\(\int x^3 (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx\) [84]

Optimal result

Integrand size = 25, antiderivative size = 438 \[ \int x^3 (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx=\frac {3 a b d^3 x}{2 c^3}-\frac {122 i b^2 d^3 x}{105 c^3}+\frac {7 b^2 d^3 x^2}{20 c^2}+\frac {44 i b^2 d^3 x^3}{315 c}-\frac {1}{20} b^2 d^3 x^4-\frac {1}{105} i b^2 c d^3 x^5+\frac {122 i b^2 d^3 \arctan (c x)}{105 c^4}+\frac {3 b^2 d^3 x \arctan (c x)}{2 c^3}+\frac {26 i b d^3 x^2 (a+b \arctan (c x))}{35 c^2}-\frac {b d^3 x^3 (a+b \arctan (c x))}{2 c}-\frac {13}{35} i b d^3 x^4 (a+b \arctan (c x))+\frac {1}{5} b c d^3 x^5 (a+b \arctan (c x))+\frac {1}{21} i b c^2 d^3 x^6 (a+b \arctan (c x))-\frac {209 d^3 (a+b \arctan (c x))^2}{140 c^4}+\frac {1}{4} d^3 x^4 (a+b \arctan (c x))^2+\frac {3}{5} i c d^3 x^5 (a+b \arctan (c x))^2-\frac {1}{2} c^2 d^3 x^6 (a+b \arctan (c x))^2-\frac {1}{7} i c^3 d^3 x^7 (a+b \arctan (c x))^2+\frac {52 i b d^3 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{35 c^4}-\frac {11 b^2 d^3 \log \left (1+c^2 x^2\right )}{10 c^4}-\frac {26 b^2 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{35 c^4} \]

[Out]

3/2*a*b*d^3*x/c^3+1/21*I*b*c^2*d^3*x^6*(a+b*arctan(c*x))+7/20*b^2*d^3*x^2/c^2+52/35*I*b*d^3*(a+b*arctan(c*x))*
ln(2/(1+I*c*x))/c^4-1/20*b^2*d^3*x^4+26/35*I*b*d^3*x^2*(a+b*arctan(c*x))/c^2-1/105*I*b^2*c*d^3*x^5+3/2*b^2*d^3
*x*arctan(c*x)/c^3-1/7*I*c^3*d^3*x^7*(a+b*arctan(c*x))^2-1/2*b*d^3*x^3*(a+b*arctan(c*x))/c-122/105*I*b^2*d^3*x
/c^3+1/5*b*c*d^3*x^5*(a+b*arctan(c*x))+3/5*I*c*d^3*x^5*(a+b*arctan(c*x))^2-209/140*d^3*(a+b*arctan(c*x))^2/c^4
+1/4*d^3*x^4*(a+b*arctan(c*x))^2+44/315*I*b^2*d^3*x^3/c-1/2*c^2*d^3*x^6*(a+b*arctan(c*x))^2+122/105*I*b^2*d^3*
arctan(c*x)/c^4-13/35*I*b*d^3*x^4*(a+b*arctan(c*x))-11/10*b^2*d^3*ln(c^2*x^2+1)/c^4-26/35*b^2*d^3*polylog(2,1-
2/(1+I*c*x))/c^4

Rubi [A] (verified)

Time = 0.98 (sec) , antiderivative size = 438, normalized size of antiderivative = 1.00, number of steps used = 62, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {4996, 4946, 5036, 272, 45, 4930, 266, 5004, 308, 209, 327, 5040, 4964, 2449, 2352} \[ \int x^3 (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx=-\frac {209 d^3 (a+b \arctan (c x))^2}{140 c^4}+\frac {52 i b d^3 \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{35 c^4}-\frac {1}{7} i c^3 d^3 x^7 (a+b \arctan (c x))^2-\frac {1}{2} c^2 d^3 x^6 (a+b \arctan (c x))^2+\frac {1}{21} i b c^2 d^3 x^6 (a+b \arctan (c x))+\frac {26 i b d^3 x^2 (a+b \arctan (c x))}{35 c^2}+\frac {3}{5} i c d^3 x^5 (a+b \arctan (c x))^2+\frac {1}{5} b c d^3 x^5 (a+b \arctan (c x))+\frac {1}{4} d^3 x^4 (a+b \arctan (c x))^2-\frac {13}{35} i b d^3 x^4 (a+b \arctan (c x))-\frac {b d^3 x^3 (a+b \arctan (c x))}{2 c}+\frac {3 a b d^3 x}{2 c^3}+\frac {122 i b^2 d^3 \arctan (c x)}{105 c^4}+\frac {3 b^2 d^3 x \arctan (c x)}{2 c^3}-\frac {26 b^2 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{35 c^4}-\frac {122 i b^2 d^3 x}{105 c^3}+\frac {7 b^2 d^3 x^2}{20 c^2}-\frac {11 b^2 d^3 \log \left (c^2 x^2+1\right )}{10 c^4}-\frac {1}{105} i b^2 c d^3 x^5+\frac {44 i b^2 d^3 x^3}{315 c}-\frac {1}{20} b^2 d^3 x^4 \]

[In]

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

[Out]

(3*a*b*d^3*x)/(2*c^3) - (((122*I)/105)*b^2*d^3*x)/c^3 + (7*b^2*d^3*x^2)/(20*c^2) + (((44*I)/315)*b^2*d^3*x^3)/
c - (b^2*d^3*x^4)/20 - (I/105)*b^2*c*d^3*x^5 + (((122*I)/105)*b^2*d^3*ArcTan[c*x])/c^4 + (3*b^2*d^3*x*ArcTan[c
*x])/(2*c^3) + (((26*I)/35)*b*d^3*x^2*(a + b*ArcTan[c*x]))/c^2 - (b*d^3*x^3*(a + b*ArcTan[c*x]))/(2*c) - ((13*
I)/35)*b*d^3*x^4*(a + b*ArcTan[c*x]) + (b*c*d^3*x^5*(a + b*ArcTan[c*x]))/5 + (I/21)*b*c^2*d^3*x^6*(a + b*ArcTa
n[c*x]) - (209*d^3*(a + b*ArcTan[c*x])^2)/(140*c^4) + (d^3*x^4*(a + b*ArcTan[c*x])^2)/4 + ((3*I)/5)*c*d^3*x^5*
(a + b*ArcTan[c*x])^2 - (c^2*d^3*x^6*(a + b*ArcTan[c*x])^2)/2 - (I/7)*c^3*d^3*x^7*(a + b*ArcTan[c*x])^2 + (((5
2*I)/35)*b*d^3*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/c^4 - (11*b^2*d^3*Log[1 + c^2*x^2])/(10*c^4) - (26*b^2*
d^3*PolyLog[2, 1 - 2/(1 + I*c*x)])/(35*c^4)

