\(\int \frac {x^3 (a+b \arctan (c x))}{d+i c d x} \, dx\) [43]

Optimal result

Integrand size = 23, antiderivative size = 196 \[ \int \frac {x^3 (a+b \arctan (c x))}{d+i c d x} \, dx=\frac {i a x}{c^3 d}-\frac {b x}{2 c^3 d}+\frac {i b x^2}{6 c^2 d}+\frac {b \arctan (c x)}{2 c^4 d}+\frac {i b x \arctan (c x)}{c^3 d}+\frac {x^2 (a+b \arctan (c x))}{2 c^2 d}-\frac {i x^3 (a+b \arctan (c x))}{3 c d}+\frac {(a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^4 d}-\frac {2 i b \log \left (1+c^2 x^2\right )}{3 c^4 d}+\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 c^4 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {4986, 4946, 272, 45, 327, 209, 4930, 266, 4964, 2449, 2352} \[ \int \frac {x^3 (a+b \arctan (c x))}{d+i c d x} \, dx=\frac {\log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c^4 d}+\frac {x^2 (a+b \arctan (c x))}{2 c^2 d}-\frac {i x^3 (a+b \arctan (c x))}{3 c d}+\frac {i a x}{c^3 d}+\frac {b \arctan (c x)}{2 c^4 d}+\frac {i b x \arctan (c x)}{c^3 d}+\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{2 c^4 d}-\frac {b x}{2 c^3 d}+\frac {i b x^2}{6 c^2 d}-\frac {2 i b \log \left (c^2 x^2+1\right )}{3 c^4 d} \]

[In]

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

[Out]

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

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 4986

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

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

Mathematica [A] (verified)

Time = 0.53 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.85 \[ \int \frac {x^3 (a+b \arctan (c x))}{d+i c d x} \, dx=-\frac {i \left (-b-6 a c x-3 i b c x+3 i a c^2 x^2-b c^2 x^2+2 a c^3 x^3+6 b \arctan (c x)^2+\arctan (c x) \left (6 a+b \left (3 i-6 c x+3 i c^2 x^2+2 c^3 x^3\right )+6 i b \log \left (1+e^{2 i \arctan (c x)}\right )\right )-3 i a \log \left (1+c^2 x^2\right )+4 b \log \left (1+c^2 x^2\right )+3 b \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )\right )}{6 c^4 d} \]

[In]

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

[Out]

((-1/6*I)*(-b - 6*a*c*x - (3*I)*b*c*x + (3*I)*a*c^2*x^2 - b*c^2*x^2 + 2*a*c^3*x^3 + 6*b*ArcTan[c*x]^2 + ArcTan
[c*x]*(6*a + b*(3*I - 6*c*x + (3*I)*c^2*x^2 + 2*c^3*x^3) + (6*I)*b*Log[1 + E^((2*I)*ArcTan[c*x])]) - (3*I)*a*L
og[1 + c^2*x^2] + 4*b*Log[1 + c^2*x^2] + 3*b*PolyLog[2, -E^((2*I)*ArcTan[c*x])]))/(c^4*d)

Maple [A] (verified)

