\(\int \frac {(d+i c d x) (a+b \arctan (c x))^2}{x^4} \, dx\) [75]

Optimal result

Integrand size = 23, antiderivative size = 224 \[ \int \frac {(d+i c d x) (a+b \arctan (c x))^2}{x^4} \, dx=-\frac {b^2 c^2 d}{3 x}-\frac {1}{3} b^2 c^3 d \arctan (c x)-\frac {b c d (a+b \arctan (c x))}{3 x^2}-\frac {i b c^2 d (a+b \arctan (c x))}{x}-\frac {1}{6} i c^3 d (a+b \arctan (c x))^2-\frac {d (a+b \arctan (c x))^2}{3 x^3}-\frac {i c d (a+b \arctan (c x))^2}{2 x^2}+i b^2 c^3 d \log (x)-\frac {1}{2} i b^2 c^3 d \log \left (1+c^2 x^2\right )-\frac {2}{3} b c^3 d (a+b \arctan (c x)) \log \left (2-\frac {2}{1-i c x}\right )+\frac {1}{3} i b^2 c^3 d \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i c x}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {4996, 4946, 5038, 331, 209, 5044, 4988, 2497, 272, 36, 29, 31, 5004} \[ \int \frac {(d+i c d x) (a+b \arctan (c x))^2}{x^4} \, dx=-\frac {1}{6} i c^3 d (a+b \arctan (c x))^2-\frac {2}{3} b c^3 d \log \left (2-\frac {2}{1-i c x}\right ) (a+b \arctan (c x))-\frac {i b c^2 d (a+b \arctan (c x))}{x}-\frac {d (a+b \arctan (c x))^2}{3 x^3}-\frac {i c d (a+b \arctan (c x))^2}{2 x^2}-\frac {b c d (a+b \arctan (c x))}{3 x^2}-\frac {1}{3} b^2 c^3 d \arctan (c x)+\frac {1}{3} i b^2 c^3 d \operatorname {PolyLog}\left (2,\frac {2}{1-i c x}-1\right )+i b^2 c^3 d \log (x)-\frac {b^2 c^2 d}{3 x}-\frac {1}{2} i b^2 c^3 d \log \left (c^2 x^2+1\right ) \]

[In]

Int[((d + I*c*d*x)*(a + b*ArcTan[c*x])^2)/x^4,x]

[Out]

-1/3*(b^2*c^2*d)/x - (b^2*c^3*d*ArcTan[c*x])/3 - (b*c*d*(a + b*ArcTan[c*x]))/(3*x^2) - (I*b*c^2*d*(a + b*ArcTa
n[c*x]))/x - (I/6)*c^3*d*(a + b*ArcTan[c*x])^2 - (d*(a + b*ArcTan[c*x])^2)/(3*x^3) - ((I/2)*c*d*(a + b*ArcTan[
c*x])^2)/x^2 + I*b^2*c^3*d*Log[x] - (I/2)*b^2*c^3*d*Log[1 + c^2*x^2] - (2*b*c^3*d*(a + b*ArcTan[c*x])*Log[2 -
2/(1 - I*c*x)])/3 + (I/3)*b^2*c^3*d*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 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 2497

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/D[u, x])]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, 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 4988

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x_Symbol] :> Simp[(a + b*ArcTan[c*x])
^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] - Dist[b*c*(p/d), Int[(a + b*ArcTan[c*x])^(p - 1)*(Log[2 - 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 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])

Rule 5004

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTan[c*x])^(p +
1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && NeQ[p, -1]

Rule 5038

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

Rule 5044

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

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

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.07 \[ \int \frac {(d+i c d x) (a+b \arctan (c x))^2}{x^4} \, dx=\frac {d \left (-2 a^2-3 i a^2 c x-2 a b c x-6 i a b c^2 x^2-2 b^2 c^2 x^2-i b^2 \left (-2 i+3 c x+c^3 x^3\right ) \arctan (c x)^2-2 b \arctan (c x) \left (b c x \left (1+3 i c x+c^2 x^2\right )+a \left (2+3 i c x+3 i c^3 x^3\right )+2 b c^3 x^3 \log \left (1-e^{2 i \arctan (c x)}\right )\right )-4 a b c^3 x^3 \log (c x)+6 i b^2 c^3 x^3 \log \left (\frac {c x}{\sqrt {1+c^2 x^2}}\right )+2 a b c^3 x^3 \log \left (1+c^2 x^2\right )+2 i b^2 c^3 x^3 \operatorname {PolyLog}\left (2,e^{2 i \arctan (c x)}\right )\right )}{6 x^3} \]

[In]

Integrate[((d + I*c*d*x)*(a + b*ArcTan[c*x])^2)/x^4,x]

[Out]

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

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 407 vs. \(2 (198 ) = 396\).