Rule 45

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])

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 266

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 272

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 308

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 327

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 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2449

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 4930

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcTan[c*x^n])^p, x] - Dist[b*c
*n*p, Int[x^n*((a + b*ArcTan[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0
] && (EqQ[n, 1] || EqQ[p, 1])

Rule 4946

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcTan[c*x^
n])^p/(m + 1)), x] - Dist[b*c*n*(p/(m + 1)), Int[x^(m + n)*((a + b*ArcTan[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))),
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1] && IntegerQ[m])) && NeQ[m, -1]

Rule 4964

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTan[c*x])^p)*(
Log[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 4996

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 5004

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 5036

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 5040

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-I)*((a + b*ArcT
an[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]

Rubi steps \begin{align*} \text {integral}& = \int \left (d^3 x^3 (a+b \arctan (c x))^2+3 i c d^3 x^4 (a+b \arctan (c x))^2-3 c^2 d^3 x^5 (a+b \arctan (c x))^2-i c^3 d^3 x^6 (a+b \arctan (c x))^2\right ) \, dx \\ & = d^3 \int x^3 (a+b \arctan (c x))^2 \, dx+\left (3 i c d^3\right ) \int x^4 (a+b \arctan (c x))^2 \, dx-\left (3 c^2 d^3\right ) \int x^5 (a+b \arctan (c x))^2 \, dx-\left (i c^3 d^3\right ) \int x^6 (a+b \arctan (c x))^2 \, dx \\ & = \frac {1}{4} d^3 x^4 (a+b \arctan (c x))^2+\frac {3}{5} i c d^3 x^5 (a+b \arctan (c x))^2-\frac {1}{2} c^2 d^3 x^6 (a+b \arctan (c x))^2-\frac {1}{7} i c^3 d^3 x^7 (a+b \arctan (c x))^2-\frac {1}{2} \left (b c d^3\right ) \int \frac {x^4 (a+b \arctan (c x))}{1+c^2 x^2} \, dx-\frac {1}{5} \left (6 i b c^2 d^3\right ) \int \frac {x^5 (a+b \arctan (c x))}{1+c^2 x^2} \, dx+\left (b c^3 d^3\right ) \int \frac {x^6 (a+b \arctan (c x))}{1+c^2 x^2} \, dx+\frac {1}{7} \left (2 i b c^4 d^3\right ) \int \frac {x^7 (a+b \arctan (c x))}{1+c^2 x^2} \, dx \\ & = \frac {1}{4} d^3 x^4 (a+b \arctan (c x))^2+\frac {3}{5} i c d^3 x^5 (a+b \arctan (c x))^2-\frac {1}{2} c^2 d^3 x^6 (a+b \arctan (c x))^2-\frac {1}{7} i c^3 d^3 x^7 (a+b \arctan (c x))^2-\frac {1}{5} \left (6 i b d^3\right ) \int x^3 (a+b \arctan (c x)) \, dx+\frac {1}{5} \left (6 i b d^3\right ) \int \frac {x^3 (a+b \arctan (c x))}{1+c^2 x^2} \, dx-\frac {\left (b d^3\right ) \int x^2 (a+b \arctan (c x)) \, dx}{2 c}+\frac {\left (b d^3\right ) \int \frac {x^2 (a+b \arctan (c x))}{1+c^2 x^2} \, dx}{2 c}+\left (b c d^3\right ) \int x^4 (a+b \arctan (c x)) \, dx-\left (b c d^3\right ) \int \frac {x^4 (a+b \arctan (c x))}{1+c^2 x^2} \, dx+\frac {1}{7} \left (2 i b c^2 d^3\right ) \int x^5 (a+b \arctan (c x)) \, dx-\frac {1}{7} \left (2 i b c^2 d^3\right ) \int \frac {x^5 (a+b \arctan (c x))}{1+c^2 x^2} \, dx \\ & = -\frac {b d^3 x^3 (a+b \arctan (c x))}{6 c}-\frac {3}{10} i b d^3 x^4 (a+b \arctan (c x))+\frac {1}{5} b c d^3 x^5 (a+b \arctan (c x))+\frac {1}{21} i b c^2 d^3 x^6 (a+b \arctan (c x))+\frac {1}{4} d^3 x^4 (a+b \arctan (c x))^2+\frac {3}{5} i c d^3 x^5 (a+b \arctan (c x))^2-\frac {1}{2} c^2 d^3 x^6 (a+b \arctan (c x))^2-\frac {1}{7} i c^3 d^3 x^7 (a+b \arctan (c x))^2-\frac {1}{7} \left (2 i b d^3\right ) \int x^3 (a+b \arctan (c x)) \, dx+\frac {1}{7} \left (2 i b d^3\right ) \int \frac {x^3 (a+b \arctan (c x))}{1+c^2 x^2} \, dx+\frac {1}{6} \left (b^2 d^3\right ) \int \frac {x^3}{1+c^2 x^2} \, dx+\frac {\left (b d^3\right ) \int (a+b \arctan (c x)) \, dx}{2 c^3}-\frac {\left (b d^3\right ) \int \frac {a+b \arctan (c x)}{1+c^2 x^2} \, dx}{2 c^3}+\frac {\left (6 i b d^3\right ) \int x (a+b \arctan (c x)) \, dx}{5 c^2}-\frac {\left (6 i b d^3\right ) \int \frac {x (a+b \arctan (c x))}{1+c^2 x^2} \, dx}{5 c^2}-\frac {\left (b d^3\right ) \int x^2 (a+b \arctan (c x)) \, dx}{c}+\frac {\left (b d^3\right ) \int \frac {x^2 (a+b \arctan (c x))}{1+c^2 x^2} \, dx}{c}+\frac {1}{10} \left (3 i b^2 c d^3\right ) \int \frac {x^4}{1+c^2 x^2} \, dx-\frac {1}{5} \left (b^2 c^2 d^3\right ) \int \frac {x^5}{1+c^2 x^2} \, dx-\frac {1}{21} \left (i b^2 c^3 d^3\right ) \int \frac {x^6}{1+c^2 x^2} \, dx \\ & = \frac {a b d^3 x}{2 c^3}+\frac {3 i b d^3 x^2 (a+b \arctan (c x))}{5 c^2}-\frac {b d^3 x^3 (a+b \arctan (c x))}{2 c}-\frac {13}{35} i b d^3 x^4 (a+b \arctan (c x))+\frac {1}{5} b c d^3 x^5 (a+b \arctan (c x))+\frac {1}{21} i b c^2 d^3 x^6 (a+b \arctan (c x))-\frac {17 d^3 (a+b \arctan (c x))^2}{20 c^4}+\frac {1}{4} d^3 x^4 (a+b \arctan (c x))^2+\frac {3}{5} i c d^3 x^5 (a+b \arctan (c x))^2-\frac {1}{2} c^2 d^3 x^6 (a+b \arctan (c x))^2-\frac {1}{7} i c^3 d^3 x^7 (a+b \arctan (c x))^2+\frac {1}{12} \left (b^2 d^3\right ) \text {Subst}\left (\int \frac {x}{1+c^2 x} \, dx,x,x^2\right )+\frac {1}{3} \left (b^2 d^3\right ) \int \frac {x^3}{1+c^2 x^2} \, dx+\frac {\left (6 i b d^3\right ) \int \frac {a+b \arctan (c x)}{i-c x} \, dx}{5 c^3}+\frac {\left (b d^3\right ) \int (a+b \arctan (c x)) \, dx}{c^3}-\frac {\left (b d^3\right ) \int \frac {a+b \arctan (c x)}{1+c^2 x^2} \, dx}{c^3}+\frac {\left (b^2 d^3\right ) \int \arctan (c x) \, dx}{2 c^3}+\frac {\left (2 i b d^3\right ) \int x (a+b \arctan (c x)) \, dx}{7 c^2}-\frac {\left (2 i b d^3\right ) \int \frac {x (a+b \arctan (c x))}{1+c^2 x^2} \, dx}{7 c^2}-\frac {\left (3 i b^2 d^3\right ) \int \frac {x^2}{1+c^2 x^2} \, dx}{5 c}+\frac {1}{14} \left (i b^2 c d^3\right ) \int \frac {x^4}{1+c^2 x^2} \, dx+\frac {1}{10} \left (3 i b^2 c d^3\right ) \int \left (-\frac {1}{c^4}+\frac {x^2}{c^2}+\frac {1}{c^4 \left (1+c^2 x^2\right )}\right ) \, dx-\frac {1}{10} \left (b^2 c^2 d^3\right ) \text {Subst}\left (\int \frac {x^2}{1+c^2 x} \, dx,x,x^2\right )-\frac {1}{21} \left (i b^2 c^3 d^3\right ) \int \left (\frac {1}{c^6}-\frac {x^2}{c^4}+\frac {x^4}{c^2}-\frac {1}{c^6 \left (1+c^2 x^2\right )}\right ) \, dx \\ & = \frac {3 a b