\(\int x^2 (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx\) [85]

Optimal result

Integrand size = 25, antiderivative size = 402 \[ \int x^2 (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx=\frac {11 i a b d^3 x}{6 c^2}+\frac {37 b^2 d^3 x}{30 c^2}+\frac {61 i b^2 d^3 x^2}{180 c}-\frac {1}{10} b^2 d^3 x^3-\frac {1}{60} i b^2 c d^3 x^4-\frac {37 b^2 d^3 \arctan (c x)}{30 c^3}+\frac {11 i b^2 d^3 x \arctan (c x)}{6 c^2}-\frac {14 b d^3 x^2 (a+b \arctan (c x))}{15 c}-\frac {11}{18} i b d^3 x^3 (a+b \arctan (c x))+\frac {3}{10} b c d^3 x^4 (a+b \arctan (c x))+\frac {1}{15} i b c^2 d^3 x^5 (a+b \arctan (c x))-\frac {37 i d^3 (a+b \arctan (c x))^2}{20 c^3}+\frac {1}{3} d^3 x^3 (a+b \arctan (c x))^2+\frac {3}{4} i c d^3 x^4 (a+b \arctan (c x))^2-\frac {3}{5} c^2 d^3 x^5 (a+b \arctan (c x))^2-\frac {1}{6} i c^3 d^3 x^6 (a+b \arctan (c x))^2-\frac {28 b d^3 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{15 c^3}-\frac {113 i b^2 d^3 \log \left (1+c^2 x^2\right )}{90 c^3}-\frac {14 i b^2 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{15 c^3} \]

[Out]

-14/15*I*b^2*d^3*polylog(2,1-2/(1+I*c*x))/c^3+37/30*b^2*d^3*x/c^2+1/15*I*b*c^2*d^3*x^5*(a+b*arctan(c*x))-1/10*
b^2*d^3*x^3+61/180*I*b^2*d^3*x^2/c-37/30*b^2*d^3*arctan(c*x)/c^3-37/20*I*d^3*(a+b*arctan(c*x))^2/c^3-14/15*b*d
^3*x^2*(a+b*arctan(c*x))/c+11/6*I*a*b*d^3*x/c^2+3/10*b*c*d^3*x^4*(a+b*arctan(c*x))-1/60*I*b^2*c*d^3*x^4-11/18*
I*b*d^3*x^3*(a+b*arctan(c*x))+1/3*d^3*x^3*(a+b*arctan(c*x))^2-1/6*I*c^3*d^3*x^6*(a+b*arctan(c*x))^2-3/5*c^2*d^
3*x^5*(a+b*arctan(c*x))^2+11/6*I*b^2*d^3*x*arctan(c*x)/c^2-28/15*b*d^3*(a+b*arctan(c*x))*ln(2/(1+I*c*x))/c^3+3
/4*I*c*d^3*x^4*(a+b*arctan(c*x))^2-113/90*I*b^2*d^3*ln(c^2*x^2+1)/c^3

Rubi [A] (verified)

Time = 0.85 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.00, number of steps used = 52, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {4996, 4946, 5036, 327, 209, 5040, 4964, 2449, 2352, 272, 45, 4930, 266, 5004, 308} \[ \int x^2 (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx=-\frac {1}{6} i c^3 d^3 x^6 (a+b \arctan (c x))^2-\frac {37 i d^3 (a+b \arctan (c x))^2}{20 c^3}-\frac {28 b d^3 \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{15 c^3}-\frac {3}{5} c^2 d^3 x^5 (a+b \arctan (c x))^2+\frac {1}{15} i b c^2 d^3 x^5 (a+b \arctan (c x))+\frac {3}{4} i c d^3 x^4 (a+b \arctan (c x))^2+\frac {3}{10} b c d^3 x^4 (a+b \arctan (c x))+\frac {1}{3} d^3 x^3 (a+b \arctan (c x))^2-\frac {11}{18} i b d^3 x^3 (a+b \arctan (c x))-\frac {14 b d^3 x^2 (a+b \arctan (c x))}{15 c}+\frac {11 i a b d^3 x}{6 c^2}-\frac {37 b^2 d^3 \arctan (c x)}{30 c^3}+\frac {11 i b^2 d^3 x \arctan (c x)}{6 c^2}-\frac {14 i b^2 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{15 c^3}+\frac {37 b^2 d^3 x}{30 c^2}-\frac {113 i b^2 d^3 \log \left (c^2 x^2+1\right )}{90 c^3}-\frac {1}{60} i b^2 c d^3 x^4+\frac {61 i b^2 d^3 x^2}{180 c}-\frac {1}{10} b^2 d^3 x^3 \]

[In]

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

[Out]

