\(\int \frac {a+b \arctan (c x)}{x^3 (d+i c d x)^3} \, dx\) [65]

Optimal result

Integrand size = 23, antiderivative size = 306 \[ \int \frac {a+b \arctan (c x)}{x^3 (d+i c d x)^3} \, dx=-\frac {b c}{2 d^3 x}-\frac {i b c^2}{8 d^3 (i-c x)^2}-\frac {13 b c^2}{8 d^3 (i-c x)}+\frac {9 b c^2 \arctan (c x)}{8 d^3}-\frac {a+b \arctan (c x)}{2 d^3 x^2}+\frac {3 i c (a+b \arctan (c x))}{d^3 x}+\frac {c^2 (a+b \arctan (c x))}{2 d^3 (i-c x)^2}-\frac {3 i c^2 (a+b \arctan (c x))}{d^3 (i-c x)}-\frac {6 a c^2 \log (x)}{d^3}-\frac {3 i b c^2 \log (x)}{d^3}-\frac {6 c^2 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{d^3}+\frac {3 i b c^2 \log \left (1+c^2 x^2\right )}{2 d^3}-\frac {3 i b c^2 \operatorname {PolyLog}(2,-i c x)}{d^3}+\frac {3 i b c^2 \operatorname {PolyLog}(2,i c x)}{d^3}-\frac {3 i b c^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{d^3} \]

[Out]

-1/2*b*c/d^3/x-1/8*I*b*c^2/d^3/(I-c*x)^2-13/8*b*c^2/d^3/(I-c*x)+9/8*b*c^2*arctan(c*x)/d^3+1/2*(-a-b*arctan(c*x
))/d^3/x^2+3*I*c*(a+b*arctan(c*x))/d^3/x+1/2*c^2*(a+b*arctan(c*x))/d^3/(I-c*x)^2-3*I*c^2*(a+b*arctan(c*x))/d^3
/(I-c*x)-6*a*c^2*ln(x)/d^3-3*I*b*c^2*ln(x)/d^3-6*c^2*(a+b*arctan(c*x))*ln(2/(1+I*c*x))/d^3+3/2*I*b*c^2*ln(c^2*
x^2+1)/d^3-3*I*b*c^2*polylog(2,-I*c*x)/d^3+3*I*b*c^2*polylog(2,I*c*x)/d^3-3*I*b*c^2*polylog(2,1-2/(1+I*c*x))/d
^3

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.696, Rules used = {4996, 4946, 331, 209, 272, 36, 29, 31, 4940, 2438, 4972, 641, 46, 4964, 2449, 2352} \[ \int \frac {a+b \arctan (c x)}{x^3 (d+i c d x)^3} \, dx=-\frac {3 i c^2 (a+b \arctan (c x))}{d^3 (-c x+i)}+\frac {c^2 (a+b \arctan (c x))}{2 d^3 (-c x+i)^2}-\frac {6 c^2 \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{d^3}-\frac {a+b \arctan (c x)}{2 d^3 x^2}+\frac {3 i c (a+b \arctan (c x))}{d^3 x}-\frac {6 a c^2 \log (x)}{d^3}+\frac {9 b c^2 \arctan (c x)}{8 d^3}-\frac {3 i b c^2 \operatorname {PolyLog}(2,-i c x)}{d^3}+\frac {3 i b c^2 \operatorname {PolyLog}(2,i c x)}{d^3}-\frac {3 i b c^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{d^3}+\frac {3 i b c^2 \log \left (c^2 x^2+1\right )}{2 d^3}-\frac {13 b c^2}{8 d^3 (-c x+i)}-\frac {i b c^2}{8 d^3 (-c x+i)^2}-\frac {3 i b c^2 \log (x)}{d^3}-\frac {b c}{2 d^3 x} \]

[In]

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

[Out]

