\(\int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^5} \, dx\) [92]

Optimal result

Integrand size = 25, antiderivative size = 293 \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^5} \, dx=-\frac {b^2 c^2 d^3}{12 x^2}-\frac {i b^2 c^3 d^3}{x}-i b^2 c^4 d^3 \arctan (c x)-\frac {b c d^3 (a+b \arctan (c x))}{6 x^3}-\frac {i b c^2 d^3 (a+b \arctan (c x))}{x^2}+\frac {7 b c^3 d^3 (a+b \arctan (c x))}{2 x}-\frac {d^3 (1+i c x)^4 (a+b \arctan (c x))^2}{4 x^4}-4 i a b c^4 d^3 \log (x)-\frac {11}{3} b^2 c^4 d^3 \log (x)-4 i b c^4 d^3 (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )+\frac {11}{6} b^2 c^4 d^3 \log \left (1+c^2 x^2\right )+2 b^2 c^4 d^3 \operatorname {PolyLog}(2,-i c x)-2 b^2 c^4 d^3 \operatorname {PolyLog}(2,i c x)-2 b^2 c^4 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right ) \]

[Out]

-1/12*b^2*c^2*d^3/x^2-I*b^2*c^3*d^3/x-I*b^2*c^4*d^3*arctan(c*x)-1/6*b*c*d^3*(a+b*arctan(c*x))/x^3-I*b*c^2*d^3*
(a+b*arctan(c*x))/x^2+7/2*b*c^3*d^3*(a+b*arctan(c*x))/x-1/4*d^3*(1+I*c*x)^4*(a+b*arctan(c*x))^2/x^4-4*I*a*b*c^
4*d^3*ln(x)-11/3*b^2*c^4*d^3*ln(x)-4*I*b*c^4*d^3*(a+b*arctan(c*x))*ln(2/(1-I*c*x))+11/6*b^2*c^4*d^3*ln(c^2*x^2
+1)+2*b^2*c^4*d^3*polylog(2,-I*c*x)-2*b^2*c^4*d^3*polylog(2,I*c*x)-2*b^2*c^4*d^3*polylog(2,1-2/(1-I*c*x))

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {37, 4994, 4946, 272, 46, 331, 209, 36, 29, 31, 4940, 2438, 4964, 2449, 2352} \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^5} \, dx=-4 i b c^4 d^3 \log \left (\frac {2}{1-i c x}\right ) (a+b \arctan (c x))+\frac {7 b c^3 d^3 (a+b \arctan (c x))}{2 x}-\frac {i b c^2 d^3 (a+b \arctan (c x))}{x^2}-\frac {d^3 (1+i c x)^4 (a+b \arctan (c x))^2}{4 x^4}-\frac {b c d^3 (a+b \arctan (c x))}{6 x^3}-4 i a b c^4 d^3 \log (x)-i b^2 c^4 d^3 \arctan (c x)+2 b^2 c^4 d^3 \operatorname {PolyLog}(2,-i c x)-2 b^2 c^4 d^3 \operatorname {PolyLog}(2,i c x)-2 b^2 c^4 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )-\frac {11}{3} b^2 c^4 d^3 \log (x)-\frac {i b^2 c^3 d^3}{x}-\frac {b^2 c^2 d^3}{12 x^2}+\frac {11}{6} b^2 c^4 d^3 \log \left (c^2 x^2+1\right ) \]

[In]

Int[((d + I*c*d*x)^3*(a + b*ArcTan[c*x])^2)/x^5,x]

[Out]

-1/12*(b^2*c^2*d^3)/x^2 - (I*b^2*c^3*d^3)/x - I*b^2*c^4*d^3*ArcTan[c*x] - (b*c*d^3*(a + b*ArcTan[c*x]))/(6*x^3
) - (I*b*c^2*d^3*(a + b*ArcTan[c*x]))/x^2 + (7*b*c^3*d^3*(a + b*ArcTan[c*x]))/(2*x) - (d^3*(1 + I*c*x)^4*(a +
b*ArcTan[c*x])^2)/(4*x^4) - (4*I)*a*b*c^4*d^3*Log[x] - (11*b^2*c^4*d^3*Log[x])/3 - (4*I)*b*c^4*d^3*(a + b*ArcT
an[c*x])*Log[2/(1 - I*c*x)] + (11*b^2*c^4*d^3*Log[1 + c^2*x^2])/6 + 2*b^2*c^4*d^3*PolyLog[2, (-I)*c*x] - 2*b^2
*c^4*d^3*PolyLog[2, I*c*x] - 2*b^2*c^4*d^3*PolyLog[2, 1 - 2/(1 - I*c*x)]

