\(\int x (d+i c d x)^2 (a+b \arctan (c x))^2 \, dx\) [78]

Optimal result

Integrand size = 23, antiderivative size = 293 \[ \int x (d+i c d x)^2 (a+b \arctan (c x))^2 \, dx=-\frac {3 a b d^2 x}{2 c}+\frac {2 i b^2 d^2 x}{3 c}-\frac {1}{12} b^2 d^2 x^2-\frac {2 i b^2 d^2 \arctan (c x)}{3 c^2}-\frac {3 b^2 d^2 x \arctan (c x)}{2 c}-\frac {2}{3} i b d^2 x^2 (a+b \arctan (c x))+\frac {1}{6} b c d^2 x^3 (a+b \arctan (c x))+\frac {17 d^2 (a+b \arctan (c x))^2}{12 c^2}+\frac {1}{2} d^2 x^2 (a+b \arctan (c x))^2+\frac {2}{3} i c d^2 x^3 (a+b \arctan (c x))^2-\frac {1}{4} c^2 d^2 x^4 (a+b \arctan (c x))^2-\frac {4 i b d^2 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{3 c^2}+\frac {5 b^2 d^2 \log \left (1+c^2 x^2\right )}{6 c^2}+\frac {2 b^2 d^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{3 c^2} \]

[Out]

-3/2*a*b*d^2*x/c+2/3*I*b^2*d^2*x/c-1/12*b^2*d^2*x^2-2/3*I*b^2*d^2*arctan(c*x)/c^2-3/2*b^2*d^2*x*arctan(c*x)/c-
2/3*I*b*d^2*x^2*(a+b*arctan(c*x))+1/6*b*c*d^2*x^3*(a+b*arctan(c*x))+17/12*d^2*(a+b*arctan(c*x))^2/c^2+1/2*d^2*
x^2*(a+b*arctan(c*x))^2+2/3*I*c*d^2*x^3*(a+b*arctan(c*x))^2-1/4*c^2*d^2*x^4*(a+b*arctan(c*x))^2-4/3*I*b*d^2*(a
+b*arctan(c*x))*ln(2/(1+I*c*x))/c^2+5/6*b^2*d^2*ln(c^2*x^2+1)/c^2+2/3*b^2*d^2*polylog(2,1-2/(1+I*c*x))/c^2

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules used = {4996, 4946, 5036, 4930, 266, 5004, 327, 209, 5040, 4964, 2449, 2352, 272, 45} \[ \int x (d+i c d x)^2 (a+b \arctan (c x))^2 \, dx=-\frac {1}{4} c^2 d^2 x^4 (a+b \arctan (c x))^2+\frac {17 d^2 (a+b \arctan (c x))^2}{12 c^2}-\frac {4 i b d^2 \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{3 c^2}+\frac {2}{3} i c d^2 x^3 (a+b \arctan (c x))^2+\frac {1}{6} b c d^2 x^3 (a+b \arctan (c x))+\frac {1}{2} d^2 x^2 (a+b \arctan (c x))^2-\frac {2}{3} i b d^2 x^2 (a+b \arctan (c x))-\frac {3 a b d^2 x}{2 c}-\frac {2 i b^2 d^2 \arctan (c x)}{3 c^2}-\frac {3 b^2 d^2 x \arctan (c x)}{2 c}+\frac {2 b^2 d^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{3 c^2}+\frac {5 b^2 d^2 \log \left (c^2 x^2+1\right )}{6 c^2}+\frac {2 i b^2 d^2 x}{3 c}-\frac {1}{12} b^2 d^2 x^2 \]

[In]

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

[Out]