-1/2*(b*c)/(d^3*x) - ((I/8)*b*c^2)/(d^3*(I - c*x)^2) - (13*b*c^2)/(8*d^3*(I - c*x)) + (9*b*c^2*ArcTan[c*x])/(8
*d^3) - (a + b*ArcTan[c*x])/(2*d^3*x^2) + ((3*I)*c*(a + b*ArcTan[c*x]))/(d^3*x) + (c^2*(a + b*ArcTan[c*x]))/(2
*d^3*(I - c*x)^2) - ((3*I)*c^2*(a + b*ArcTan[c*x]))/(d^3*(I - c*x)) - (6*a*c^2*Log[x])/d^3 - ((3*I)*b*c^2*Log[
x])/d^3 - (6*c^2*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/d^3 + (((3*I)/2)*b*c^2*Log[1 + c^2*x^2])/d^3 - ((3*I)
*b*c^2*PolyLog[2, (-I)*c*x])/d^3 + ((3*I)*b*c^2*PolyLog[2, I*c*x])/d^3 - ((3*I)*b*c^2*PolyLog[2, 1 - 2/(1 + I*
c*x)])/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 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && 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 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^3}-\frac {3 i c (a+b \arctan (c x))}{d^3 x^2}-\frac {6 c^2 (a+b \arctan (c x))}{d^3 x}-\frac {c^3 (a+b \arctan (c x))}{d^3 (-i+c x)^3}-\frac {3 i c^3 (a+b \arctan (c x))}{d^3 (-i+c x)^2}+\frac {6 c^3 (a+b \arctan (c x))}{d^3 (-i+c x)}\right ) \, dx \\ & = \frac {\int \frac {a+b \arctan (c x)}{x^3} \, dx}{d^3}-\frac {(3 i c) \int \frac {a+b \arctan (c x)}{x^2} \, dx}{d^3}-\frac {\left (6 c^2\right ) \int \frac {a+b \arctan (c x)}{x} \, dx}{d^3}-\frac {\left (3 i c^3\right ) \int \frac {a+b \arctan (c x)}{(-i+c x)^2} \, dx}{d^3}-\frac {c^3 \int \frac {a+b \arctan (c x)}{(-i+c x)^3} \, dx}{d^3}+\frac {\left (6 c^3\right ) \int \frac {a+b \arctan (c x)}{-i+c x} \, dx}{d^3} \\ & = -\frac {a+b \arctan (c x)}{2 d^3 x^2}+\frac {3 i c (a+b \arctan (c x))}{d^3 x}+\frac {c^2 (a+b \arctan (c x))}{2 d^3 (i-c x)^2}-\frac {3 i c^2 (a+b \arctan (c x))}{d^3 (i-c x)}-\frac {6 a c^2 \log (x)}{d^3}-\frac {6 c^2 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{d^3}+\frac {(b c) \int \frac {1}{x^2 \left (1+c^2 x^2\right )} \, dx}{2 d^3}-\frac {\left (3 i b c^2\right ) \int \frac {1}{x \left (1+c^2 x^2\right )} \, dx}{d^3}-\frac {\left (3 i b c^2\right ) \int \frac {\log (1-i c x)}{x} \, dx}{d^3}+\frac {\left (3 i b c^2\right ) \int \frac {\log (1+i c x)}{x} \, dx}{d^3}-\frac {\left (3 i b c^3\right ) \int \frac {1}{(-i+c x) \left (1+c^2 x^2\right )} \, dx}{d^3}-\frac {\left (b c^3\right ) \int \frac {1}{(-i+c x)^2 \left (1+c^2 x^2\right )} \, dx}{2 d^3}+\frac {\left (6 b c^3\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d^3} \\ & = -\frac {b c}{2 d^3 x}-\frac {a+b \arctan (c x)}{2 d^3 x^2}+\frac {3 i c (a+b \arctan (c x))}{d^3 x}+\frac {c^2 (a+b \arctan (c x))}{2 d^3 (i-c x)^2}-\frac {3 i c^2 (a+b \arctan (c x))}{d^3 (i-c x)}-\frac {6 a c^2 \log (x)}{d^3}-\frac {6 c^2 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{d^3}-\frac {3 i b c^2 \operatorname {PolyLog}(2,-i c x)}{d^3}+\frac {3 i b c^2 \operatorname {PolyLog}(2,i c x)}{d^3}-\frac {\left (3 i b c^2\right ) \text {Subst}\left (\int \frac {1}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )}{2 d^3}-\frac {\left (6 i b c^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{d^3}-\frac {\left (3 i b c^3\right ) \int \frac {1}{(-i+c x)^2 (i+c x)} \, dx}{d^3}-\frac {\left (b c^3\right ) \int \frac {1}{(-i+c x)^3 (i+c x)} \, dx}{2 d^3}-\frac {\left (b c^3\right ) \int \frac {1}{1+c^2 x^2} \, dx}{2 d^3} \\ & = -\frac {b c}{2 d^3 