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 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

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 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 4994

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x_))^(q_), x_Symbol] :> With[{u
= IntHide[(f*x)^m*(d + e*x)^q, x]}, Dist[(a + b*ArcTan[c*x])^p, u, x] - Dist[b*c*p, Int[ExpandIntegrand[(a +
b*ArcTan[c*x])^(p - 1), u/(1 + c^2*x^2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, q}, x] && IGtQ[p, 1] && EqQ[c
^2*d^2 + e^2, 0] && IntegersQ[m, q] && NeQ[m, -1] && NeQ[q, -1] && ILtQ[m + q + 1, 0] && LtQ[m*q, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {d^3 (1+i c x)^4 (a+b \arctan (c x))^2}{4 x^4}-(2 b c) \int \left (-\frac {d^3 (a+b \arctan (c x))}{4 x^4}-\frac {i c d^3 (a+b \arctan (c x))}{x^3}+\frac {7 c^2 d^3 (a+b \arctan (c x))}{4 x^2}+\frac {2 i c^3 d^3 (a+b \arctan (c x))}{x}-\frac {2 i c^4 d^3 (a+b \arctan (c x))}{i+c x}\right ) \, dx \\ & = -\frac {d^3 (1+i c x)^4 (a+b \arctan (c x))^2}{4 x^4}+\frac {1}{2} \left (b c d^3\right ) \int \frac {a+b \arctan (c x)}{x^4} \, dx+\left (2 i b c^2 d^3\right ) \int \frac {a+b \arctan (c x)}{x^3} \, dx-\frac {1}{2} \left (7 b c^3 d^3\right ) \int \frac {a+b \arctan (c x)}{x^2} \, dx-\left (4 i b c^4 d^3\right ) \int \frac {a+b \arctan (c x)}{x} \, dx+\left (4 i b c^5 d^3\right ) \int \frac {a+b \arctan (c x)}{i+c x} \, dx \\ & = -\frac {b c d^3 (a+b \arctan (c x))}{6 x^3}-\frac {i b c^2 d^3 (a+b \arctan (c x))}{x^2}+\frac {7 b c^3 d^3 (a+b \arctan (c x))}{2 x}-\frac {d^3 (1+i c x)^4 (a+b \arctan (c x))^2}{4 x^4}-4 i a b c^4 d^3 \log (x)-4 i b c^4 d^3 (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )+\frac {1}{6} \left (b^2 c^2 d^3\right ) \int \frac {1}{x^3 \left (1+c^2 x^2\right )} \, dx+\left (i b^2 c^3 d^3\right ) \int \frac {1}{x^2 \left (1+c^2 x^2\right )} \, dx+\left (2 b^2 c^4 d^3\right ) \int \frac {\log (1-i c x)}{x} \, dx-\left (2 b^2 c^4 d^3\right ) \int \frac {\log (1+i c x)}{x} \, dx-\frac {1}{2} \left (7 b^2 c^4 d^3\right ) \int \frac {1}{x \left (1+c^2 x^2\right )} \, dx+\left (4 i b^2 c^5 d^3\right ) \int \frac {\log \left (\frac {2}{1-i c x}\right )}{1+c^2 x^2} \, dx \\ & = -\frac {i b^2 c^3 d^3}{x}-\frac {b c d^3 (a+b \arctan (c x))}{6 x^3}-\frac {i b c^2 d^3 (a+b \arctan (c x))}{x^2}+\frac {7 b c^3 d^3 (a+b \arctan (c x))}{2 x}-\frac {d^3 (1+i c x)^4 (a+b \arctan (c x))^2}{4 x^4}-4 i a b c^4 d^3 \log (x)-4 i b c^4 d^3 (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )+2 b^2 c^4 d^3 \operatorname {PolyLog}(2,-i c x)-2 b^2 c^4 d^3 \operatorname {PolyLog}(2,i c x)+\frac {1}{12} \left (b^2 c^2 d^3\right ) \text {Subst}\left (\int \frac {1}{x^2 \left (1+c^2 x\right )} \, dx,x,x^2\right )-\frac {1}{4} \left (7 b^2 c^4 d^3\right ) \text {Subst}\left (\int \frac {1}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )-\left (4 b^2 c^4 d^3\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-i c x}\right )-\left (i b^2 c^5 d^3\right ) \int \frac {1}{1+c^2 x^2} \, dx \\ & = -\frac {i b^2 c^3 d^3}{x}-i b^2 c^4 d^3 \arctan (c x)-\frac {b c d^3 (a+b \arctan (c x))}{6 x^3}-\frac {i b c^2 d^3 (a+b \arctan (c x))}{x^2}+\frac {7 b c^3 d^3 (a+b \arctan (c x))}{2 x}-\frac {d^3 (1+i c x)^4 (a+b \arctan (c x))^2}{4 x^4}-4 i a b c^4 d^3 \log (x)-4 i b c^4 d^3 (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )+2 b^2 c^4 d^3 \operatorname {PolyLog}(2,-i c x)-2 b^2 c^4 d^3 \operatorname {PolyLog}(2,i c x)-2 b^2 c^4 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )+\frac {1}{12} \left (b^2 c^2 d^3\right ) \text {Subst}\left (\int \left (\frac {1}{x^2}-\frac {c^2}{x}+\frac {c^4}{1+c^2 x}\right ) \, dx,x,x^2\right )-\frac {1}{4} \left (7 b^2 c^4 d^3\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )+\frac {1}{4} \left (7 b^2 c^6 d^3\right ) \text {Subst}\left (\int \frac {1}{1+c^2 x} \, dx,x,x^2\right ) \\ & = -\frac {b^2 c^2 d^3}{12 x^2}-\frac {i b^2 c^3 d^3}{x}-i b^2 c^4 d^3 \arctan (c x)-\frac {b c d^3 (a+b \arctan (c x))}{6 x^3}-\frac {i b c^2 d^3 (a+b \arctan (c x))}{x^2}+\frac {7 b c^3 d^3 (a+b \arctan (c x))}{2 x}-\frac {d^3 (1+i c x)^4 (a+b \arctan (c x))^2}{4 x^4}-4 i a b c^4 d^3 \log (x)-\frac {11}{3} b^2 c^4 d^3 \log (x)-4 i b c^4 d^3 (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )+\frac {11}{6} b^2 c^4 d^3 \log \left (1+c^2 x^2\right )+2 b^2 c^4 d^3 \operatorname {PolyLog}(2,-i c x)-2 b^2 c^4 d^3 \operatorname {PolyLog}(2,i c x)-2 b^2 c^4 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.69 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.10 \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^5} \, dx=\frac {d^3 \left (-3 a^2-12 i a^2 c x-2 a b c x+18 a^2 c^2 x^2-12 i a b c^2 x^2-b^2 c^2 x^2+12 i a^2 c^3 x^3+42 a b c^3 x^3-12 i b^2 c^3 x^3-b^2 c^4 x^4-3 b^2 (-i+c x)^4 \arctan (c x)^2+2 b \arctan (c x) \left (b c x \left (-1-6 i c x+21 c^2 x^2-6 i c^3 x^3\right )+3 a \left (-1-4 i c x+6 c^2 x^2+4 i c^3 x^3+7 c^4 x^4\right )-24 i b c^4 x^4 \log \left (1-e^{2 i \arctan (c x)}\right )\right )-48 i a b c^4 x^4 \log (c x)-44 b^2 c^4 x^4 \log \left (\frac {c x}{\sqrt {1+c^2 x^2}}\right )+24 i a b c^4 x^4 \log \left (1+c^2 x^2\right )-24 b^2 c^4 x^4 \operatorname {PolyLog}\left (2,e^{2 i \arctan (c x)}\right )\right )}{12 x^4} \]

[In]

Integrate[((d + I*c*d*x)^3*(a + b*ArcTan[c*x])^2)/x^5,x]

[Out]