(-3*a*b*d^2*x)/(2*c) + (((2*I)/3)*b^2*d^2*x)/c - (b^2*d^2*x^2)/12 - (((2*I)/3)*b^2*d^2*ArcTan[c*x])/c^2 - (3*b
^2*d^2*x*ArcTan[c*x])/(2*c) - ((2*I)/3)*b*d^2*x^2*(a + b*ArcTan[c*x]) + (b*c*d^2*x^3*(a + b*ArcTan[c*x]))/6 +
(17*d^2*(a + b*ArcTan[c*x])^2)/(12*c^2) + (d^2*x^2*(a + b*ArcTan[c*x])^2)/2 + ((2*I)/3)*c*d^2*x^3*(a + b*ArcTa
n[c*x])^2 - (c^2*d^2*x^4*(a + b*ArcTan[c*x])^2)/4 - (((4*I)/3)*b*d^2*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/c
^2 + (5*b^2*d^2*Log[1 + c^2*x^2])/(6*c^2) + (2*b^2*d^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/(3*c^2)

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 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^2 x (a+b \arctan (c x))^2+2 i c d^2 x^2 (a+b \arctan (c x))^2-c^2 d^2 x^3 (a+b \arctan (c x))^2\right ) \, dx \\ & = d^2 \int x (a+b \arctan (c x))^2 \, dx+\left (2 i c d^2\right ) \int x^2 (a+b \arctan (c x))^2 \, dx-\left (c^2 d^2\right ) \int x^3 (a+b \arctan (c x))^2 \, dx \\ & = \frac {1}{2} d^2 x^2 (a+b \arctan (c x))^2+\frac {2}{3} i c d^2 x^3 (a+b \arctan (c x))^2-\frac {1}{4} c^2 d^2 x^4 (a+b \arctan (c x))^2-\left (b c d^2\right ) \int \frac {x^2 (a+b \arctan (c x))}{1+c^2 x^2} \, dx-\frac {1}{3} \left (4 i b c^2 d^2\right ) \int \frac {x^3 (a+b \arctan (c x))}{1+c^2 x^2} \, dx+\frac {1}{2} \left (b c^3 d^2\right ) \int \frac {x^4 (a+b \arctan (c x))}{1+c^2 x^2} \, dx \\ & = \frac {1}{2} d^2 x^2 (a+b \arctan (c x))^2+\frac {2}{3} i c d^2 x^3 (a+b \arctan (c x))^2-\frac {1}{4} c^2 d^2 x^4 (a+b \arctan (c x))^2-\frac {1}{3} \left (4 i b d^2\right ) \int x (a+b \arctan (c x)) \, dx+\frac {1}{3} \left (4 i b d^2\right ) \int \frac {x (a+b \arctan (c x))}{1+c^2 x^2} \, dx-\frac {\left (b d^2\right ) \int (a+b \arctan (c x)) \, dx}{c}+\frac {\left (b d^2\right ) \int \frac {a+b \arctan (c x)}{1+c^2 x^2} \, dx}{c}+\frac {1}{2} \left (b c d^2\right ) \int x^2 (a+b \arctan (c x)) \, dx-\frac {1}{2} \left (b c d^2\right ) \int \frac {x^2 (a+b \arctan (c x))}{1+c^2 x^2} \, dx \\ & = -\frac {a b d^2 x}{c}-\frac {2}{3} i b d^2 x^2 (a+b \arctan (c x))+\frac {1}{6} b c d^2 x^3 (a+b \arctan (c x))+\frac {7 d^2 (a+b \arctan (c x))^2}{6 c^2}+\frac {1}{2} d^2 x^2 (a+b \arctan (c x))^2+\frac {2}{3} i c d^2 x^3 (a+b \arctan (c x))^2-\frac {1}{4} c^2 d^2 x^4 (a+b \arctan (c x))^2-\frac {\left (4 i b d^2\right ) \int \frac {a+b \arctan (c x)}{i-c x} \, dx}{3 c}-\frac {\left (b d^2\right ) \int (a+b \arctan (c x)) \, dx}{2 c}+\frac {\left (b d^2\right ) \int \frac {a+b \arctan (c x)}{1+c^2 x^2} \, dx}{2 c}-\frac {\left (b^2 d^2\right ) \int \arctan (c x) \, dx}{c}+\frac {1}{3} \left (2 i b^2 c d^2\right ) \int \frac {x^2}{1+c^2 x^2} \, dx-\frac {1}{6} \left (b^2 c^2 d^2\right ) \int \frac {x^3}{1+c^2 x^2} \, dx \\ & = -\frac {3 a b d^2 x}{2 c}+\frac {2 i b^2 d^2 x}{3 c}-\frac {b^2 d^2 x \arctan (c x)}{c}-\frac {2}{3} i