^{5} + {\left (36 \, a b^{2} - 3 i \, b^{3}\right )} c^{4} d^{3} x^{4} - 12 \, {\left (2 i \, a b^{2} + b^{3}\right )} c^{3} d^{3} x^{3} + {\left (24 \, a b^{2} + 18 i \, b^{3}\right )} c^{2} d^{3} x^{2} - 12 \, a b^{2} d^{3} - 12 \, {\left (3 i \, a b^{2} - b^{3}\right )} c d^{3} x\right )} \log \left (-\frac {c x + i}{c x - i}\right )^{2} + {\left (24 \, a^{2} b c^{5} d^{3} x^{5} - 72 i \, a^{2} b c^{4} d^{3} x^{4} - 48 \, a^{2} b c^{3} d^{3} x^{3} - 48 i \, a^{2} b c^{2} d^{3} x^{2} - 72 \, a^{2} b c d^{3} x + 24 i \, a^{2} b d^{3}\right )} \log \left (-\frac {c x + i}{c x - i}\right )}{16 \, {\left (c^{2} x^{2} + 1\right )}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/32*(b^3*c^3*d^3*x^4 - 4*I*b^3*c^2*d^3*x^3 - 6*b^3*c*d^3*x^2 + 4*I*b^3*d^3*x)*log(-(c*x + I)/(c*x - I))^3 +
integral(1/16*(-16*I*a^3*c^5*d^3*x^5 - 48*a^3*c^4*d^3*x^4 + 32*I*a^3*c^3*d^3*x^3 - 32*a^3*c^2*d^3*x^2 + 48*I*a
^3*c*d^3*x + 16*a^3*d^3 + (12*I*a*b^2*c^5*d^3*x^5 + (36*a*b^2 - 3*I*b^3)*c^4*d^3*x^4 - 12*(2*I*a*b^2 + b^3)*c^
3*d^3*x^3 + (24*a*b^2 + 18*I*b^3)*c^2*d^3*x^2 - 12*a*b^2*d^3 - 12*(3*I*a*b^2 - b^3)*c*d^3*x)*log(-(c*x + I)/(c
*x - I))^2 + (24*a^2*b*c^5*d^3*x^5 - 72*I*a^2*b*c^4*d^3*x^4 - 48*a^2*b*c^3*d^3*x^3 - 48*I*a^2*b*c^2*d^3*x^2 -
72*a^2*b*c*d^3*x + 24*I*a^2*b*d^3)*log(-(c*x + I)/(c*x - I)))/(c^2*x^2 + 1), 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((d+I*c*d*x)^3*(a+b*arctan(c*x))^3,x, algorithm="giac")

[Out]