(((11*I)/6)*a*b*d^3*x)/c^2 + (37*b^2*d^3*x)/(30*c^2) + (((61*I)/180)*b^2*d^3*x^2)/c - (b^2*d^3*x^3)/10 - (I/60
)*b^2*c*d^3*x^4 - (37*b^2*d^3*ArcTan[c*x])/(30*c^3) + (((11*I)/6)*b^2*d^3*x*ArcTan[c*x])/c^2 - (14*b*d^3*x^2*(
a + b*ArcTan[c*x]))/(15*c) - ((11*I)/18)*b*d^3*x^3*(a + b*ArcTan[c*x]) + (3*b*c*d^3*x^4*(a + b*ArcTan[c*x]))/1
0 + (I/15)*b*c^2*d^3*x^5*(a + b*ArcTan[c*x]) - (((37*I)/20)*d^3*(a + b*ArcTan[c*x])^2)/c^3 + (d^3*x^3*(a + b*A
rcTan[c*x])^2)/3 + ((3*I)/4)*c*d^3*x^4*(a + b*ArcTan[c*x])^2 - (3*c^2*d^3*x^5*(a + b*ArcTan[c*x])^2)/5 - (I/6)
*c^3*d^3*x^6*(a + b*ArcTan[c*x])^2 - (28*b*d^3*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(15*c^3) - (((113*I)/90
)*b^2*d^3*Log[1 + c^2*x^2])/c^3 - (((14*I)/15)*b^2*d^3*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^3

