\(\int \frac {a+b \arctan (c x)}{x (d+i c d x)^3} \, dx\) [63]

Optimal result

Integrand size = 23, antiderivative size = 195 \[ \int \frac {a+b \arctan (c x)}{x (d+i c d x)^3} \, dx=\frac {i b}{8 d^3 (i-c x)^2}+\frac {5 b}{8 d^3 (i-c x)}-\frac {5 b \arctan (c x)}{8 d^3}-\frac {a+b \arctan (c x)}{2 d^3 (i-c x)^2}+\frac {i (a+b \arctan (c x))}{d^3 (i-c x)}+\frac {a \log (x)}{d^3}+\frac {(a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{d^3}+\frac {i b \operatorname {PolyLog}(2,-i c x)}{2 d^3}-\frac {i b \operatorname {PolyLog}(2,i c x)}{2 d^3}+\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 d^3} \]

[Out]

1/8*I*b/d^3/(I-c*x)^2+5/8*b/d^3/(I-c*x)-5/8*b*arctan(c*x)/d^3+1/2*(-a-b*arctan(c*x))/d^3/(I-c*x)^2+I*(a+b*arct
an(c*x))/d^3/(I-c*x)+a*ln(x)/d^3+(a+b*arctan(c*x))*ln(2/(1+I*c*x))/d^3+1/2*I*b*polylog(2,-I*c*x)/d^3-1/2*I*b*p
olylog(2,I*c*x)/d^3+1/2*I*b*polylog(2,1-2/(1+I*c*x))/d^3

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 195, 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, 4940, 2438, 4972, 641, 46, 209, 4964, 2449, 2352} \[ \int \frac {a+b \arctan (c x)}{x (d+i c d x)^3} \, dx=\frac {i (a+b \arctan (c x))}{d^3 (-c x+i)}-\frac {a+b \arctan (c x)}{2 d^3 (-c x+i)^2}+\frac {\log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{d^3}+\frac {a \log (x)}{d^3}-\frac {5 b \arctan (c x)}{8 d^3}+\frac {i b \operatorname {PolyLog}(2,-i c x)}{2 d^3}-\frac {i b \operatorname {PolyLog}(2,i c x)}{2 d^3}+\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{2 d^3}+\frac {5 b}{8 d^3 (-c x+i)}+\frac {i b}{8 d^3 (-c x+i)^2} \]

[In]

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

[Out]

((I/8)*b)/(d^3*(I - c*x)^2) + (5*b)/(8*d^3*(I - c*x)) - (5*b*ArcTan[c*x])/(8*d^3) - (a + b*ArcTan[c*x])/(2*d^3
*(I - c*x)^2) + (I*(a + b*ArcTan[c*x]))/(d^3*(I - c*x)) + (a*Log[x])/d^3 + ((a + b*ArcTan[c*x])*Log[2/(1 + I*c
*x)])/d^3 + ((I/2)*b*PolyLog[2, (-I)*c*x])/d^3 - ((I/2)*b*PolyLog[2, I*c*x])/d^3 + ((I/2)*b*PolyLog[2, 1 - 2/(
1 + I*c*x)])/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 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 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

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 4940

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] + (Dist[I*(b/2), Int[Log[1 - I*c*x
]/x, x], x] - Dist[I*(b/2), Int[Log[1 + I*c*x]/x, x], x]) /; FreeQ[{a, b, c}, x]

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

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.83 \[ \int \frac {a+b \arctan (c x)}{x (d+i c d x)^3} \, dx=\frac {\frac {5 b}{i-c x}+\frac {i b}{(-i+c x)^2}-5 b \arctan (c x)-\frac {4 (a+b \arctan (c x))}{(-i+c x)^2}-\frac {8 i (a+b \arctan (c x))}{-i+c x}+8 a \log (x)+8 (a+b \arctan (c x)) \log \left (\frac {2 i}{i-c x}\right )+4 i b \operatorname {PolyLog}(2,-i c x)-4 i b \operatorname {PolyLog}(2,i c x)+4 i b \operatorname {PolyLog}\left (2,\frac {i+c x}{-i+c x}\right )}{8 d^3} \]

[In]

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

[Out]

((5*b)/(I - c*x) + (I*b)/(-I + c*x)^2 - 5*b*ArcTan[c*x] - (4*(a + b*ArcTan[c*x]))/(-I + c*x)^2 - ((8*I)*(a + b
*ArcTan[c*x]))/(-I + c*x) + 8*a*Log[x] + 8*(a + b*ArcTan[c*x])*Log[(2*I)/(I - c*x)] + (4*I)*b*PolyLog[2, (-I)*
c*x] - (4*I)*b*PolyLog[2, I*c*x] + (4*I)*b*PolyLog[2, (I + c*x)/(-I + c*x)])/(8*d^3)

Maple [A] (verified)

