\(\int x^2 (d+i c d x)^2 (a+b \arctan (c x))^2 \, dx\) [77]

Optimal result

Integrand size = 25, antiderivative size = 333 \[ \int x^2 (d+i c d x)^2 (a+b \arctan (c x))^2 \, dx=\frac {i a b d^2 x}{c^2}+\frac {19 b^2 d^2 x}{30 c^2}+\frac {i b^2 d^2 x^2}{6 c}-\frac {1}{30} b^2 d^2 x^3-\frac {19 b^2 d^2 \arctan (c x)}{30 c^3}+\frac {i b^2 d^2 x \arctan (c x)}{c^2}-\frac {8 b d^2 x^2 (a+b \arctan (c x))}{15 c}-\frac {1}{3} i b d^2 x^3 (a+b \arctan (c x))+\frac {1}{10} b c d^2 x^4 (a+b \arctan (c x))-\frac {31 i d^2 (a+b \arctan (c x))^2}{30 c^3}+\frac {1}{3} d^2 x^3 (a+b \arctan (c x))^2+\frac {1}{2} i c d^2 x^4 (a+b \arctan (c x))^2-\frac {1}{5} c^2 d^2 x^5 (a+b \arctan (c x))^2-\frac {16 b d^2 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{15 c^3}-\frac {2 i b^2 d^2 \log \left (1+c^2 x^2\right )}{3 c^3}-\frac {8 i b^2 d^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{15 c^3} \]

[Out]

1/2*I*c*d^2*x^4*(a+b*arctan(c*x))^2+19/30*b^2*d^2*x/c^2+1/6*I*b^2*d^2*x^2/c-1/30*b^2*d^2*x^3-19/30*b^2*d^2*arc
tan(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-1/3*I*b*d^2*x^3*(a
+b*arctan(c*x))+1/10*b*c*d^2*x^4*(a+b*arctan(c*x))-2/3*I*b^2*d^2*ln(c^2*x^2+1)/c^3+1/3*d^2*x^3*(a+b*arctan(c*x
))^2+I*a*b*d^2*x/c^2-1/5*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+I*b
^2*d^2*x*arctan(c*x)/c^2-31/30*I*d^2*(a+b*arctan(c*x))^2/c^3

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.00, number of steps used = 36, 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} \[ \int x^2 (d+i c d x)^2 (a+b \arctan (c x))^2 \, dx=-\frac {31 i d^2 (a+b \arctan (c x))^2}{30 c^3}-\frac {16 b d^2 \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{15 c^3}-\frac {1}{5} c^2 d^2 x^5 (a+b \arctan (c x))^2+\frac {1}{2} i c d^2 x^4 (a+b \arctan (c x))^2+\frac {1}{10} b c d^2 x^4 (a+b \arctan (c x))+\frac {1}{3} d^2 x^3 (a+b \arctan (c x))^2-\frac {1}{3} i b d^2 x^3 (a+b \arctan (c x))-\frac {8 b d^2 x^2 (a+b \arctan (c x))}{15 c}+\frac {i a b d^2 x}{c^2}-\frac {19 b^2 d^2 \arctan (c x)}{30 c^3}+\frac {i b^2 d^2 x \arctan (c x)}{c^2}-\frac {8 i b^2 d^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{15 c^3}+\frac {19 b^2 d^2 x}{30 c^2}-\frac {2 i b^2 d^2 \log \left (c^2 x^2+1\right )}{3 c^3}+\frac {i b^2 d^2 x^2}{6 c}-\frac {1}{30} b^2 d^2 x^3 \]

