3.77 \(\int x^2 (d+i c d x)^2 (a+b \tan ^{-1}(c x))^2 \, dx\)

Optimal. Leaf size=333 \[ -\frac {31 i d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{30 c^3}-\frac {16 b d^2 \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{15 c^3}-\frac {1}{5} c^2 d^2 x^5 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {i a b d^2 x}{c^2}+\frac {1}{2} i c d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{10} b c d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{3} d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {1}{3} i b d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )-\frac {8 b d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{15 c}-\frac {8 i b^2 d^2 \text {Li}_2\left (1-\frac {2}{i c x+1}\right )}{15 c^3}-\frac {19 b^2 d^2 \tan ^{-1}(c x)}{30 c^3}+\frac {19 b^2 d^2 x}{30 c^2}+\frac {i b^2 d^2 x \tan ^{-1}(c x)}{c^2}-\frac {2 i b^2 d^2 \log \left (c^2 x^2+1\right )}{3 c^3}+\frac {i b^2 d^2 x^2}{6 c}-\frac {1}{30} b^2 d^2 x^3 \]

[Out]

-31/30*I*d^2*(a+b*arctan(c*x))^2/c^3+19/30*b^2*d^2*x/c^2+I*a*b*d^2*x/c^2-1/30*b^2*d^2*x^3-19/30*b^2*d^2*arctan
(c*x)/c^3-8/15*I*b^2*d^2*polylog(2,1-2/(1+I*c*x))/c^3-8/15*b*d^2*x^2*(a+b*arctan(c*x))/c+1/2*I*c*d^2*x^4*(a+b*
arctan(c*x))^2+1/10*b*c*d^2*x^4*(a+b*arctan(c*x))+1/6*I*b^2*d^2*x^2/c+1/3*d^2*x^3*(a+b*arctan(c*x))^2-1/3*I*b*
d^2*x^3*(a+b*arctan(c*x))-1/5*c^2*d^2*x^5*(a+b*arctan(c*x))^2-16/15*b*d^2*(a+b*arctan(c*x))*ln(2/(1+I*c*x))/c^
3-2/3*I*b^2*d^2*ln(c^2*x^2+1)/c^3+I*b^2*d^2*x*arctan(c*x)/c^2

________________________________________________________________________________________

Rubi [A]  time = 0.86, antiderivative size = 333, normalized size of antiderivative = 1.00, number of steps used = 36, number of rules used = 15, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {4876, 4852, 4916, 321, 203, 4920, 4854, 2402, 2315, 266, 43, 4846, 260, 4884, 302} \[ -\frac {8 i b^2 d^2 \text {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{15 c^3}-\frac {1}{5} c^2 d^2 x^5 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {i a b d^2 x}{c^2}-\frac {31 i d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{30 c^3}-\frac {16 b d^2 \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{15 c^3}+\frac {1}{2} i c d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{10} b c d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{3} d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {1}{3} i b d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )-\frac {8 b d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{15 c}-\frac {2 i b^2 d^2 \log \left (c^2 x^2+1\right )}{3 c^3}+\frac {19 b^2 d^2 x}{30 c^2}+\frac {i b^2 d^2 x \tan ^{-1}(c x)}{c^2}-\frac {19 b^2 d^2 \tan ^{-1}(c x)}{30 c^3}+\frac {i b^2 d^2 x^2}{6 c}-\frac {1}{30} b^2 d^2 x^3 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(I*a*b*d^2*x)/c^2 + (19*b^2*d^2*x)/(30*c^2) + ((I/6)*b^2*d^2*x^2)/c - (b^2*d^2*x^3)/30 - (19*b^2*d^2*ArcTan[c*
x])/(30*c^3) + (I*b^2*d^2*x*ArcTan[c*x])/c^2 - (8*b*d^2*x^2*(a + b*ArcTan[c*x]))/(15*c) - (I/3)*b*d^2*x^3*(a +
b*ArcTan[c*x]) + (b*c*d^2*x^4*(a + b*ArcTan[c*x]))/10 - (((31*I)/30)*d^2*(a + b*ArcTan[c*x])^2)/c^3 + (d^2*x^
3*(a + b*ArcTan[c*x])^2)/3 + (I/2)*c*d^2*x^4*(a + b*ArcTan[c*x])^2 - (c^2*d^2*x^5*(a + b*ArcTan[c*x])^2)/5 - (
16*b*d^2*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(15*c^3) - (((2*I)/3)*b^2*d^2*Log[1 + c^2*x^2])/c^3 - (((8*I)
/15)*b^2*d^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^3