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^2 (a+b \arctan (c x))^2+3 i c d^3 x^3 (a+b \arctan (c x))^2-3 c^2 d^3 x^4 (a+b \arctan (c x))^2-i c^3 d^3 x^5 (a+b \arctan (c x))^2\right ) \, dx \\ & = d^3 \int x^2 (a+b \arctan (c x))^2 \, dx+\left (3 i c d^3\right ) \int x^3 (a+b \arctan (c x))^2 \, dx-\left (3 c^2 d^3\right ) \int x^4 (a+b \arctan (c x))^2 \, dx-\left (i c^3 d^3\right ) \int x^5 (a+b \arctan (c x))^2 \, dx \\ & = \frac {1}{3} d^3 x^3 (a+b \arctan (c x))^2+\frac {3}{4} i c d^3 x^4 (a+b \arctan (c x))^2-\frac {3}{5} c^2 d^3 x^5 (a+b \arctan (c x))^2-\frac {1}{6} i c^3 d^3 x^6 (a+b \arctan (c x))^2-\frac {1}{3} \left (2 b c d^3\right ) \int \frac {x^3 (a+b \arctan (c x))}{1+c^2 x^2} \, dx-\frac {1}{2} \left (3 i b c^2 d^3\right ) \int \frac {x^4 (a+b \arctan (c x))}{1+c^2 x^2} \, dx+\frac {1}{5} \left (6 b c^3 d^3\right ) \int \frac {x^5 (a+b \arctan (c x))}{1+c^2 x^2} \, dx+\frac {1}{3} \left (i b c^4 d^3\right ) \int \frac {x^6 (a+b \arctan (c x))}{1+c^2 x^2} \, dx \\ & = \frac {1}{3} d^3 x^3 (a+b \arctan (c x))^2+\frac {3}{4} i c d^3 x^4 (a+b \arctan (c x))^2-\frac {3}{5} c^2 d^3 x^5 (a+b \arctan (c x))^2-\frac {1}{6} i c^3 d^3 x^6 (a+b \arctan (c x))^2-\frac {1}{2} \left (3 i b d^3\right ) \int x^2 (a+b \arctan (c x)) \, dx+\frac {1}{2} \left (3 i b d^3\right ) \int \frac {x^2 (a+b \arctan (c x))}{1+c^2 x^2} \, dx-\frac {\left (2 b d^3\right ) \int x (a+b \arctan (c x)) \, dx}{3 c}+\frac {\left (2 b d^3\right ) \int \frac {x (a+b \arctan (c x))}{1+c^2 x^2} \, dx}{3 c}+\frac {1}{5} \left (6 b c d^3\right ) \int x^3 (a+b \arctan (c x)) \, dx-\frac {1}{5} \left (6 b c d^3\right ) \int \frac {x^3 (a+b \arctan (c x))}{1+c^2 x^2} \, dx+\frac {1}{3} \left (i b c^2 d^3\right ) \int x^4 (a+b \arctan (c x)) \, dx-\frac {1}{3} \left (i b c^2 d^3\right ) \int \frac {x^4 (a+b \arctan (c x))}{1+c^2 x^2} \, dx \\ & = -\frac {b d^3 x^2 (a+b \arctan (c x))}{3 c}-\frac {1}{2} i b d^3 x^3 (a+b \arctan (c x))+\frac {3}{10} b c d^3 x^4 (a+b \arctan (c x))+\frac {1}{15} i b c^2 d^3 x^5 (a+b \arctan (c x))-\frac {i d^3 (a+b \arctan (c x))^2}{3 c^3}+\frac {1}{3} d^3 x^3 (a+b \arctan (c x))^2+\frac {3}{4} i c d^3 x^4 (a+b \arctan (c x))^2-\frac {3}{5} c^2 d^3 x^5 (a+b \arctan (c x))^2-\frac {1}{6} i c^3 d^3 x^6 (a+b \arctan (c x))^2-\frac {1}{3} \left (i b d^3\right ) \int x^2 (a+b \arctan (c x)) \, dx+\frac {1}{3} \left (i b d^3\right ) \int \frac {x^2 (a+b \arctan (c x))}{1+c^2 x^2} \, dx+\frac {1}{3} \left (b^2 d^3\right ) \int \frac {x^2}{1+c^2 x^2} \, dx+\frac {\left (3 i b d^3\right ) \int (a+b \arctan (c x)) \, dx}{2 c^2}-\frac {\left (3 i b d^3\right ) \int \frac {a+b \arctan (c x)}{1+c^2 x^2} \, dx}{2 c^2}-\frac {\left (2 b d^3\right ) \int \frac {a+b \arctan (c x)}{i-c x} \, dx}{3 c^2}-\frac {\left (6 b d^3\right ) \int x (a+b \arctan (c x)) \, dx}{5 c}+\frac {\left (6 b d^3\right ) \int \frac {x (a+b \arctan (c x))}{1+c^2 x^2} \, dx}{5 c}+\frac {1}{2} \left (i b^2 c d^3\right ) \int \frac {x^3}{1+c^2 x^2} \, dx-\frac {1}{10} \left (3 b^2 c^2 d^3\right ) \int \frac {x^4}{1+c^2 x^2} \, dx-\frac {1}{15} \left (i b^2 c^3 d^3\right ) \int \frac {x^5}{1+c^2 x^2} \, dx \\ & = \frac {3 i a b d^3 x}{2 c^2}+\frac {b^2 d^3 x}{3 c^2}-\frac {14 b d^3 x^2 (a+b \arctan (c x))}{15 c}-\frac {11}{18} i b d^3 x^3 (a+b \arctan (c x))+\frac {3}{10} b c d^3 x^4 (a+b \arctan (c x))+\frac {1}{15} i b c^2 d^3 x^5 (a+b \arctan (c x))-\frac {101 i d^3 (a+b \arctan (c x))^2}{60 c^3}+\frac {1}{3} d^3 x^3 (a+b \arctan (c x))^2+\frac {3}{4} i c d^3 x^4 (a+b \arctan (c x))^2-\frac {3}{5} c^2 d^3 x^5 (a+b \arctan (c x))^2-\frac {1}{6} i c^3 d^3 x^6 (a+b \arctan (c x))^2-\frac {2 b d^3 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{3 c^3}+\frac {1}{5} \left (3 b^2 d^3\right ) \int \frac {x^2}{1+c^2 x^2} \, dx+\frac {\left (i b d^3\right ) \int (a+b \arctan (c x)) \, dx}{3 c^2}-\frac {\left (i b d^3\right ) \int \frac {a+b \arctan (c x)}{1+c^2 x^2} \, dx}{3 c^2}-\frac {\left (6 b d^3\right ) \int \frac {a+b \arctan (c x)}{i-c x} \, dx}{5 c^2}+\frac {\left (3 i b^2 d^3\right ) \int \arctan (c x) \, dx}{2 c^2}-\frac {\left (b^2 d^3\right ) \int \frac {1}{1+c^2 x^2} \, dx}{3 c^2}+\frac {\left (2 b^2 d^3\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{3 c^2}+\frac {1}{9} \left (i b^2 c d^3\right ) \int \frac {x^3}{1+c^2 x^2} \, dx+\frac {1}{4} \left (i b^2 c d^3\right ) \text {Subst}\left (\int \frac {x}{1+c^2 x} \, dx,x,x^2\right )-\frac {1}{10} \left (3 b^2 c^2 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}{30} \left (i b^2 c^3 d^3\right ) \text {Subst}\left (\int \frac {x^2}{1+c^2 x} \, dx,x,x^2\right ) \\ & = \frac {11 i a b d^3 x}{6 c^2}+\frac {37 b^2 d^3 x}{30 c^2}-\frac {1}{10} b^2 d^3 x^3-\frac {b^2 d^3 \arctan (c x)}{3 c^3}+\frac {3 i b^2 d^3 x \arctan (c x)}{2 c^2}-\frac {14 b d^3 x^2 (a+b \arctan (c x))}{15 c}-\frac {11}{18} i b d^3 x^3 (a+b \arctan (c x))+\frac {3}{10} b c d^3 x^4 (a+b \arctan (c x))+\frac {1}{15} i b c^2 d^3 x^5 (a+b \arctan (c x))-\frac {37 i d^3 (a+b \arctan (c x))^2}{20 c^3}+\frac {1}{3} d^3 x^3 (a+b \arctan (c x))^2+\frac {3}{4} i c d^3 x^4 (a+b \arctan (c x))^2-\frac {3}{5} c^2 d^3 x^5 (a+b \arctan (c x))^2-\frac {1}{6} i c^3 d^3 x^6 (a+b \arctan (c x))^2-\frac {28 b d^3 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{15 c^3}-\frac {\left (2 i 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 )}{3 c^3}+\frac {\left (i b^2 d^3\right ) \int \arctan (c x) \, dx}{3 c^2}-\frac {\left (3 b^2 d^3\right ) \int \frac {1}{1+c^2 x^2} \, dx}{10 c^2}-\frac {\left (3 b^2 d^3\right ) \int \frac {1}{1+c^2 x^2} \, dx}{5 c^2}+\frac {\left (6 b^2 d^3\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{5 c^2}-\frac {\left (3 i b^2 d^3\right ) \int \frac {x}{1+c^2 x^2} \, dx}{2 c}+\frac {1}{18} \left (i b^2 c d^3\right ) \text {Subst}\left (\int \frac {x}{1+c^2 x} \, dx,x,x^2\right )+\frac {1}{4} \left (i b^2 c 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}{30} \left (i b^2 c^3 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 {11 i a b d^3 x}{6 c^2}+\frac {37 b^2 d^3 x}{30 c^2}+\frac {17 i b^2 d^3 x^2}{60 c}-\frac {1}{10} b^2 d^3 x^3-\frac {1}{60} i b^2 c d^3 x^4-\frac {37 b^2 d^3 \arctan (c x)}{30 c^3}+\frac {11 i b^2 d^3 x \arctan (c x)}{6 c^2}-\frac {14 b d^3 x^2 (a+b \arctan (c x))}{15 c}-\frac {11}{18} i b d^3 x^3 (a+b \arctan (c x))+\frac {3}{10} b c d^3 x^4 (a+b \arctan (c x))+\frac {1}{15} i b c^2 d^3 x^5 (a+b \arctan (c x))-\frac {37 i d^3 (a+b \arctan (c x))^2}{20 c^3}+\frac {1}{3} d^3 x^3 (a+b \arctan (c x))^2+\frac {3}{4} i c d^3 x^4 (a+b \arctan (c x))^2-\frac {3}{5} c^2 d^3 x^5 (a+b \arctan (c x))^2-\frac {1}{6} i c^3 d^3 x^6 (a+b \arctan (c x))^2-\frac {28 b d^3 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{15 c^3}-\frac {31 i b^2 d^3 \log \left (1+c^2 x^2\right )}{30 c^3}-\frac {i b^2 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{3 c^3}-\frac {\left (6 i 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^3}-\frac {\left (i b^2 d^3\right ) \int \frac {x}{1+c^2 x^2} \, dx}{3 c}+\frac {1}{18} \left (i b^2 c 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 {11 i a b d^3 x}{6 c^2}+\frac {37 b^2 d^3 x}{30 c^2}+\frac {61 i b^2 d^3 x^2}{180 c}-\frac {1}{10} b^2 d^3 x^3-\frac {1}{60} i b^2 c d^3 x^4-\frac {37 b^2 d^3 \arctan (c x)}{30 c^3}+\frac {11 i b^2 d^3 x \arctan (c x)}{6 c^2}-\frac {14 b d^3 x^2 (a+b \arctan (c x))}{15 c}-\frac {11}{18} i b d^3 x^3 (a+b \arctan (c x))+\frac {3}{10} b c d^3 x^4 (a+b \arctan (c x))+\frac {1}{15} i b c^2 d^3 x^5 (a+b \arctan (c x))-\frac {37 i d^3 (a+b \arctan (c x))^2}{20 c^3}+\frac {1}{3} d^3 x^3 (a+b \arctan (c x))^2+\frac {3}{4} i c d^3 x^4 (a+b \arctan (c x))^2-\frac {3}{5} c^2 d^3 x^5 (a+b \arctan (c x))^2-\frac {1}{6} i c^3 d^3 x^6 (a+b \arctan (c x))^2-\frac {28 b d^3 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{15 c^3}-\frac {113 i b^2 d^3 \log \left (1+c^2 x^2\right )}{90 c^3}-\frac {14 i b^2 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{15 c^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.13 (sec) , antiderivative size = 369, normalized size of antiderivative = 0.92 \[ \int x^2 (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx=\frac {d^3 \left (-162 a b+64 i b^2+330 i a b c x+222 b^2 c x-168 a b c^2 x^2+61 i b^2 c^2 x^2+60 a^2 c^3 x^3-110 i a b c^3 x^3-18 b^2 c^3 x^3+135 i a^2 c^4 x^4+54 a b c^4 x^4-3 i b^2 c^4 x^4-108 a^2 c^5 x^5+12 i a b c^5 x^5-30 i a^2 c^6 x^6+3 b^2 (-i+c x)^4 \left (i+4 c x-10 i c^2 x^2\right ) \arctan (c x)^2+2 b \arctan (c x) \left (b \left (-111+165 i c x-84 c^2 x^2-55 i c^3 x^3+27 c^4 x^4+6 i c^5 x^5\right )+3 a \left (-55 i+20 c^3 x^3+45 i c^4 x^4-36 c^5 x^5-10 i c^6 x^6\right )-168 b \log \left (1+e^{2 i \arctan (c x)}\right )\right )+168 a b \log \left (1+c^2 x^2\right )-226 i b^2 \log \left (1+c^2 x^2\right )+168 i b^2 \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )\right )}{180 c^3} \]

