\(\int \frac {x^3 (a+b \arctan (c x))}{(d+i c d x)^3} \, dx\) [59]

Optimal result

Integrand size = 23, antiderivative size = 225 \[ \int \frac {x^3 (a+b \arctan (c x))}{(d+i c d x)^3} \, dx=\frac {i a x}{c^3 d^3}+\frac {i b}{8 c^4 d^3 (i-c x)^2}-\frac {11 b}{8 c^4 d^3 (i-c x)}+\frac {11 b \arctan (c x)}{8 c^4 d^3}+\frac {i b x \arctan (c x)}{c^3 d^3}-\frac {a+b \arctan (c x)}{2 c^4 d^3 (i-c x)^2}-\frac {3 i (a+b \arctan (c x))}{c^4 d^3 (i-c x)}+\frac {3 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^4 d^3}-\frac {i b \log \left (1+c^2 x^2\right )}{2 c^4 d^3}+\frac {3 i b \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 c^4 d^3} \]

[Out]

I*a*x/c^3/d^3+1/8*I*b/c^4/d^3/(I-c*x)^2-11/8*b/c^4/d^3/(I-c*x)+11/8*b*arctan(c*x)/c^4/d^3+I*b*x*arctan(c*x)/c^
3/d^3+1/2*(-a-b*arctan(c*x))/c^4/d^3/(I-c*x)^2-3*I*(a+b*arctan(c*x))/c^4/d^3/(I-c*x)+3*(a+b*arctan(c*x))*ln(2/
(1+I*c*x))/c^4/d^3-1/2*I*b*ln(c^2*x^2+1)/c^4/d^3+3/2*I*b*polylog(2,1-2/(1+I*c*x))/c^4/d^3

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {4996, 4930, 266, 4972, 641, 46, 209, 4964, 2449, 2352} \[ \int \frac {x^3 (a+b \arctan (c x))}{(d+i c d x)^3} \, dx=-\frac {3 i (a+b \arctan (c x))}{c^4 d^3 (-c x+i)}-\frac {a+b \arctan (c x)}{2 c^4 d^3 (-c x+i)^2}+\frac {3 \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c^4 d^3}+\frac {i a x}{c^3 d^3}+\frac {11 b \arctan (c x)}{8 c^4 d^3}+\frac {i b x \arctan (c x)}{c^3 d^3}+\frac {3 i b \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{2 c^4 d^3}-\frac {11 b}{8 c^4 d^3 (-c x+i)}+\frac {i b}{8 c^4 d^3 (-c x+i)^2}-\frac {i b \log \left (c^2 x^2+1\right )}{2 c^4 d^3} \]

[In]

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

[Out]

(I*a*x)/(c^3*d^3) + ((I/8)*b)/(c^4*d^3*(I - c*x)^2) - (11*b)/(8*c^4*d^3*(I - c*x)) + (11*b*ArcTan[c*x])/(8*c^4
*d^3) + (I*b*x*ArcTan[c*x])/(c^3*d^3) - (a + b*ArcTan[c*x])/(2*c^4*d^3*(I - c*x)^2) - ((3*I)*(a + b*ArcTan[c*x
]))/(c^4*d^3*(I - c*x)) + (3*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(c^4*d^3) - ((I/2)*b*Log[1 + c^2*x^2])/(c
^4*d^3) + (((3*I)/2)*b*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^4*d^3)