(d^3*(-3*a^2 - (12*I)*a^2*c*x - 2*a*b*c*x + 18*a^2*c^2*x^2 - (12*I)*a*b*c^2*x^2 - b^2*c^2*x^2 + (12*I)*a^2*c^3
*x^3 + 42*a*b*c^3*x^3 - (12*I)*b^2*c^3*x^3 - b^2*c^4*x^4 - 3*b^2*(-I + c*x)^4*ArcTan[c*x]^2 + 2*b*ArcTan[c*x]*
(b*c*x*(-1 - (6*I)*c*x + 21*c^2*x^2 - (6*I)*c^3*x^3) + 3*a*(-1 - (4*I)*c*x + 6*c^2*x^2 + (4*I)*c^3*x^3 + 7*c^4
*x^4) - (24*I)*b*c^4*x^4*Log[1 - E^((2*I)*ArcTan[c*x])]) - (48*I)*a*b*c^4*x^4*Log[c*x] - 44*b^2*c^4*x^4*Log[(c
*x)/Sqrt[1 + c^2*x^2]] + (24*I)*a*b*c^4*x^4*Log[1 + c^2*x^2] - 24*b^2*c^4*x^4*PolyLog[2, E^((2*I)*ArcTan[c*x])
]))/(12*x^4)

Maple [A] (verified)

Time = 4.46 (sec) , antiderivative size = 501, normalized size of antiderivative = 1.71