[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 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^2 x^2 (a+b \arctan (c x))^2+2 i c d^2 x^3 (a+b \arctan (c x))^2-c^2 d^2 x^4 (a+b \arctan (c x))^2\right ) \, dx \\ & = d^2 \int x^2 (a+b \arctan (c x))^2 \, dx+\left (2 i c d^2\right ) \int x^3 (a+b \arctan (c x))^2 \, dx-\left (c^2 d^2\right ) \int x^4 (a+b \arctan (c x))^2 \, dx \\ & = \frac {1}{3} d^2 x^3 (a+b \arctan (c x))^2+\frac {1}{2} i c d^2 x^4 (a+b \arctan (c x))^2-\frac {1}{5} c^2 d^2 x^5 (a+b \arctan (c x))^2-\frac {1}{3} \left (2 b c d^2\right ) \int \frac {x^3 (a+b \arctan (c x))}{1+c^2 x^2} \, dx-\left (i b c^2 d^2\right ) \int \frac {x^4 (a+b \arctan (c x))}{1+c^2 x^2} \, dx+\frac {1}{5} \left (2 b c^3 d^2\right ) \int \frac {x^5 (a+b \arctan (c x))}{1+c^2 x^2} \, dx \\ & = \frac {1}{3} d^2 x^3 (a+b \arctan (c x))^2+\frac {1}{2} i c d^2 x^4 (a+b \arctan (c x))^2-\frac {1}{5} c^2 d^2 x^5 (a+b \arctan (c x))^2-\left (i b d^2\right ) \int x^2 (a+b \arctan (c x)) \, dx+\left (i b d^2\right ) \int \frac {x^2 (a+b \arctan (c x))}{1+c^2 x^2} \, dx-\frac {\left (2 b d^2\right ) \int x (a+b \arctan (c x)) \, dx}{3 c}+\frac {\left (2 b d^2\right ) \int \frac {x (a+b \arctan (c x))}{1+c^2 x^2} \, dx}{3 c}+\frac {1}{5} \left (2 b c d^2\right ) \int x^3 (a+b \arctan (c x)) \, dx-\frac {1}{5} \left (2 b c d^2\right ) \int \frac {x^3 (a+b \arctan (c x))}{1+c^2 x^2} \, dx \\ & = -\frac {b d^2 x^2 (a+b \arctan (c x))}{3 c}-\frac {1}{3} i b d^2 x^3 (a+b \arctan (c x))+\frac {1}{10} b c d^2 x^4 (a+b \arctan (c x))-\frac {i d^2 (a+b \arctan (c x))^2}{3 c^3}+\frac {1}{3} d^2 x^3 (a+b \arctan (c x))^2+\frac {1}{2} i c d^2 x^4 (a+b \arctan (c x))^2-\frac {1}{5} c^2 d^2 x^5 (a+b \arctan (c x))^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 (a+b \arctan (c x)) \, dx}{c^2}-\frac {\left (i b d^2\right ) \int \frac {a+b \arctan (c x)}{1+c^2 x^2} \, dx}{c^2}-\frac {\left (2 b d^2\right ) \int \frac {a+b \arctan (c x)}{i-c x} \, dx}{3 c^2}-\frac {\left (2 b d^2\right ) \int x (a+b \arctan (c x)) \, dx}{5 c}+\frac {\left (2 b d^2\right ) \int \frac {x (a+b \arctan (c x))}{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 (a+b \arctan (c x))}{15 c}-\frac {1}{3} i b d^2 x^3 (a+b \arctan (c x))+\frac {1}{10} b c d^2 x^4 (a+b \arctan (c x))-\frac {31 i d^2 (a+b \arctan (c x))^2}{30 c^3}+\frac {1}{3} d^2 x^3 (a+b \arctan (c x))^2+\frac {1}{2} i c d^2 x^4 (a+b \arctan (c x))^2-\frac {1}{5} c^2 d^2 x^5 (a+b \arctan (c x))^2-\frac {2 b d^2 (a+b \arctan (c x)) \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 \arctan (c x)}{i-c x} \, dx}{5 c^2}+\frac {\left (i b^2 d^2\right ) \int \arctan (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 ) \text {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 \arctan (c x)}{3 c^3}+\frac {i b^2 d^2 x \arctan (c x)}{c^2}-\frac {8 b d^2 x^2 (a+b \arctan (c x))}{15 c}-\frac {1}{3} i b d^2 x^3 (a+b \arctan (c x))+\frac {1}{10} b c d^2 x^4 (a+b \arctan (c x))-\frac {31 i d^2 (a+b \arctan (c x))^2}{30 c^3}+\frac {1}{3} d^2 x^3 (a+b \arctan (c x))^2+\frac {1}{2} i c d^2 x^4 (a+b \arctan (c x))^2-\frac {1}{5} c^2 d^2 x^5 (a+b \arctan (c x))^2-\frac {16 b d^2 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{15 c^3}-\frac {\left (2 i 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^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 ) \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 {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 \arctan (c x)}{30 c^3}+\frac {i b^2 d^2 x \arctan (c x)}{c^2}-\frac {8 b d^2 x^2 (a+b \arctan (c x))}{15 c}-\frac {1}{3} i b d^2 x^3 (a+b \arctan (c x))+\frac {1}{10} b c d^2 x^4 (a+b \arctan (c x))-\frac {31 i d^2 (a+b \arctan (c x))^2}{30 c^3}+\frac {1}{3} d^2 x^3 (a+b \arctan (c x))^2+\frac {1}{2} i c d^2 x^4 (a+b \arctan (c x))^2-\frac {1}{5} c^2 d^2 x^5 (a+b \arctan (c x))^2-\frac {16 b d^2 (a+b \arctan (c x)) \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 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{3 c^3}-\frac {\left (2 i 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 )}{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 \arctan (c x)}{30 c^3}+\frac {i b^2 d^2 x \arctan (c x)}{c^2}-\frac {8 b d^2 x^2 (a+b \arctan (c x))}{15 c}-\frac {1}{3} i b d^2 x^3 (a+b \arctan (c x))+\frac {1}{10} b c d^2 x^4 (a+b \arctan (c x))-\frac {31 i d^2 (a+b \arctan (c x))^2}{30 c^3}+\frac {1}{3} d^2 x^3 (a+b \arctan (c x))^2+\frac {1}{2} i c d^2 x^4 (a+b \arctan (c x))^2-\frac {1}{5} c^2 d^2 x^5 (a+b \arctan (c x))^2-\frac {16 b d^2 (a+b \arctan (c x)) \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 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{15 c^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.97 (sec) , antiderivative size = 306, normalized size of antiderivative = 0.92 \[ \int x^2 (d+i c d x)^2 (a+b \arctan (c x))^2 \, dx=-\frac {d^2 \left (9 a b-5 i b^2-30 i a b c x-19 b^2 c x+16 a b c^2 x^2-5 i b^2 c^2 x^2-10 a^2 c^3 x^3+10 i a b c^3 x^3+b^2 c^3 x^3-15 i a^2 c^4 x^4-3 a b c^4 x^4+6 a^2 c^5 x^5+b^2 (-i+c x)^3 \left (-1+3 i c x+6 c^2 x^2\right ) \arctan (c x)^2+b \arctan (c x) \left (b \left (19-30 i c x+16 c^2 x^2+10 i c^3 x^3-3 c^4 x^4\right )+2 a \left (15 i-10 c^3 x^3-15 i c^4 x^4+6 c^5 x^5\right )+32 b \log \left (1+e^{2 i \arctan (c x)}\right )\right )-16 a b \log \left (1+c^2 x^2\right )+20 i b^2 \log \left (1+c^2 x^2\right )-16 i b^2 \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )\right )}{30 c^3} \]