Time = 2.04 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.59

method	result	size
derivativedivides	\(\frac {-\frac {5 i b \ln \left (c^{4} x^{4}+10 c^{2} x^{2}+9\right )}{48 d}-\frac {i b \arctan \left (c x \right ) c^{3} x^{3}}{3 d}+\frac {a \,c^{2} x^{2}}{2 d}-\frac {a \ln \left (c^{2} x^{2}+1\right )}{2 d}+\frac {i b \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{2 d}+\frac {i b \,c^{2} x^{2}}{6 d}+\frac {i b \ln \left (-\frac {i \left (c x +i\right )}{2}\right ) \ln \left (c x -i\right )}{2 d}+\frac {b \arctan \left (c x \right ) c^{2} x^{2}}{2 d}-\frac {b \arctan \left (c x \right ) \ln \left (c x -i\right )}{d}-\frac {11 i b \ln \left (c^{2} x^{2}+1\right )}{24 d}-\frac {i a \,c^{3} x^{3}}{3 d}-\frac {i b \ln \left (c x -i\right )^{2}}{4 d}-\frac {b c x}{2 d}+\frac {i a c x}{d}+\frac {2 i b}{3 d}-\frac {i a \arctan \left (c x \right )}{d}+\frac {5 b \arctan \left (\frac {c x}{2}\right )}{24 d}-\frac {5 b \arctan \left (\frac {1}{6} c^{3} x^{3}+\frac {7}{6} c x \right )}{24 d}-\frac {5 b \arctan \left (\frac {c x}{2}-\frac {i}{2}\right )}{12 d}+\frac {i b \arctan \left (c x \right ) c x}{d}+\frac {11 b \arctan \left (c x \right )}{12 d}}{c^{4}}\)	\(312\)
default	\(\frac {-\frac {5 i b \ln \left (c^{4} x^{4}+10 c^{2} x^{2}+9\right )}{48 d}-\frac {i b \arctan \left (c x \right ) c^{3} x^{3}}{3 d}+\frac {a \,c^{2} x^{2}}{2 d}-\frac {a \ln \left (c^{2} x^{2}+1\right )}{2 d}+\frac {i b \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{2 d}+\frac {i b \,c^{2} x^{2}}{6 d}+\frac {i b \ln \left (-\frac {i \left (c x +i\right )}{2}\right ) \ln \left (c x -i\right )}{2 d}+\frac {b \arctan \left (c x \right ) c^{2} x^{2}}{2 d}-\frac {b \arctan \left (c x \right ) \ln \left (c x -i\right )}{d}-\frac {11 i b \ln \left (c^{2} x^{2}+1\right )}{24 d}-\frac {i a \,c^{3} x^{3}}{3 d}-\frac {i b \ln \left (c x -i\right )^{2}}{4 d}-\frac {b c x}{2 d}+\frac {i a c x}{d}+\frac {2 i b}{3 d}-\frac {i a \arctan \left (c x \right )}{d}+\frac {5 b \arctan \left (\frac {c x}{2}\right )}{24 d}-\frac {5 b \arctan \left (\frac {1}{6} c^{3} x^{3}+\frac {7}{6} c x \right )}{24 d}-\frac {5 b \arctan \left (\frac {c x}{2}-\frac {i}{2}\right )}{12 d}+\frac {i b \arctan \left (c x \right ) c x}{d}+\frac {11 b \arctan \left (c x \right )}{12 d}}{c^{4}}\)	\(312\)
risch	\(\frac {i a x}{c^{3} d}-\frac {b \left (\frac {1}{3} c^{2} x^{3}+\frac {1}{2} i c \,x^{2}-x \right ) \ln \left (i c x +1\right )}{2 c^{3} d}+\frac {i b \,x^{2}}{6 c^{2} d}-\frac {5 a}{6 d \,c^{4}}+\frac {a \,x^{2}}{2 d \,c^{2}}-\frac {11 i b \ln \left (c^{2} x^{2}+1\right )}{24 c^{4} d}+\frac {b \,x^{3} \ln \left (-i c x +1\right )}{6 d c}+\frac {i b \ln \left (i c x +1\right )^{2}}{4 d \,c^{4}}-\frac {b x \ln \left (-i c x +1\right )}{2 d \,c^{3}}-\frac {i a \,x^{3}}{3 d c}-\frac {a \ln \left (c^{2} x^{2}+1\right )}{2 d \,c^{4}}-\frac {5 i \ln \left (-i c x +1\right ) b}{12 d \,c^{4}}-\frac {b x}{2 c^{3} d}-\frac {i a \arctan \left (c x \right )}{d \,c^{4}}-\frac {i b \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (-i c x +1\right )}{2 d \,c^{4}}+\frac {i b \operatorname {dilog}\left (\frac {1}{2}-\frac {i c x}{2}\right )}{2 d \,c^{4}}+\frac {31 i b}{72 d \,c^{4}}+\frac {i x^{2} b \ln \left (-i c x +1\right )}{4 d \,c^{2}}+\frac {i b \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (\frac {1}{2}-\frac {i c x}{2}\right )}{2 d \,c^{4}}+\frac {11 b \arctan \left (c x \right )}{12 c^{4} d}\)	\(349\)
parts	\(\frac {i a x}{c^{3} d}+\frac {a \,x^{2}}{2 d \,c^{2}}-\frac {i b \arctan \left (c x \right ) x^{3}}{3 d c}-\frac {a \ln \left (c^{2} x^{2}+1\right )}{2 d \,c^{4}}+\frac {i b \,x^{2}}{6 c^{2} d}-\frac {5 i b \ln \left (c^{4} x^{4}+10 c^{2} x^{2}+9\right )}{48 d \,c^{4}}+\frac {i b \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{2 d \,c^{4}}+\frac {b \arctan \left (c x \right ) x^{2}}{2 d \,c^{2}}-\frac {b \arctan \left (c x \right ) \ln \left (c x -i\right )}{d \,c^{4}}-\frac {11 i b \ln \left (c^{2} x^{2}+1\right )}{24 c^{4} d}-\frac {i b \ln \left (c x -i\right )^{2}}{4 d \,c^{4}}-\frac {i a \,x^{3}}{3 d c}-\frac {b x}{2 c^{3} d}-\frac {i a \arctan \left (c x \right )}{d \,c^{4}}+\frac {i b \ln \left (-\frac {i \left (c x +i\right )}{2}\right ) \ln \left (c x -i\right )}{2 d \,c^{4}}+\frac {2 i b}{3 d \,c^{4}}+\frac {5 b \arctan \left (\frac {c x}{2}\right )}{24 d \,c^{4}}-\frac {5 b \arctan \left (\frac {1}{6} c^{3} x^{3}+\frac {7}{6} c x \right )}{24 d \,c^{4}}-\frac {5 b \arctan \left (\frac {c x}{2}-\frac {i}{2}\right )}{12 d \,c^{4}}+\frac {i b x \arctan \left (c x \right )}{c^{3} d}+\frac {11 b \arctan \left (c x \right )}{12 c^{4} d}\)	\(353\)

