\(\int \frac {x^3 (a+b \arctan (c x))}{(d+i c d x)^2} \, dx\) [51]

Optimal result

Integrand size = 23, antiderivative size = 203 \[ \int \frac {x^3 (a+b \arctan (c x))}{(d+i c d x)^2} \, dx=-\frac {2 i a x}{c^3 d^2}+\frac {b x}{2 c^3 d^2}+\frac {b}{2 c^4 d^2 (i-c x)}-\frac {b \arctan (c x)}{c^4 d^2}-\frac {2 i b x \arctan (c x)}{c^3 d^2}-\frac {x^2 (a+b \arctan (c x))}{2 c^2 d^2}+\frac {i (a+b \arctan (c x))}{c^4 d^2 (i-c x)}-\frac {3 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^4 d^2}+\frac {i b \log \left (1+c^2 x^2\right )}{c^4 d^2}-\frac {3 i b \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 c^4 d^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {4996, 4930, 266, 4946, 327, 209, 4972, 641, 46, 4964, 2449, 2352} \[ \int \frac {x^3 (a+b \arctan (c x))}{(d+i c d x)^2} \, dx=\frac {i (a+b \arctan (c x))}{c^4 d^2 (-c x+i)}-\frac {3 \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c^4 d^2}-\frac {x^2 (a+b \arctan (c x))}{2 c^2 d^2}-\frac {2 i a x}{c^3 d^2}-\frac {b \arctan (c x)}{c^4 d^2}-\frac {2 i b x \arctan (c x)}{c^3 d^2}-\frac {3 i b \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{2 c^4 d^2}+\frac {b}{2 c^4 d^2 (-c x+i)}+\frac {b x}{2 c^3 d^2}+\frac {i b \log \left (c^2 x^2+1\right )}{c^4 d^2} \]

[In]

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

[Out]

((-2*I)*a*x)/(c^3*d^2) + (b*x)/(2*c^3*d^2) + b/(2*c^4*d^2*(I - c*x)) - (b*ArcTan[c*x])/(c^4*d^2) - ((2*I)*b*x*
ArcTan[c*x])/(c^3*d^2) - (x^2*(a + b*ArcTan[c*x]))/(2*c^2*d^2) + (I*(a + b*ArcTan[c*x]))/(c^4*d^2*(I - c*x)) -
(3*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(c^4*d^2) + (I*b*Log[1 + c^2*x^2])/(c^4*d^2) - (((3*I)/2)*b*PolyLo
g[2, 1 - 2/(1 + I*c*x)])/(c^4*d^2)

Rule 46

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[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 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 641

Int[((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a/d + (c/e)*x)^
p, x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && I
ntegerQ[m + p]))

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 4972

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)^(q + 1)*((a + b*
ArcTan[c*x])/(e*(q + 1))), x] - Dist[b*(c/(e*(q + 1))), Int[(d + e*x)^(q + 1)/(1 + c^2*x^2), x], x] /; FreeQ[{
a, b, c, d, e, q}, x] && NeQ[q, -1]

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])