[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

Maple [A] (verified)

Time = 2.52 (sec) , antiderivative size = 420, normalized size of antiderivative = 1.26

method	result	size
parts	\(a^{2} d^{2} \left (-\frac {1}{5} c^{2} x^{5}+\frac {1}{2} i c \,x^{4}+\frac {1}{3} x^{3}\right )+\frac {b^{2} d^{2} \left (-\frac {\arctan \left (c x \right )^{2} c^{5} x^{5}}{5}-\frac {2 i \ln \left (c^{2} x^{2}+1\right )}{3}+\frac {c^{3} x^{3} \arctan \left (c x \right )^{2}}{3}+\frac {4 i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{15}+\frac {c^{4} x^{4} \arctan \left (c x \right )}{10}-\frac {4 i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{15}-\frac {8 c^{2} x^{2} \arctan \left (c x \right )}{15}+\frac {8 \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{15}-\frac {i \arctan \left (c x \right )^{2}}{2}+\frac {19 c x}{30}-\frac {c^{3} x^{3}}{30}+\frac {i \arctan \left (c x \right )^{2} c^{4} x^{4}}{2}-\frac {i \arctan \left (c x \right ) c^{3} x^{3}}{3}-\frac {19 \arctan \left (c x \right )}{30}+i \arctan \left (c x \right ) c x +\frac {i c^{2} x^{2}}{6}\right )}{c^{3}}+\frac {2 a \,d^{2} b \left (-\frac {c^{5} x^{5} \arctan \left (c x \right )}{5}+\frac {i \arctan \left (c x \right ) c^{4} x^{4}}{2}+\frac {c^{3} x^{3} \arctan \left (c x \right )}{3}+\frac {i c x}{2}+\frac {c^{4} x^{4}}{20}-\frac {i c^{3} x^{3}}{6}-\frac {4 c^{2} x^{2}}{15}+\frac {4 \ln \left (c^{2} x^{2}+1\right )}{15}-\frac {i \arctan \left (c x \right )}{2}\right )}{c^{3}}\)	\(420\)
derivativedivides	\(\frac {a^{2} d^{2} \left (-\frac {1}{5} c^{5} x^{5}+\frac {1}{2} i c^{4} x^{4}+\frac {1}{3} c^{3} x^{3}\right )+b^{2} d^{2} \left (-\frac {\arctan \left (c x \right )^{2} c^{5} x^{5}}{5}-\frac {2 i \ln \left (c^{2} x^{2}+1\right )}{3}+\frac {c^{3} x^{3} \arctan \left (c x \right )^{2}}{3}+\frac {4 i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{15}+\frac {c^{4} x^{4} \arctan \left (c x \right )}{10}-\frac {4 i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{15}-\frac {8 c^{2} x^{2} \arctan \left (c x \right )}{15}+\frac {8 \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{15}-\frac {i \arctan \left (c x \right )^{2}}{2}+\frac {19 c x}{30}-\frac {c^{3} x^{3}}{30}+\frac {i \arctan \left (c x \right )^{2} c^{4} x^{4}}{2}-\frac {i \arctan \left (c x \right ) c^{3} x^{3}}{3}-\frac {19 \arctan \left (c x \right )}{30}+i \arctan \left (c x \right ) c x +\frac {i c^{2} x^{2}}{6}\right )+2 a \,d^{2} b \left (-\frac {c^{5} x^{5} \arctan \left (c x \right )}{5}+\frac {i \arctan \left (c x \right ) c^{4} x^{4}}{2}+\frac {c^{3} x^{3} \arctan \left (c x \right )}{3}+\frac {i c x}{2}+\frac {c^{4} x^{4}}{20}-\frac {i c^{3} x^{3}}{6}-\frac {4 c^{2} x^{2}}{15}+\frac {4 \ln \left (c^{2} x^{2}+1\right )}{15}-\frac {i \arctan \left (c x \right )}{2}\right )}{c^{3}}\)	\(423\)