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 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 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 {i (a+b \arctan (c x))}{c^3 d^3}+\frac {a+b \arctan (c x)}{c^3 d^3 (-i+c x)^3}-\frac {3 i (a+b \arctan (c x))}{c^3 d^3 (-i+c x)^2}-\frac {3 (a+b \arctan (c x))}{c^3 d^3 (-i+c x)}\right ) \, dx \\ & = \frac {i \int (a+b \arctan (c x)) \, dx}{c^3 d^3}-\frac {(3 i) \int \frac {a+b \arctan (c x)}{(-i+c x)^2} \, dx}{c^3 d^3}+\frac {\int \frac {a+b \arctan (c x)}{(-i+c x)^3} \, dx}{c^3 d^3}-\frac {3 \int \frac {a+b \arctan (c x)}{-i+c x} \, dx}{c^3 d^3} \\ & = \frac {i a x}{c^3 d^3}-\frac {a+b \arctan (c x)}{2 c^4 d^3 (i-c x)^2}-\frac {3 i (a+b \arctan (c x))}{c^4 d^3 (i-c x)}+\frac {3 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^4 d^3}+\frac {(i b) \int \arctan (c x) \, dx}{c^3 d^3}-\frac {(3 i b) \int \frac {1}{(-i+c x) \left (1+c^2 x^2\right )} \, dx}{c^3 d^3}+\frac {b \int \frac {1}{(-i+c x)^2 \left (1+c^2 x^2\right )} \, dx}{2 c^3 d^3}-\frac {(3 b) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{c^3 d^3} \\ & = \frac {i a x}{c^3 d^3}+\frac {i b x \arctan (c x)}{c^3 d^3}-\frac {a+b \arctan (c x)}{2 c^4 d^3 (i-c x)^2}-\frac {3 i (a+b \arctan (c x))}{c^4 d^3 (i-c x)}+\frac {3 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^4 d^3}+\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^3}-\frac {(3 i b) \int \frac {1}{(-i+c x)^2 (i+c x)} \, dx}{c^3 d^3}+\frac {b \int \frac {1}{(-i+c x)^3 (i+c x)} \, dx}{2 c^3 d^3}-\frac {(i b) \int \frac {x}{1+c^2 x^2} \, dx}{c^2 d^3} \\ & = \frac {i a x}{c^3 d^3}+\frac {i b x \arctan (c x)}{c^3 d^3}-\frac {a+b \arctan (c x)}{2 c^4 d^3 (i-c x)^2}-\frac {3 i (a+b \arctan (c x))}{c^4 d^3 (i-c x)}+\frac {3 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^4 d^3}-\frac {i b \log \left (1+c^2 x^2\right )}{2 c^4 d^3}+\frac {3 i b \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 c^4 d^3}-\frac {(3 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^3}+\frac {b \int \left (-\frac {i}{2 (-i+c x)^3}+\frac {1}{4 (-i+c x)^2}-\frac {1}{4 \left (1+c^2 x^2\right )}\right ) \, dx}{2 c^3 d^3} \\ & = \frac {i a x}{c^3 d^3}+\frac {i b}{8 c^4 d^3 (i-c x)^2}-\frac {11 b}{8 c^4 d^3 (i-c x)}+\frac {i b x \arctan (c x)}{c^3 d^3}-\frac {a+b \arctan (c x)}{2 c^4 d^3 (i-c x)^2}-\frac {3 i (a+b \arctan (c x))}{c^4 d^3 (i-c x)}+\frac {3 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^4 d^3}-\frac {i b \log \left (1+c^2 x^2\right )}{2 c^4 d^3}+\frac {3 i b \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 c^4 d^3}-\frac {b \int \frac {1}{1+c^2 x^2} \, dx}{8 c^3 d^3}+\frac {(3 b) \int \frac {1}{1+c^2 x^2} \, dx}{2 c^3 d^3} \\ & = \frac {i a x}{c^3 d^3}+\frac {i b}{8 c^4 d^3 (i-c x)^2}-\frac {11 b}{8 c^4 d^3 (i-c x)}+\frac {11 b \arctan (c x)}{8 c^4 d^3}+\frac {i b x \arctan (c x)}{c^3 d^3}-\frac {a+b \arctan (c x)}{2 c^4 d^3 (i-c x)^2}-\frac {3 i (a+b \arctan (c x))}{c^4 d^3 (i-c x)}+\frac {3 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^4 d^3}-\frac {i b \log \left (1+c^2 x^2\right )}{2 c^4 d^3}+\frac {3 i b \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 c^4 d^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.72 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.96 \[ \int \frac {x^3 (a+b \arctan (c x))}{(d+i c d x)^3} \, dx=\frac {32 i a c x-\frac {16 a}{(-i+c x)^2}+\frac {96 i a}{-i+c x}-96 i a \arctan (c x)-48 a \log \left (1+c^2 x^2\right )+i b \left (-96 \arctan (c x)^2+20 \cos (2 \arctan (c x))-\cos (4 \arctan (c x))-16 \log \left (1+c^2 x^2\right )-48 \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )-20 i \sin (2 \arctan (c x))+4 \arctan (c x) \left (8 c x+10 i \cos (2 \arctan (c x))-i \cos (4 \arctan (c x))-24 i \log \left (1+e^{2 i \arctan (c x)}\right )+10 \sin (2 \arctan (c x))-\sin (4 \arctan (c x))\right )+i \sin (4 \arctan (c x))\right )}{32 c^4 d^3} \]