Rule 43

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 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 266

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 302

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 321

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

Rule 2315

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

Rule 2402

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

Rule 4846

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

Rule 4852

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

Rule 4854

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcTan[c*x])^p*Lo
g[2/(1 + (e*x)/d)])/e, x] + Dist[(b*c*p)/e, Int[((a + b*ArcTan[c*x])^(p - 1)*Log[2/(1 + (e*x)/d)])/(1 + c^2*x^
2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0]

Rule 4876

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

Rule 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]

Rubi steps

\begin {align*} \int x^2 (d+i c d x)^2 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx &=\int \left (d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )^2+2 i c d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2-c^2 d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )^2\right ) \, dx\\ &=d^2 \int x^2 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx+\left (2 i c d^2\right ) \int x^3 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx-\left (c^2 d^2\right ) \int x^4 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx\\ &=\frac {1}{3} d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{2} i c d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {1}{5} c^2 d^2 x^5 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {1}{3} \left (2 b c d^2\right ) \int \frac {x^3 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx-\left (i b c^2 d^2\right ) \int \frac {x^4 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx+\frac {1}{5} \left (2 b c^3 d^2\right ) \int \frac {x^5 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx\\ &=\frac {1}{3} d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{2} i c d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {1}{5} c^2 d^2 x^5 \left (a+b \tan ^{-1}(c x)\right )^2-\left (i b d^2\right ) \int x^2 \left (a+b \tan ^{-1}(c x)\right ) \, dx+\left (i b d^2\right ) \int \frac {x^2 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx-\frac {\left (2 b d^2\right ) \int x \left (a+b \tan ^{-1}(c x)\right ) \, dx}{3 c}+\frac {\left (2 b d^2\right ) \int \frac {x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{3 c}+\frac {1}{5} \left (2 b c d^2\right ) \int x^3 \left (a+b \tan ^{-1}(c x)\right ) \, dx-\frac {1}{5} \left (2 b c d^2\right ) \int \frac {x^3 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx\\ &=-\frac {b d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c}-\frac {1}{3} i b d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{10} b c d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )-\frac {i d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^3}+\frac {1}{3} d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{2} i c d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {1}{5} c^2 d^2 x^5 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{3} \left (b^2 d^2\right ) \int \frac {x^2}{1+c^2 x^2} \, dx+\frac {\left (i b d^2\right ) \int \left (a+b \tan ^{-1}(c x)\right ) \, dx}{c^2}-\frac {\left (i b d^2\right ) \int \frac {a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{c^2}-\frac {\left (2 b d^2\right ) \int \frac {a+b \tan ^{-1}(c x)}{i-c x} \, dx}{3 c^2}-\frac {\left (2 b d^2\right ) \int x \left (a+b \tan ^{-1}(c x)\right ) \, dx}{5 c}+\frac {\left (2 b d^2\right ) \int \frac {x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{5 c}+\frac {1}{3} \left (i b^2 c d^2\right ) \int \frac {x^3}{1+c^2 x^2} \, dx-\frac {1}{10} \left (b^2 c^2 d^2\right ) \int \frac {x^4}{1+c^2 x^2} \, dx\\ &=\frac {i a b d^2 x}{c^2}+\frac {b^2 d^2 x}{3 c^2}-\frac {8 b d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{15 c}-\frac {1}{3} i b d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{10} b c d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )-\frac {31 i d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{30 c^3}+\frac {1}{3} d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{2} i c d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {1}{5} c^2 d^2 x^5 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {2 b d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{3 c^3}+\frac {1}{5} \left (b^2 d^2\right ) \int \frac {x^2}{1+c^2 x^2} \, dx-\frac {\left (2 b d^2\right ) \int \frac {a+b \tan ^{-1}(c x)}{i-c x} \, dx}{5 c^2}+\frac {\left (i b^2 d^2\right ) \int \tan ^{-1}(c