[In]

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

[Out]

(d^3*(-162*a*b + (64*I)*b^2 + (330*I)*a*b*c*x + 222*b^2*c*x - 168*a*b*c^2*x^2 + (61*I)*b^2*c^2*x^2 + 60*a^2*c^
3*x^3 - (110*I)*a*b*c^3*x^3 - 18*b^2*c^3*x^3 + (135*I)*a^2*c^4*x^4 + 54*a*b*c^4*x^4 - (3*I)*b^2*c^4*x^4 - 108*
a^2*c^5*x^5 + (12*I)*a*b*c^5*x^5 - (30*I)*a^2*c^6*x^6 + 3*b^2*(-I + c*x)^4*(I + 4*c*x - (10*I)*c^2*x^2)*ArcTan
[c*x]^2 + 2*b*ArcTan[c*x]*(b*(-111 + (165*I)*c*x - 84*c^2*x^2 - (55*I)*c^3*x^3 + 27*c^4*x^4 + (6*I)*c^5*x^5) +
3*a*(-55*I + 20*c^3*x^3 + (45*I)*c^4*x^4 - 36*c^5*x^5 - (10*I)*c^6*x^6) - 168*b*Log[1 + E^((2*I)*ArcTan[c*x])
]) + 168*a*b*Log[1 + c^2*x^2] - (226*I)*b^2*Log[1 + c^2*x^2] + (168*I)*b^2*PolyLog[2, -E^((2*I)*ArcTan[c*x])])
)/(180*c^3)

Maple [A] (verified)

Time = 2.84 (sec) , antiderivative size = 488, normalized size of antiderivative = 1.21

