\(\int \frac {a+b \arctan (c x)}{x^2 (d+i c d x)^3} \, dx\) [64]

Optimal result

Integrand size = 23, antiderivative size = 250 \[ \int \frac {a+b \arctan (c x)}{x^2 (d+i c d x)^3} \, dx=\frac {b c}{8 d^3 (i-c x)^2}-\frac {9 i b c}{8 d^3 (i-c x)}+\frac {9 i b c \arctan (c x)}{8 d^3}-\frac {a+b \arctan (c x)}{d^3 x}+\frac {i c (a+b \arctan (c x))}{2 d^3 (i-c x)^2}+\frac {2 c (a+b \arctan (c x))}{d^3 (i-c x)}-\frac {3 i a c \log (x)}{d^3}+\frac {b c \log (x)}{d^3}-\frac {3 i c (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{d^3}-\frac {b c \log \left (1+c^2 x^2\right )}{2 d^3}+\frac {3 b c \operatorname {PolyLog}(2,-i c x)}{2 d^3}-\frac {3 b c \operatorname {PolyLog}(2,i c x)}{2 d^3}+\frac {3 b c \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 d^3} \]

[Out]

1/8*b*c/d^3/(I-c*x)^2-9/8*I*b*c/d^3/(I-c*x)+9/8*I*b*c*arctan(c*x)/d^3+(-a-b*arctan(c*x))/d^3/x+1/2*I*c*(a+b*ar
ctan(c*x))/d^3/(I-c*x)^2+2*c*(a+b*arctan(c*x))/d^3/(I-c*x)-3*I*a*c*ln(x)/d^3+b*c*ln(x)/d^3-3*I*c*(a+b*arctan(c
*x))*ln(2/(1+I*c*x))/d^3-1/2*b*c*ln(c^2*x^2+1)/d^3+3/2*b*c*polylog(2,-I*c*x)/d^3-3/2*b*c*polylog(2,I*c*x)/d^3+
3/2*b*c*polylog(2,1-2/(1+I*c*x))/d^3

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.652, Rules used = {4996, 4946, 272, 36, 29, 31, 4940, 2438, 4972, 641, 46, 209, 4964, 2449, 2352} \[ \int \frac {a+b \arctan (c x)}{x^2 (d+i c d x)^3} \, dx=\frac {2 c (a+b \arctan (c x))}{d^3 (-c x+i)}+\frac {i c (a+b \arctan (c x))}{2 d^3 (-c x+i)^2}-\frac {a+b \arctan (c x)}{d^3 x}-\frac {3 i c \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{d^3}-\frac {3 i a c \log (x)}{d^3}+\frac {9 i b c \arctan (c x)}{8 d^3}-\frac {b c \log \left (c^2 x^2+1\right )}{2 d^3}+\frac {3 b c \operatorname {PolyLog}(2,-i c x)}{2 d^3}-\frac {3 b c \operatorname {PolyLog}(2,i c x)}{2 d^3}+\frac {3 b c \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{2 d^3}-\frac {9 i b c}{8 d^3 (-c x+i)}+\frac {b c}{8 d^3 (-c x+i)^2}+\frac {b c \log (x)}{d^3} \]

[In]

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

[Out]

(b*c)/(8*d^3*(I - c*x)^2) - (((9*I)/8)*b*c)/(d^3*(I - c*x)) + (((9*I)/8)*b*c*ArcTan[c*x])/d^3 - (a + b*ArcTan[
c*x])/(d^3*x) + ((I/2)*c*(a + b*ArcTan[c*x]))/(d^3*(I - c*x)^2) + (2*c*(a + b*ArcTan[c*x]))/(d^3*(I - c*x)) -
((3*I)*a*c*Log[x])/d^3 + (b*c*Log[x])/d^3 - ((3*I)*c*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/d^3 - (b*c*Log[1
+ c^2*x^2])/(2*d^3) + (3*b*c*PolyLog[2, (-I)*c*x])/(2*d^3) - (3*b*c*PolyLog[2, I*c*x])/(2*d^3) + (3*b*c*PolyLo
g[2, 1 - 2/(1 + I*c*x)])/(2*d^3)