x}-\frac {b c^2 \arctan (c x)}{2 d^3}-\frac {a+b \arctan (c x)}{2 d^3 x^2}+\frac {3 i c (a+b \arctan (c x))}{d^3 x}+\frac {c^2 (a+b \arctan (c x))}{2 d^3 (i-c x)^2}-\frac {3 i c^2 (a+b \arctan (c x))}{d^3 (i-c x)}-\frac {6 a c^2 \log (x)}{d^3}-\frac {6 c^2 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{d^3}-\frac {3 i b c^2 \operatorname {PolyLog}(2,-i c x)}{d^3}+\frac {3 i b c^2 \operatorname {PolyLog}(2,i c x)}{d^3}-\frac {3 i b c^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{d^3}-\frac {\left (3 i b c^2\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )}{2 d^3}-\frac {\left (3 i b c^3\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 ) \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 (3 i b c^4\right ) \text {Subst}\left (\int \frac {1}{1+c^2 x} \, dx,x,x^2\right )}{2 d^3} \\ & = -\frac {b c}{2 d^3 x}-\frac {i b c^2}{8 d^3 (i-c x)^2}-\frac {13 b c^2}{8 d^3 (i-c x)}-\frac {b c^2 \arctan (c x)}{2 d^3}-\frac {a+b \arctan (c x)}{2 d^3 x^2}+\frac {3 i c (a+b \arctan (c x))}{d^3 x}+\frac {c^2 (a+b \arctan (c x))}{2 d^3 (i-c x)^2}-\frac {3 i c^2 (a+b \arctan (c x))}{d^3 (i-c x)}-\frac {6 a c^2 \log (x)}{d^3}-\frac {3 i b c^2 \log (x)}{d^3}-\frac {6 c^2 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{d^3}+\frac {3 i b c^2 \log \left (1+c^2 x^2\right )}{2 d^3}-\frac {3 i b c^2 \operatorname {PolyLog}(2,-i c x)}{d^3}+\frac {3 i b c^2 \operatorname {PolyLog}(2,i c x)}{d^3}-\frac {3 i b c^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{d^3}+\frac {\left (b c^3\right ) \int \frac {1}{1+c^2 x^2} \, dx}{8 d^3}+\frac {\left (3 b c^3\right ) \int \frac {1}{1+c^2 x^2} \, dx}{2 d^3} \\ & = -\frac {b c}{2 d^3 x}-\frac {i b c^2}{8 d^3 (i-c x)^2}-\frac {13 b c^2}{8 d^3 (i-c x)}+\frac {9 b c^2 \arctan (c x)}{8 d^3}-\frac {a+b \arctan (c x)}{2 d^3 x^2}+\frac {3 i c (a+b \arctan (c x))}{d^3 x}+\frac {c^2 (a+b \arctan (c x))}{2 d^3 (i-c x)^2}-\frac {3 i c^2 (a+b \arctan (c x))}{d^3 (i-c x)}-\frac {6 a c^2 \log (x)}{d^3}-\frac {3 i b c^2 \log (x)}{d^3}-\frac {6 c^2 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{d^3}+\frac {3 i b c^2 \log \left (1+c^2 x^2\right )}{2 d^3}-\frac {3 i b c^2 \operatorname {PolyLog}(2,-i c x)}{d^3}+\frac {3 i b c^2 \operatorname {PolyLog}(2,i c x)}{d^3}-\frac {3 i b c^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{d^3} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.41 (sec) , antiderivative size = 285, normalized size of antiderivative = 0.93 \[ \int \frac {a+b \arctan (c x)}{x^3 (d+i c d x)^3} \, dx=-\frac {12 b c^2 \left (\frac {1}{i-c x}-\arctan (c x)\right )+\frac {4 (a+b \arctan (c x))}{x^2}-\frac {24 i c (a+b \arctan (c x))}{x}-\frac {4 c^2 (a+b \arctan (c x))}{(-i+c x)^2}-\frac {24 i c^2 (a+b \arctan (c x))}{-i+c x}-\frac {b c^2 \left (-2 i+c x+(-i+c x)^2 \arctan (c x)\right )}{(-i+c x)^2}+\frac {4 b c \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-c^2 x^2\right )}{x}+48 a c^2 \log (x)+48 c^2 (a+b \arctan (c x)) \log \left (\frac {2 i}{i-c x}\right )+12 i b c^2 \left (2 \log (x)-\log \left (1+c^2 x^2\right )\right )+24 i b c^2 \operatorname {PolyLog}(2,-i c x)-24 i b c^2 \operatorname {PolyLog}(2,i c x)+24 i b c^2 \operatorname {PolyLog}\left (2,\frac {i+c x}{-i+c x}\right )}{8 d^3} \]