default	\(\frac {a^{2} d^{2} \left (-\frac {1}{5} c^{5} x^{5}+\frac {1}{2} i c^{4} x^{4}+\frac {1}{3} c^{3} x^{3}\right )+b^{2} d^{2} \left (-\frac {\arctan \left (c x \right )^{2} c^{5} x^{5}}{5}-\frac {2 i \ln \left (c^{2} x^{2}+1\right )}{3}+\frac {c^{3} x^{3} \arctan \left (c x \right )^{2}}{3}+\frac {4 i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{15}+\frac {c^{4} x^{4} \arctan \left (c x \right )}{10}-\frac {4 i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{15}-\frac {8 c^{2} x^{2} \arctan \left (c x \right )}{15}+\frac {8 \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{15}-\frac {i \arctan \left (c x \right )^{2}}{2}+\frac {19 c x}{30}-\frac {c^{3} x^{3}}{30}+\frac {i \arctan \left (c x \right )^{2} c^{4} x^{4}}{2}-\frac {i \arctan \left (c x \right ) c^{3} x^{3}}{3}-\frac {19 \arctan \left (c x \right )}{30}+i \arctan \left (c x \right ) c x +\frac {i c^{2} x^{2}}{6}\right )+2 a \,d^{2} b \left (-\frac {c^{5} x^{5} \arctan \left (c x \right )}{5}+\frac {i \arctan \left (c x \right ) c^{4} x^{4}}{2}+\frac {c^{3} x^{3} \arctan \left (c x \right )}{3}+\frac {i c x}{2}+\frac {c^{4} x^{4}}{20}-\frac {i c^{3} x^{3}}{6}-\frac {4 c^{2} x^{2}}{15}+\frac {4 \ln \left (c^{2} x^{2}+1\right )}{15}-\frac {i \arctan \left (c x \right )}{2}\right )}{c^{3}}\)	\(423\)
risch	\(\frac {19 b^{2} d^{2} x}{30 c^{2}}-\frac {2537 b^{2} d^{2} \arctan \left (c x \right )}{3600 c^{3}}-\frac {b^{2} d^{2} x^{3}}{30}+\frac {a^{2} d^{2} x^{3}}{3}-\frac {a^{2} c^{2} d^{2} x^{5}}{5}+\frac {a b c \,d^{2} x^{4}}{10}-\frac {59 a b \,d^{2}}{30 c^{3}}+\frac {i a b \,d^{2} x}{c^{2}}-\frac {8 i d^{2} b^{2} \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 {8 i d^{2} b^{2} \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (-i c x +1\right )}{15 c^{3}}+\frac {8 a b \,d^{2} \ln \left (c^{2} x^{2}+1\right )}{15 c^{3}}-\frac {8 d^{2} x^{2} a b}{15 c}+\frac {5 i d^{2} b^{2}}{6 c^{3}}-\frac {31 i d^{2} a^{2}}{30 c^{3}}+\frac {b^{2} d^{2} \ln \left (-i c x +1\right ) x^{3}}{6}-\frac {d^{2} b^{2} \ln \left (-i c x +1\right )^{2} x^{3}}{12}-\frac {i a b \,d^{2} \arctan \left (c x \right )}{c^{3}}-\frac {4 i d^{2} b^{2} \ln \left (-i c x +1\right ) x^{2}}{15 c}+\frac {i d^{2} c \,b^{2} \ln \left (-i c x +1\right ) x^{4}}{20}-\frac {d^{2} c a b \ln \left (-i c x +1\right ) x^{4}}{2}-\frac {i d^{2} c \,b^{2} \ln \left (-i c x +1\right )^{2} x^{4}}{8}+\frac {i d^{2} a b \ln \left (-i c x +1\right ) x^{3}}{3}+\left (-\frac {b^{2} d^{2} \left (6 c^{2} x^{5}-15 i c \,x^{4}-10 x^{3}\right ) \ln \left (-i c x +1\right )}{60}-\frac {i b \,d^{2} \left (-12 a \,c^{5} x^{5}+3 b \,c^{4} x^{4}+30 i a \,c^{4} x^{4}+20 a \,c^{3} x^{3}-10 i b \,c^{3} x^{3}-16 b \,c^{2} x^{2}+31 b \ln \left (-i c x +1\right )+30 i b c x \right )}{60 c^{3}}\right ) \ln \left (i c x +1\right )-\frac {i d^{2} c^{2} b a \ln \left (-i c x +1\right ) x^{5}}{5}+\frac {i a^{2} c \,d^{2} x^{4}}{2}+\frac {i b^{2} d^{2} x^{2}}{6 c}-\frac {5057 i d^{2} b^{2} \ln \left (c^{2} x^{2}+1\right )}{7200 c^{3}}-\frac {i a b \,d^{2} x^{3}}{3}-\frac {8 i d^{2} b^{2} \operatorname {dilog}\left (\frac {1}{2}-\frac {i c x}{2}\right )}{15 c^{3}}+\frac {257 i d^{2} b^{2} \ln \left (-i c x +1\right )}{3600 c^{3}}+\frac {31 i d^{2} b^{2} \ln \left (-i c x +1\right )^{2}}{120 c^{3}}+\frac {b^{2} d^{2} \left (6 c^{5} x^{5}-15 i c^{4} x^{4}-10 c^{3} x^{3}-i\right ) \ln \left (i c x +1\right )^{2}}{120 c^{3}}-\frac {d^{2} b^{2} \ln \left (-i c x +1\right ) x}{2 c^{2}}+\frac {d^{2} c^{2} b^{2} \ln \left (-i c x +1\right )^{2} x^{5}}{20}\)	\(740\)