d^3 x}{2 c^3}-\frac {199 i b^2 d^3 x}{210 c^3}+\frac {73 i b^2 d^3 x^3}{630 c}-\frac {1}{105} i b^2 c d^3 x^5+\frac {b^2 d^3 x \arctan (c x)}{2 c^3}+\frac {26 i b d^3 x^2 (a+b \arctan (c x))}{35 c^2}-\frac {b d^3 x^3 (a+b \arctan (c x))}{2 c}-\frac {13}{35} i b d^3 x^4 (a+b \arctan (c x))+\frac {1}{5} b c d^3 x^5 (a+b \arctan (c x))+\frac {1}{21} i b c^2 d^3 x^6 (a+b \arctan (c x))-\frac {209 d^3 (a+b \arctan (c x))^2}{140 c^4}+\frac {1}{4} d^3 x^4 (a+b \arctan (c x))^2+\frac {3}{5} i c d^3 x^5 (a+b \arctan (c x))^2-\frac {1}{2} c^2 d^3 x^6 (a+b \arctan (c x))^2-\frac {1}{7} i c^3 d^3 x^7 (a+b \arctan (c x))^2+\frac {6 i b d^3 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{5 c^4}+\frac {1}{12} \left (b^2 d^3\right ) \text {Subst}\left (\int \left (\frac {1}{c^2}-\frac {1}{c^2 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )+\frac {1}{6} \left (b^2 d^3\right ) \text {Subst}\left (\int \frac {x}{1+c^2 x} \, dx,x,x^2\right )+\frac {\left (2 i b d^3\right ) \int \frac {a+b \arctan (c x)}{i-c x} \, dx}{7 c^3}+\frac {\left (i b^2 d^3\right ) \int \frac {1}{1+c^2 x^2} \, dx}{21 c^3}+\frac {\left (3 i b^2 d^3\right ) \int \frac {1}{1+c^2 x^2} \, dx}{10 c^3}+\frac {\left (3 i b^2 d^3\right ) \int \frac {1}{1+c^2 x^2} \, dx}{5 c^3}-\frac {\left (6 i b^2 d^3\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{5 c^3}+\frac {\left (b^2 d^3\right ) \int \arctan (c x) \, dx}{c^3}-\frac {\left (b^2 d^3\right ) \int \frac {x}{1+c^2 x^2} \, dx}{2 c^2}-\frac {\left (i b^2 d^3\right ) \int \frac {x^2}{1+c^2 x^2} \, dx}{7 c}+\frac {1}{14} \left (i b^2 c d^3\right ) \int \left (-\frac {1}{c^4}+\frac {x^2}{c^2}+\frac {1}{c^4 \left (1+c^2 x^2\right )}\right ) \, dx-\frac {1}{10} \left (b^2 c^2 d^3\right ) \text {Subst}\left (\int \left (-\frac {1}{c^4}+\frac {x}{c^2}+\frac {1}{c^4 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right ) \\ & = \frac {3 a b d^3 x}{2 c^3}-\frac {122 i b^2 d^3 x}{105 c^3}+\frac {11 b^2 d^3 x^2}{60 c^2}+\frac {44 i b^2 d^3 x^3}{315 c}-\frac {1}{20} b^2 d^3 x^4-\frac {1}{105} i b^2 c d^3 x^5+\frac {199 i b^2 d^3 \arctan (c x)}{210 c^4}+\frac {3 b^2 d^3 x \arctan (c x)}{2 c^3}+\frac {26 i b d^3 x^2 (a+b \arctan (c x))}{35 c^2}-\frac {b d^3 x^3 (a+b \arctan (c x))}{2 c}-\frac {13}{35} i b d^3 x^4 (a+b \arctan (c x))+\frac {1}{5} b c d^3 x^5 (a+b \arctan (c x))+\frac {1}{21} i b c^2 d^3 x^6 (a+b \arctan (c x))-\frac {209 d^3 (a+b \arctan (c x))^2}{140 c^4}+\frac {1}{4} d^3 x^4 (a+b \arctan (c x))^2+\frac {3}{5} i c d^3 x^5 (a+b \arctan (c x))^2-\frac {1}{2} c^2 d^3 x^6 (a+b \arctan (c x))^2-\frac {1}{7} i c^3 d^3 x^7 (a+b \arctan (c x))^2+\frac {52 i b d^3 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{35 c^4}-\frac {13 b^2 d^3 \log \left (1+c^2 x^2\right )}{30 c^4}+\frac {1}{6} \left (b^2 d^3\right ) \text {Subst}\left (\int \left (\frac {1}{c^2}-\frac {1}{c^2 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )-\frac {\left (6 b^2 d^3\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{5 c^4}+\frac {\left (i b^2 d^3\right ) \int \frac {1}{1+c^2 x^2} \, dx}{14 c^3}+\frac {\left (i b^2 d^3\right ) \int \frac {1}{1+c^2 x^2} \, dx}{7 c^3}-\frac {\left (2 i b^2 d^3\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{7 c^3}-\frac {\left (b^2 d^3\right ) \int \frac {x}{1+c^2 x^2} \, dx}{c^2} \\ & = \frac {3 a b d^3 x}{2 c^3}-\frac {122 i b^2 d^3 x}{105 c^3}+\frac {7 b^2 d^3 x^2}{20 c^2}+\frac {44 i b^2 d^3 x^3}{315 c}-\frac {1}{20} b^2 d^3 x^4-\frac {1}{105} i b^2 c d^3 x^5+\frac {122 i b^2 d^3 \arctan (c x)}{105 c^4}+\frac {3 b^2 d^3 x \arctan (c x)}{2 c^3}+\frac {26 i b d^3 x^2 (a+b \arctan (c x))}{35 c^2}-\frac {b d^3 x^3 (a+b \arctan (c x))}{2 c}-\frac {13}{35} i b d^3 x^4 (a+b \arctan (c x))+\frac {1}{5} b c d^3 x^5 (a+b \arctan (c x))+\frac {1}{21} i b c^2 d^3 x^6 (a+b \arctan (c x))-\frac {209 d^3 (a+b \arctan (c x))^2}{140 c^4}+\frac {1}{4} d^3 x^4 (a+b \arctan (c x))^2+\frac {3}{5} i c d^3 x^5 (a+b \arctan (c x))^2-\frac {1}{2} c^2 d^3 x^6 (a+b \arctan (c x))^2-\frac {1}{7} i c^3 d^3 x^7 (a+b \arctan (c x))^2+\frac {52 i b d^3 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{35 c^4}-\frac {11 b^2 d^3 \log \left (1+c^2 x^2\right )}{10 c^4}-\frac {3 b^2 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{5 c^4}-\frac {\left (2 b^2 d^3\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{7 c^4} \\ & = \frac {3 a b d^3 x}{2 c^3}-\frac {122 i b^2 d^3 x}{105 c^3}+\frac {7 b^2 d^3 x^2}{20 c^2}+\frac {44 i b^2 d^3 x^3}{315 c}-\frac {1}{20} b^2 d^3 x^4-\frac {1}{105} i b^2 c d^3 x^5+\frac {122 i b^2 d^3 \arctan (c x)}{105 c^4}+\frac {3 b^2 d^3 x \arctan (c x)}{2 c^3}+\frac {26 i b d^3 x^2 (a+b \arctan (c x))}{35 c^2}-\frac {b d^3 x^3 (a+b \arctan (c x))}{2 c}-\frac {13}{35} i b d^3 x^4 (a+b \arctan (c x))+\frac {1}{5} b c d^3 x^5 (a+b \arctan (c x))+\frac {1}{21} i b c^2 d^3 x^6 (a+b \arctan (c x))-\frac {209 d^3 (a+b \arctan (c x))^2}{140 c^4}+\frac {1}{4} d^3 x^4 (a+b \arctan (c x))^2+\frac {3}{5} i c d^3 x^5 (a+b \arctan (c x))^2-\frac {1}{2} c^2 d^3 x^6 (a+b \arctan (c x))^2-\frac {1}{7} i c^3 d^3 x^7 (a+b \arctan (c x))^2+\frac {52 i b d^3 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{35 c^4}-\frac {11 b^2 d^3 \log \left (1+c^2 x^2\right )}{10 c^4}-\frac {26 b^2 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{35 c^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.51 (sec) , antiderivative size = 408, normalized size of antiderivative = 0.93 \[ \int x^3 (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx=\frac {d^3 \left (1464 i a b+504 b^2+1890 a b c x-1464 i b^2 c x+936 i a b c^2 x^2+441 b^2 c^2 x^2-630 a b c^3 x^3+176 i b^2 c^3 x^3+315 a^2 c^4 x^4-468 i a b c^4 x^4-63 b^2 c^4 x^4+756 i a^2 c^5 x^5+252 a b c^5 x^5-12 i b^2 c^5 x^5-630 a^2 c^6 x^6+60 i a b c^6 x^6-180 i a^2 c^7 x^7+9 b^2 (-i+c x)^4 \left (-1+4 i c x+10 c^2 x^2-20 i c^3 x^3\right ) \arctan (c x)^2+6 b \arctan (c x) \left (b \left (244 i+315 c x+156 i c^2 x^2-105 c^3 x^3-78 i c^4 x^4+42 c^5 x^5+10 i c^6 x^6\right )+3 a \left (-105+35 c^4 x^4+84 i c^5 x^5-70 c^6 x^6-20 i c^7 x^7\right )+312 i b \log \left (1+e^{2 i \arctan (c x)}\right )\right )-936 i a b \log \left (1+c^2 x^2\right )-1386 b^2 \log \left (1+c^2 x^2\right )+936 b^2 \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )\right )}{1260 c^4} \]