b d^2 x^2 (a+b \arctan (c x))+\frac {1}{6} b c d^2 x^3 (a+b \arctan (c x))+\frac {17 d^2 (a+b \arctan (c x))^2}{12 c^2}+\frac {1}{2} d^2 x^2 (a+b \arctan (c x))^2+\frac {2}{3} i c d^2 x^3 (a+b \arctan (c x))^2-\frac {1}{4} c^2 d^2 x^4 (a+b \arctan (c x))^2-\frac {4 i b d^2 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{3 c^2}+\left (b^2 d^2\right ) \int \frac {x}{1+c^2 x^2} \, dx-\frac {\left (2 i b^2 d^2\right ) \int \frac {1}{1+c^2 x^2} \, dx}{3 c}+\frac {\left (4 i b^2 d^2\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{3 c}-\frac {\left (b^2 d^2\right ) \int \arctan (c x) \, dx}{2 c}-\frac {1}{12} \left (b^2 c^2 d^2\right ) \text {Subst}\left (\int \frac {x}{1+c^2 x} \, dx,x,x^2\right ) \\ & = -\frac {3 a b d^2 x}{2 c}+\frac {2 i b^2 d^2 x}{3 c}-\frac {2 i b^2 d^2 \arctan (c x)}{3 c^2}-\frac {3 b^2 d^2 x \arctan (c x)}{2 c}-\frac {2}{3} i b d^2 x^2 (a+b \arctan (c x))+\frac {1}{6} b c d^2 x^3 (a+b \arctan (c x))+\frac {17 d^2 (a+b \arctan (c x))^2}{12 c^2}+\frac {1}{2} d^2 x^2 (a+b \arctan (c x))^2+\frac {2}{3} i c d^2 x^3 (a+b \arctan (c x))^2-\frac {1}{4} c^2 d^2 x^4 (a+b \arctan (c x))^2-\frac {4 i b d^2 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{3 c^2}+\frac {b^2 d^2 \log \left (1+c^2 x^2\right )}{2 c^2}+\frac {1}{2} \left (b^2 d^2\right ) \int \frac {x}{1+c^2 x^2} \, dx+\frac {\left (4 b^2 d^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{3 c^2}-\frac {1}{12} \left (b^2 c^2 d^2\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 {3 a b d^2 x}{2 c}+\frac {2 i b^2 d^2 x}{3 c}-\frac {1}{12} b^2 d^2 x^2-\frac {2 i b^2 d^2 \arctan (c x)}{3 c^2}-\frac {3 b^2 d^2 x \arctan (c x)}{2 c}-\frac {2}{3} i b d^2 x^2 (a+b \arctan (c x))+\frac {1}{6} b c d^2 x^3 (a+b \arctan (c x))+\frac {17 d^2 (a+b \arctan (c x))^2}{12 c^2}+\frac {1}{2} d^2 x^2 (a+b \arctan (c x))^2+\frac {2}{3} i c d^2 x^3 (a+b \arctan (c x))^2-\frac {1}{4} c^2 d^2 x^4 (a+b \arctan (c x))^2-\frac {4 i b d^2 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{3 c^2}+\frac {5 b^2 d^2 \log \left (1+c^2 x^2\right )}{6 c^2}+\frac {2 b^2 d^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{3 c^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.00 (sec) , antiderivative size = 257, normalized size of antiderivative = 0.88 \[ \int x (d+i c d x)^2 (a+b \arctan (c x))^2 \, dx=-\frac {d^2 \left (b^2+18 a b c x-8 i b^2 c x-6 a^2 c^2 x^2+8 i a b c^2 x^2+b^2 c^2 x^2-8 i a^2 c^3 x^3-2 a b c^3 x^3+3 a^2 c^4 x^4+b^2 (-i+c x)^3 (i+3 c x) \arctan (c x)^2+2 b \arctan (c x) \left (b \left (4 i+9 c x+4 i c^2 x^2-c^3 x^3\right )+a \left (-9-6 c^2 x^2-8 i c^3 x^3+3 c^4 x^4\right )+8 i b \log \left (1+e^{2 i \arctan (c x)}\right )\right )-8 i a b \log \left (1+c^2 x^2\right )-10 b^2 \log \left (1+c^2 x^2\right )+8 b^2 \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )\right )}{12 c^2} \]