sage0*x

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maple [C]  time = 12.78, size = 2004, normalized size = 5.25 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*a*b^2*d^3*x-3*b^3*d^3*x*arctan(c*x)+x*a^3*d^3-3/2*I/c*d^3*b^3*Pi*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+1/4*I*
b^3*d^3*x+3/2*c*d^3*a^2*b*x^2-3/c*d^3*a^2*b*ln(c^2*x^2+1)+3/c*d^3*a*b^2*arctan(c*x)-3/c*d^3*b^3*arctan(c*x)^2*
ln(c^2*x^2+1)+6/c*d^3*b^3*arctan(c*x)^2*ln((1+I*c*x)/(c^2*x^2+1)^(1/2))+3/2*c*d^3*b^3*arctan(c*x)^2*x^2-c^2*d^
3*b^3*arctan(c*x)^3*x^3+6/c*d^3*b^3*ln(2)*arctan(c*x)^2+3*d^3*a*b^2*arctan(c*x)^2*x+3*d^3*a^2*b*arctan(c*x)*x+
11/4*I/c*d^3*b^3*arctan(c*x)-1/4*I*c^3*x^4*a^3*d^3+3/2*I*c*x^2*a^3*d^3-1/4*I/c*d^3*b^3*arctan(c*x)^3-21/4*I*d^
3*b^3*arctan(c*x)^2*x-21/4*I*d^3*a^2*b*x-3*c^2*d^3*a*b^2*arctan(c*x)^2*x^3+3*c*d^3*a*b^2*arctan(c*x)*x^2-3*c^2
*d^3*a^2*b*arctan(c*x)*x^3-6/c*d^3*a*b^2*arctan(c*x)*ln(c^2*x^2+1)-6*I/c*d^3*b^3*arctan(c*x)*polylog(2,-(1+I*c
*x)^2/(c^2*x^2+1))-11*I/c*d^3*b^3*arctan(c*x)*ln(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-11*I/c*d^3*b^3*arctan(c*x)*l
n(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+3/2*I/c*d^3*a*b^2*ln(c*x-I)^2-3*I/c*d^3*a*b^2*dilog(1/2*I*(c*x-I))+21/4*I/c
*d^3*a^2*b*arctan(c*x)+1/4*I*c^2*d^3*a^2*b*x^3-1/4*I*c*d^3*b^3*arctan(c*x)*x^2-1/4*I*c^3*d^3*b^3*arctan(c*x)^3
*x^4+3/2*I*c*d^3*b^3*arctan(c*x)^3*x^2+1/4*I*c^2*d^3*b^3*arctan(c*x)^2*x^3-3/2*I/c*d^3*a*b^2*ln(I+c*x)^2+3*I/c
*d^3*a*b^2*dilog(-1/2*I*(I+c*x))+11/2*I/c*d^3*a*b^2*ln(c^2*x^2+1)+21/4*I/c*d^3*a*b^2*arctan(c*x)^2-1/4*I*c*d^3
*a*b^2*x^2-21/2*I*d^3*a*b^2*arctan(c*x)*x-1/4*I/c*d^3*a^3-c^2*x^3*a^3*d^3+d^3*b^3*arctan(c*x)^3*x-11/c*d^3*b^3
*dilog(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-3/c*d^3*b^3*ln((1+I*c*x)^2/(c^2*x^2+1)+1)-4/c*d^3*b^3*arctan(c*x)^2-11
/c*d^3*b^3*dilog(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+3/c*d^3*b^3*polylog(3,-(1+I*c*x)^2/(c^2*x^2+1))+9/2*I*c*d^3*
a*b^2*arctan(c*x)^2*x^2+1/2*I*c^2*d^3*a*b^2*arctan(c*x)*x^3-3/4*I*c^3*d^3*a*b^2*arctan(c*x)^2*x^4-3/2*I/c*d^3*
b^3*Pi*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))^3*arctan(c*x)^2-3/2*I/c*d^3*b^3*Pi*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*I/c*d^3*a*b^2*ln(I+c*x)*ln(c^2*x^2+1)+3*I/c*d^3*a*b^2*ln(c*x-I)*ln
(-1/2*I*(I+c*x))-3*I/c*d^3*a*b^2*ln(c*x-I)*ln(c^2*x^2+1)-3*I/c*d^3*a*b^2*ln(I+c*x)*ln(1/2*I*(c*x-I))+3/2*I/c*d
^3*b^3*Pi*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)+1)^2)^3*arctan(c*x)^2-3/4*I*c^3*d^3*a^2*b*arctan(c*x)*x^4+9/2*I*c*d^
3*a^2*b*arctan(c*x)*x^2-1/4/c*d^3*b^3+3*I/c*d^3*b^3*Pi*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+3/2*I/c*d^3*b^3*Pi*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*d^3*b^3*Pi*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*d^3*b^3*Pi*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*I/c*d^3*b^3*Pi*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+3/2*I/c*d^3*b^3*Pi*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

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

[Out]