x) \, dx}{c^2}-\frac {\left (b^2 d^2\right ) \int \frac {1}{1+c^2 x^2} \, dx}{3 c^2}+\frac {\left (2 b^2 d^2\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{3 c^2}+\frac {1}{6} \left (i b^2 c d^2\right ) \operatorname {Subst}\left (\int \frac {x}{1+c^2 x} \, dx,x,x^2\right )-\frac {1}{10} \left (b^2 c^2 d^2\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 {i a b d^2 x}{c^2}+\frac {19 b^2 d^2 x}{30 c^2}-\frac {1}{30} b^2 d^2 x^3-\frac {b^2 d^2 \tan ^{-1}(c x)}{3 c^3}+\frac {i b^2 d^2 x \tan ^{-1}(c x)}{c^2}-\frac {8 b d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{15 c}-\frac {1}{3} i b d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{10} b c d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )-\frac {31 i d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{30 c^3}+\frac {1}{3} d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{2} i c d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {1}{5} c^2 d^2 x^5 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {16 b d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{15 c^3}-\frac {\left (2 i b^2 d^2\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{3 c^3}-\frac {\left (b^2 d^2\right ) \int \frac {1}{1+c^2 x^2} \, dx}{10 c^2}-\frac {\left (b^2 d^2\right ) \int \frac {1}{1+c^2 x^2} \, dx}{5 c^2}+\frac {\left (2 b^2 d^2\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{5 c^2}-\frac {\left (i b^2 d^2\right ) \int \frac {x}{1+c^2 x^2} \, dx}{c}+\frac {1}{6} \left (i b^2 c d^2\right ) \operatorname {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 {i a b d^2 x}{c^2}+\frac {19 b^2 d^2 x}{30 c^2}+\frac {i b^2 d^2 x^2}{6 c}-\frac {1}{30} b^2 d^2 x^3-\frac {19 b^2 d^2 \tan ^{-1}(c x)}{30 c^3}+\frac {i b^2 d^2 x \tan ^{-1}(c x)}{c^2}-\frac {8 b d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{15 c}-\frac {1}{3} i b d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{10} b c d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )-\frac {31 i d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{30 c^3}+\frac {1}{3} d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{2} i c d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {1}{5} c^2 d^2 x^5 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {16 b d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{15 c^3}-\frac {2 i b^2 d^2 \log \left (1+c^2 x^2\right )}{3 c^3}-\frac {i b^2 d^2 \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{3 c^3}-\frac {\left (2 i b^2 d^2\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{5 c^3}\\ &=\frac {i a b d^2 x}{c^2}+\frac {19 b^2 d^2 x}{30 c^2}+\frac {i b^2 d^2 x^2}{6 c}-\frac {1}{30} b^2 d^2 x^3-\frac {19 b^2 d^2 \tan ^{-1}(c x)}{30 c^3}+\frac {i b^2 d^2 x \tan ^{-1}(c x)}{c^2}-\frac {8 b d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{15 c}-\frac {1}{3} i b d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{10} b c d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )-\frac {31 i d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{30 c^3}+\frac {1}{3} d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{2} i c d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {1}{5} c^2 d^2 x^5 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {16 b d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{15 c^3}-\frac {2 i b^2 d^2 \log \left (1+c^2 x^2\right )}{3 c^3}-\frac {8 i b^2 d^2 \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{15 c^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 1.25, size = 306, normalized size = 0.92 \[ -\frac {d^2 \left (6 a^2 c^5 x^5-15 i a^2 c^4 x^4-10 a^2 c^3 x^3-3 a b c^4 x^4+10 i a b c^3 x^3+16 a b c^2 x^2-16 a b \log \left (c^2 x^2+1\right )+b \tan ^{-1}(c x) \left (2 a \left (6 c^5 x^5-15 i c^4 x^4-10 c^3 x^3+15 i\right )+b \left (-3 c^4 x^4+10 i c^3 x^3+16 c^2 x^2-30 i c x+19\right )+32 b \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )\right )-30 i a b c x+9 a b+b^2 c^3 x^3-5 i b^2 c^2 x^2+20 i b^2 \log \left (c^2 x^2+1\right )+b^2 (c x-i)^3 \left (6 c^2 x^2+3 i c x-1\right ) \tan ^{-1}(c x)^2-16 i b^2 \text {Li}_2\left (-e^{2 i \tan ^{-1}(c x)}\right )-19 b^2 c x-5 i b^2\right )}{30 c^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/30*(d^2*(9*a*b - (5*I)*b^2 - (30*I)*a*b*c*x - 19*b^2*c*x + 16*a*b*c^2*x^2 - (5*I)*b^2*c^2*x^2 - 10*a^2*c^3*
x^3 + (10*I)*a*b*c^3*x^3 + b^2*c^3*x^3 - (15*I)*a^2*c^4*x^4 - 3*a*b*c^4*x^4 + 6*a^2*c^5*x^5 + b^2*(-I + c*x)^3
*(-1 + (3*I)*c*x + 6*c^2*x^2)*ArcTan[c*x]^2 + b*ArcTan[c*x]*(b*(19 - (30*I)*c*x + 16*c^2*x^2 + (10*I)*c^3*x^3
- 3*c^4*x^4) + 2*a*(15*I - 10*c^3*x^3 - (15*I)*c^4*x^4 + 6*c^5*x^5) + 32*b*Log[1 + E^((2*I)*ArcTan[c*x])]) - 1
6*a*b*Log[1 + c^2*x^2] + (20*I)*b^2*Log[1 + c^2*x^2] - (16*I)*b^2*PolyLog[2, -E^((2*I)*ArcTan[c*x])]))/c^3