method	result	size
parts	\(d^{3} a^{2} \left (\frac {3 c^{2}}{2 x^{2}}-\frac {1}{4 x^{4}}+\frac {i c^{3}}{x}-\frac {i c}{x^{3}}\right )+b^{2} d^{3} c^{4} \left (\frac {3 \arctan \left (c x \right )^{2}}{2 c^{2} x^{2}}+\frac {11 \ln \left (c^{2} x^{2}+1\right )}{6}+\frac {7 \arctan \left (c x \right )^{2}}{4}-\frac {11 \ln \left (c x \right )}{3}-\frac {1}{12 c^{2} x^{2}}-\frac {\arctan \left (c x \right )^{2}}{4 c^{4} x^{4}}+\frac {\ln \left (c x -i\right )^{2}}{2}+\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )+\frac {7 \arctan \left (c x \right )}{2 c x}-2 \operatorname {dilog}\left (-i c x +1\right )+2 \operatorname {dilog}\left (i c x +1\right )-\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )+\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )+\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )-i \arctan \left (c x \right )+2 \ln \left (c x \right ) \ln \left (i c x +1\right )-2 \ln \left (c x \right ) \ln \left (-i c x +1\right )-4 i \arctan \left (c x \right ) \ln \left (c x \right )-\frac {\arctan \left (c x \right )}{6 c^{3} x^{3}}-\frac {i}{c x}-\frac {i \arctan \left (c x \right )}{c^{2} x^{2}}+2 i \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )+\frac {i \arctan \left (c x \right )^{2}}{c x}-\frac {i \arctan \left (c x \right )^{2}}{c^{3} x^{3}}\right )+2 a \,d^{3} b \,c^{4} \left (-\frac {\arctan \left (c x \right )}{4 c^{4} x^{4}}-\frac {i \arctan \left (c x \right )}{c^{3} x^{3}}+\frac {i \arctan \left (c x \right )}{c x}+\frac {3 \arctan \left (c x \right )}{2 c^{2} x^{2}}-\frac {i}{2 c^{2} x^{2}}-2 i \ln \left (c x \right )-\frac {1}{12 c^{3} x^{3}}+\frac {7}{4 c x}+i \ln \left (c^{2} x^{2}+1\right )+\frac {7 \arctan \left (c x \right )}{4}\right )\)	\(501\)
derivativedivides	\(c^{4} \left (d^{3} a^{2} \left (-\frac {1}{4 c^{4} x^{4}}-\frac {i}{c^{3} x^{3}}+\frac {i}{c x}+\frac {3}{2 c^{2} x^{2}}\right )+b^{2} d^{3} \left (\frac {3 \arctan \left (c x \right )^{2}}{2 c^{2} x^{2}}+\frac {11 \ln \left (c^{2} x^{2}+1\right )}{6}+\frac {7 \arctan \left (c x \right )^{2}}{4}-\frac {11 \ln \left (c x \right )}{3}-\frac {1}{12 c^{2} x^{2}}-\frac {\arctan \left (c x \right )^{2}}{4 c^{4} x^{4}}+\frac {\ln \left (c x -i\right )^{2}}{2}+\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )+\frac {7 \arctan \left (c x \right )}{2 c x}-2 \operatorname {dilog}\left (-i c x +1\right )+2 \operatorname {dilog}\left (i c x +1\right )-\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )+\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )+\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )-i \arctan \left (c x \right )+2 \ln \left (c x \right ) \ln \left (i c x +1\right )-2 \ln \left (c x \right ) \ln \left (-i c x +1\right )-4 i \arctan \left (c x \right ) \ln \left (c x \right )-\frac {\arctan \left (c x \right )}{6 c^{3} x^{3}}-\frac {i}{c x}-\frac {i \arctan \left (c x \right )}{c^{2} x^{2}}+2 i \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )+\frac {i \arctan \left (c x \right )^{2}}{c x}-\frac {i \arctan \left (c x \right )^{2}}{c^{3} x^{3}}\right )+2 a \,d^{3} b \left (-\frac {\arctan \left (c x \right )}{4 c^{4} x^{4}}-\frac {i \arctan \left (c x \right )}{c^{3} x^{3}}+\frac {i \arctan \left (c x \right )}{c x}+\frac {3 \arctan \left (c x \right )}{2 c^{2} x^{2}}-\frac {i}{2 c^{2} x^{2}}-2 i \ln \left (c x \right )-\frac {1}{12 c^{3} x^{3}}+\frac {7}{4 c x}+i \ln \left (c^{2} x^{2}+1\right )+\frac {7 \arctan \left (c x \right )}{4}\right )\right )\)	\(504\)
default	\(c^{4} \left (d^{3} a^{2} \left (-\frac {1}{4 c^{4} x^{4}}-\frac {i}{c^{3} x^{3}}+\frac {i}{c x}+\frac {3}{2 c^{2} x^{2}}\right )+b^{2} d^{3} \left (\frac {3 \arctan \left (c x \right )^{2}}{2 c^{2} x^{2}}+\frac {11 \ln \left (c^{2} x^{2}+1\right )}{6}+\frac {7 \arctan \left (c x \right )^{2}}{4}-\frac {11 \ln \left (c x \right )}{3}-\frac {1}{12 c^{2} x^{2}}-\frac {\arctan \left (c x \right )^{2}}{4 c^{4} x^{4}}+\frac {\ln \left (c x -i\right )^{2}}{2}+\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )+\frac {7 \arctan \left (c x \right )}{2 c x}-2 \operatorname {dilog}\left (-i c x +1\right )+2 \operatorname {dilog}\left (i c x +1\right )-\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )+\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )+\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )-i \arctan \left (c x \right )+2 \ln \left (c x \right ) \ln \left (i c x +1\right )-2 \ln \left (c x \right ) \ln \left (-i c x +1\right )-4 i \arctan \left (c x \right ) \ln \left (c x \right )-\frac {\arctan \left (c x \right )}{6 c^{3} x^{3}}-\frac {i}{c x}-\frac {i \arctan \left (c x \right )}{c^{2} x^{2}}+2 i \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )+\frac {i \arctan \left (c x \right )^{2}}{c x}-\frac {i \arctan \left (c x \right )^{2}}{c^{3} x^{3}}\right )+2 a \,d^{3} b \left (-\frac {\arctan \left (c x \right )}{4 c^{4} x^{4}}-\frac {i \arctan \left (c x \right )}{c^{3} x^{3}}+\frac {i \arctan \left (c x \right )}{c x}+\frac {3 \arctan \left (c x \right )}{2 c^{2} x^{2}}-\frac {i}{2 c^{2} x^{2}}-2 i \ln \left (c x \right )-\frac {1}{12 c^{3} x^{3}}+\frac {7}{4 c x}+i \ln \left (c^{2} x^{2}+1\right )+\frac {7 \arctan \left (c x \right )}{4}\right )\right )\)	\(504\)

[In]

int((d+I*c*d*x)^3*(a+b*arctan(c*x))^2/x^5,x,method=_RETURNVERBOSE)