-1/4*I*a^3*c^3*d^3*x^4 - 24*b^3*c^5*d^3*integrate(1/128*x^5*arctan(c*x)^2*log(c^2*x^2 + 1)/(c^2*x^2 + 1), x) +
2*b^3*c^5*d^3*integrate(1/128*x^5*log(c^2*x^2 + 1)^3/(c^2*x^2 + 1), x) - 12*b^3*c^5*d^3*integrate(1/128*x^5*a
rctan(c*x)^2/(c^2*x^2 + 1), x) + 3*b^3*c^5*d^3*integrate(1/128*x^5*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) - a^3*
c^2*d^3*x^3 - 336*b^3*c^4*d^3*integrate(1/128*x^4*arctan(c*x)^3/(c^2*x^2 + 1), x) - 36*b^3*c^4*d^3*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^4*d^3*integrate(1/128*x^4*arctan(c*x)
^2/(c^2*x^2 + 1), x) - 60*b^3*c^4*d^3*integrate(1/128*x^4*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^2 + 1), x) - 1/4
*I*(3*x^4*arctan(c*x) - c*((c^2*x^3 - 3*x)/c^4 + 3*arctan(c*x)/c^5))*a^2*b*c^3*d^3 + 48*b^3*c^3*d^3*integrate(
1/128*x^3*arctan(c*x)^2*log(c^2*x^2 + 1)/(c^2*x^2 + 1), x) - 4*b^3*c^3*d^3*integrate(1/128*x^3*log(c^2*x^2 + 1
)^3/(c^2*x^2 + 1), x) + 120*b^3*c^3*d^3*integrate(1/128*x^3*arctan(c*x)^2/(c^2*x^2 + 1), x) - 30*b^3*c^3*d^3*i
ntegrate(1/128*x^3*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) - 3/2*(2*x^3*arctan(c*x) - c*(x^2/c^2 - log(c^2*x^2 +
1)/c^4))*a^2*b*c^2*d^3 + 3/2*I*a^3*c*d^3*x^2 + 7/32*b^3*d^3*arctan(c*x)^4/c - 224*b^3*c^2*d^3*integrate(1/128*
x^2*arctan(c*x)^3/(c^2*x^2 + 1), x) - 24*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) - 768*a*b^2*c^2*d^3*integrate(1/128*x^2*arctan(c*x)^2/(c^2*x^2 + 1), x) + 120*b^3*c^2*d^3*integrat
e(1/128*x^2*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^2 + 1), x) + 9/2*I*(x^2*arctan(c*x) - c*(x/c^2 - arctan(c*x)/c
^3))*a^2*b*c*d^3 + a*b^2*d^3*arctan(c*x)^3/c + 72*b^3*c*d^3*integrate(1/128*x*arctan(c*x)^2*log(c^2*x^2 + 1)/(
c^2*x^2 + 1), x) - 6*b^3*c*d^3*integrate(1/128*x*log(c^2*x^2 + 1)^3/(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)
+ a^3*d^3*x + 12*b^3*d^3*integrate(1/128*arctan(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 + 1/256*(-8*I*b^3*c^3*d^3*x^4 - 32*b^3*c^2*d^3*x^3 + 48*I*b^3*c*d^3*x^2
+ 32*b^3*d^3*x)*arctan(c*x)^3 + 1/256*(12*b^3*c^3*d^3*x^4 - 48*I*b^3*c^2*d^3*x^3 - 72*b^3*c*d^3*x^2 + 48*I*b^3
*d^3*x)*arctan(c*x)^2*log(c^2*x^2 + 1) + 1/256*(6*I*b^3*c^3*d^3*x^4 + 24*b^3*c^2*d^3*x^3 - 36*I*b^3*c*d^3*x^2
- 24*b^3*d^3*x)*arctan(c*x)*log(c^2*x^2 + 1)^2 - 1/256*(b^3*c^3*d^3*x^4 - 4*I*b^3*c^2*d^3*x^3 - 6*b^3*c*d^3*x^
2 + 4*I*b^3*d^3*x)*log(c^2*x^2 + 1)^3 - I*integrate(1/128*(112*(b^3*c^5*d^3*x^5 - 2*b^3*c^3*d^3*x^3 - 3*b^3*c*
d^3*x)*arctan(c*x)^3 + 2*(3*b^3*c^4*d^3*x^4 + 2*b^3*c^2*d^3*x^2 - b^3*d^3)*log(c^2*x^2 + 1)^3 + 12*(32*a*b^2*c
^5*d^3*x^5 - 5*b^3*c^4*d^3*x^4 - 64*a*b^2*c^3*d^3*x^3 + 10*b^3*c^2*d^3*x^2 - 96*a*b^2*c*d^3*x)*arctan(c*x)^2 +
3*(5*b^3*c^4*d^3*x^4 - 10*b^3*c^2*d^3*x^2 + 4*(b^3*c^5*d^3*x^5 - 2*b^3*c^3*d^3*x^3 - 3*b^3*c*d^3*x)*arctan(c*
x))*log(c^2*x^2 + 1)^2 - 12*(2*(3*b^3*c^4*d^3*x^4 + 2*b^3*c^2*d^3*x^2 - b^3*d^3)*arctan(c*x)^2 - (b^3*c^5*d^3*
x^5 - 10*b^3*c^3*d^3*x^3 + 4*b^3*c*d^3*x)*arctan(c*x))*log(c^2*x^2 + 1))/(c^2*x^2 + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atan(c*x))^3*(d + c*d*x*1i)^3,x)

[Out]

int((a + b*atan(c*x))^3*(d + c*d*x*1i)^3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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