[In]

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

[Out]

(d^3*((1464*I)*a*b + 504*b^2 + 1890*a*b*c*x - (1464*I)*b^2*c*x + (936*I)*a*b*c^2*x^2 + 441*b^2*c^2*x^2 - 630*a
*b*c^3*x^3 + (176*I)*b^2*c^3*x^3 + 315*a^2*c^4*x^4 - (468*I)*a*b*c^4*x^4 - 63*b^2*c^4*x^4 + (756*I)*a^2*c^5*x^
5 + 252*a*b*c^5*x^5 - (12*I)*b^2*c^5*x^5 - 630*a^2*c^6*x^6 + (60*I)*a*b*c^6*x^6 - (180*I)*a^2*c^7*x^7 + 9*b^2*
(-I + c*x)^4*(-1 + (4*I)*c*x + 10*c^2*x^2 - (20*I)*c^3*x^3)*ArcTan[c*x]^2 + 6*b*ArcTan[c*x]*(b*(244*I + 315*c*
x + (156*I)*c^2*x^2 - 105*c^3*x^3 - (78*I)*c^4*x^4 + 42*c^5*x^5 + (10*I)*c^6*x^6) + 3*a*(-105 + 35*c^4*x^4 + (
84*I)*c^5*x^5 - 70*c^6*x^6 - (20*I)*c^7*x^7) + (312*I)*b*Log[1 + E^((2*I)*ArcTan[c*x])]) - (936*I)*a*b*Log[1 +
c^2*x^2] - 1386*b^2*Log[1 + c^2*x^2] + 936*b^2*PolyLog[2, -E^((2*I)*ArcTan[c*x])]))/(1260*c^4)

Maple [A] (verified)

Time = 2.93 (sec) , antiderivative size = 511, normalized size of antiderivative = 1.17