[In]

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

[Out]

-1/12*(d^2*(b^2 + 18*a*b*c*x - (8*I)*b^2*c*x - 6*a^2*c^2*x^2 + (8*I)*a*b*c^2*x^2 + b^2*c^2*x^2 - (8*I)*a^2*c^3
*x^3 - 2*a*b*c^3*x^3 + 3*a^2*c^4*x^4 + b^2*(-I + c*x)^3*(I + 3*c*x)*ArcTan[c*x]^2 + 2*b*ArcTan[c*x]*(b*(4*I +
9*c*x + (4*I)*c^2*x^2 - c^3*x^3) + a*(-9 - 6*c^2*x^2 - (8*I)*c^3*x^3 + 3*c^4*x^4) + (8*I)*b*Log[1 + E^((2*I)*A
rcTan[c*x])]) - (8*I)*a*b*Log[1 + c^2*x^2] - 10*b^2*Log[1 + c^2*x^2] + 8*b^2*PolyLog[2, -E^((2*I)*ArcTan[c*x])
]))/c^2

Maple [A] (verified)

Time = 1.84 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.31

method	result	size
parts	\(a^{2} d^{2} \left (-\frac {1}{4} c^{2} x^{4}+\frac {2}{3} i c \,x^{3}+\frac {1}{2} x^{2}\right )+\frac {b^{2} d^{2} \left (-\frac {c^{4} x^{4} \arctan \left (c x \right )^{2}}{4}+\frac {2 i \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{3}+\frac {c^{2} x^{2} \arctan \left (c x \right )^{2}}{2}-\frac {2 i \arctan \left (c x \right ) c^{2} x^{2}}{3}+\frac {c^{3} x^{3} \arctan \left (c x \right )}{6}+\frac {2 i \arctan \left (c x \right )^{2} c^{3} x^{3}}{3}+\frac {3 \arctan \left (c x \right )^{2}}{4}-\frac {3 c x \arctan \left (c x \right )}{2}-\frac {\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )}{3}+\frac {\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )}{3}+\frac {\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{3}+\frac {\ln \left (c x -i\right )^{2}}{6}-\frac {\ln \left (c x +i\right )^{2}}{6}-\frac {\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )}{3}+\frac {\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{3}-\frac {\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )}{3}-\frac {2 i \arctan \left (c x \right )}{3}-\frac {c^{2} x^{2}}{12}+\frac {5 \ln \left (c^{2} x^{2}+1\right )}{6}+\frac {2 i c x}{3}\right )}{c^{2}}+\frac {2 a \,d^{2} b \left (-\frac {c^{4} x^{4} \arctan \left (c x \right )}{4}+\frac {2 i \arctan \left (c x \right ) c^{3} x^{3}}{3}+\frac {c^{2} x^{2} \arctan \left (c x \right )}{2}-\frac {3 c x}{4}+\frac {c^{3} x^{3}}{12}-\frac {i c^{2} x^{2}}{3}+\frac {i \ln \left (c^{2} x^{2}+1\right )}{3}+\frac {3 \arctan \left (c x \right )}{4}\right )}{c^{2}}\)	\(384\)
derivativedivides	\(\frac {a^{2} d^{2} \left (-\frac {1}{4} c^{4} x^{4}+\frac {2}{3} i c^{3} x^{3}+\frac {1}{2} c^{2} x^{2}\right )+b^{2} d^{2} \left (-\frac {c^{4} x^{4} \arctan \left (c x \right )^{2}}{4}+\frac {2 i \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{3}+\frac {c^{2} x^{2} \arctan \left (c x \right )^{2}}{2}-\frac {2 i \arctan \left (c x \right ) c^{2} x^{2}}{3}+\frac {c^{3} x^{3} \arctan \left (c x \right )}{6}+\frac {2 i \arctan \left (c x \right )^{2} c^{3} x^{3}}{3}+\frac {3 \arctan \left (c x \right )^{2}}{4}-\frac {3 c x \arctan \left (c x \right )}{2}-\frac {\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )}{3}+\frac {\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )}{3}+\frac {\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{3}+\frac {\ln \left (c x -i\right )^{2}}{6}-\frac {\ln \left (c x +i\right )^{2}}{6}-\frac {\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )}{3}+\frac {\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{3}-\frac {\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )}{3}-\frac {2 i \arctan \left (c x \right )}{3}-\frac {c^{2} x^{2}}{12}+\frac {5 \ln \left (c^{2} x^{2}+1\right )}{6}+\frac {2 i c x}{3}\right )+2 a \,d^{2} b \left (-\frac {c^{4} x^{4} \arctan \left (c x \right )}{4}+\frac {2 i \arctan \left (c x \right ) c^{3} x^{3}}{3}+\frac {c^{2} x^{2} \arctan \left (c x \right )}{2}-\frac {3 c x}{4}+\frac {c^{3} x^{3}}{12}-\frac {i c^{2} x^{2}}{3}+\frac {i \ln \left (c^{2} x^{2}+1\right )}{3}+\frac {3 \arctan \left (c x \right )}{4}\right )}{c^{2}}\)	\(387\)