Time = 1.41 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.34

method	result	size
derivativedivides	\(\frac {a \ln \left (c x \right )}{d^{3}}-\frac {a}{2 d^{3} \left (c x -i\right )^{2}}-\frac {i a}{d^{3} \left (c x -i\right )}-\frac {a \ln \left (c^{2} x^{2}+1\right )}{2 d^{3}}-\frac {i a \arctan \left (c x \right )}{d^{3}}+\frac {b \left (\arctan \left (c x \right ) \ln \left (c x \right )-\frac {\arctan \left (c x \right )}{2 \left (c x -i\right )^{2}}-\frac {i \arctan \left (c x \right )}{c x -i}-\arctan \left (c x \right ) \ln \left (c x -i\right )-\frac {5 \arctan \left (c x \right )}{8}+\frac {i}{8 \left (c x -i\right )^{2}}-\frac {5}{8 \left (c x -i\right )}-\frac {i \left (\operatorname {dilog}\left (-i \left (c x +i\right )\right )+\ln \left (c x \right ) \ln \left (-i \left (c x +i\right )\right )\right )}{2}+\frac {i \left (\left (\ln \left (c x \right )-\ln \left (-i c x \right )\right ) \ln \left (-i \left (-c x +i\right )\right )-\operatorname {dilog}\left (-i c x \right )\right )}{2}+\frac {i \left (\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )+\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{2}-\frac {i \ln \left (c x -i\right )^{2}}{4}\right )}{d^{3}}\)	\(261\)
default	\(\frac {a \ln \left (c x \right )}{d^{3}}-\frac {a}{2 d^{3} \left (c x -i\right )^{2}}-\frac {i a}{d^{3} \left (c x -i\right )}-\frac {a \ln \left (c^{2} x^{2}+1\right )}{2 d^{3}}-\frac {i a \arctan \left (c x \right )}{d^{3}}+\frac {b \left (\arctan \left (c x \right ) \ln \left (c x \right )-\frac {\arctan \left (c x \right )}{2 \left (c x -i\right )^{2}}-\frac {i \arctan \left (c x \right )}{c x -i}-\arctan \left (c x \right ) \ln \left (c x -i\right )-\frac {5 \arctan \left (c x \right )}{8}+\frac {i}{8 \left (c x -i\right )^{2}}-\frac {5}{8 \left (c x -i\right )}-\frac {i \left (\operatorname {dilog}\left (-i \left (c x +i\right )\right )+\ln \left (c x \right ) \ln \left (-i \left (c x +i\right )\right )\right )}{2}+\frac {i \left (\left (\ln \left (c x \right )-\ln \left (-i c x \right )\right ) \ln \left (-i \left (-c x +i\right )\right )-\operatorname {dilog}\left (-i c x \right )\right )}{2}+\frac {i \left (\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )+\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{2}-\frac {i \ln \left (c x -i\right )^{2}}{4}\right )}{d^{3}}\)	\(261\)
parts	\(\frac {a \ln \left (x \right )}{d^{3}}-\frac {a}{2 d^{3} \left (-c x +i\right )^{2}}+\frac {i a}{d^{3} \left (-c x +i\right )}-\frac {a \ln \left (c^{2} x^{2}+1\right )}{2 d^{3}}-\frac {i a \arctan \left (c x \right )}{d^{3}}+\frac {b \left (\arctan \left (c x \right ) \ln \left (c x \right )-\frac {\arctan \left (c x \right )}{2 \left (c x -i\right )^{2}}-\frac {i \arctan \left (c x \right )}{c x -i}-\arctan \left (c x \right ) \ln \left (c x -i\right )-\frac {5 \arctan \left (c x \right )}{8}+\frac {i}{8 \left (c x -i\right )^{2}}-\frac {5}{8 \left (c x -i\right )}-\frac {i \left (\operatorname {dilog}\left (-i \left (c x +i\right )\right )+\ln \left (c x \right ) \ln \left (-i \left (c x +i\right )\right )\right )}{2}+\frac {i \left (\left (\ln \left (c x \right )-\ln \left (-i c x \right )\right ) \ln \left (-i \left (-c x +i\right )\right )-\operatorname {dilog}\left (-i c x \right )\right )}{2}+\frac {i \left (\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )+\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{2}-\frac {i \ln \left (c x -i\right )^{2}}{4}\right )}{d^{3}}\)	\(261\)
risch	\(\frac {i b \operatorname {dilog}\left (\frac {1}{2}-\frac {i c x}{2}\right )}{2 d^{3}}-\frac {i b \ln \left (i c x +1\right )}{2 d^{3} \left (i c x +1\right )}+\frac {i b \ln \left (-i c x +1\right ) c^{2} x^{2}}{16 d^{3} \left (-i c x -1\right )^{2}}-\frac {i a \arctan \left (c x \right )}{d^{3}}+\frac {i \ln \left (\frac {1}{2}-\frac {i c x}{2}\right ) \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) b}{2 d^{3}}-\frac {i b \ln \left (i c x +1\right )}{4 d^{3} \left (i c x +1\right )^{2}}+\frac {a \ln \left (-i c x \right )}{d^{3}}+\frac {a}{2 d^{3} \left (-i c x -1\right )^{2}}-\frac {a}{d^{3} \left (-i c x -1\right )}+\frac {5 i b \ln \left (c^{2} x^{2}+1\right )}{32 d^{3}}-\frac {b \ln \left (-i c x +1\right ) c x}{4 d^{3} \left (-i c x -1\right )}+\frac {i b \ln \left (i c x +1\right )^{2}}{4 d^{3}}-\frac {i \operatorname {dilog}\left (-i c x +1\right ) b}{2 d^{3}}+\frac {3 i b \ln \left (-i c x +1\right )}{16 d^{3} \left (-i c x -1\right )^{2}}-\frac {i b}{8 d^{3} \left (i c x +1\right )^{2}}-\frac {i b}{2 d^{3} \left (i c x +1\right )}+\frac {b \ln \left (-i c x +1\right ) c x}{8 d^{3} \left (-i c x -1\right )^{2}}-\frac {a \ln \left (c^{2} x^{2}+1\right )}{2 d^{3}}-\frac {i b \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (-i c x +1\right )}{2 d^{3}}+\frac {i b \operatorname {dilog}\left (i c x +1\right )}{2 d^{3}}-\frac {i b \ln \left (-i c x +1\right )}{4 d^{3} \left (-i c x -1\right )}-\frac {5 b \arctan \left (c x \right )}{16 d^{3}}+\frac {i b}{8 d^{3} \left (-i c x -1\right )}\)	\(429\)