method	result	size
parts	\(d^{3} a^{2} \left (-\frac {1}{6} i c^{3} x^{6}-\frac {3}{5} c^{2} x^{5}+\frac {3}{4} i c \,x^{4}+\frac {1}{3} x^{3}\right )+\frac {b^{2} d^{3} \left (-\frac {113 i \ln \left (c^{2} x^{2}+1\right )}{90}-\frac {3 \arctan \left (c x \right )^{2} c^{5} x^{5}}{5}+\frac {i \arctan \left (c x \right ) c^{5} x^{5}}{15}+\frac {c^{3} x^{3} \arctan \left (c x \right )^{2}}{3}-\frac {11 i \arctan \left (c x \right ) c^{3} x^{3}}{18}+\frac {11 i \arctan \left (c x \right ) c x}{6}+\frac {3 c^{4} x^{4} \arctan \left (c x \right )}{10}+\frac {61 i c^{2} x^{2}}{180}-\frac {14 c^{2} x^{2} \arctan \left (c x \right )}{15}+\frac {14 \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{15}+\frac {7 i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{15}-\frac {7 i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{15}-\frac {11 i \arctan \left (c x \right )^{2}}{12}+\frac {3 i \arctan \left (c x \right )^{2} c^{4} x^{4}}{4}+\frac {37 c x}{30}-\frac {i c^{4} x^{4}}{60}-\frac {c^{3} x^{3}}{10}-\frac {i \arctan \left (c x \right )^{2} c^{6} x^{6}}{6}-\frac {37 \arctan \left (c x \right )}{30}\right )}{c^{3}}+\frac {2 a \,d^{3} b \left (-\frac {i \arctan \left (c x \right ) c^{6} x^{6}}{6}-\frac {3 c^{5} x^{5} \arctan \left (c x \right )}{5}+\frac {3 i \arctan \left (c x \right ) c^{4} x^{4}}{4}+\frac {c^{3} x^{3} \arctan \left (c x \right )}{3}+\frac {11 i c x}{12}+\frac {i c^{5} x^{5}}{30}+\frac {3 c^{4} x^{4}}{20}-\frac {11 i c^{3} x^{3}}{36}-\frac {7 c^{2} x^{2}}{15}+\frac {7 \ln \left (c^{2} x^{2}+1\right )}{15}-\frac {11 i \arctan \left (c x \right )}{12}\right )}{c^{3}}\)	\(488\)
derivativedivides	\(\frac {d^{3} a^{2} \left (-\frac {1}{6} i c^{6} x^{6}-\frac {3}{5} c^{5} x^{5}+\frac {3}{4} i c^{4} x^{4}+\frac {1}{3} c^{3} x^{3}\right )+b^{2} d^{3} \left (-\frac {113 i \ln \left (c^{2} x^{2}+1\right )}{90}-\frac {3 \arctan \left (c x \right )^{2} c^{5} x^{5}}{5}+\frac {i \arctan \left (c x \right ) c^{5} x^{5}}{15}+\frac {c^{3} x^{3} \arctan \left (c x \right )^{2}}{3}-\frac {11 i \arctan \left (c x \right ) c^{3} x^{3}}{18}+\frac {11 i \arctan \left (c x \right ) c x}{6}+\frac {3 c^{4} x^{4} \arctan \left (c x \right )}{10}+\frac {61 i c^{2} x^{2}}{180}-\frac {14 c^{2} x^{2} \arctan \left (c x \right )}{15}+\frac {14 \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{15}+\frac {7 i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{15}-\frac {7 i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{15}-\frac {11 i \arctan \left (c x \right )^{2}}{12}+\frac {3 i \arctan \left (c x \right )^{2} c^{4} x^{4}}{4}+\frac {37 c x}{30}-\frac {i c^{4} x^{4}}{60}-\frac {c^{3} x^{3}}{10}-\frac {i \arctan \left (c x \right )^{2} c^{6} x^{6}}{6}-\frac {37 \arctan \left (c x \right )}{30}\right )+2 a \,d^{3} b \left (-\frac {i \arctan \left (c x \right ) c^{6} x^{6}}{6}-\frac {3 c^{5} x^{5} \arctan \left (c x \right )}{5}+\frac {3 i \arctan \left (c x \right ) c^{4} x^{4}}{4}+\frac {c^{3} x^{3} \arctan \left (c x \right )}{3}+\frac {11 i c x}{12}+\frac {i c^{5} x^{5}}{30}+\frac {3 c^{4} x^{4}}{20}-\frac {11 i c^{3} x^{3}}{36}-\frac {7 c^{2} x^{2}}{15}+\frac {7 \ln \left (c^{2} x^{2}+1\right )}{15}-\frac {11 i \arctan \left (c x \right )}{12}\right )}{c^{3}}\)	\(491\)
default	\(\frac {d^{3} a^{2} \left (-\frac {1}{6} i c^{6} x^{6}-\frac {3}{5} c^{5} x^{5}+\frac {3}{4} i c^{4} x^{4}+\frac {1}{3} c^{3} x^{3}\right )+b^{2} d^{3} \left (-\frac {113 i \ln \left (c^{2} x^{2}+1\right )}{90}-\frac {3 \arctan \left (c x \right )^{2} c^{5} x^{5}}{5}+\frac {i \arctan \left (c x \right ) c^{5} x^{5}}{15}+\frac {c^{3} x^{3} \arctan \left (c x \right )^{2}}{3}-\frac {11 i \arctan \left (c x \right ) c^{3} x^{3}}{18}+\frac {11 i \arctan \left (c x \right ) c x}{6}+\frac {3 c^{4} x^{4} \arctan \left (c x \right )}{10}+\frac {61 i c^{2} x^{2}}{180}-\frac {14 c^{2} x^{2} \arctan \left (c x \right )}{15}+\frac {14 \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{15}+\frac {7 i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{15}-\frac {7 i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{15}-\frac {11 i \arctan \left (c x \right )^{2}}{12}+\frac {3 i \arctan \left (c x \right )^{2} c^{4} x^{4}}{4}+\frac {37 c x}{30}-\frac {i c^{4} x^{4}}{60}-\frac {c^{3} x^{3}}{10}-\frac {i \arctan \left (c x \right )^{2} c^{6} x^{6}}{6}-\frac {37 \arctan \left (c x \right )}{30}\right )+2 a \,d^{3} b \left (-\frac {i \arctan \left (c x \right ) c^{6} x^{6}}{6}-\frac {3 c^{5} x^{5} \arctan \left (c x \right )}{5}+\frac {3 i \arctan \left (c x \right ) c^{4} x^{4}}{4}+\frac {c^{3} x^{3} \arctan \left (c x \right )}{3}+\frac {11 i c x}{12}+\frac {i c^{5} x^{5}}{30}+\frac {3 c^{4} x^{4}}{20}-\frac {11 i c^{3} x^{3}}{36}-\frac {7 c^{2} x^{2}}{15}+\frac {7 \ln \left (c^{2} x^{2}+1\right )}{15}-\frac {11 i \arctan \left (c x \right )}{12}\right )}{c^{3}}\)	\(491\)
risch	\(-\frac {b^{2} d^{3} x^{3}}{10}+\frac {37 b^{2} d^{3} x}{30 c^{2}}-\frac {37 b^{2} d^{3} \arctan \left (c x \right )}{30 c^{3}}+\frac {3 a b c \,d^{3} x^{4}}{10}-\frac {3 a^{2} c^{2} d^{3} x^{5}}{5}-\frac {337 a b \,d^{3}}{90 c^{3}}+\frac {a^{2} d^{3} x^{3}}{3}+\frac {14 a b \,d^{3} \ln \left (c^{2} x^{2}+1\right )}{15 c^{3}}-\frac {14 d^{3} a b \,x^{2}}{15 c}-\frac {11 i a b \,d^{3} \arctan \left (c x \right )}{6 c^{3}}-\frac {3 d^{3} c a b \ln \left (-i c x +1\right ) x^{4}}{4}+\frac {d^{3} c^{3} b a \ln \left (-i c x +1\right ) x^{6}}{6}-\frac {7 i b^{2} d^{3} \ln \left (-i c x +1\right ) x^{2}}{15 c}+\frac {3 i b^{2} d^{3} c \ln \left (-i c x +1\right ) x^{4}}{20}+\frac {i b \,d^{3} c^{2} x^{5} a}{15}+\frac {i b^{2} d^{3} \left (10 c^{6} x^{6}-36 i c^{5} x^{5}-45 c^{4} x^{4}+20 i c^{3} x^{3}-1\right ) \ln \left (i c x +1\right )^{2}}{240 c^{3}}-\frac {3 i d^{3} c \,b^{2} \ln \left (-i c x +1\right )^{2} x^{4}}{16}+\frac {i d^{3} a b \ln \left (-i c x +1\right ) x^{3}}{3}+\frac {i d^{3} c^{3} b^{2} \ln \left (-i c x +1\right )^{2} x^{6}}{24}+\frac {14 i b^{2} d^{3} \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (-i c x +1\right )}{15 c^{3}}-\frac {14 i 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 )}{15 c^{3}}+\frac {11 i a b \,d^{3} x}{6 c^{2}}+\left (-\frac {i b^{2} d^{3} \left (10 c^{3} x^{6}-36 i c^{2} x^{5}-45 c \,x^{4}+20 i x^{3}\right ) \ln \left (-i c x +1\right )}{120}-\frac {b \,d^{3} \left (60 a \,c^{6} x^{6}-216 i a \,c^{5} x^{5}-12 b \,c^{5} x^{5}+54 i b \,c^{4} x^{4}-270 a \,c^{4} x^{4}+120 i a \,c^{3} x^{3}+110 b \,c^{3} x^{3}-168 i b \,c^{2} x^{2}+333 i b \ln \left (-i c x +1\right )-330 x b c \right )}{360 c^{3}}\right ) \ln \left (i c x +1\right )-\frac {3 i d^{3} c^{2} a b \ln \left (-i c x +1\right ) x^{5}}{5}-\frac {i b^{2} c \,d^{3} x^{4}}{60}-\frac {113 i b^{2} d^{3} \ln \left (c^{2} x^{2}+1\right )}{90 c^{3}}+\frac {61 i b^{2} d^{3} x^{2}}{180 c}-\frac {11 i b \,d^{3} x^{3} a}{18}-\frac {d^{3} c^{2} b^{2} \ln \left (-i c x +1\right ) x^{5}}{30}+\frac {3 d^{3} c^{2} b^{2} \ln \left (-i c x +1\right )^{2} x^{5}}{20}+\frac {37 i d^{3} b^{2} \ln \left (-i c x +1\right )^{2}}{80 c^{3}}-\frac {11 d^{3} b^{2} \ln \left (-i c x +1\right ) x}{12 c^{2}}-\frac {14 i b^{2} d^{3} \operatorname {dilog}\left (\frac {1}{2}-\frac {i c x}{2}\right )}{15 c^{3}}-\frac {i a^{2} c^{3} d^{3} x^{6}}{6}+\frac {3 i d^{3} c \,x^{4} a^{2}}{4}+\frac {76 i b^{2} d^{3}}{45 c^{3}}-\frac {d^{3} b^{2} \ln \left (-i c x +1\right )^{2} x^{3}}{12}+\frac {11 d^{3} b^{2} \ln \left (-i c x +1\right ) x^{3}}{36}-\frac {37 i d^{3} a^{2}}{20 c^{3}}\)	\(868\)