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {2 i (a+b \arctan (c x))}{c^3 d^2}-\frac {x (a+b \arctan (c x))}{c^2 d^2}+\frac {i (a+b \arctan (c x))}{c^3 d^2 (-i+c x)^2}+\frac {3 (a+b \arctan (c x))}{c^3 d^2 (-i+c x)}\right ) \, dx \\ & = \frac {i \int \frac {a+b \arctan (c x)}{(-i+c x)^2} \, dx}{c^3 d^2}-\frac {(2 i) \int (a+b \arctan (c x)) \, dx}{c^3 d^2}+\frac {3 \int \frac {a+b \arctan (c x)}{-i+c x} \, dx}{c^3 d^2}-\frac {\int x (a+b \arctan (c x)) \, dx}{c^2 d^2} \\ & = -\frac {2 i a x}{c^3 d^2}-\frac {x^2 (a+b \arctan (c x))}{2 c^2 d^2}+\frac {i (a+b \arctan (c x))}{c^4 d^2 (i-c x)}-\frac {3 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^4 d^2}+\frac {(i b) \int \frac {1}{(-i+c x) \left (1+c^2 x^2\right )} \, dx}{c^3 d^2}-\frac {(2 i b) \int \arctan (c x) \, dx}{c^3 d^2}+\frac {(3 b) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{c^3 d^2}+\frac {b \int \frac {x^2}{1+c^2 x^2} \, dx}{2 c d^2} \\ & = -\frac {2 i a x}{c^3 d^2}+\frac {b x}{2 c^3 d^2}-\frac {2 i b x \arctan (c x)}{c^3 d^2}-\frac {x^2 (a+b \arctan (c x))}{2 c^2 d^2}+\frac {i (a+b \arctan (c x))}{c^4 d^2 (i-c x)}-\frac {3 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^4 d^2}-\frac {(3 i b) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{c^4 d^2}+\frac {(i b) \int \frac {1}{(-i+c x)^2 (i+c x)} \, dx}{c^3 d^2}-\frac {b \int \frac {1}{1+c^2 x^2} \, dx}{2 c^3 d^2}+\frac {(2 i b) \int \frac {x}{1+c^2 x^2} \, dx}{c^2 d^2} \\ & = -\frac {2 i a x}{c^3 d^2}+\frac {b x}{2 c^3 d^2}-\frac {b \arctan (c x)}{2 c^4 d^2}-\frac {2 i b x \arctan (c x)}{c^3 d^2}-\frac {x^2 (a+b \arctan (c x))}{2 c^2 d^2}+\frac {i (a+b \arctan (c x))}{c^4 d^2 (i-c x)}-\frac {3 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^4 d^2}+\frac {i b \log \left (1+c^2 x^2\right )}{c^4 d^2}-\frac {3 i b \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 c^4 d^2}+\frac {(i b) \int \left (-\frac {i}{2 (-i+c x)^2}+\frac {i}{2 \left (1+c^2 x^2\right )}\right ) \, dx}{c^3 d^2} \\ & = -\frac {2 i a x}{c^3 d^2}+\frac {b x}{2 c^3 d^2}+\frac {b}{2 c^4 d^2 (i-c x)}-\frac {b \arctan (c x)}{2 c^4 d^2}-\frac {2 i b x \arctan (c x)}{c^3 d^2}-\frac {x^2 (a+b \arctan (c x))}{2 c^2 d^2}+\frac {i (a+b \arctan (c x))}{c^4 d^2 (i-c x)}-\frac {3 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^4 d^2}+\frac {i b \log \left (1+c^2 x^2\right )}{c^4 d^2}-\frac {3 i b \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 c^4 d^2}-\frac {b \int \frac {1}{1+c^2 x^2} \, dx}{2 c^3 d^2} \\ & = -\frac {2 i a x}{c^3 d^2}+\frac {b x}{2 c^3 d^2}+\frac {b}{2 c^4 d^2 (i-c x)}-\frac {b \arctan (c x)}{c^4 d^2}-\frac {2 i b x \arctan (c x)}{c^3 d^2}-\frac {x^2 (a+b \arctan (c x))}{2 c^2 d^2}+\frac {i (a+b \arctan (c x))}{c^4 d^2 (i-c x)}-\frac {3 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^4 d^2}+\frac {i b \log \left (1+c^2 x^2\right )}{c^4 d^2}-\frac {3 i b \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 c^4 d^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.90 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.92 \[ \int \frac {x^3 (a+b \arctan (c x))}{(d+i c d x)^2} \, dx=-\frac {8 i a c x+2 a c^2 x^2+\frac {4 i a}{-i+c x}-12 i a \arctan (c x)-6 a \log \left (1+c^2 x^2\right )+b \left (-2 c x-12 i \arctan (c x)^2+i \cos (2 \arctan (c x))-4 i \log \left (1+c^2 x^2\right )-6 i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )+2 \arctan (c x) \left (1+4 i c x+c^2 x^2-\cos (2 \arctan (c x))+6 \log \left (1+e^{2 i \arctan (c x)}\right )+i \sin (2 \arctan (c x))\right )+\sin (2 \arctan (c x))\right )}{4 c^4 d^2} \]

[In]

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

[Out]

-1/4*((8*I)*a*c*x + 2*a*c^2*x^2 + ((4*I)*a)/(-I + c*x) - (12*I)*a*ArcTan[c*x] - 6*a*Log[1 + c^2*x^2] + b*(-2*c
*x - (12*I)*ArcTan[c*x]^2 + I*Cos[2*ArcTan[c*x]] - (4*I)*Log[1 + c^2*x^2] - (6*I)*PolyLog[2, -E^((2*I)*ArcTan[
c*x])] + 2*ArcTan[c*x]*(1 + (4*I)*c*x + c^2*x^2 - Cos[2*ArcTan[c*x]] + 6*Log[1 + E^((2*I)*ArcTan[c*x])] + I*Si
n[2*ArcTan[c*x]]) + Sin[2*ArcTan[c*x]]))/(c^4*d^2)

Maple [A] (verified)