[In]

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

[Out]

-1/8*(12*b*c^2*((I - c*x)^(-1) - ArcTan[c*x]) + (4*(a + b*ArcTan[c*x]))/x^2 - ((24*I)*c*(a + b*ArcTan[c*x]))/x
- (4*c^2*(a + b*ArcTan[c*x]))/(-I + c*x)^2 - ((24*I)*c^2*(a + b*ArcTan[c*x]))/(-I + c*x) - (b*c^2*(-2*I + c*x
+ (-I + c*x)^2*ArcTan[c*x]))/(-I + c*x)^2 + (4*b*c*Hypergeometric2F1[-1/2, 1, 1/2, -(c^2*x^2)])/x + 48*a*c^2*
Log[x] + 48*c^2*(a + b*ArcTan[c*x])*Log[(2*I)/(I - c*x)] + (12*I)*b*c^2*(2*Log[x] - Log[1 + c^2*x^2]) + (24*I)
*b*c^2*PolyLog[2, (-I)*c*x] - (24*I)*b*c^2*PolyLog[2, I*c*x] + (24*I)*b*c^2*PolyLog[2, (I + c*x)/(-I + c*x)])/
d^3

Maple [A] (verified)

Time = 1.10 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.13

method	result	size
derivativedivides	\(c^{2} \left (-\frac {a}{2 d^{3} c^{2} x^{2}}+\frac {3 i a}{d^{3} c x}-\frac {6 a \ln \left (c x \right )}{d^{3}}+\frac {3 i a}{d^{3} \left (c x -i\right )}+\frac {a}{2 d^{3} \left (c x -i\right )^{2}}+\frac {3 a \ln \left (c^{2} x^{2}+1\right )}{d^{3}}+\frac {6 i a \arctan \left (c x \right )}{d^{3}}+\frac {b \left (-\frac {\arctan \left (c x \right )}{2 c^{2} x^{2}}+\frac {3 i \ln \left (c^{2} x^{2}+1\right )}{2}-6 \arctan \left (c x \right ) \ln \left (c x \right )+\frac {3 i \ln \left (c x -i\right )^{2}}{2}+\frac {\arctan \left (c x \right )}{2 \left (c x -i\right )^{2}}+6 \arctan \left (c x \right ) \ln \left (c x -i\right )-3 i \left (\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )+\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )+\frac {9 \arctan \left (c x \right )}{8}-3 i \left (\left (\ln \left (c x \right )-\ln \left (-i c x \right )\right ) \ln \left (-i \left (-c x +i\right )\right )-\operatorname {dilog}\left (-i c x \right )\right )-\frac {1}{2 c x}-3 i \ln \left (c x \right )+\frac {13}{8 \left (c x -i\right )}-\frac {i}{8 \left (c x -i\right )^{2}}+3 i \left (\operatorname {dilog}\left (-i \left (c x +i\right )\right )+\ln \left (c x \right ) \ln \left (-i \left (c x +i\right )\right )\right )+\frac {3 i \arctan \left (c x \right )}{c x}+\frac {3 i \arctan \left (c x \right )}{c x -i}\right )}{d^{3}}\right )\)	\(345\)
default	\(c^{2} \left (-\frac {a}{2 d^{3} c^{2} x^{2}}+\frac {3 i a}{d^{3} c x}-\frac {6 a \ln \left (c x \right )}{d^{3}}+\frac {3 i a}{d^{3} \left (c x -i\right )}+\frac {a}{2 d^{3} \left (c x -i\right )^{2}}+\frac {3 a \ln \left (c^{2} x^{2}+1\right )}{d^{3}}+\frac {6 i a \arctan \left (c x \right )}{d^{3}}+\frac {b \left (-\frac {\arctan \left (c x \right )}{2 c^{2} x^{2}}+\frac {3 i \ln \left (c^{2} x^{2}+1\right )}{2}-6 \arctan \left (c x \right ) \ln \left (c x \right )+\frac {3 i \ln \left (c x -i\right )^{2}}{2}+\frac {\arctan \left (c x \right )}{2 \left (c x -i\right )^{2}}+6 \arctan \left (c x \right ) \ln \left (c x -i\right )-3 i \left (\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )+\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )+\frac {9 \arctan \left (c x \right )}{8}-3 i \left (\left (\ln \left (c x \right )-\ln \left (-i c x \right )\right ) \ln \left (-i \left (-c x +i\right )\right )-\operatorname {dilog}\left (-i c x \right )\right )-\frac {1}{2 c x}-3 i \ln \left (c x \right )+\frac {13}{8 \left (c x -i\right )}-\frac {i}{8 \left (c x -i\right )^{2}}+3 i \left (\operatorname {dilog}\left (-i \left (c x +i\right )\right )+\ln \left (c x \right ) \ln \left (-i \left (c x +i\right )\right )\right )+\frac {3 i \arctan \left (c x \right )}{c x}+\frac {3 i \arctan \left (c x \right )}{c x -i}\right )}{d^{3}}\right )\)	\(345\)