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

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 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 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 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 {a+b \arctan (c x)}{d^3 x^2}-\frac {3 i c (a+b \arctan (c x))}{d^3 x}-\frac {i c^2 (a+b \arctan (c x))}{d^3 (-i+c x)^3}+\frac {2 c^2 (a+b \arctan (c x))}{d^3 (-i+c x)^2}+\frac {3 i c^2 (a+b \arctan (c x))}{d^3 (-i+c x)}\right ) \, dx \\ & = \frac {\int \frac {a+b \arctan (c x)}{x^2} \, dx}{d^3}-\frac {(3 i c) \int \frac {a+b \arctan (c x)}{x} \, dx}{d^3}-\frac {\left (i c^2\right ) \int \frac {a+b \arctan (c x)}{(-i+c x)^3} \, dx}{d^3}+\frac {\left (3 i c^2\right ) \int \frac {a+b \arctan (c x)}{-i+c x} \, dx}{d^3}+\frac {\left (2 c^2\right ) \int \frac {a+b \arctan (c x)}{(-i+c x)^2} \, dx}{d^3} \\ & = -\frac {a+b \arctan (c x)}{d^3 x}+\frac {i c (a+b \arctan (c x))}{2 d^3 (i-c x)^2}+\frac {2 c (a+b \arctan (c x))}{d^3 (i-c x)}-\frac {3 i a c \log (x)}{d^3}-\frac {3 i c (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{d^3}+\frac {(b c) \int \frac {1}{x \left (1+c^2 x^2\right )} \, dx}{d^3}+\frac {(3 b c) \int \frac {\log (1-i c x)}{x} \, dx}{2 d^3}-\frac {(3 b c) \int \frac {\log (1+i c x)}{x} \, dx}{2 d^3}-\frac {\left (i b c^2\right ) \int \frac {1}{(-i+c x)^2 \left (1+c^2 x^2\right )} \, dx}{2 d^3}+\frac {\left (3 i b c^2\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d^3}+\frac {\left (2 b c^2\right ) \int \frac {1}{(-i+c x) \left (1+c^2 x^2\right )} \, dx}{d^3} \\ & = -\frac {a+b \arctan (c x)}{d^3 x}+\frac {i c (a+b \arctan (c x))}{2 d^3 (i-c x)^2}+\frac {2 c (a+b \arctan (c x))}{d^3 (i-c x)}-\frac {3 i a c \log (x)}{d^3}-\frac {3 i c (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{d^3}+\frac {3 b c \operatorname {PolyLog}(2,-i c x)}{2 d^3}-\frac {3 b c \operatorname {PolyLog}(2,i c x)}{2 d^3}+\frac {(b c) \text {Subst}\left (\int \frac {1}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )}{2 d^3}+\frac {(3 b c) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{d^3}-\frac {\left (i b c^2\right ) \int \frac {1}{(-i+c x)^3 (i+c x)} \, dx}{2 d^3}+\frac {\left (2 b c^2\right ) \int \frac {1}{(-i+c x)^2 (i+c x)} \, dx}{d^3} \\ & = -\frac {a+b \arctan (c x)}{d^3 x}+\frac {i c (a+b \arctan (c x))}{2 d^3 (i-c x)^2}+\frac {2 c (a+b \arctan (c x))}{d^3 (i-c x)}-\frac {3 i a c \log (x)}{d^3}-\frac {3 i c (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{d^3}+\frac {3 b c \operatorname {PolyLog}(2,-i c x)}{2 d^3}-\frac {3 b c \operatorname {PolyLog}(2,i c x)}{2 d^3}+\frac {3 b c \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 d^3}+\frac {(b c) \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )}{2 d^3}-\frac {\left (i b c^2\right ) \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 {\left (2 b c^2\right ) \int \left (-\frac {i}{2 (-i+c x)^2}+\frac {i}{2 \left (1+c^2 x^2\right )}\right ) \, dx}{d^3}-\frac {\left (b c^3\right ) \text {Subst}\left (\int \frac {1}{1+c^2 x} \, dx,x,x^2\right )}{2 d^3} \\ & = \frac {b c}{8 d^3 (i-c x)^2}-\frac {9 i b c}{8 d^3 (i-c x)}-\frac {a+b \arctan (c x)}{d^3 x}+\frac {i c (a+b \arctan (c x))}{2 d^3 (i-c x)^2}+\frac {2 c (a+b \arctan (c x))}{d^3 (i-c x)}-\frac {3 i a c \log (x)}{d^3}+\frac {b c \log (x)}{d^3}-\frac {3 i c (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{d^3}-\frac {b c \log \left (1+c^2 x^2\right )}{2 d^3}+\frac {3 b c \operatorname {PolyLog}(2,-i c x)}{2 d^3}-\frac {3 b c \operatorname {PolyLog}(2,i c x)}{2 d^3}+\frac {3 b c \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 d^3}+\frac {\left (i b c^2\right ) \int \frac {1}{1+c^2 x^2} \, dx}{8 d^3}+\frac {\left (i b c^2\right ) \int \frac {1}{1+c^2 x^2} \, dx}{d^3} \\ & = \frac {b c}{8 d^3 (i-c x)^2}-\frac {9 i b c}{8 d^3 (i-c x)}+\frac {9 i b c \arctan (c x)}{8 d^3}-\frac {a+b \arctan (c x)}{d^3 x}+\frac {i c (a+b \arctan (c x))}{2 d^3 (i-c x)^2}+\frac {2 c (a+b \arctan (c x))}{d^3 (i-c x)}-\frac {3 i a c \log (x)}{d^3}+\frac {b c \log (x)}{d^3}-\frac {3 i c (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{d^3}-\frac {b c \log \left (1+c^2 x^2\right )}{2 d^3}+\frac {3 b c \operatorname {PolyLog}(2,-i c x)}{2 d^3}-\frac {3 b c \operatorname {PolyLog}(2,i c x)}{2 d^3}+\frac {3 b c \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 d^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.91 \[ \int \frac {a+b \arctan (c x)}{x^2 (d+i c d x)^3} \, dx=\frac {-8 i b c \left (\frac {1}{i-c x}-\arctan (c x)\right )-\frac {8 (a+b \arctan (c x))}{x}+\frac {4 i c (a+b \arctan (c x))}{(-i+c x)^2}-\frac {16 c (a+b \arctan (c x))}{-i+c x}+\frac {b c \left (2+i c x+i (-i+c x)^2 \arctan (c x)\right )}{(-i+c x)^2}-24 i a c \log (x)-24 i c (a+b \arctan (c x)) \log \left (\frac {2 i}{i-c x}\right )+4 b c \left (2 \log (x)-\log \left (1+c^2 x^2\right )\right )+12 b c \operatorname {PolyLog}(2,-i c x)-12 b c \operatorname {PolyLog}(2,i c x)+12 b c \operatorname {PolyLog}\left (2,\frac {i+c x}{-i+c x}\right )}{8 d^3} \]