method	result	size
parts	\(d^{3} a^{2} \left (-\frac {1}{7} i c^{3} x^{7}-\frac {1}{2} c^{2} x^{6}+\frac {3}{5} i c \,x^{5}+\frac {1}{4} x^{4}\right )+\frac {b^{2} d^{3} \left (\frac {3 c x \arctan \left (c x \right )}{2}-\frac {c^{3} x^{3} \arctan \left (c x \right )}{2}+\frac {c^{5} x^{5} \arctan \left (c x \right )}{5}-\frac {11 \ln \left (c^{2} x^{2}+1\right )}{10}+\frac {7 c^{2} x^{2}}{20}-\frac {3 \arctan \left (c x \right )^{2}}{4}-\frac {c^{4} x^{4}}{20}-\frac {122 i c x}{105}-\frac {13 \ln \left (c x -i\right )^{2}}{70}-\frac {13 \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{35}+\frac {13 \ln \left (c x +i\right )^{2}}{70}+\frac {13 \operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )}{35}+\frac {c^{4} x^{4} \arctan \left (c x \right )^{2}}{4}+\frac {13 \ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )}{35}-\frac {13 \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{35}-\frac {13 \ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )}{35}+\frac {13 \ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )}{35}-\frac {\arctan \left (c x \right )^{2} c^{6} x^{6}}{2}-\frac {i c^{5} x^{5}}{105}-\frac {13 i \arctan \left (c x \right ) c^{4} x^{4}}{35}+\frac {122 i \arctan \left (c x \right )}{105}+\frac {44 i c^{3} x^{3}}{315}+\frac {26 i \arctan \left (c x \right ) c^{2} x^{2}}{35}+\frac {i \arctan \left (c x \right ) c^{6} x^{6}}{21}+\frac {3 i \arctan \left (c x \right )^{2} c^{5} x^{5}}{5}-\frac {26 i \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{35}-\frac {i \arctan \left (c x \right )^{2} c^{7} x^{7}}{7}\right )}{c^{4}}+\frac {2 a \,d^{3} b \left (-\frac {i \arctan \left (c x \right ) c^{7} x^{7}}{7}-\frac {\arctan \left (c x \right ) c^{6} x^{6}}{2}+\frac {3 i \arctan \left (c x \right ) c^{5} x^{5}}{5}+\frac {c^{4} x^{4} \arctan \left (c x \right )}{4}+\frac {3 c x}{4}+\frac {i c^{6} x^{6}}{42}+\frac {c^{5} x^{5}}{10}-\frac {13 i c^{4} x^{4}}{70}-\frac {c^{3} x^{3}}{4}+\frac {13 i c^{2} x^{2}}{35}-\frac {13 i \ln \left (c^{2} x^{2}+1\right )}{35}-\frac {3 \arctan \left (c x \right )}{4}\right )}{c^{4}}\)	\(511\)
derivativedivides	\(\frac {d^{3} a^{2} \left (-\frac {1}{7} i c^{7} x^{7}-\frac {1}{2} c^{6} x^{6}+\frac {3}{5} i c^{5} x^{5}+\frac {1}{4} c^{4} x^{4}\right )+b^{2} d^{3} \left (\frac {3 c x \arctan \left (c x \right )}{2}-\frac {c^{3} x^{3} \arctan \left (c x \right )}{2}+\frac {c^{5} x^{5} \arctan \left (c x \right )}{5}-\frac {11 \ln \left (c^{2} x^{2}+1\right )}{10}+\frac {7 c^{2} x^{2}}{20}-\frac {3 \arctan \left (c x \right )^{2}}{4}-\frac {c^{4} x^{4}}{20}-\frac {122 i c x}{105}-\frac {13 \ln \left (c x -i\right )^{2}}{70}-\frac {13 \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{35}+\frac {13 \ln \left (c x +i\right )^{2}}{70}+\frac {13 \operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )}{35}+\frac {c^{4} x^{4} \arctan \left (c x \right )^{2}}{4}+\frac {13 \ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )}{35}-\frac {13 \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{35}-\frac {13 \ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )}{35}+\frac {13 \ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )}{35}-\frac {\arctan \left (c x \right )^{2} c^{6} x^{6}}{2}-\frac {i c^{5} x^{5}}{105}-\frac {13 i \arctan \left (c x \right ) c^{4} x^{4}}{35}+\frac {122 i \arctan \left (c x \right )}{105}+\frac {44 i c^{3} x^{3}}{315}+\frac {26 i \arctan \left (c x \right ) c^{2} x^{2}}{35}+\frac {i \arctan \left (c x \right ) c^{6} x^{6}}{21}+\frac {3 i \arctan \left (c x \right )^{2} c^{5} x^{5}}{5}-\frac {26 i \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{35}-\frac {i \arctan \left (c x \right )^{2} c^{7} x^{7}}{7}\right )+2 a \,d^{3} b \left (-\frac {i \arctan \left (c x \right ) c^{7} x^{7}}{7}-\frac {\arctan \left (c x \right ) c^{6} x^{6}}{2}+\frac {3 i \arctan \left (c x \right ) c^{5} x^{5}}{5}+\frac {c^{4} x^{4} \arctan \left (c x \right )}{4}+\frac {3 c x}{4}+\frac {i c^{6} x^{6}}{42}+\frac {c^{5} x^{5}}{10}-\frac {13 i c^{4} x^{4}}{70}-\frac {c^{3} x^{3}}{4}+\frac {13 i c^{2} x^{2}}{35}-\frac {13 i \ln \left (c^{2} x^{2}+1\right )}{35}-\frac {3 \arctan \left (c x \right )}{4}\right )}{c^{4}}\)	\(514\)