[In]

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

[Out]

((32*I)*a*c*x - (16*a)/(-I + c*x)^2 + ((96*I)*a)/(-I + c*x) - (96*I)*a*ArcTan[c*x] - 48*a*Log[1 + c^2*x^2] + I
*b*(-96*ArcTan[c*x]^2 + 20*Cos[2*ArcTan[c*x]] - Cos[4*ArcTan[c*x]] - 16*Log[1 + c^2*x^2] - 48*PolyLog[2, -E^((
2*I)*ArcTan[c*x])] - (20*I)*Sin[2*ArcTan[c*x]] + 4*ArcTan[c*x]*(8*c*x + (10*I)*Cos[2*ArcTan[c*x]] - I*Cos[4*Ar
cTan[c*x]] - (24*I)*Log[1 + E^((2*I)*ArcTan[c*x])] + 10*Sin[2*ArcTan[c*x]] - Sin[4*ArcTan[c*x]]) + I*Sin[4*Arc
Tan[c*x]]))/(32*c^4*d^3)

Maple [A] (verified)

Time = 0.90 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.43

method	result	size
derivativedivides	\(\frac {-\frac {3 i b \ln \left (c x -i\right )^{2}}{4 d^{3}}-\frac {a}{2 d^{3} \left (c x -i\right )^{2}}+\frac {i b}{8 d^{3} \left (c x -i\right )^{2}}-\frac {3 a \ln \left (c^{2} x^{2}+1\right )}{2 d^{3}}+\frac {i a c x}{d^{3}}+\frac {3 i b \ln \left (c^{4} x^{4}+10 c^{2} x^{2}+9\right )}{64 d^{3}}-\frac {b \arctan \left (c x \right )}{2 d^{3} \left (c x -i\right )^{2}}-\frac {3 i a \arctan \left (c x \right )}{d^{3}}-\frac {3 b \arctan \left (c x \right ) \ln \left (c x -i\right )}{d^{3}}+\frac {3 i b \arctan \left (c x \right )}{d^{3} \left (c x -i\right )}-\frac {3 b \arctan \left (\frac {c x}{2}\right )}{32 d^{3}}+\frac {3 b \arctan \left (\frac {1}{6} c^{3} x^{3}+\frac {7}{6} c x \right )}{32 d^{3}}+\frac {3 b \arctan \left (\frac {c x}{2}-\frac {i}{2}\right )}{16 d^{3}}+\frac {3 i a}{d^{3} \left (c x -i\right )}+\frac {3 i b \ln \left (-\frac {i \left (c x +i\right )}{2}\right ) \ln \left (c x -i\right )}{2 d^{3}}+\frac {19 b \arctan \left (c x \right )}{16 d^{3}}+\frac {11 b}{8 d^{3} \left (c x -i\right )}+\frac {3 i b \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{2 d^{3}}-\frac {19 i b \ln \left (c^{2} x^{2}+1\right )}{32 d^{3}}+\frac {i b \arctan \left (c x \right ) c x}{d^{3}}}{c^{4}}\)	\(321\)
default	\(\frac {-\frac {3 i b \ln \left (c x -i\right )^{2}}{4 d^{3}}-\frac {a}{2 d^{3} \left (c x -i\right )^{2}}+\frac {i b}{8 d^{3} \left (c x -i\right )^{2}}-\frac {3 a \ln \left (c^{2} x^{2}+1\right )}{2 d^{3}}+\frac {i a c x}{d^{3}}+\frac {3 i b \ln \left (c^{4} x^{4}+10 c^{2} x^{2}+9\right )}{64 d^{3}}-\frac {b \arctan \left (c x \right )}{2 d^{3} \left (c x -i\right )^{2}}-\frac {3 i a \arctan \left (c x \right )}{d^{3}}-\frac {3 b \arctan \left (c x \right ) \ln \left (c x -i\right )}{d^{3}}+\frac {3 i b \arctan \left (c x \right )}{d^{3} \left (c x -i\right )}-\frac {3 b \arctan \left (\frac {c x}{2}\right )}{32 d^{3}}+\frac {3 b \arctan \left (\frac {1}{6} c^{3} x^{3}+\frac {7}{6} c x \right )}{32 d^{3}}+\frac {3 b \arctan \left (\frac {c x}{2}-\frac {i}{2}\right )}{16 d^{3}}+\frac {3 i a}{d^{3} \left (c x -i\right )}+\frac {3 i b \ln \left (-\frac {i \left (c x +i\right )}{2}\right ) \ln \left (c x -i\right )}{2 d^{3}}+\frac {19 b \arctan \left (c x \right )}{16 d^{3}}+\frac {11 b}{8 d^{3} \left (c x -i\right )}+\frac {3 i b \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{2 d^{3}}-\frac {19 i b \ln \left (c^{2} x^{2}+1\right )}{32 d^{3}}+\frac {i b \arctan \left (c x \right ) c x}{d^{3}}}{c^{4}}\)	\(321\)