Time = 0.94 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.56

method	result	size
derivativedivides	\(\frac {-\frac {2 i b \arctan \left (c x \right ) c x}{d^{2}}-\frac {a \,c^{2} x^{2}}{2 d^{2}}-\frac {i a}{d^{2} \left (c x -i\right )}+\frac {3 a \ln \left (c^{2} x^{2}+1\right )}{2 d^{2}}+\frac {3 i b \ln \left (c x -i\right )^{2}}{4 d^{2}}+\frac {3 i a \arctan \left (c x \right )}{d^{2}}-\frac {b \arctan \left (c x \right ) c^{2} x^{2}}{2 d^{2}}+\frac {i b \ln \left (c^{4} x^{4}+10 c^{2} x^{2}+9\right )}{8 d^{2}}+\frac {3 b \arctan \left (c x \right ) \ln \left (c x -i\right )}{d^{2}}-\frac {3 i b \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{2 d^{2}}-\frac {i b}{2 d^{2}}-\frac {2 i a c x}{d^{2}}+\frac {b c x}{2 d^{2}}-\frac {i b \arctan \left (c x \right )}{d^{2} \left (c x -i\right )}+\frac {3 i b \ln \left (c^{2} x^{2}+1\right )}{4 d^{2}}-\frac {b \arctan \left (\frac {c x}{2}\right )}{4 d^{2}}+\frac {b \arctan \left (\frac {1}{6} c^{3} x^{3}+\frac {7}{6} c x \right )}{4 d^{2}}+\frac {b \arctan \left (\frac {c x}{2}-\frac {i}{2}\right )}{2 d^{2}}-\frac {b}{2 d^{2} \left (c x -i\right )}-\frac {3 i b \ln \left (-\frac {i \left (c x +i\right )}{2}\right ) \ln \left (c x -i\right )}{2 d^{2}}-\frac {3 b \arctan \left (c x \right )}{2 d^{2}}}{c^{4}}\)	\(317\)
default	\(\frac {-\frac {2 i b \arctan \left (c x \right ) c x}{d^{2}}-\frac {a \,c^{2} x^{2}}{2 d^{2}}-\frac {i a}{d^{2} \left (c x -i\right )}+\frac {3 a \ln \left (c^{2} x^{2}+1\right )}{2 d^{2}}+\frac {3 i b \ln \left (c x -i\right )^{2}}{4 d^{2}}+\frac {3 i a \arctan \left (c x \right )}{d^{2}}-\frac {b \arctan \left (c x \right ) c^{2} x^{2}}{2 d^{2}}+\frac {i b \ln \left (c^{4} x^{4}+10 c^{2} x^{2}+9\right )}{8 d^{2}}+\frac {3 b \arctan \left (c x \right ) \ln \left (c x -i\right )}{d^{2}}-\frac {3 i b \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{2 d^{2}}-\frac {i b}{2 d^{2}}-\frac {2 i a c x}{d^{2}}+\frac {b c x}{2 d^{2}}-\frac {i b \arctan \left (c x \right )}{d^{2} \left (c x -i\right )}+\frac {3 i b \ln \left (c^{2} x^{2}+1\right )}{4 d^{2}}-\frac {b \arctan \left (\frac {c x}{2}\right )}{4 d^{2}}+\frac {b \arctan \left (\frac {1}{6} c^{3} x^{3}+\frac {7}{6} c x \right )}{4 d^{2}}+\frac {b \arctan \left (\frac {c x}{2}-\frac {i}{2}\right )}{2 d^{2}}-\frac {b}{2 d^{2} \left (c x -i\right )}-\frac {3 i b \ln \left (-\frac {i \left (c x +i\right )}{2}\right ) \ln \left (c x -i\right )}{2 d^{2}}-\frac {3 b \arctan \left (c x \right )}{2 d^{2}}}{c^{4}}\)	\(317\)
parts	\(-\frac {x^{2} a}{2 d^{2} c^{2}}+\frac {i b \ln \left (c^{4} x^{4}+10 c^{2} x^{2}+9\right )}{8 d^{2} c^{4}}-\frac {3 i b \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{2 d^{2} c^{4}}+\frac {3 a \ln \left (c^{2} x^{2}+1\right )}{2 d^{2} c^{4}}-\frac {i b}{2 d^{2} c^{4}}-\frac {b \arctan \left (c x \right ) x^{2}}{2 d^{2} c^{2}}-\frac {2 i b x \arctan \left (c x \right )}{c^{3} d^{2}}+\frac {3 i b \ln \left (c^{2} x^{2}+1\right )}{4 c^{4} d^{2}}+\frac {3 b \arctan \left (c x \right ) \ln \left (c x -i\right )}{d^{2} c^{4}}-\frac {2 i a x}{c^{3} d^{2}}+\frac {3 i b \ln \left (c x -i\right )^{2}}{4 d^{2} c^{4}}-\frac {i b \arctan \left (c x \right )}{d^{2} c^{4} \left (c x -i\right )}+\frac {b x}{2 c^{3} d^{2}}+\frac {i a}{d^{2} c^{4} \left (-c x +i\right )}-\frac {3 i b \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{2 d^{2} c^{4}}-\frac {b \arctan \left (\frac {c x}{2}\right )}{4 d^{2} c^{4}}+\frac {b \arctan \left (\frac {1}{6} c^{3} x^{3}+\frac {7}{6} c x \right )}{4 d^{2} c^{4}}+\frac {b \arctan \left (\frac {c x}{2}-\frac {i}{2}\right )}{2 d^{2} c^{4}}+\frac {3 i a \arctan \left (c x \right )}{d^{2} c^{4}}-\frac {3 b \arctan \left (c x \right )}{2 c^{4} d^{2}}-\frac {b}{2 c^{4} d^{2} \left (c x -i\right )}\)	\(368\)
risch	\(\frac {b x}{2 c^{3} d^{2}}+\frac {b}{2 c^{4} d^{2} \left (-c x +i\right )}-\frac {i \ln \left (-i c x +1\right ) b \,x^{2}}{4 d^{2} c^{2}}-\frac {3 b \arctan \left (c x \right )}{2 c^{4} d^{2}}-\frac {i b \ln \left (-i c x +1\right )}{4 d^{2} c^{4} \left (-i c x -1\right )}-\frac {b \ln \left (-i c x +1\right ) x}{4 d^{2} c^{3} \left (-i c x -1\right )}-\frac {3 i b \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (\frac {1}{2}-\frac {i c x}{2}\right )}{2 d^{2} c^{4}}-\frac {5 i b}{8 d^{2} c^{4}}+\frac {3 i b \ln \left (c^{2} x^{2}+1\right )}{4 c^{4} d^{2}}-\frac {2 i a x}{c^{3} d^{2}}-\frac {3 i b \ln \left (i c x +1\right )^{2}}{4 d^{2} c^{4}}+\frac {3 a}{2 d^{2} c^{4}}+\left (\frac {i b \left (\frac {1}{2} c \,x^{2}+2 i x \right )}{2 c^{3} d^{2}}-\frac {b}{2 c^{4} d^{2} \left (c x -i\right )}\right ) \ln \left (i c x +1\right )+\frac {\ln \left (-i c x +1\right ) b x}{d^{2} c^{3}}+\frac {3 i b \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (-i c x +1\right )}{2 d^{2} c^{4}}-\frac {3 i b \operatorname {dilog}\left (\frac {1}{2}-\frac {i c x}{2}\right )}{2 d^{2} c^{4}}+\frac {3 i a \arctan \left (c x \right )}{d^{2} c^{4}}-\frac {x^{2} a}{2 d^{2} c^{2}}+\frac {3 a \ln \left (c^{2} x^{2}+1\right )}{2 d^{2} c^{4}}-\frac {a}{d^{2} c^{4} \left (-i c x -1\right )}+\frac {3 i \ln \left (-i c x +1\right ) b}{4 d^{2} c^{4}}\)	\(404\)