________________________________________________________________________________________

fricas [F]  time = 0.60, size = 0, normalized size = 0.00 \[ \frac {1}{120} \, {\left (6 \, b^{2} c^{2} d^{2} x^{5} - 15 i \, b^{2} c d^{2} x^{4} - 10 \, b^{2} d^{2} x^{3}\right )} \log \left (-\frac {c x + i}{c x - i}\right )^{2} + {\rm integral}\left (-\frac {30 \, a^{2} c^{4} d^{2} x^{6} - 60 i \, a^{2} c^{3} d^{2} x^{5} - 60 i \, a^{2} c d^{2} x^{3} - 30 \, a^{2} d^{2} x^{2} - {\left (-30 i \, a b c^{4} d^{2} x^{6} - {\left (60 \, a b - 6 i \, b^{2}\right )} c^{3} d^{2} x^{5} + 15 \, b^{2} c^{2} d^{2} x^{4} - {\left (60 \, a b + 10 i \, b^{2}\right )} c d^{2} x^{3} + 30 i \, a b d^{2} x^{2}\right )} \log \left (-\frac {c x + i}{c x - i}\right )}{30 \, {\left (c^{2} x^{2} + 1\right )}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/120*(6*b^2*c^2*d^2*x^5 - 15*I*b^2*c*d^2*x^4 - 10*b^2*d^2*x^3)*log(-(c*x + I)/(c*x - I))^2 + integral(-1/30*(
30*a^2*c^4*d^2*x^6 - 60*I*a^2*c^3*d^2*x^5 - 60*I*a^2*c*d^2*x^3 - 30*a^2*d^2*x^2 - (-30*I*a*b*c^4*d^2*x^6 - (60
*a*b - 6*I*b^2)*c^3*d^2*x^5 + 15*b^2*c^2*d^2*x^4 - (60*a*b + 10*I*b^2)*c*d^2*x^3 + 30*I*a*b*d^2*x^2)*log(-(c*x
+ I)/(c*x - I)))/(c^2*x^2 + 1), x)