parts	\(\frac {i b}{8 c^{4} d^{3} \left (c x -i\right )^{2}}+\frac {i a x}{c^{3} d^{3}}-\frac {3 a \ln \left (c^{2} x^{2}+1\right )}{2 d^{3} c^{4}}+\frac {3 i b \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{2 c^{4} d^{3}}-\frac {a}{2 d^{3} c^{4} \left (-c x +i\right )^{2}}-\frac {3 i b \ln \left (c x -i\right )^{2}}{4 c^{4} d^{3}}+\frac {i b x \arctan \left (c x \right )}{c^{3} d^{3}}-\frac {b \arctan \left (c x \right )}{2 c^{4} d^{3} \left (c x -i\right )^{2}}-\frac {3 b \arctan \left (c x \right ) \ln \left (c x -i\right )}{c^{4} d^{3}}-\frac {19 i b \ln \left (c^{2} x^{2}+1\right )}{32 c^{4} d^{3}}-\frac {3 b \arctan \left (\frac {c x}{2}\right )}{32 c^{4} d^{3}}+\frac {3 b \arctan \left (\frac {1}{6} c^{3} x^{3}+\frac {7}{6} c x \right )}{32 c^{4} d^{3}}+\frac {3 b \arctan \left (\frac {c x}{2}-\frac {i}{2}\right )}{16 c^{4} d^{3}}-\frac {3 i a}{d^{3} c^{4} \left (-c x +i\right )}+\frac {3 i b \ln \left (c^{4} x^{4}+10 c^{2} x^{2}+9\right )}{64 c^{4} d^{3}}+\frac {19 b \arctan \left (c x \right )}{16 c^{4} d^{3}}+\frac {11 b}{8 c^{4} d^{3} \left (c x -i\right )}-\frac {3 i a \arctan \left (c x \right )}{d^{3} c^{4}}+\frac {3 i b \arctan \left (c x \right )}{c^{4} d^{3} \left (c x -i\right )}+\frac {3 i b \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{2 c^{4} d^{3}}\)	\(377\)
risch	\(-\frac {3 b}{2 c^{4} d^{3} \left (-c x +i\right )}+\frac {19 b \arctan \left (c x \right )}{16 c^{4} d^{3}}+\frac {i a x}{c^{3} d^{3}}+\frac {i b}{8 c^{4} d^{3} \left (-c x +i\right )^{2}}-\frac {3 i a \arctan \left (c x \right )}{d^{3} c^{4}}-\frac {i b \ln \left (-i c x +1\right )}{2 d^{3} c^{4}}+\frac {3 i b \operatorname {dilog}\left (\frac {1}{2}-\frac {i c x}{2}\right )}{2 d^{3} c^{4}}+\frac {i b}{8 d^{3} c^{4} \left (-i c x -1\right )}-\frac {19 i b \ln \left (c^{2} x^{2}+1\right )}{32 c^{4} d^{3}}-\frac {b \ln \left (-i c x +1\right ) x}{2 d^{3} c^{3}}+\left (\frac {b x}{2 c^{3} d^{3}}-\frac {-3 b \,d^{3} x +\frac {5 i b \,d^{3}}{2 c}}{2 c^{3} d^{6} \left (c x -i\right )^{2}}\right ) \ln \left (i c x +1\right )+\frac {3 a}{d^{3} c^{4} \left (-i c x -1\right )}+\frac {a}{2 d^{3} c^{4} \left (-i c x -1\right )^{2}}-\frac {3 a \ln \left (c^{2} x^{2}+1\right )}{2 d^{3} c^{4}}+\frac {i b}{2 c^{4} d^{3}}-\frac {a}{d^{3} c^{4}}+\frac {i b \ln \left (-i c x +1\right ) x^{2}}{16 d^{3} c^{2} \left (-i c x -1\right )^{2}}+\frac {3 b \ln \left (-i c x +1\right ) x}{4 d^{3} c^{3} \left (-i c x -1\right )}+\frac {b \ln \left (-i c x +1\right ) x}{8 d^{3} c^{3} \left (-i c x -1\right )^{2}}+\frac {3 i \ln \left (\frac {1}{2}-\frac {i c x}{2}\right ) \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) b}{2 d^{3} c^{4}}-\frac {3 i b \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (-i c x +1\right )}{2 d^{3} c^{4}}+\frac {3 i b \ln \left (-i c x +1\right )}{4 d^{3} c^{4} \left (-i c x -1\right )}+\frac {3 i b \ln \left (-i c x +1\right )}{16 d^{3} c^{4} \left (-i c x -1\right )^{2}}+\frac {3 i b \ln \left (i c x +1\right )^{2}}{4 c^{4} d^{3}}\)	\(508\)