default	\(\frac {a^{2} d^{2} \left (-\frac {1}{4} c^{4} x^{4}+\frac {2}{3} i c^{3} x^{3}+\frac {1}{2} c^{2} x^{2}\right )+b^{2} d^{2} \left (-\frac {c^{4} x^{4} \arctan \left (c x \right )^{2}}{4}+\frac {2 i \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{3}+\frac {c^{2} x^{2} \arctan \left (c x \right )^{2}}{2}-\frac {2 i \arctan \left (c x \right ) c^{2} x^{2}}{3}+\frac {c^{3} x^{3} \arctan \left (c x \right )}{6}+\frac {2 i \arctan \left (c x \right )^{2} c^{3} x^{3}}{3}+\frac {3 \arctan \left (c x \right )^{2}}{4}-\frac {3 c x \arctan \left (c x \right )}{2}-\frac {\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )}{3}+\frac {\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )}{3}+\frac {\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{3}+\frac {\ln \left (c x -i\right )^{2}}{6}-\frac {\ln \left (c x +i\right )^{2}}{6}-\frac {\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )}{3}+\frac {\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{3}-\frac {\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )}{3}-\frac {2 i \arctan \left (c x \right )}{3}-\frac {c^{2} x^{2}}{12}+\frac {5 \ln \left (c^{2} x^{2}+1\right )}{6}+\frac {2 i c x}{3}\right )+2 a \,d^{2} b \left (-\frac {c^{4} x^{4} \arctan \left (c x \right )}{4}+\frac {2 i \arctan \left (c x \right ) c^{3} x^{3}}{3}+\frac {c^{2} x^{2} \arctan \left (c x \right )}{2}-\frac {3 c x}{4}+\frac {c^{3} x^{3}}{12}-\frac {i c^{2} x^{2}}{3}+\frac {i \ln \left (c^{2} x^{2}+1\right )}{3}+\frac {3 \arctan \left (c x \right )}{4}\right )}{c^{2}}\)	\(387\)
risch	\(-\frac {b^{2} d^{2} x^{2}}{12}+\frac {503 b^{2} d^{2} \ln \left (c^{2} x^{2}+1\right )}{576 c^{2}}-\frac {3 a b \,d^{2} x}{2 c}-\frac {3 b^{2} d^{2}}{4 c^{2}}-\frac {2 i d^{2} x^{2} a b}{3}+\frac {d^{2} b^{2} \ln \left (-i c x +1\right ) x^{2}}{3}+\frac {a b c \,d^{2} x^{3}}{6}+\frac {a^{2} d^{2} x^{2}}{2}-\frac {a^{2} c^{2} d^{2} x^{4}}{4}+\frac {i d^{2} c \,b^{2} \ln \left (-i c x +1\right ) x^{3}}{12}-\frac {3 i d^{2} b^{2} \ln \left (-i c x +1\right ) x}{4 c}+\frac {2 i b \,d^{2} a \ln \left (c^{2} x^{2}+1\right )}{3 c^{2}}-\frac {2 d^{2} c a b \ln \left (-i c x +1\right ) x^{3}}{3}+\frac {i d^{2} a b \ln \left (-i c x +1\right ) x^{2}}{2}-\frac {i d^{2} c \,b^{2} \ln \left (-i c x +1\right )^{2} x^{3}}{6}+\frac {3 b \,d^{2} a \arctan \left (c x \right )}{2 c^{2}}+\frac {17 d^{2} a^{2}}{12 c^{2}}-\frac {23 d^{2} \ln \left (-i c x +1\right ) b^{2}}{288 c^{2}}+\frac {2 b^{2} d^{2} \operatorname {dilog}\left (\frac {1}{2}-\frac {i c x}{2}\right )}{3 c^{2}}-\frac {d^{2} b^{2} \ln \left (-i c x +1\right )^{2} x^{2}}{8}-\frac {17 d^{2} \ln \left (-i c x +1\right )^{2} b^{2}}{48 c^{2}}-\frac {i d^{2} c^{2} b a \ln \left (-i c x +1\right ) x^{4}}{4}+\left (-\frac {b^{2} d^{2} \left (3 c^{2} x^{4}-8 i c \,x^{3}-6 x^{2}\right ) \ln \left (-i c x +1\right )}{24}-\frac {b \,d^{2} \left (-6 i a \,c^{4} x^{4}+2 i b \,c^{3} x^{3}-16 a \,c^{3} x^{3}+12 i a \,c^{2} x^{2}+8 b \,c^{2} x^{2}-18 i b c x -17 b \ln \left (-i c x +1\right )\right )}{24 c^{2}}\right ) \ln \left (i c x +1\right )+\frac {2 i b^{2} d^{2} x}{3 c}-\frac {7 i d^{2} a b}{3 c^{2}}-\frac {215 i b^{2} d^{2} \arctan \left (c x \right )}{288 c^{2}}+\frac {2 b^{2} d^{2} \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (\frac {1}{2}-\frac {i c x}{2}\right )}{3 c^{2}}-\frac {2 b^{2} d^{2} \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (-i c x +1\right )}{3 c^{2}}+\frac {b^{2} d^{2} \left (3 c^{4} x^{4}-8 i c^{3} x^{3}-6 c^{2} x^{2}-1\right ) \ln \left (i c x +1\right )^{2}}{48 c^{2}}+\frac {d^{2} c^{2} b^{2} \ln \left (-i c x +1\right )^{2} x^{4}}{16}+\frac {2 i a^{2} c \,d^{2} x^{3}}{3}\)	\(674\)