parts	\(-\frac {a}{2 d^{3} x^{2}}+\frac {3 i c a}{d^{3} x}-\frac {6 a \,c^{2} \ln \left (x \right )}{d^{3}}-\frac {3 i a \,c^{2}}{d^{3} \left (-c x +i\right )}+\frac {a \,c^{2}}{2 d^{3} \left (-c x +i\right )^{2}}+\frac {6 i c^{2} a \arctan \left (c x \right )}{d^{3}}+\frac {3 c^{2} a \ln \left (c^{2} x^{2}+1\right )}{d^{3}}+\frac {b \,c^{2} \left (-\frac {\arctan \left (c x \right )}{2 c^{2} x^{2}}+\frac {3 i \ln \left (c^{2} x^{2}+1\right )}{2}-6 \arctan \left (c x \right ) \ln \left (c x \right )+\frac {3 i \ln \left (c x -i\right )^{2}}{2}+\frac {\arctan \left (c x \right )}{2 \left (c x -i\right )^{2}}+6 \arctan \left (c x \right ) \ln \left (c x -i\right )-3 i \left (\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )+\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )+\frac {9 \arctan \left (c x \right )}{8}-3 i \left (\left (\ln \left (c x \right )-\ln \left (-i c x \right )\right ) \ln \left (-i \left (-c x +i\right )\right )-\operatorname {dilog}\left (-i c x \right )\right )-\frac {1}{2 c x}-3 i \ln \left (c x \right )+\frac {13}{8 \left (c x -i\right )}-\frac {i}{8 \left (c x -i\right )^{2}}+3 i \left (\operatorname {dilog}\left (-i \left (c x +i\right )\right )+\ln \left (c x \right ) \ln \left (-i \left (c x +i\right )\right )\right )+\frac {3 i \arctan \left (c x \right )}{c x}+\frac {3 i \arctan \left (c x \right )}{c x -i}\right )}{d^{3}}\)	\(354\)
risch	\(-\frac {b c}{2 d^{3} x}-\frac {15 b \,c^{2} \arctan \left (c x \right )}{16 d^{3}}+\frac {3 i c^{2} b \ln \left (-i c x +1\right )}{4 d^{3} \left (-i c x -1\right )}-\frac {3 i c^{2} b \ln \left (-i c x +1\right )}{16 d^{3} \left (-i c x -1\right )^{2}}+\frac {3 c^{3} b \ln \left (-i c x +1\right ) x}{4 d^{3} \left (-i c x -1\right )}-\frac {c^{3} b \ln \left (-i c x +1\right ) x}{8 d^{3} \left (-i c x -1\right )^{2}}-\frac {3 i c^{2} \ln \left (\frac {1}{2}-\frac {i c x}{2}\right ) \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) b}{d^{3}}+\frac {3 i c^{2} b \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (-i c x +1\right )}{d^{3}}+\frac {i c^{2} b \ln \left (i c x +1\right )}{4 d^{3} \left (i c x +1\right )^{2}}+\frac {3 i c^{2} b \ln \left (i c x +1\right )}{2 d^{3} \left (i c x +1\right )}+\frac {3 c^{2} a \ln \left (c^{2} x^{2}+1\right )}{d^{3}}-\frac {a}{2 d^{3} x^{2}}-\frac {6 c^{2} a \ln \left (-i c x \right )}{d^{3}}-\frac {c^{2} a}{2 d^{3} \left (-i c x -1\right )^{2}}+\frac {3 c^{2} a}{d^{3} \left (-i c x -1\right )}-\frac {i c^{4} b \ln \left (-i c x +1\right ) x^{2}}{16 d^{3} \left (-i c x -1\right )^{2}}+\frac {i c^{2} b}{8 d^{3} \left (i c x +1\right )^{2}}-\frac {3 i c^{2} b \ln \left (i c x +1\right )^{2}}{2 d^{3}}-\frac {3 i c^{2} b \operatorname {dilog}\left (i c x +1\right )}{d^{3}}-\frac {7 i c^{2} b \ln \left (i c x \right )}{4 d^{3}}+\frac {i b \ln \left (i c x +1\right )}{4 d^{3} x^{2}}+\frac {3 i c^{2} b}{2 d^{3} \left (i c x +1\right )}+\frac {3 i c^{2} \operatorname {dilog}\left (-i c x +1\right ) b}{d^{3}}-\frac {5 i c^{2} b \ln \left (-i c x \right )}{4 d^{3}}+\frac {5 i c^{2} b \ln \left (-i c x +1\right )}{4 d^{3}}-\frac {3 i c^{2} b \operatorname {dilog}\left (\frac {1}{2}-\frac {i c x}{2}\right )}{d^{3}}-\frac {i b \ln \left (-i c x +1\right )}{4 d^{3} x^{2}}-\frac {i c^{2} b}{8 d^{3} \left (-i c x -1\right )}+\frac {15 i c^{2} b \ln \left (c^{2} x^{2}+1\right )}{32 d^{3}}-\frac {3 c b \ln \left (-i c x +1\right )}{2 d^{3} x}+\frac {3 c b \ln \left (i c x +1\right )}{2 d^{3} x}+\frac {3 i c a}{d^{3} x}+\frac {6 i c^{2} a \arctan \left (c x \right )}{d^{3}}\)	\(646\)