[In]

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

[Out]

a/d^3*ln(c*x)-1/2*a/d^3/(c*x-I)^2-I*a/d^3/(c*x-I)-1/2*a/d^3*ln(c^2*x^2+1)-I*a/d^3*arctan(c*x)+b/d^3*(arctan(c*
x)*ln(c*x)-1/2*arctan(c*x)/(c*x-I)^2-I*arctan(c*x)/(c*x-I)-arctan(c*x)*ln(c*x-I)-5/8*arctan(c*x)+1/8*I/(c*x-I)
^2-5/8/(c*x-I)-1/2*I*(dilog(-I*(c*x+I))+ln(c*x)*ln(-I*(c*x+I)))+1/2*I*((ln(c*x)-ln(-I*c*x))*ln(-I*(-c*x+I))-di
log(-I*c*x))+1/2*I*(dilog(-1/2*I*(c*x+I))+ln(c*x-I)*ln(-1/2*I*(c*x+I)))-1/4*I*ln(c*x-I)^2)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.03 \[ \int \frac {a+b \arctan (c x)}{x (d+i c d x)^3} \, dx=-\frac {2 \, {\left (8 i \, a + 5 \, b\right )} c x + 8 \, {\left (i \, b c^{2} x^{2} + 2 \, b c x - i \, b\right )} {\rm Li}_2\left (\frac {c x + i}{c x - i} + 1\right ) - 16 \, {\left (a c^{2} x^{2} - 2 i \, a c x - a\right )} \log \left (x\right ) - 4 \, {\left (2 \, b c x - 3 i \, b\right )} \log \left (-\frac {c x + i}{c x - i}\right ) + 5 \, {\left (i \, b c^{2} x^{2} + 2 \, b c x - i \, b\right )} \log \left (\frac {c x + i}{c}\right ) + {\left ({\left (16 \, a - 5 i \, b\right )} c^{2} x^{2} + 2 \, {\left (-16 i \, a - 5 \, b\right )} c x - 16 \, a + 5 i \, b\right )} \log \left (\frac {c x - i}{c}\right ) + 24 \, a - 12 i \, b}{16 \, {\left (c^{2} d^{3} x^{2} - 2 i \, c d^{3} x - d^{3}\right )}} \]

[In]

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

[Out]

-1/16*(2*(8*I*a + 5*b)*c*x + 8*(I*b*c^2*x^2 + 2*b*c*x - I*b)*dilog((c*x + I)/(c*x - I) + 1) - 16*(a*c^2*x^2 -
2*I*a*c*x - a)*log(x) - 4*(2*b*c*x - 3*I*b)*log(-(c*x + I)/(c*x - I)) + 5*(I*b*c^2*x^2 + 2*b*c*x - I*b)*log((c
*x + I)/c) + ((16*a - 5*I*b)*c^2*x^2 + 2*(-16*I*a - 5*b)*c*x - 16*a + 5*I*b)*log((c*x - I)/c) + 24*a - 12*I*b)
/(c^2*d^3*x^2 - 2*I*c*d^3*x - d^3)

Sympy [F(-2)]

Exception generated. \[ \int \frac {a+b \arctan (c x)}{x (d+i c d x)^3} \, dx=\text {Exception raised: RecursionError} \]

[In]

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

[Out]

Exception raised: RecursionError >> maximum recursion depth exceeded in comparison

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 406 vs. \(2 (151) = 302\).