________________________________________________________________________________________

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

[Out]

sage0*x

________________________________________________________________________________________

maple [B]  time = 0.11, size = 612, normalized size = 1.84 \[ \frac {d^{2} a^{2} x^{3}}{3}+\frac {i b^{2} d^{2} x \arctan \left (c x \right )}{c^{2}}+\frac {19 b^{2} d^{2} x}{30 c^{2}}+\frac {i c \,d^{2} b^{2} \arctan \left (c x \right )^{2} x^{4}}{2}+\frac {4 i d^{2} b^{2} \ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )}{15 c^{3}}-\frac {i d^{2} a b \arctan \left (c x \right )}{c^{3}}+\frac {i a b \,d^{2} x}{c^{2}}-\frac {4 i d^{2} b^{2} \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{15 c^{3}}-\frac {4 i d^{2} b^{2} \ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )}{15 c^{3}}+\frac {4 i d^{2} b^{2} \ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )}{15 c^{3}}+i c \,d^{2} a b \arctan \left (c x \right ) x^{4}-\frac {2 i d^{2} b^{2} \ln \left (c x -i\right )^{2}}{15 c^{3}}-\frac {i d^{2} a b \,x^{3}}{3}-\frac {i d^{2} b^{2} \arctan \left (c x \right )^{2}}{2 c^{3}}+\frac {2 i d^{2} b^{2} \ln \left (c x +i\right )^{2}}{15 c^{3}}-\frac {4 i d^{2} b^{2} \dilog \left (-\frac {i \left (c x +i\right )}{2}\right )}{15 c^{3}}-\frac {i d^{2} b^{2} \arctan \left (c x \right ) x^{3}}{3}+\frac {i c \,d^{2} a^{2} x^{4}}{2}-\frac {2 i b^{2} d^{2} \ln \left (c^{2} x^{2}+1\right )}{3 c^{3}}+\frac {4 i d^{2} b^{2} \dilog \left (\frac {i \left (c x -i\right )}{2}\right )}{15 c^{3}}+\frac {i b^{2} d^{2} x^{2}}{6 c}+\frac {8 d^{2} a b \ln \left (c^{2} x^{2}+1\right )}{15 c^{3}}-\frac {8 d^{2} a b \,x^{2}}{15 c}+\frac {c \,d^{2} a b \,x^{4}}{10}+\frac {2 d^{2} a b \arctan \left (c x \right ) x^{3}}{3}+\frac {8 d^{2} b^{2} \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{15 c^{3}}-\frac {8 d^{2} b^{2} \arctan \left (c x \right ) x^{2}}{15 c}-\frac {c^{2} d^{2} b^{2} \arctan \left (c x \right )^{2} x^{5}}{5}+\frac {c \,d^{2} b^{2} \arctan \left (c x \right ) x^{4}}{10}-\frac {c^{2} d^{2} a^{2} x^{5}}{5}-\frac {b^{2} d^{2} x^{3}}{30}-\frac {19 b^{2} d^{2} \arctan \left (c x \right )}{30 c^{3}}+\frac {d^{2} b^{2} \arctan \left (c x \right )^{2} x^{3}}{3}-\frac {2 c^{2} d^{2} a b \arctan \left (c x \right ) x^{5}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(d+I*c*d*x)^2*(a+b*arctan(c*x))^2,x)

[Out]