[In]

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

[Out]

c^2*(-1/2*a/d^3/c^2/x^2+3*I*a/d^3/c/x-6*a/d^3*ln(c*x)+3*I*a/d^3/(c*x-I)+1/2*a/d^3/(c*x-I)^2+3*a/d^3*ln(c^2*x^2
+1)+6*I*a/d^3*arctan(c*x)+b/d^3*(-1/2/c^2/x^2*arctan(c*x)+3/2*I*ln(c^2*x^2+1)-6*arctan(c*x)*ln(c*x)+3/2*I*ln(c
*x-I)^2+1/2*arctan(c*x)/(c*x-I)^2+6*arctan(c*x)*ln(c*x-I)-3*I*(dilog(-1/2*I*(c*x+I))+ln(c*x-I)*ln(-1/2*I*(c*x+
I)))+9/8*arctan(c*x)-3*I*((ln(c*x)-ln(-I*c*x))*ln(-I*(-c*x+I))-dilog(-I*c*x))-1/2/c/x-3*I*ln(c*x)+13/8/(c*x-I)
-1/8*I/(c*x-I)^2+3*I*(dilog(-I*(c*x+I))+ln(c*x)*ln(-I*(c*x+I)))+3*I*arctan(c*x)/c/x+3*I*arctan(c*x)/(c*x-I)))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.02 \[ \int \frac {a+b \arctan (c x)}{x^3 (d+i c d x)^3} \, dx=-\frac {6 \, {\left (-16 i \, a - 3 \, b\right )} c^{3} x^{3} - 12 \, {\left (12 \, a - i \, b\right )} c^{2} x^{2} + 8 \, {\left (4 i \, a - b\right )} c x + 48 \, {\left (-i \, b c^{4} x^{4} - 2 \, b c^{3} x^{3} + i \, b c^{2} x^{2}\right )} {\rm Li}_2\left (\frac {c x + i}{c x - i} + 1\right ) + 48 \, {\left ({\left (2 \, a + i \, b\right )} c^{4} x^{4} + 2 \, {\left (-2 i \, a + b\right )} c^{3} x^{3} - {\left (2 \, a + i \, b\right )} c^{2} x^{2}\right )} \log \left (x\right ) + 4 \, {\left (12 \, b c^{3} x^{3} - 18 i \, b c^{2} x^{2} - 4 \, b c x - i \, b\right )} \log \left (-\frac {c x + i}{c x - i}\right ) + 33 \, {\left (-i \, b c^{4} x^{4} - 2 \, b c^{3} x^{3} + i \, b c^{2} x^{2}\right )} \log \left (\frac {c x + i}{c}\right ) - 3 \, {\left ({\left (32 \, a + 5 i \, b\right )} c^{4} x^{4} - 2 \, {\left (32 i \, a - 5 \, b\right )} c^{3} x^{3} - {\left (32 \, a + 5 i \, b\right )} c^{2} x^{2}\right )} \log \left (\frac {c x - i}{c}\right ) - 8 \, a}{16 \, {\left (c^{2} d^{3} x^{4} - 2 i \, c d^{3} x^{3} - d^{3} x^{2}\right )}} \]