[In]

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

[Out]

((-8*I)*b*c*((I - c*x)^(-1) - ArcTan[c*x]) - (8*(a + b*ArcTan[c*x]))/x + ((4*I)*c*(a + b*ArcTan[c*x]))/(-I + c
*x)^2 - (16*c*(a + b*ArcTan[c*x]))/(-I + c*x) + (b*c*(2 + I*c*x + I*(-I + c*x)^2*ArcTan[c*x]))/(-I + c*x)^2 -
(24*I)*a*c*Log[x] - (24*I)*c*(a + b*ArcTan[c*x])*Log[(2*I)/(I - c*x)] + 4*b*c*(2*Log[x] - Log[1 + c^2*x^2]) +
12*b*c*PolyLog[2, (-I)*c*x] - 12*b*c*PolyLog[2, I*c*x] + 12*b*c*PolyLog[2, (I + c*x)/(-I + c*x)])/(8*d^3)

Maple [A] (verified)

Time = 1.72 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.21

method	result	size
derivativedivides	\(c \left (-\frac {a}{d^{3} c x}-\frac {3 i a \ln \left (c x \right )}{d^{3}}+\frac {i a}{2 d^{3} \left (c x -i\right )^{2}}+\frac {3 i a \ln \left (c^{2} x^{2}+1\right )}{2 d^{3}}-\frac {3 a \arctan \left (c x \right )}{d^{3}}-\frac {2 a}{d^{3} \left (c x -i\right )}+\frac {b \left (-\frac {\arctan \left (c x \right )}{c x}+\frac {9 i}{8 \left (c x -i\right )}+\frac {i \arctan \left (c x \right )}{2 \left (c x -i\right )^{2}}-3 i \arctan \left (c x \right ) \ln \left (c x \right )-\frac {2 \arctan \left (c x \right )}{c x -i}-\frac {\ln \left (c^{2} x^{2}+1\right )}{2}+\frac {9 i \arctan \left (c x \right )}{8}+\ln \left (c x \right )+3 i \arctan \left (c x \right ) \ln \left (c x -i\right )+\frac {1}{8 \left (c x -i\right )^{2}}-\frac {3 \operatorname {dilog}\left (-i \left (c x +i\right )\right )}{2}-\frac {3 \ln \left (c x \right ) \ln \left (-i \left (c x +i\right )\right )}{2}+\frac {3 \left (\ln \left (c x \right )-\ln \left (-i c x \right )\right ) \ln \left (-i \left (-c x +i\right )\right )}{2}-\frac {3 \operatorname {dilog}\left (-i c x \right )}{2}+\frac {3 \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{2}+\frac {3 \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{2}-\frac {3 \ln \left (c x -i\right )^{2}}{4}\right )}{d^{3}}\right )\)	\(303\)
default	\(c \left (-\frac {a}{d^{3} c x}-\frac {3 i a \ln \left (c x \right )}{d^{3}}+\frac {i a}{2 d^{3} \left (c x -i\right )^{2}}+\frac {3 i a \ln \left (c^{2} x^{2}+1\right )}{2 d^{3}}-\frac {3 a \arctan \left (c x \right )}{d^{3}}-\frac {2 a}{d^{3} \left (c x -i\right )}+\frac {b \left (-\frac {\arctan \left (c x \right )}{c x}+\frac {9 i}{8 \left (c x -i\right )}+\frac {i \arctan \left (c x \right )}{2 \left (c x -i\right )^{2}}-3 i \arctan \left (c x \right ) \ln \left (c x \right )-\frac {2 \arctan \left (c x \right )}{c x -i}-\frac {\ln \left (c^{2} x^{2}+1\right )}{2}+\frac {9 i \arctan \left (c x \right )}{8}+\ln \left (c x \right )+3 i \arctan \left (c x \right ) \ln \left (c x -i\right )+\frac {1}{8 \left (c x -i\right )^{2}}-\frac {3 \operatorname {dilog}\left (-i \left (c x +i\right )\right )}{2}-\frac {3 \ln \left (c x \right ) \ln \left (-i \left (c x +i\right )\right )}{2}+\frac {3 \left (\ln \left (c x \right )-\ln \left (-i c x \right )\right ) \ln \left (-i \left (-c x +i\right )\right )}{2}-\frac {3 \operatorname {dilog}\left (-i c x \right )}{2}+\frac {3 \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{2}+\frac {3 \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{2}-\frac {3 \ln \left (c x -i\right )^{2}}{4}\right )}{d^{3}}\right )\)	\(303\)
parts	\(-\frac {a}{d^{3} x}-\frac {3 i a c \ln \left (x \right )}{d^{3}}+\frac {i a c}{2 d^{3} \left (-c x +i\right )^{2}}+\frac {3 i c a \ln \left (c^{2} x^{2}+1\right )}{2 d^{3}}-\frac {3 c a \arctan \left (c x \right )}{d^{3}}+\frac {2 a c}{d^{3} \left (-c x +i\right )}+\frac {b c \left (-\frac {\arctan \left (c x \right )}{c x}+\frac {9 i}{8 \left (c x -i\right )}+\frac {i \arctan \left (c x \right )}{2 \left (c x -i\right )^{2}}-3 i \arctan \left (c x \right ) \ln \left (c x \right )-\frac {2 \arctan \left (c x \right )}{c x -i}-\frac {\ln \left (c^{2} x^{2}+1\right )}{2}+\frac {9 i \arctan \left (c x \right )}{8}+\ln \left (c x \right )+3 i \arctan \left (c x \right ) \ln \left (c x -i\right )+\frac {1}{8 \left (c x -i\right )^{2}}-\frac {3 \operatorname {dilog}\left (-i \left (c x +i\right )\right )}{2}-\frac {3 \ln \left (c x \right ) \ln \left (-i \left (c x +i\right )\right )}{2}+\frac {3 \left (\ln \left (c x \right )-\ln \left (-i c x \right )\right ) \ln \left (-i \left (-c x +i\right )\right )}{2}-\frac {3 \operatorname {dilog}\left (-i c x \right )}{2}+\frac {3 \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{2}+\frac {3 \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{2}-\frac {3 \ln \left (c x -i\right )^{2}}{4}\right )}{d^{3}}\)	\(304\)
risch	\(-\frac {a}{d^{3} x}+\frac {9 b c \ln \left (c^{2} x^{2}+1\right )}{32 d^{3}}-\frac {3 c a \arctan \left (c x \right )}{d^{3}}+\frac {i c^{2} b \ln \left (-i c x +1\right ) x}{2 d^{3} \left (-i c x -1\right )}-\frac {i c^{2} b \ln \left (-i c x +1\right ) x}{8 d^{3} \left (-i c x -1\right )^{2}}+\frac {3 c \ln \left (\frac {1}{2}-\frac {i c x}{2}\right ) \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) b}{2 d^{3}}-\frac {3 c b \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (-i c x +1\right )}{2 d^{3}}-\frac {c b \ln \left (-i c x +1\right )}{2 d^{3} \left (-i c x -1\right )}+\frac {3 c b \ln \left (-i c x +1\right )}{16 d^{3} \left (-i c x -1\right )^{2}}-\frac {3 i c a \ln \left (-i c x \right )}{d^{3}}-\frac {i c a}{2 d^{3} \left (-i c x -1\right )^{2}}+\frac {2 i c a}{d^{3} \left (-i c x -1\right )}-\frac {i b \ln \left (-i c x +1\right )}{2 d^{3} x}+\frac {9 i b \arctan \left (c x \right ) c}{16 d^{3}}-\frac {b c \ln \left (i c x +1\right )}{d^{3} \left (i c x +1\right )}+\frac {i b \ln \left (i c x +1\right )}{2 d^{3} x}-\frac {b c \ln \left (i c x +1\right )}{4 d^{3} \left (i c x +1\right )^{2}}+\frac {c^{3} b \ln \left (-i c x +1\right ) x^{2}}{16 d^{3} \left (-i c x -1\right )^{2}}+\frac {3 i c a \ln \left (c^{2} x^{2}+1\right )}{2 d^{3}}-\frac {b c}{d^{3} \left (i c x +1\right )}+\frac {c b}{8 d^{3} \left (-i c x -1\right )}-\frac {3 c \operatorname {dilog}\left (-i c x +1\right ) b}{2 d^{3}}+\frac {c b \ln \left (-i c x \right )}{2 d^{3}}-\frac {c b \ln \left (-i c x +1\right )}{2 d^{3}}+\frac {3 c b \operatorname {dilog}\left (\frac {1}{2}-\frac {i c x}{2}\right )}{2 d^{3}}-\frac {b c}{8 d^{3} \left (i c x +1\right )^{2}}+\frac {3 b c \ln \left (i c x +1\right )^{2}}{4 d^{3}}+\frac {3 b c \operatorname {dilog}\left (i c x +1\right )}{2 d^{3}}+\frac {b c \ln \left (i c x \right )}{2 d^{3}}-\frac {b c \ln \left (i c x +1\right )}{2 d^{3}}\)	\(546\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.05 \[ \int \frac {a+b \arctan (c x)}{x^2 (d+i c d x)^3} \, dx=-\frac {6 \, {\left (8 \, a - 3 i \, b\right )} c^{2} x^{2} + 4 \, {\left (-18 i \, a - 5 \, b\right )} c x + 24 \, {\left (b c^{3} x^{3} - 2 i \, b c^{2} x^{2} - b c x\right )} {\rm Li}_2\left (\frac {c x + i}{c x - i} + 1\right ) + 16 \, {\left ({\left (3 i \, a - b\right )} c^{3} x^{3} + 2 \, {\left (3 \, a + i \, b\right )} c^{2} x^{2} + {\left (-3 i \, a + b\right )} c x\right )} \log \left (x\right ) + 4 \, {\left (6 i \, b c^{2} x^{2} + 9 \, b c x - 2 i \, b\right )} \log \left (-\frac {c x + i}{c x - i}\right ) + 17 \, {\left (b c^{3} x^{3} - 2 i \, b c^{2} x^{2} - b c x\right )} \log \left (\frac {c x + i}{c}\right ) - {\left ({\left (48 i \, a + b\right )} c^{3} x^{3} + 2 \, {\left (48 \, a - i \, b\right )} c^{2} x^{2} + {\left (-48 i \, a - b\right )} c x\right )} \log \left (\frac {c x - i}{c}\right ) - 16 \, a}{16 \, {\left (c^{2} d^{3} x^{3} - 2 i \, c d^{3} x^{2} - d^{3} x\right )}} \]

[In]

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

[Out]

-1/16*(6*(8*a - 3*I*b)*c^2*x^2 + 4*(-18*I*a - 5*b)*c*x + 24*(b*c^3*x^3 - 2*I*b*c^2*x^2 - b*c*x)*dilog((c*x + I
)/(c*x - I) + 1) + 16*((3*I*a - b)*c^3*x^3 + 2*(3*a + I*b)*c^2*x^2 + (-3*I*a + b)*c*x)*log(x) + 4*(6*I*b*c^2*x
^2 + 9*b*c*x - 2*I*b)*log(-(c*x + I)/(c*x - I)) + 17*(b*c^3*x^3 - 2*I*b*c^2*x^2 - b*c*x)*log((c*x + I)/c) - ((
48*I*a + b)*c^3*x^3 + 2*(48*a - I*b)*c^2*x^2 + (-48*I*a - b)*c*x)*log((c*x - I)/c) - 16*a)/(c^2*d^3*x^3 - 2*I*
c*d^3*x^2 - d^3*x)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {a+b \arctan (c x)}{x^2 (d+i c d x)^3} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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