Time = 0.29 (sec) , antiderivative size = 406, normalized size of antiderivative = 2.08 \[ \int \frac {a+b \arctan (c x)}{x (d+i c d x)^3} \, dx=\frac {2 \, {\left (-8 i \, a - 5 \, b\right )} c x + 4 \, {\left (-i \, b c^{2} x^{2} - 2 \, b c x + i \, b\right )} \arctan \left (c x\right )^{2} - {\left (i \, b c^{2} x^{2} + 2 \, b c x - i \, b\right )} \log \left (c^{2} x^{2} + 1\right )^{2} - 4 \, {\left (b c^{2} x^{2} - 2 i \, b c x - b\right )} \arctan \left (c x\right ) \log \left (\frac {1}{4} \, c^{2} x^{2} + \frac {1}{4}\right ) + 16 \, {\left (b c^{2} x^{2} - 2 i \, b c x - b\right )} \arctan \left (c x\right ) \log \left (c x\right ) - {\left ({\left (16 i \, a + 5 \, b\right )} c^{2} x^{2} + 2 \, {\left (16 \, a + 3 i \, b\right )} c x - 16 i \, a + 19 \, b\right )} \arctan \left (c x\right ) + 5 \, {\left (b c^{2} x^{2} - 2 i \, b c x - b\right )} \arctan \left (c x, -1\right ) + 8 \, {\left (-i \, b c^{2} x^{2} - 2 \, b c x + i \, b\right )} {\rm Li}_2\left (i \, c x + 1\right ) + 8 \, {\left (i \, b c^{2} x^{2} + 2 \, b c x - i \, b\right )} {\rm Li}_2\left (\frac {1}{2} i \, c x + \frac {1}{2}\right ) + 8 \, {\left (i \, b c^{2} x^{2} + 2 \, b c x - i \, b\right )} {\rm Li}_2\left (-i \, c x + 1\right ) - 2 \, {\left (2 \, {\left (\pi b + 2 \, a\right )} c^{2} x^{2} - 4 \, {\left (i \, \pi b + 2 i \, a\right )} c x - 2 \, \pi b - {\left (i \, b c^{2} x^{2} + 2 \, b c x - i \, b\right )} \log \left (\frac {1}{4} \, c^{2} x^{2} + \frac {1}{4}\right ) - 4 \, a\right )} \log \left (c^{2} x^{2} + 1\right ) + 16 \, {\left (a c^{2} x^{2} - 2 i \, a c x - a\right )} \log \left (x\right ) - 24 \, a + 12 i \, b}{16 \, {\left (c^{2} d^{3} x^{2} - 2 i \, c d^{3} x - d^{3}\right )}} \]

[In]

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

[Out]

1/16*(2*(-8*I*a - 5*b)*c*x + 4*(-I*b*c^2*x^2 - 2*b*c*x + I*b)*arctan(c*x)^2 - (I*b*c^2*x^2 + 2*b*c*x - I*b)*lo
g(c^2*x^2 + 1)^2 - 4*(b*c^2*x^2 - 2*I*b*c*x - b)*arctan(c*x)*log(1/4*c^2*x^2 + 1/4) + 16*(b*c^2*x^2 - 2*I*b*c*
x - b)*arctan(c*x)*log(c*x) - ((16*I*a + 5*b)*c^2*x^2 + 2*(16*a + 3*I*b)*c*x - 16*I*a + 19*b)*arctan(c*x) + 5*
(b*c^2*x^2 - 2*I*b*c*x - b)*arctan2(c*x, -1) + 8*(-I*b*c^2*x^2 - 2*b*c*x + I*b)*dilog(I*c*x + 1) + 8*(I*b*c^2*
x^2 + 2*b*c*x - I*b)*dilog(1/2*I*c*x + 1/2) + 8*(I*b*c^2*x^2 + 2*b*c*x - I*b)*dilog(-I*c*x + 1) - 2*(2*(pi*b +
2*a)*c^2*x^2 - 4*(I*pi*b + 2*I*a)*c*x - 2*pi*b - (I*b*c^2*x^2 + 2*b*c*x - I*b)*log(1/4*c^2*x^2 + 1/4) - 4*a)*
log(c^2*x^2 + 1) + 16*(a*c^2*x^2 - 2*I*a*c*x - a)*log(x) - 24*a + 12*I*b)/(c^2*d^3*x^2 - 2*I*c*d^3*x - d^3)

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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