[In]

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

[Out]

-1/16*(6*(-16*I*a - 3*b)*c^3*x^3 - 12*(12*a - I*b)*c^2*x^2 + 8*(4*I*a - b)*c*x + 48*(-I*b*c^4*x^4 - 2*b*c^3*x^
3 + I*b*c^2*x^2)*dilog((c*x + I)/(c*x - I) + 1) + 48*((2*a + I*b)*c^4*x^4 + 2*(-2*I*a + b)*c^3*x^3 - (2*a + I*
b)*c^2*x^2)*log(x) + 4*(12*b*c^3*x^3 - 18*I*b*c^2*x^2 - 4*b*c*x - I*b)*log(-(c*x + I)/(c*x - I)) + 33*(-I*b*c^
4*x^4 - 2*b*c^3*x^3 + I*b*c^2*x^2)*log((c*x + I)/c) - 3*((32*a + 5*I*b)*c^4*x^4 - 2*(32*I*a - 5*b)*c^3*x^3 - (
32*a + 5*I*b)*c^2*x^2)*log((c*x - I)/c) - 8*a)/(c^2*d^3*x^4 - 2*I*c*d^3*x^3 - d^3*x^2)

Sympy [F]

\[ \int \frac {a+b \arctan (c x)}{x^3 (d+i c d x)^3} \, dx=\frac {i \left (\int \frac {a}{c^{3} x^{6} - 3 i c^{2} x^{5} - 3 c x^{4} + i x^{3}}\, dx + \int \frac {b \operatorname {atan}{\left (c x \right )}}{c^{3} x^{6} - 3 i c^{2} x^{5} - 3 c x^{4} + i x^{3}}\, dx\right )}{d^{3}} \]

[In]

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

[Out]

I*(Integral(a/(c**3*x**6 - 3*I*c**2*x**5 - 3*c*x**4 + I*x**3), x) + Integral(b*atan(c*x)/(c**3*x**6 - 3*I*c**2
*x**5 - 3*c*x**4 + I*x**3), x))/d**3

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. 590 vs. \(2 (253) = 506\).