[In]

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

[Out]

a^2*d^2*(-1/4*c^2*x^4+2/3*I*c*x^3+1/2*x^2)+b^2*d^2/c^2*(-1/4*c^4*x^4*arctan(c*x)^2+2/3*I*arctan(c*x)*ln(c^2*x^
2+1)+1/2*c^2*x^2*arctan(c*x)^2-2/3*I*arctan(c*x)*c^2*x^2+1/6*c^3*x^3*arctan(c*x)+2/3*I*arctan(c*x)^2*c^3*x^3+3
/4*arctan(c*x)^2-3/2*c*x*arctan(c*x)-1/3*ln(c*x-I)*ln(c^2*x^2+1)+1/3*ln(c*x+I)*ln(c^2*x^2+1)+1/3*ln(c*x-I)*ln(
-1/2*I*(c*x+I))+1/6*ln(c*x-I)^2-1/6*ln(c*x+I)^2-1/3*ln(c*x+I)*ln(1/2*I*(c*x-I))+1/3*dilog(-1/2*I*(c*x+I))-1/3*
dilog(1/2*I*(c*x-I))-2/3*I*arctan(c*x)-1/12*c^2*x^2+5/6*ln(c^2*x^2+1)+2/3*I*c*x)+2*a*d^2*b/c^2*(-1/4*c^4*x^4*a
rctan(c*x)+2/3*I*arctan(c*x)*c^3*x^3+1/2*c^2*x^2*arctan(c*x)-3/4*c*x+1/12*c^3*x^3-1/3*I*c^2*x^2+1/3*I*ln(c^2*x
^2+1)+3/4*arctan(c*x))