default	\(\frac {d^{3} a^{2} \left (-\frac {1}{7} i c^{7} x^{7}-\frac {1}{2} c^{6} x^{6}+\frac {3}{5} i c^{5} x^{5}+\frac {1}{4} c^{4} x^{4}\right )+b^{2} d^{3} \left (\frac {3 c x \arctan \left (c x \right )}{2}-\frac {c^{3} x^{3} \arctan \left (c x \right )}{2}+\frac {c^{5} x^{5} \arctan \left (c x \right )}{5}-\frac {11 \ln \left (c^{2} x^{2}+1\right )}{10}+\frac {7 c^{2} x^{2}}{20}-\frac {3 \arctan \left (c x \right )^{2}}{4}-\frac {c^{4} x^{4}}{20}-\frac {122 i c x}{105}-\frac {13 \ln \left (c x -i\right )^{2}}{70}-\frac {13 \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{35}+\frac {13 \ln \left (c x +i\right )^{2}}{70}+\frac {13 \operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )}{35}+\frac {c^{4} x^{4} \arctan \left (c x \right )^{2}}{4}+\frac {13 \ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )}{35}-\frac {13 \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{35}-\frac {13 \ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )}{35}+\frac {13 \ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )}{35}-\frac {\arctan \left (c x \right )^{2} c^{6} x^{6}}{2}-\frac {i c^{5} x^{5}}{105}-\frac {13 i \arctan \left (c x \right ) c^{4} x^{4}}{35}+\frac {122 i \arctan \left (c x \right )}{105}+\frac {44 i c^{3} x^{3}}{315}+\frac {26 i \arctan \left (c x \right ) c^{2} x^{2}}{35}+\frac {i \arctan \left (c x \right ) c^{6} x^{6}}{21}+\frac {3 i \arctan \left (c x \right )^{2} c^{5} x^{5}}{5}-\frac {26 i \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{35}-\frac {i \arctan \left (c x \right )^{2} c^{7} x^{7}}{7}\right )+2 a \,d^{3} b \left (-\frac {i \arctan \left (c x \right ) c^{7} x^{7}}{7}-\frac {\arctan \left (c x \right ) c^{6} x^{6}}{2}+\frac {3 i \arctan \left (c x \right ) c^{5} x^{5}}{5}+\frac {c^{4} x^{4} \arctan \left (c x \right )}{4}+\frac {3 c x}{4}+\frac {i c^{6} x^{6}}{42}+\frac {c^{5} x^{5}}{10}-\frac {13 i c^{4} x^{4}}{70}-\frac {c^{3} x^{3}}{4}+\frac {13 i c^{2} x^{2}}{35}-\frac {13 i \ln \left (c^{2} x^{2}+1\right )}{35}-\frac {3 \arctan \left (c x \right )}{4}\right )}{c^{4}}\)	\(514\)
risch	\(-\frac {b^{2} d^{3} x^{4}}{20}+\frac {7 b^{2} d^{3} x^{2}}{20 c^{2}}-\frac {11 b^{2} d^{3} \ln \left (c^{2} x^{2}+1\right )}{10 c^{4}}+\frac {3 a b \,d^{3} x}{2 c^{3}}+\frac {77 b^{2} d^{3}}{45 c^{4}}+\frac {d^{3} x^{4} a^{2}}{4}-\frac {3 d^{3} b a \arctan \left (c x \right )}{2 c^{4}}-\frac {d^{3} a b \,x^{3}}{2 c}-\frac {d^{3} c^{2} a^{2} x^{6}}{2}-\frac {209 d^{3} a^{2}}{140 c^{4}}+\frac {d^{3} c b a \,x^{5}}{5}+\frac {d^{3} c^{3} b a \ln \left (-i c x +1\right ) x^{7}}{7}+\frac {3 i b^{2} d^{3} \ln \left (-i c x +1\right ) x}{4 c^{3}}-\frac {3 i d^{3} c \,b^{2} \ln \left (-i c x +1\right )^{2} x^{5}}{20}+\frac {i d^{3} c^{3} b^{2} \ln \left (-i c x +1\right )^{2} x^{7}}{28}+\frac {i b^{2} d^{3} \left (20 c^{7} x^{7}-70 i c^{6} x^{6}-84 c^{5} x^{5}+35 i c^{4} x^{4}-i\right ) \ln \left (i c x +1\right )^{2}}{560 c^{4}}+\frac {i d^{3} a b \ln \left (-i c x +1\right ) x^{4}}{4}-\frac {26 i d^{3} b a \ln \left (c^{2} x^{2}+1\right )}{35 c^{4}}-\frac {i b^{2} d^{3} \ln \left (-i c x +1\right ) x^{3}}{4 c}+\frac {i b^{2} d^{3} c \ln \left (-i c x +1\right ) x^{5}}{10}+\frac {26 i b \,d^{3} x^{2} a}{35 c^{2}}+\frac {i b \,d^{3} c^{2} x^{6} a}{21}-\frac {3 d^{3} c a b \ln \left (-i c x +1\right ) x^{5}}{5}-\frac {13 i b \,d^{3} x^{4} a}{35}+\frac {353 i b \,d^{3} a}{105 c^{4}}+\frac {26 b^{2} d^{3} \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (-i c x +1\right )}{35 c^{4}}-\frac {26 b^{2} d^{3} \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (\frac {1}{2}-\frac {i c x}{2}\right )}{35 c^{4}}-\frac {d^{3} c^{2} b^{2} \ln \left (-i c x +1\right ) x^{6}}{42}-\frac {13 d^{3} b^{2} \ln \left (-i c x +1\right ) x^{2}}{35 c^{2}}+\frac {d^{3} c^{2} b^{2} \ln \left (-i c x +1\right )^{2} x^{6}}{8}-\frac {122 i b^{2} d^{3} x}{105 c^{3}}-\frac {i b^{2} c \,d^{3} x^{5}}{105}+\frac {122 i b^{2} d^{3} \arctan \left (c x \right )}{105 c^{4}}+\frac {44 i b^{2} d^{3} x^{3}}{315 c}+\frac {3 i d^{3} c \,x^{5} a^{2}}{5}-\frac {i d^{3} c^{3} a^{2} x^{7}}{7}+\frac {13 d^{3} b^{2} \ln \left (-i c x +1\right ) x^{4}}{70}-\frac {d^{3} b^{2} \ln \left (-i c x +1\right )^{2} x^{4}}{16}+\frac {209 d^{3} b^{2} \ln \left (-i c x +1\right )^{2}}{560 c^{4}}-\frac {26 b^{2} d^{3} \operatorname {dilog}\left (\frac {1}{2}-\frac {i c x}{2}\right )}{35 c^{4}}+\left (-\frac {i b^{2} d^{3} \left (20 c^{3} x^{7}-70 i c^{2} x^{6}-84 x^{5} c +35 i x^{4}\right ) \ln \left (-i c x +1\right )}{280}+\frac {b \,d^{3} \left (-120 a \,c^{7} x^{7}+420 i a \,c^{6} x^{6}+20 b \,c^{6} x^{6}-84 i b \,c^{5} x^{5}+504 a \,c^{5} x^{5}-210 i a \,c^{4} x^{4}-156 b \,c^{4} x^{4}+210 i b \,c^{3} x^{3}+312 b \,c^{2} x^{2}-630 i b c x -627 b \ln \left (-i c x +1\right )\right )}{840 c^{4}}\right ) \ln \left (i c x +1\right )-\frac {i d^{3} c^{2} a b \ln \left (-i c x +1\right ) x^{6}}{2}\)	\(924\)