Time = 0.31 (sec) , antiderivative size = 590, normalized size of antiderivative = 1.93 \[ \int \frac {a+b \arctan (c x)}{x^3 (d+i c d x)^3} \, dx=-\frac {33 \, b c^{4} x^{4} \arctan \left (1, c x\right ) + 6 \, {\left (b {\left (-11 i \, \arctan \left (1, c x\right ) - 3\right )} - 16 i \, a\right )} c^{3} x^{3} - 3 \, {\left (b {\left (11 \, \arctan \left (1, c x\right ) - 4 i\right )} + 48 \, a\right )} c^{2} x^{2} + 8 \, {\left (4 i \, a - b\right )} c x + 24 \, {\left (-i \, b c^{4} x^{4} - 2 \, b c^{3} x^{3} + i \, b c^{2} x^{2}\right )} \arctan \left (c x\right )^{2} + 6 \, {\left (-i \, b c^{4} x^{4} - 2 \, b c^{3} x^{3} + i \, b c^{2} x^{2}\right )} \log \left (c^{2} x^{2} + 1\right )^{2} - 24 \, {\left (b c^{4} x^{4} - 2 i \, b c^{3} x^{3} - b c^{2} x^{2}\right )} \arctan \left (c x\right ) \log \left (\frac {1}{4} \, c^{2} x^{2} + \frac {1}{4}\right ) + 96 \, {\left (b c^{4} x^{4} - 2 i \, b c^{3} x^{3} - b c^{2} x^{2}\right )} \arctan \left (c x\right ) \log \left (c x\right ) + {\left (3 \, {\left (-32 i \, a + 5 \, b\right )} c^{4} x^{4} - 6 \, {\left (32 \, a + 21 i \, b\right )} c^{3} x^{3} + 3 \, {\left (32 i \, a - 53 \, b\right )} c^{2} x^{2} + 32 i \, b c x - 8 \, b\right )} \arctan \left (c x\right ) + 48 \, {\left (-i \, b c^{4} x^{4} - 2 \, b c^{3} x^{3} + i \, b c^{2} x^{2}\right )} {\rm Li}_2\left (i \, c x + 1\right ) + 48 \, {\left (i \, b c^{4} x^{4} + 2 \, b c^{3} x^{3} - i \, b c^{2} x^{2}\right )} {\rm Li}_2\left (\frac {1}{2} i \, c x + \frac {1}{2}\right ) + 48 \, {\left (i \, b c^{4} x^{4} + 2 \, b c^{3} x^{3} - i \, b c^{2} x^{2}\right )} {\rm Li}_2\left (-i \, c x + 1\right ) - 12 \, {\left (2 \, {\left ({\left (\pi + i\right )} b + 2 \, a\right )} c^{4} x^{4} - 4 \, {\left ({\left (i \, \pi - 1\right )} b + 2 i \, a\right )} c^{3} x^{3} - 2 \, {\left ({\left (\pi + i\right )} b + 2 \, a\right )} c^{2} x^{2} - {\left (i \, b c^{4} x^{4} + 2 \, b c^{3} x^{3} - i \, b c^{2} x^{2}\right )} \log \left (\frac {1}{4} \, c^{2} x^{2} + \frac {1}{4}\right )\right )} \log \left (c^{2} x^{2} + 1\right ) + 48 \, {\left ({\left (2 \, a + i \, b\right )} c^{4} x^{4} + 2 \, {\left (-2 i \, a + b\right )} c^{3} x^{3} - {\left (2 \, a + i \, b\right )} c^{2} x^{2}\right )} \log \left (x\right ) - 8 \, a}{16 \, {\left (c^{2} d^{3} x^{4} - 2 i \, c d^{3} x^{3} - d^{3} x^{2}\right )}} \]

[In]

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

[Out]

-1/16*(33*b*c^4*x^4*arctan2(1, c*x) + 6*(b*(-11*I*arctan2(1, c*x) - 3) - 16*I*a)*c^3*x^3 - 3*(b*(11*arctan2(1,
c*x) - 4*I) + 48*a)*c^2*x^2 + 8*(4*I*a - b)*c*x + 24*(-I*b*c^4*x^4 - 2*b*c^3*x^3 + I*b*c^2*x^2)*arctan(c*x)^2
+ 6*(-I*b*c^4*x^4 - 2*b*c^3*x^3 + I*b*c^2*x^2)*log(c^2*x^2 + 1)^2 - 24*(b*c^4*x^4 - 2*I*b*c^3*x^3 - b*c^2*x^2
)*arctan(c*x)*log(1/4*c^2*x^2 + 1/4) + 96*(b*c^4*x^4 - 2*I*b*c^3*x^3 - b*c^2*x^2)*arctan(c*x)*log(c*x) + (3*(-
32*I*a + 5*b)*c^4*x^4 - 6*(32*a + 21*I*b)*c^3*x^3 + 3*(32*I*a - 53*b)*c^2*x^2 + 32*I*b*c*x - 8*b)*arctan(c*x)
+ 48*(-I*b*c^4*x^4 - 2*b*c^3*x^3 + I*b*c^2*x^2)*dilog(I*c*x + 1) + 48*(I*b*c^4*x^4 + 2*b*c^3*x^3 - I*b*c^2*x^2
)*dilog(1/2*I*c*x + 1/2) + 48*(I*b*c^4*x^4 + 2*b*c^3*x^3 - I*b*c^2*x^2)*dilog(-I*c*x + 1) - 12*(2*((pi + I)*b
+ 2*a)*c^4*x^4 - 4*((I*pi - 1)*b + 2*I*a)*c^3*x^3 - 2*((pi + I)*b + 2*a)*c^2*x^2 - (I*b*c^4*x^4 + 2*b*c^3*x^3
- I*b*c^2*x^2)*log(1/4*c^2*x^2 + 1/4))*log(c^2*x^2 + 1) + 48*((2*a + I*b)*c^4*x^4 + 2*(-2*I*a + b)*c^3*x^3 - (
2*a + I*b)*c^2*x^2)*log(x) - 8*a)/(c^2*d^3*x^4 - 2*I*c*d^3*x^3 - d^3*x^2)

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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