[Out]

d^3*a^2*(3/2*c^2/x^2-1/4/x^4+I*c^3/x-I*c/x^3)+b^2*d^3*c^4*(3/2/c^2/x^2*arctan(c*x)^2+11/6*ln(c^2*x^2+1)+7/4*ar
ctan(c*x)^2-11/3*ln(c*x)+dilog(-1/2*I*(c*x+I))-1/12/c^2/x^2-1/4/c^4/x^4*arctan(c*x)^2-dilog(1/2*I*(c*x-I))+7/2
/c/x*arctan(c*x)+2*dilog(1+I*c*x)-2*dilog(1-I*c*x)-I*arctan(c*x)+1/2*ln(c*x-I)^2-1/2*ln(c*x+I)^2-ln(c*x-I)*ln(
c^2*x^2+1)+ln(c*x-I)*ln(-1/2*I*(c*x+I))+ln(c*x+I)*ln(c^2*x^2+1)-ln(c*x+I)*ln(1/2*I*(c*x-I))+2*ln(c*x)*ln(1+I*c
*x)-2*ln(c*x)*ln(1-I*c*x)-I/c/x-1/6*arctan(c*x)/c^3/x^3-I*arctan(c*x)/c^2/x^2-4*I*arctan(c*x)*ln(c*x)+2*I*arct
an(c*x)*ln(c^2*x^2+1)+I*arctan(c*x)^2/c/x-I*arctan(c*x)^2/c^3/x^3)+2*a*d^3*b*c^4*(-1/4*arctan(c*x)/c^4/x^4-I*a
rctan(c*x)/c^3/x^3+I*arctan(c*x)/c/x+3/2/c^2/x^2*arctan(c*x)-1/2*I/c^2/x^2-2*I*ln(c*x)-1/12/c^3/x^3+7/4/c/x+I*
ln(c^2*x^2+1)+7/4*arctan(c*x))

Fricas [F]

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

[In]

integrate((d+I*c*d*x)^3*(a+b*arctan(c*x))^2/x^5,x, algorithm="fricas")

[Out]

1/16*(16*x^4*integral(1/4*(-4*I*a^2*c^5*d^3*x^5 - 12*a^2*c^4*d^3*x^4 + 8*I*a^2*c^3*d^3*x^3 - 8*a^2*c^2*d^3*x^2
+ 12*I*a^2*c*d^3*x + 4*a^2*d^3 + (4*a*b*c^5*d^3*x^5 - 4*(3*I*a*b - b^2)*c^4*d^3*x^4 - 2*(4*a*b + 3*I*b^2)*c^3
*d^3*x^3 - 4*(2*I*a*b + b^2)*c^2*d^3*x^2 - (12*a*b - I*b^2)*c*d^3*x + 4*I*a*b*d^3)*log(-(c*x + I)/(c*x - I)))/
(c^2*x^7 + x^5), x) + (-4*I*b^2*c^3*d^3*x^3 - 6*b^2*c^2*d^3*x^2 + 4*I*b^2*c*d^3*x + b^2*d^3)*log(-(c*x + I)/(c
*x - I))^2)/x^4

Sympy [F(-1)]

Timed out. \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^5} \, dx=\text {Timed out} \]

[In]

integrate((d+I*c*d*x)**3*(a+b*atan(c*x))**2/x**5,x)

[Out]

Timed out

Maxima [F]

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

[In]

integrate((d+I*c*d*x)^3*(a+b*arctan(c*x))^2/x^5,x, algorithm="maxima")

[Out]