1/3*d^2*a^2*x^3-4/15*I/c^3*d^2*b^2*ln(c*x-I)*ln(-1/2*I*(I+c*x))-4/15*I/c^3*d^2*b^2*ln(I+c*x)*ln(c^2*x^2+1)+4/1
5*I/c^3*d^2*b^2*ln(c*x-I)*ln(c^2*x^2+1)+1/2*I*c*d^2*b^2*arctan(c*x)^2*x^4+4/15*I/c^3*d^2*b^2*ln(I+c*x)*ln(1/2*
I*(c*x-I))+1/6*I*b^2*d^2*x^2/c+19/30*b^2*d^2*x/c^2-I/c^3*d^2*a*b*arctan(c*x)+I*a*b*d^2*x/c^2+I*b^2*d^2*x*arcta
n(c*x)/c^2-2/3*I*b^2*d^2*ln(c^2*x^2+1)/c^3+I*c*d^2*a*b*arctan(c*x)*x^4+8/15/c^3*d^2*a*b*ln(c^2*x^2+1)-8/15/c*d
^2*a*b*x^2+4/15*I/c^3*d^2*b^2*dilog(1/2*I*(c*x-I))-1/3*I*d^2*b^2*arctan(c*x)*x^3+1/2*I*c*d^2*a^2*x^4+1/10*c*d^
2*a*b*x^4+2/3*d^2*a*b*arctan(c*x)*x^3+8/15/c^3*d^2*b^2*arctan(c*x)*ln(c^2*x^2+1)-8/15/c*d^2*b^2*arctan(c*x)*x^
2-1/5*c^2*d^2*b^2*arctan(c*x)^2*x^5+1/10*c*d^2*b^2*arctan(c*x)*x^4-1/2*I/c^3*d^2*b^2*arctan(c*x)^2-4/15*I/c^3*
d^2*b^2*dilog(-1/2*I*(I+c*x))+2/15*I/c^3*d^2*b^2*ln(I+c*x)^2-2/15*I/c^3*d^2*b^2*ln(c*x-I)^2-1/5*c^2*d^2*a^2*x^
5-1/30*b^2*d^2*x^3-1/3*I*d^2*a*b*x^3-19/30*b^2*d^2*arctan(c*x)/c^3+1/3*d^2*b^2*arctan(c*x)^2*x^3-2/5*c^2*d^2*a
*b*arctan(c*x)*x^5