Fricas [F]

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

[In]

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

[Out]

1/48*(3*b^2*c^2*d^2*x^4 - 8*I*b^2*c*d^2*x^3 - 6*b^2*d^2*x^2)*log(-(c*x + I)/(c*x - I))^2 + integral(-1/12*(12*
a^2*c^4*d^2*x^5 - 24*I*a^2*c^3*d^2*x^4 - 24*I*a^2*c*d^2*x^2 - 12*a^2*d^2*x - (-12*I*a*b*c^4*d^2*x^5 - 3*(8*a*b
- I*b^2)*c^3*d^2*x^4 + 8*b^2*c^2*d^2*x^3 - 6*(4*a*b + I*b^2)*c*d^2*x^2 + 12*I*a*b*d^2*x)*log(-(c*x + I)/(c*x
- I)))/(c^2*x^2 + 1), x)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

-1/4*a^2*c^2*d^2*x^4 + 2/3*I*a^2*c*d^2*x^3 + 1/2*b^2*d^2*x^2*arctan(c*x)^2 - 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*c^2*d^2 + 2/3*I*(2*x^3*arctan(c*x) - c*(x^2/c^2 - log(c^2*x^2 + 1)/c^
4))*a*b*c*d^2 + 1/2*a^2*d^2*x^2 + (x^2*arctan(c*x) - c*(x/c^2 - arctan(c*x)/c^3))*a*b*d^2 - 1/2*(2*c*(x/c^2 -
arctan(c*x)/c^3)*arctan(c*x) + (arctan(c*x)^2 - log(c^2*x^2 + 1))/c^2)*b^2*d^2 - 1/48*(3*b^2*c^2*d^2*x^4 - 8*I
*b^2*c*d^2*x^3)*arctan(c*x)^2 + 1/48*(-3*I*b^2*c^2*d^2*x^4 - 8*b^2*c*d^2*x^3)*arctan(c*x)*log(c^2*x^2 + 1) + 1
/192*(3*b^2*c^2*d^2*x^4 - 8*I*b^2*c*d^2*x^3)*log(c^2*x^2 + 1)^2 - integrate(-1/48*(22*b^2*c^3*d^2*x^4*arctan(c
*x) - 36*(b^2*c^4*d^2*x^5 + b^2*c^2*d^2*x^3)*arctan(c*x)^2 - 3*(b^2*c^4*d^2*x^5 + b^2*c^2*d^2*x^3)*log(c^2*x^2
+ 1)^2 - (3*b^2*c^4*d^2*x^5 - 8*b^2*c^2*d^2*x^3 - 24*(b^2*c^3*d^2*x^4 + b^2*c*d^2*x^2)*arctan(c*x))*log(c^2*x
^2 + 1))/(c^2*x^2 + 1), x) + I*integrate(1/48*(72*(b^2*c^3*d^2*x^4 + b^2*c*d^2*x^2)*arctan(c*x)^2 + 6*(b^2*c^3
*d^2*x^4 + b^2*c*d^2*x^2)*log(c^2*x^2 + 1)^2 + 2*(3*b^2*c^4*d^2*x^5 - 8*b^2*c^2*d^2*x^3)*arctan(c*x) + (11*b^2
*c^3*d^2*x^4 + 12*(b^2*c^4*d^2*x^5 + b^2*c^2*d^2*x^3)*arctan(c*x))*log(c^2*x^2 + 1))/(c^2*x^2 + 1), x)

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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