[In]

int(x^3*(d+I*c*d*x)^3*(a+b*arctan(c*x))^2,x,method=_RETURNVERBOSE)

[Out]

d^3*a^2*(-1/7*I*c^3*x^7-1/2*c^2*x^6+3/5*I*c*x^5+1/4*x^4)+b^2*d^3/c^4*(3/2*c*x*arctan(c*x)-1/2*c^3*x^3*arctan(c
*x)+1/5*c^5*x^5*arctan(c*x)-11/10*ln(c^2*x^2+1)+7/20*c^2*x^2-3/4*arctan(c*x)^2-1/20*c^4*x^4-13/35*dilog(-1/2*I
*(c*x+I))+1/4*c^4*x^4*arctan(c*x)^2+13/35*dilog(1/2*I*(c*x-I))-13/70*ln(c*x-I)^2+13/70*ln(c*x+I)^2+13/35*ln(c*
x-I)*ln(c^2*x^2+1)-13/35*ln(c*x-I)*ln(-1/2*I*(c*x+I))-13/35*ln(c*x+I)*ln(c^2*x^2+1)+13/35*ln(c*x+I)*ln(1/2*I*(
c*x-I))-1/2*arctan(c*x)^2*c^6*x^6-13/35*I*arctan(c*x)*c^4*x^4+26/35*I*arctan(c*x)*c^2*x^2+1/21*I*arctan(c*x)*c
^6*x^6-1/7*I*arctan(c*x)^2*c^7*x^7+3/5*I*arctan(c*x)^2*c^5*x^5+122/105*I*arctan(c*x)-122/105*I*c*x-1/105*I*c^5
*x^5+44/315*I*c^3*x^3-26/35*I*arctan(c*x)*ln(c^2*x^2+1))+2*a*d^3*b/c^4*(-1/7*I*arctan(c*x)*c^7*x^7-1/2*arctan(
c*x)*c^6*x^6+3/5*I*arctan(c*x)*c^5*x^5+1/4*c^4*x^4*arctan(c*x)+3/4*c*x+1/42*I*c^6*x^6+1/10*c^5*x^5-13/70*I*c^4
*x^4-1/4*c^3*x^3+13/35*I*c^2*x^2-13/35*I*ln(c^2*x^2+1)-3/4*arctan(c*x))