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{5} \, a^{2} c^{2} d^{2} x^{5} + \frac {1}{2} i \, a^{2} c d^{2} x^{4} - \frac {1}{10} \, {\left (4 \, x^{5} \arctan \left (c x\right ) - c {\left (\frac {c^{2} x^{4} - 2 \, x^{2}}{c^{4}} + \frac {2 \, \log \left (c^{2} x^{2} + 1\right )}{c^{6}}\right )}\right )} a b c^{2} d^{2} + \frac {1}{3} \, a^{2} d^{2} x^{3} + \frac {1}{3} i \, {\left (3 \, x^{4} \arctan \left (c x\right ) - c {\left (\frac {c^{2} x^{3} - 3 \, x}{c^{4}} + \frac {3 \, \arctan \left (c x\right )}{c^{5}}\right )}\right )} a b c d^{2} + \frac {1}{3} \, {\left (2 \, x^{3} \arctan \left (c x\right ) - c {\left (\frac {x^{2}}{c^{2}} - \frac {\log \left (c^{2} x^{2} + 1\right )}{c^{4}}\right )}\right )} a b d^{2} - \frac {1}{480} \, {\left (24 \, b^{2} c^{2} d^{2} x^{5} - 60 i \, b^{2} c d^{2} x^{4} - 40 \, b^{2} d^{2} x^{3}\right )} \arctan \left (c x\right )^{2} - \frac {1}{480} \, {\left (24 i \, b^{2} c^{2} d^{2} x^{5} + 60 \, b^{2} c d^{2} x^{4} - 40 i \, b^{2} d^{2} x^{3}\right )} \arctan \left (c x\right ) \log \left (c^{2} x^{2} + 1\right ) + \frac {1}{480} \, {\left (6 \, b^{2} c^{2} d^{2} x^{5} - 15 i \, b^{2} c d^{2} x^{4} - 10 \, b^{2} d^{2} x^{3}\right )} \log \left (c^{2} x^{2} + 1\right )^{2} - \int \frac {180 \, {\left (b^{2} c^{4} d^{2} x^{6} - b^{2} d^{2} x^{2}\right )} \arctan \left (c x\right )^{2} + 15 \, {\left (b^{2} c^{4} d^{2} x^{6} - b^{2} d^{2} x^{2}\right )} \log \left (c^{2} x^{2} + 1\right )^{2} - 4 \, {\left (21 \, b^{2} c^{3} d^{2} x^{5} - 10 \, b^{2} c d^{2} x^{3}\right )} \arctan \left (c x\right ) + 2 \, {\left (6 \, b^{2} c^{4} d^{2} x^{6} - 25 \, b^{2} c^{2} d^{2} x^{4} - 60 \, {\left (b^{2} c^{3} d^{2} x^{5} + b^{2} c d^{2} x^{3}\right )} \arctan \left (c x\right )\right )} \log \left (c^{2} x^{2} + 1\right )}{240 \, {\left (c^{2} x^{2} + 1\right )}}\,{d x} + i \, \int \frac {180 \, {\left (b^{2} c^{3} d^{2} x^{5} + b^{2} c d^{2} x^{3}\right )} \arctan \left (c x\right )^{2} + 15 \, {\left (b^{2} c^{3} d^{2} x^{5} + b^{2} c d^{2} x^{3}\right )} \log \left (c^{2} x^{2} + 1\right )^{2} + 2 \, {\left (6 \, b^{2} c^{4} d^{2} x^{6} - 25 \, b^{2} c^{2} d^{2} x^{4}\right )} \arctan \left (c x\right ) + {\left (21 \, b^{2} c^{3} d^{2} x^{5} - 10 \, b^{2} c d^{2} x^{3} + 30 \, {\left (b^{2} c^{4} d^{2} x^{6} - b^{2} d^{2} x^{2}\right )} \arctan \left (c x\right )\right )} \log \left (c^{2} x^{2} + 1\right )}{120 \, {\left (c^{2} x^{2} + 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*a^2*c^2*d^2*x^5 + 1/2*I*a^2*c*d^2*x^4 - 1/10*(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^2 + 1/3*a^2*d^2*x^3 + 1/3*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^2 + 1/3*(2*x^3*arctan(c*x) - c*(x^2/c^2 - log(c^2*x^2 + 1)/c^4))*a*b*d^2 - 1/480*(24*b^2*c^2*d^
2*x^5 - 60*I*b^2*c*d^2*x^4 - 40*b^2*d^2*x^3)*arctan(c*x)^2 - 1/480*(24*I*b^2*c^2*d^2*x^5 + 60*b^2*c*d^2*x^4 -
40*I*b^2*d^2*x^3)*arctan(c*x)*log(c^2*x^2 + 1) + 1/480*(6*b^2*c^2*d^2*x^5 - 15*I*b^2*c*d^2*x^4 - 10*b^2*d^2*x^
3)*log(c^2*x^2 + 1)^2 - integrate(1/240*(180*(b^2*c^4*d^2*x^6 - b^2*d^2*x^2)*arctan(c*x)^2 + 15*(b^2*c^4*d^2*x
^6 - b^2*d^2*x^2)*log(c^2*x^2 + 1)^2 - 4*(21*b^2*c^3*d^2*x^5 - 10*b^2*c*d^2*x^3)*arctan(c*x) + 2*(6*b^2*c^4*d^
2*x^6 - 25*b^2*c^2*d^2*x^4 - 60*(b^2*c^3*d^2*x^5 + b^2*c*d^2*x^3)*arctan(c*x))*log(c^2*x^2 + 1))/(c^2*x^2 + 1)
, x) + I*integrate(1/120*(180*(b^2*c^3*d^2*x^5 + b^2*c*d^2*x^3)*arctan(c*x)^2 + 15*(b^2*c^3*d^2*x^5 + b^2*c*d^
2*x^3)*log(c^2*x^2 + 1)^2 + 2*(6*b^2*c^4*d^2*x^6 - 25*b^2*c^2*d^2*x^4)*arctan(c*x) + (21*b^2*c^3*d^2*x^5 - 10*
b^2*c*d^2*x^3 + 30*(b^2*c^4*d^2*x^6 - b^2*d^2*x^2)*arctan(c*x))*log(c^2*x^2 + 1))/(c^2*x^2 + 1), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int x^2\,{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^2 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________