[In]

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

[Out]

d^3*a^2*(-1/6*I*c^3*x^6-3/5*c^2*x^5+3/4*I*c*x^4+1/3*x^3)+b^2*d^3/c^3*(-113/90*I*ln(c^2*x^2+1)-3/5*arctan(c*x)^
2*c^5*x^5+1/15*I*arctan(c*x)*c^5*x^5+1/3*c^3*x^3*arctan(c*x)^2-11/18*I*arctan(c*x)*c^3*x^3+11/6*I*arctan(c*x)*
c*x+3/10*c^4*x^4*arctan(c*x)+61/180*I*c^2*x^2-14/15*c^2*x^2*arctan(c*x)+14/15*arctan(c*x)*ln(c^2*x^2+1)+7/15*I
*(ln(c*x-I)*ln(c^2*x^2+1)-dilog(-1/2*I*(c*x+I))-ln(c*x-I)*ln(-1/2*I*(c*x+I))-1/2*ln(c*x-I)^2)-7/15*I*(ln(c*x+I
)*ln(c^2*x^2+1)-dilog(1/2*I*(c*x-I))-ln(c*x+I)*ln(1/2*I*(c*x-I))-1/2*ln(c*x+I)^2)-11/12*I*arctan(c*x)^2+3/4*I*
arctan(c*x)^2*c^4*x^4+37/30*c*x-1/60*I*c^4*x^4-1/10*c^3*x^3-1/6*I*arctan(c*x)^2*c^6*x^6-37/30*arctan(c*x))+2*a
*d^3*b/c^3*(-1/6*I*arctan(c*x)*c^6*x^6-3/5*c^5*x^5*arctan(c*x)+3/4*I*arctan(c*x)*c^4*x^4+1/3*c^3*x^3*arctan(c*
x)+11/12*I*c*x+1/30*I*c^5*x^5+3/20*c^4*x^4-11/36*I*c^3*x^3-7/15*c^2*x^2+7/15*ln(c^2*x^2+1)-11/12*I*arctan(c*x)
)