[In]

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

[Out]

a^2*d^2*(-1/5*c^2*x^5+1/2*I*x^4*c+1/3*x^3)+b^2*d^2/c^3*(-1/5*arctan(c*x)^2*c^5*x^5-2/3*I*ln(c^2*x^2+1)+1/3*c^3
*x^3*arctan(c*x)^2+4/15*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)+1/10*c^4*x^4*arctan(c*x)-4/15*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)-8/15*c^2*x^2*arctan(c*x)+8/15*arctan(c*x)*ln(c^2*x^2+1)-1/2*I*arctan(c*x)^2+19/30*c*x-1/
30*c^3*x^3+1/2*I*arctan(c*x)^2*c^4*x^4-1/3*I*arctan(c*x)*c^3*x^3-19/30*arctan(c*x)+I*arctan(c*x)*c*x+1/6*I*c^2
*x^2)+2*a*d^2*b/c^3*(-1/5*c^5*x^5*arctan(c*x)+1/2*I*arctan(c*x)*c^4*x^4+1/3*c^3*x^3*arctan(c*x)+1/2*I*c*x+1/20
*c^4*x^4-1/6*I*c^3*x^3-4/15*c^2*x^2+4/15*ln(c^2*x^2+1)-1/2*I*arctan(c*x))

Fricas [F]

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

[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 - 6*(
10*a*b - I*b^2)*c^3*d^2*x^5 + 15*b^2*c^2*d^2*x^4 - 10*(6*a*b + 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)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[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 - 1/120*(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)*arctan(c*x)^2 + 1/120*(-6*I*b^2*c^2*d^2*x^5 - 15*b^2*c*d^2*x^4 + 1
0*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)

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[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)