[In]

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

[Out]

1/c^4*(-2*I*b/d^2*arctan(c*x)*c*x-1/2*a/d^2*c^2*x^2-I*a/d^2/(c*x-I)+3/2*a/d^2*ln(c^2*x^2+1)+3/4*I*b/d^2*ln(c*x
-I)^2+3*I*a/d^2*arctan(c*x)-1/2*b/d^2*arctan(c*x)*c^2*x^2+1/8*I*b/d^2*ln(c^4*x^4+10*c^2*x^2+9)+3*b/d^2*arctan(
c*x)*ln(c*x-I)-3/2*I*b/d^2*dilog(-1/2*I*(c*x+I))-1/2*I*b/d^2-2*I*a/d^2*c*x+1/2*b/d^2*c*x-I*b/d^2*arctan(c*x)/(
c*x-I)+3/4*I*b/d^2*ln(c^2*x^2+1)-1/4*b/d^2*arctan(1/2*c*x)+1/4*b/d^2*arctan(1/6*c^3*x^3+7/6*c*x)+1/2*b/d^2*arc
tan(1/2*c*x-1/2*I)-1/2*b/d^2/(c*x-I)-3/2*I*b/d^2*ln(-1/2*I*(c*x+I))*ln(c*x-I)-3/2*b*arctan(c*x)/d^2)