[In]

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

[Out]

1/c^4*(-5/48*I/d*b*ln(c^4*x^4+10*c^2*x^2+9)-1/3*I/d*b*arctan(c*x)*c^3*x^3+1/2/d*a*c^2*x^2-1/2/d*a*ln(c^2*x^2+1
)+1/2*I/d*b*dilog(-1/2*I*(c*x+I))+1/6*I/d*b*c^2*x^2+1/2*I/d*b*ln(-1/2*I*(c*x+I))*ln(c*x-I)+1/2/d*b*arctan(c*x)
*c^2*x^2-1/d*b*arctan(c*x)*ln(c*x-I)-11/24*I/d*b*ln(c^2*x^2+1)-1/3*I/d*a*c^3*x^3-1/4*I/d*b*ln(c*x-I)^2-1/2/d*b
*c*x+I/d*a*c*x+2/3*I/d*b-I/d*a*arctan(c*x)+5/24/d*b*arctan(1/2*c*x)-5/24/d*b*arctan(1/6*c^3*x^3+7/6*c*x)-5/12/
d*b*arctan(1/2*c*x-1/2*I)+I/d*b*arctan(c*x)*c*x+11/12/d*b*arctan(c*x))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {x^3 (a+b \arctan (c x))}{d+i c d x} \, dx=- \frac {i \left (\int \frac {6 i b \log {\left (i c x + 1 \right )}}{c^{2} x^{2} + 1}\, dx + \int \frac {12 a c^{4} x^{4}}{c^{2} x^{2} + 1}\, dx + \int \frac {6 b c x}{c^{2} x^{2} + 1}\, dx + \int \frac {b c^{3} x^{3}}{c^{2} x^{2} + 1}\, dx + \int \frac {12 i a c^{3} x^{3}}{c^{2} x^{2} + 1}\, dx + \int \frac {3 i b c^{2} x^{2}}{c^{2} x^{2} + 1}\, dx + \int \left (- \frac {2 i b c^{4} x^{4}}{c^{2} x^{2} + 1}\right )\, dx + \int \left (- \frac {6 b c x \log {\left (i c x + 1 \right )}}{c^{2} x^{2} + 1}\right )\, dx + \int \frac {6 b c^{3} x^{3} \log {\left (i c x + 1 \right )}}{c^{2} x^{2} + 1}\, dx + \int \left (- \frac {6 i b c^{4} x^{4} \log {\left (i c x + 1 \right )}}{c^{2} x^{2} + 1}\right )\, dx\right )}{12 c^{3} d} + \frac {\left (2 b c^{3} x^{3} + 3 i b c^{2} x^{2} - 6 b c x - 6 i b \log {\left (i c x + 1 \right )}\right ) \log {\left (- i c x + 1 \right )}}{12 c^{4} d} \]