Fricas [F]

\[ \int x^3 (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx=\int { {\left (i \, c d x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}^{2} x^{3} \,d x } \]

[In]

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

[Out]

1/560*(20*I*b^2*c^3*d^3*x^7 + 70*b^2*c^2*d^3*x^6 - 84*I*b^2*c*d^3*x^5 - 35*b^2*d^3*x^4)*log(-(c*x + I)/(c*x -
I))^2 + integral(1/140*(-140*I*a^2*c^5*d^3*x^8 - 420*a^2*c^4*d^3*x^7 + 280*I*a^2*c^3*d^3*x^6 - 280*a^2*c^2*d^3
*x^5 + 420*I*a^2*c*d^3*x^4 + 140*a^2*d^3*x^3 + (140*a*b*c^5*d^3*x^8 - 20*(21*I*a*b + b^2)*c^4*d^3*x^7 - 70*(4*
a*b - I*b^2)*c^3*d^3*x^6 - 28*(10*I*a*b - 3*b^2)*c^2*d^3*x^5 - 35*(12*a*b + I*b^2)*c*d^3*x^4 + 140*I*a*b*d^3*x
^3)*log(-(c*x + I)/(c*x - I)))/(c^2*x^2 + 1), x)

Sympy [F(-1)]

Timed out. \[ \int x^3 (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int x^3 (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx=\int { {\left (i \, c d x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}^{2} x^{3} \,d x } \]

[In]

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

[Out]

-1/7*I*a^2*c^3*d^3*x^7 - 1/2*a^2*c^2*d^3*x^6 + 3/5*I*a^2*c*d^3*x^5 + 1/4*b^2*d^3*x^4*arctan(c*x)^2 - 1/42*I*(1
2*x^7*arctan(c*x) - c*((2*c^4*x^6 - 3*c^2*x^4 + 6*x^2)/c^6 - 6*log(c^2*x^2 + 1)/c^8))*a*b*c^3*d^3 + 1/4*a^2*d^
3*x^4 - 1/15*(15*x^6*arctan(c*x) - c*((3*c^4*x^5 - 5*c^2*x^3 + 15*x)/c^6 - 15*arctan(c*x)/c^7))*a*b*c^2*d^3 +
3/10*I*(4*x^5*arctan(c*x) - c*((c^2*x^4 - 2*x^2)/c^4 + 2*log(c^2*x^2 + 1)/c^6))*a*b*c*d^3 + 1/6*(3*x^4*arctan(
c*x) - c*((c^2*x^3 - 3*x)/c^4 + 3*arctan(c*x)/c^5))*a*b*d^3 - 1/12*(2*c*((c^2*x^3 - 3*x)/c^4 + 3*arctan(c*x)/c
^5)*arctan(c*x) - (c^2*x^2 + 3*arctan(c*x)^2 - 4*log(c^2*x^2 + 1))/c^4)*b^2*d^3 + 1/280*(-10*I*b^2*c^3*d^3*x^7
- 35*b^2*c^2*d^3*x^6 + 42*I*b^2*c*d^3*x^5)*arctan(c*x)^2 + 1/280*(10*b^2*c^3*d^3*x^7 - 35*I*b^2*c^2*d^3*x^6 -
42*b^2*c*d^3*x^5)*arctan(c*x)*log(c^2*x^2 + 1) - 1/1120*(-10*I*b^2*c^3*d^3*x^7 - 35*b^2*c^2*d^3*x^6 + 42*I*b^
2*c*d^3*x^5)*log(c^2*x^2 + 1)^2 - I*integrate(1/560*(420*(b^2*c^5*d^3*x^8 - 2*b^2*c^3*d^3*x^6 - 3*b^2*c*d^3*x^
4)*arctan(c*x)^2 + 35*(b^2*c^5*d^3*x^8 - 2*b^2*c^3*d^3*x^6 - 3*b^2*c*d^3*x^4)*log(c^2*x^2 + 1)^2 - 12*(15*b^2*
c^4*d^3*x^7 - 14*b^2*c^2*d^3*x^5)*arctan(c*x) + 2*(10*b^2*c^5*d^3*x^8 - 77*b^2*c^3*d^3*x^6 - 210*(b^2*c^4*d^3*
x^7 + b^2*c^2*d^3*x^5)*arctan(c*x))*log(c^2*x^2 + 1))/(c^2*x^2 + 1), x) - integrate(1/560*(1260*(b^2*c^4*d^3*x
^7 + b^2*c^2*d^3*x^5)*arctan(c*x)^2 + 105*(b^2*c^4*d^3*x^7 + b^2*c^2*d^3*x^5)*log(c^2*x^2 + 1)^2 + 4*(10*b^2*c
^5*d^3*x^8 - 77*b^2*c^3*d^3*x^6)*arctan(c*x) + 2*(45*b^2*c^4*d^3*x^7 - 42*b^2*c^2*d^3*x^5 + 70*(b^2*c^5*d^3*x^
8 - 2*b^2*c^3*d^3*x^6 - 3*b^2*c*d^3*x^4)*arctan(c*x))*log(c^2*x^2 + 1))/(c^2*x^2 + 1), x)

Giac [F]

\[ \int x^3 (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx=\int { {\left (i \, c d x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}^{2} x^{3} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int x^3 (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx=\int x^3\,{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^3 \,d x \]

[In]

int(x^3*(a + b*atan(c*x))^2*(d + c*d*x*1i)^3,x)

[Out]

int(x^3*(a + b*atan(c*x))^2*(d + c*d*x*1i)^3, x)