I*(c*(log(c^2*x^2 + 1) - log(x^2)) + 2*arctan(c*x)/x)*a*b*c^3*d^3 + 3*((c*arctan(c*x) + 1/x)*c + arctan(c*x)/x
^2)*a*b*c^2*d^3 + I*((c^2*log(c^2*x^2 + 1) - c^2*log(x^2) - 1/x^2)*c - 2*arctan(c*x)/x^3)*a*b*c*d^3 + I*a^2*c^
3*d^3/x + 1/6*((3*c^3*arctan(c*x) + (3*c^2*x^2 - 1)/x^3)*c - 3*arctan(c*x)/x^4)*a*b*d^3 + 1/12*(2*(3*c^3*arcta
n(c*x) + (3*c^2*x^2 - 1)/x^3)*c*arctan(c*x) - (3*c^2*x^2*arctan(c*x)^2 - 4*c^2*x^2*log(c^2*x^2 + 1) + 8*c^2*x^
2*log(x) + 1)*c^2/x^2)*b^2*d^3 + 3/2*a^2*c^2*d^3/x^2 - I*a^2*c*d^3/x^3 - 1/4*b^2*d^3*arctan(c*x)^2/x^4 - 1/4*a
^2*d^3/x^4 - 1/32*(8*I*(b^2*c^4*d^3*arctan(c*x)^3 + 4*b^2*c^5*d^3*integrate(1/16*x^4*log(c^2*x^2 + 1)^2/(c^2*x
^6 + x^4), x) - 16*b^2*c^5*d^3*integrate(1/16*x^4*log(c^2*x^2 + 1)/(c^2*x^6 + x^4), x) - 48*b^2*c^4*d^3*integr
ate(1/16*x^3*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^6 + x^4), x) + 80*b^2*c^4*d^3*integrate(1/16*x^3*arctan(c*x)/
(c^2*x^6 + x^4), x) - 96*b^2*c^3*d^3*integrate(1/16*x^2*arctan(c*x)^2/(c^2*x^6 + x^4), x) - 8*b^2*c^3*d^3*inte
grate(1/16*x^2*log(c^2*x^2 + 1)^2/(c^2*x^6 + x^4), x) + 40*b^2*c^3*d^3*integrate(1/16*x^2*log(c^2*x^2 + 1)/(c^
2*x^6 + x^4), x) - 48*b^2*c^2*d^3*integrate(1/16*x*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^6 + x^4), x) - 32*b^2*c
^2*d^3*integrate(1/16*x*arctan(c*x)/(c^2*x^6 + x^4), x) - 144*b^2*c*d^3*integrate(1/16*arctan(c*x)^2/(c^2*x^6
+ x^4), x) - 12*b^2*c*d^3*integrate(1/16*log(c^2*x^2 + 1)^2/(c^2*x^6 + x^4), x))*x^3 - 8*(b^2*c^4*d^3*arctan(c
*x)^2 - 16*b^2*c^5*d^3*integrate(1/16*x^4*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^6 + x^4), x) - 144*b^2*c^4*d^3*i
ntegrate(1/16*x^3*arctan(c*x)^2/(c^2*x^6 + x^4), x) - 12*b^2*c^4*d^3*integrate(1/16*x^3*log(c^2*x^2 + 1)^2/(c^
2*x^6 + x^4), x) + 40*b^2*c^4*d^3*integrate(1/16*x^3*log(c^2*x^2 + 1)/(c^2*x^6 + x^4), x) + 32*b^2*c^3*d^3*int
egrate(1/16*x^2*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^6 + x^4), x) - 80*b^2*c^3*d^3*integrate(1/16*x^2*arctan(c*
x)/(c^2*x^6 + x^4), x) - 144*b^2*c^2*d^3*integrate(1/16*x*arctan(c*x)^2/(c^2*x^6 + x^4), x) - 12*b^2*c^2*d^3*i
ntegrate(1/16*x*log(c^2*x^2 + 1)^2/(c^2*x^6 + x^4), x) - 16*b^2*c^2*d^3*integrate(1/16*x*log(c^2*x^2 + 1)/(c^2
*x^6 + x^4), x) + 48*b^2*c*d^3*integrate(1/16*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^6 + x^4), x))*x^3 - 4*(2*I*b
^2*c^3*d^3*x^2 + 3*b^2*c^2*d^3*x - 2*I*b^2*c*d^3)*arctan(c*x)^2 + 4*(2*b^2*c^3*d^3*x^2 - 3*I*b^2*c^2*d^3*x - 2
*b^2*c*d^3)*arctan(c*x)*log(c^2*x^2 + 1) + (2*I*b^2*c^3*d^3*x^2 + 3*b^2*c^2*d^3*x - 2*I*b^2*c*d^3)*log(c^2*x^2
+ 1)^2)/x^3

Giac [F(-1)]

Timed out. \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^5} \, dx=\text {Timed out} \]

[In]

integrate((d+I*c*d*x)^3*(a+b*arctan(c*x))^2/x^5,x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^5} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^3}{x^5} \,d x \]

[In]

int(((a + b*atan(c*x))^2*(d + c*d*x*1i)^3)/x^5,x)

[Out]

int(((a + b*atan(c*x))^2*(d + c*d*x*1i)^3)/x^5, x)