[In]

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

[Out]

1/c^4*(-3/4*I*b/d^3*ln(c*x-I)^2-1/2*a/d^3/(c*x-I)^2+1/8*I*b/d^3/(c*x-I)^2-3/2*a/d^3*ln(c^2*x^2+1)+I*a/d^3*c*x+
3/64*I*b/d^3*ln(c^4*x^4+10*c^2*x^2+9)-1/2*b/d^3*arctan(c*x)/(c*x-I)^2-3*I*a/d^3*arctan(c*x)-3*b/d^3*arctan(c*x
)*ln(c*x-I)+3*I*b/d^3*arctan(c*x)/(c*x-I)-3/32*b/d^3*arctan(1/2*c*x)+3/32*b/d^3*arctan(1/6*c^3*x^3+7/6*c*x)+3/
16*b/d^3*arctan(1/2*c*x-1/2*I)+3*I*a/d^3/(c*x-I)+3/2*I*b/d^3*ln(-1/2*I*(c*x+I))*ln(c*x-I)+19/16*b*arctan(c*x)/
d^3+11/8*b/d^3/(c*x-I)+3/2*I*b/d^3*dilog(-1/2*I*(c*x+I))-19/32*I*b/d^3*ln(c^2*x^2+1)+I*b/d^3*arctan(c*x)*c*x)