Fricas [F]

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

[In]

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

[Out]

1/240*(10*I*b^2*c^3*d^3*x^6 + 36*b^2*c^2*d^3*x^5 - 45*I*b^2*c*d^3*x^4 - 20*b^2*d^3*x^3)*log(-(c*x + I)/(c*x -
I))^2 + integral(1/60*(-60*I*a^2*c^5*d^3*x^7 - 180*a^2*c^4*d^3*x^6 + 120*I*a^2*c^3*d^3*x^5 - 120*a^2*c^2*d^3*x
^4 + 180*I*a^2*c*d^3*x^3 + 60*a^2*d^3*x^2 + (60*a*b*c^5*d^3*x^7 - 10*(18*I*a*b + b^2)*c^4*d^3*x^6 - 12*(10*a*b
- 3*I*b^2)*c^3*d^3*x^5 - 15*(8*I*a*b - 3*b^2)*c^2*d^3*x^4 - 20*(9*a*b + I*b^2)*c*d^3*x^3 + 60*I*a*b*d^3*x^2)*
log(-(c*x + I)/(c*x - I)))/(c^2*x^2 + 1), x)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

-1/6*I*a^2*c^3*d^3*x^6 - 3/5*a^2*c^2*d^3*x^5 + 3/4*I*a^2*c*d^3*x^4 - 1/45*I*(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^3*d^3 - 3/10*(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^2*d^3 + 1/3*a^2*d^3*x^3 + 1/2*I*(3*x^4*arctan(c*x) - c*((c^2*x^3 - 3*x)/c
^4 + 3*arctan(c*x)/c^5))*a*b*c*d^3 + 1/3*(2*x^3*arctan(c*x) - c*(x^2/c^2 - log(c^2*x^2 + 1)/c^4))*a*b*d^3 + 1/
240*(-10*I*b^2*c^3*d^3*x^6 - 36*b^2*c^2*d^3*x^5 + 45*I*b^2*c*d^3*x^4 + 20*b^2*d^3*x^3)*arctan(c*x)^2 + 1/240*(
10*b^2*c^3*d^3*x^6 - 36*I*b^2*c^2*d^3*x^5 - 45*b^2*c*d^3*x^4 + 20*I*b^2*d^3*x^3)*arctan(c*x)*log(c^2*x^2 + 1)
- 1/960*(-10*I*b^2*c^3*d^3*x^6 - 36*b^2*c^2*d^3*x^5 + 45*I*b^2*c*d^3*x^4 + 20*b^2*d^3*x^3)*log(c^2*x^2 + 1)^2
- I*integrate(1/240*(180*(b^2*c^5*d^3*x^7 - 2*b^2*c^3*d^3*x^5 - 3*b^2*c*d^3*x^3)*arctan(c*x)^2 + 15*(b^2*c^5*d
^3*x^7 - 2*b^2*c^3*d^3*x^5 - 3*b^2*c*d^3*x^3)*log(c^2*x^2 + 1)^2 - 2*(46*b^2*c^4*d^3*x^6 - 65*b^2*c^2*d^3*x^4)
*arctan(c*x) + (10*b^2*c^5*d^3*x^7 - 81*b^2*c^3*d^3*x^5 + 20*b^2*c*d^3*x^3 - 60*(3*b^2*c^4*d^3*x^6 + 2*b^2*c^2
*d^3*x^4 - b^2*d^3*x^2)*arctan(c*x))*log(c^2*x^2 + 1))/(c^2*x^2 + 1), x) - integrate(1/240*(180*(3*b^2*c^4*d^3
*x^6 + 2*b^2*c^2*d^3*x^4 - b^2*d^3*x^2)*arctan(c*x)^2 + 15*(3*b^2*c^4*d^3*x^6 + 2*b^2*c^2*d^3*x^4 - b^2*d^3*x^
2)*log(c^2*x^2 + 1)^2 + 2*(10*b^2*c^5*d^3*x^7 - 81*b^2*c^3*d^3*x^5 + 20*b^2*c*d^3*x^3)*arctan(c*x) + (46*b^2*c
^4*d^3*x^6 - 65*b^2*c^2*d^3*x^4 + 60*(b^2*c^5*d^3*x^7 - 2*b^2*c^3*d^3*x^5 - 3*b^2*c*d^3*x^3)*arctan(c*x))*log(
c^2*x^2 + 1))/(c^2*x^2 + 1), x)

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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