[In]

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

[Out]

-I*(Integral(6*I*b*log(I*c*x + 1)/(c**2*x**2 + 1), x) + Integral(12*a*c**4*x**4/(c**2*x**2 + 1), x) + Integral
(6*b*c*x/(c**2*x**2 + 1), x) + Integral(b*c**3*x**3/(c**2*x**2 + 1), x) + Integral(12*I*a*c**3*x**3/(c**2*x**2
+ 1), x) + Integral(3*I*b*c**2*x**2/(c**2*x**2 + 1), x) + Integral(-2*I*b*c**4*x**4/(c**2*x**2 + 1), x) + Int
egral(-6*b*c*x*log(I*c*x + 1)/(c**2*x**2 + 1), x) + Integral(6*b*c**3*x**3*log(I*c*x + 1)/(c**2*x**2 + 1), x)
+ Integral(-6*I*b*c**4*x**4*log(I*c*x + 1)/(c**2*x**2 + 1), x))/(12*c**3*d) + (2*b*c**3*x**3 + 3*I*b*c**2*x**2
- 6*b*c*x - 6*I*b*log(I*c*x + 1))*log(-I*c*x + 1)/(12*c**4*d)

Maxima [F]

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

[In]

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

[Out]

-1/6*a*(I*(2*c^2*x^3 + 3*I*c*x^2 - 6*x)/(c^3*d) + 6*log(I*c*x + 1)/(c^4*d)) - 1/72*(432*I*c^8*d*integrate(1/12
*x^4*arctan(c*x)/(c^5*d*x^2 + c^3*d), x) + 216*c^8*d*integrate(1/12*x^4*log(c^2*x^2 + 1)/(c^5*d*x^2 + c^3*d),
x) - 432*c^7*d*integrate(1/12*x^3*arctan(c*x)/(c^5*d*x^2 + c^3*d), x) + 216*I*c^7*d*integrate(1/12*x^3*log(c^2
*x^2 + 1)/(c^5*d*x^2 + c^3*d), x) + 432*c^5*d*integrate(1/12*x*arctan(c*x)/(c^5*d*x^2 + c^3*d), x) - 216*I*c^5
*d*integrate(1/12*x*log(c^2*x^2 + 1)/(c^5*d*x^2 + c^3*d), x) + 4*c^3*x^3 - 216*c^4*d*integrate(1/12*log(c^2*x^
2 + 1)/(c^5*d*x^2 + c^3*d), x) + 3*I*c^2*x^2 - 30*c*x - 6*(-2*I*c^3*x^3 + 3*c^2*x^2 + 6*I*c*x - 5)*arctan(c*x)
+ 18*I*arctan(c*x)^2 - 3*(2*c^3*x^3 + 3*I*c^2*x^2 - 6*c*x + I)*log(c^2*x^2 + 1) + 9*I*log(c^2*x^2 + 1)^2 + 18
*I*log(12*c^5*d*x^2 + 12*c^3*d))*b/(c^4*d)

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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