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.48 \[ \int \frac {x^3 (a+b \arctan (c x))}{(d+i c d x)^3} \, dx=-\frac {-16 i \, a c^{3} x^{3} - 32 \, a c^{2} x^{2} - 2 \, {\left (16 i \, a + 11 \, b\right )} c x - 12 \, {\left (-i \, b c^{2} x^{2} - 2 \, b c x + i \, b\right )} \arctan \left (c x\right )^{2} - 3 \, {\left (-i \, b c^{2} x^{2} - 2 \, b c x + i \, b\right )} \log \left (c^{2} x^{2} + 1\right )^{2} + 12 \, {\left (b c^{2} x^{2} - 2 i \, b c x - b\right )} \arctan \left (c x\right ) \log \left (\frac {1}{4} \, c^{2} x^{2} + \frac {1}{4}\right ) + {\left (-16 i \, b c^{3} x^{3} - 3 \, {\left (-16 i \, a + 17 \, b\right )} c^{2} x^{2} + 6 \, {\left (16 \, a + i \, b\right )} c x - 48 i \, a - 21 \, b\right )} \arctan \left (c x\right ) + 3 \, {\left (b c^{2} x^{2} - 2 i \, b c x - b\right )} \arctan \left (c x, -1\right ) - 24 \, {\left (i \, b c^{2} x^{2} + 2 \, b c x - i \, b\right )} {\rm Li}_2\left (\frac {1}{2} i \, c x + \frac {1}{2}\right ) + 2 \, {\left (4 \, {\left (3 \, a + i \, b\right )} c^{2} x^{2} - 8 \, {\left (3 i \, a - b\right )} c x - 3 \, {\left (i \, b c^{2} x^{2} + 2 \, b c x - i \, b\right )} \log \left (\frac {1}{4} \, c^{2} x^{2} + \frac {1}{4}\right ) - 12 \, a - 4 i \, b\right )} \log \left (c^{2} x^{2} + 1\right ) - 40 \, a + 20 i \, b}{16 \, {\left (c^{6} d^{3} x^{2} - 2 i \, c^{5} d^{3} x - c^{4} d^{3}\right )}} \]

[In]

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

[Out]

-1/16*(-16*I*a*c^3*x^3 - 32*a*c^2*x^2 - 2*(16*I*a + 11*b)*c*x - 12*(-I*b*c^2*x^2 - 2*b*c*x + I*b)*arctan(c*x)^
2 - 3*(-I*b*c^2*x^2 - 2*b*c*x + I*b)*log(c^2*x^2 + 1)^2 + 12*(b*c^2*x^2 - 2*I*b*c*x - b)*arctan(c*x)*log(1/4*c
^2*x^2 + 1/4) + (-16*I*b*c^3*x^3 - 3*(-16*I*a + 17*b)*c^2*x^2 + 6*(16*a + I*b)*c*x - 48*I*a - 21*b)*arctan(c*x
) + 3*(b*c^2*x^2 - 2*I*b*c*x - b)*arctan2(c*x, -1) - 24*(I*b*c^2*x^2 + 2*b*c*x - I*b)*dilog(1/2*I*c*x + 1/2) +
2*(4*(3*a + I*b)*c^2*x^2 - 8*(3*I*a - b)*c*x - 3*(I*b*c^2*x^2 + 2*b*c*x - I*b)*log(1/4*c^2*x^2 + 1/4) - 12*a
- 4*I*b)*log(c^2*x^2 + 1) - 40*a + 20*I*b)/(c^6*d^3*x^2 - 2*I*c^5*d^3*x - c^4*d^3)

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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