Fricas [F]

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

[In]

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

[Out]

integral(1/2*(-I*b*x^3*log(-(c*x + I)/(c*x - I)) - 2*a*x^3)/(c^2*d^2*x^2 - 2*I*c*d^2*x - d^2), x)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

-1/2*a*(2*I/(c^5*d^2*x - I*c^4*d^2) + (c*x^2 + 4*I*x)/(c^3*d^2) - 6*log(c*x - I)/(c^4*d^2)) + 1/8*(I*c^3*x^3 -
5*c^2*x^2 - 2*c*x*(arctan2(1, c*x) - 3*I) - 12*(-I*c*x - 1)*arctan(c*x)^2 - 3*(-I*c*x - 1)*log(c^2*x^2 + 1)^2
- 3*(c^5*d^2*x - I*c^4*d^2)*((c*(x/(c^7*d^2*x^2 + c^5*d^2) + arctan(c*x)/(c^6*d^2)) - 2*arctan(c*x)/(c^7*d^2*
x^2 + c^5*d^2))*c + 16*integrate(1/8*log(c^2*x^2 + 1)/(c^7*d^2*x^4 + 2*c^5*d^2*x^2 + c^3*d^2), x)) + 3*(-I*c^5
*d^2*x - c^4*d^2)*(c*(c^2/(c^9*d^2*x^2 + c^7*d^2) + log(c^2*x^2 + 1)/(c^7*d^2*x^2 + c^5*d^2)) + 32*integrate(1
/8*arctan(c*x)/(c^7*d^2*x^4 + 2*c^5*d^2*x^2 + c^3*d^2), x)) + 6*(c^6*d^2*x - I*c^5*d^2)*(c*(x/(c^7*d^2*x^2 + c
^5*d^2) + arctan(c*x)/(c^6*d^2)) - 16*c*integrate(1/8*x^2*log(c^2*x^2 + 1)/(c^7*d^2*x^4 + 2*c^5*d^2*x^2 + c^3*
d^2), x) - 2*arctan(c*x)/(c^7*d^2*x^2 + c^5*d^2)) - 6*(I*c^6*d^2*x + c^5*d^2)*(32*c*integrate(1/8*x^2*arctan(c
*x)/(c^7*d^2*x^4 + 2*c^5*d^2*x^2 + c^3*d^2), x) - c^2/(c^9*d^2*x^2 + c^7*d^2) - log(c^2*x^2 + 1)/(c^7*d^2*x^2
+ c^5*d^2)) - 2*(c^3*x^3 + 3*I*c^2*x^2 + 4*c*x + 2*I)*arctan(c*x) - 16*(c^9*d^2*x - I*c^8*d^2)*integrate(1/8*(
2*c*x^5*arctan(c*x) + x^4*log(c^2*x^2 + 1))/(c^7*d^2*x^4 + 2*c^5*d^2*x^2 + c^3*d^2), x) - 16*(-I*c^9*d^2*x - c
^8*d^2)*integrate(1/8*(c*x^5*log(c^2*x^2 + 1) - 2*x^4*arctan(c*x))/(c^7*d^2*x^4 + 2*c^5*d^2*x^2 + c^3*d^2), x)
- 16*(I*c^8*d^2*x + c^7*d^2)*integrate(1/8*(2*c*x^4*arctan(c*x) + x^3*log(c^2*x^2 + 1))/(c^7*d^2*x^4 + 2*c^5*
d^2*x^2 + c^3*d^2), x) - 16*(c^8*d^2*x - I*c^7*d^2)*integrate(1/8*(c*x^4*log(c^2*x^2 + 1) - 2*x^3*arctan(c*x))
/(c^7*d^2*x^4 + 2*c^5*d^2*x^2 + c^3*d^2), x) + 48*(c^7*d^2*x - I*c^6*d^2)*integrate(1/8*(2*c*x^3*arctan(c*x) +
x^2*log(c^2*x^2 + 1))/(c^7*d^2*x^4 + 2*c^5*d^2*x^2 + c^3*d^2), x) - 48*(I*c^7*d^2*x + c^6*d^2)*integrate(1/8*
(c*x^3*log(c^2*x^2 + 1) - 2*x^2*arctan(c*x))/(c^7*d^2*x^4 + 2*c^5*d^2*x^2 + c^3*d^2), x) + (-I*c^3*x^3 + 3*c^2
*x^2 - I*c*x + 5)*log(c^2*x^2 + 1) + 2*I*arctan2(1, c*x))*b/(c^5*d^2*x - I*c^4*d^2)

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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