Time = 3.14 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.82

method	result	size
parts	\(a^{2} d \left (-\frac {i c}{2 x^{2}}-\frac {1}{3 x^{3}}\right )+d \,b^{2} c^{3} \left (-\frac {\arctan \left (c x \right )^{2}}{3 c^{3} x^{3}}-\frac {i \ln \left (c^{2} x^{2}+1\right )}{2}+i \ln \left (c x \right )-\frac {\arctan \left (c x \right )}{3 c^{2} x^{2}}-\frac {2 \arctan \left (c x \right ) \ln \left (c x \right )}{3}+\frac {\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{3}+\frac {i \ln \left (c x \right ) \ln \left (-i c x +1\right )}{3}-\frac {i \arctan \left (c x \right )}{c x}-\frac {i \ln \left (c x \right ) \ln \left (i c x +1\right )}{3}-\frac {1}{3 c x}-\frac {i \arctan \left (c x \right )^{2}}{2 c^{2} x^{2}}-\frac {\arctan \left (c x \right )}{3}+\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{6}-\frac {i \operatorname {dilog}\left (i c x +1\right )}{3}-\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{6}-\frac {i \arctan \left (c x \right )^{2}}{2}+\frac {i \operatorname {dilog}\left (-i c x +1\right )}{3}\right )+2 a b d \,c^{3} \left (-\frac {\arctan \left (c x \right )}{3 c^{3} x^{3}}-\frac {i \arctan \left (c x \right )}{2 c^{2} x^{2}}-\frac {i}{2 c x}-\frac {1}{6 c^{2} x^{2}}-\frac {\ln \left (c x \right )}{3}+\frac {\ln \left (c^{2} x^{2}+1\right )}{6}-\frac {i \arctan \left (c x \right )}{2}\right )\)	\(408\)
derivativedivides	\(c^{3} \left (a^{2} d \left (-\frac {1}{3 c^{3} x^{3}}-\frac {i}{2 c^{2} x^{2}}\right )+d \,b^{2} \left (-\frac {\arctan \left (c x \right )^{2}}{3 c^{3} x^{3}}-\frac {i \ln \left (c^{2} x^{2}+1\right )}{2}+i \ln \left (c x \right )-\frac {\arctan \left (c x \right )}{3 c^{2} x^{2}}-\frac {2 \arctan \left (c x \right ) \ln \left (c x \right )}{3}+\frac {\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{3}+\frac {i \ln \left (c x \right ) \ln \left (-i c x +1\right )}{3}-\frac {i \arctan \left (c x \right )}{c x}-\frac {i \ln \left (c x \right ) \ln \left (i c x +1\right )}{3}-\frac {1}{3 c x}-\frac {i \arctan \left (c x \right )^{2}}{2 c^{2} x^{2}}-\frac {\arctan \left (c x \right )}{3}+\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{6}-\frac {i \operatorname {dilog}\left (i c x +1\right )}{3}-\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{6}-\frac {i \arctan \left (c x \right )^{2}}{2}+\frac {i \operatorname {dilog}\left (-i c x +1\right )}{3}\right )+2 a b d \left (-\frac {\arctan \left (c x \right )}{3 c^{3} x^{3}}-\frac {i \arctan \left (c x \right )}{2 c^{2} x^{2}}-\frac {i}{2 c x}-\frac {1}{6 c^{2} x^{2}}-\frac {\ln \left (c x \right )}{3}+\frac {\ln \left (c^{2} x^{2}+1\right )}{6}-\frac {i \arctan \left (c x \right )}{2}\right )\right )\)	\(411\)
default	\(c^{3} \left (a^{2} d \left (-\frac {1}{3 c^{3} x^{3}}-\frac {i}{2 c^{2} x^{2}}\right )+d \,b^{2} \left (-\frac {\arctan \left (c x \right )^{2}}{3 c^{3} x^{3}}-\frac {i \ln \left (c^{2} x^{2}+1\right )}{2}+i \ln \left (c x \right )-\frac {\arctan \left (c x \right )}{3 c^{2} x^{2}}-\frac {2 \arctan \left (c x \right ) \ln \left (c x \right )}{3}+\frac {\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{3}+\frac {i \ln \left (c x \right ) \ln \left (-i c x +1\right )}{3}-\frac {i \arctan \left (c x \right )}{c x}-\frac {i \ln \left (c x \right ) \ln \left (i c x +1\right )}{3}-\frac {1}{3 c x}-\frac {i \arctan \left (c x \right )^{2}}{2 c^{2} x^{2}}-\frac {\arctan \left (c x \right )}{3}+\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{6}-\frac {i \operatorname {dilog}\left (i c x +1\right )}{3}-\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{6}-\frac {i \arctan \left (c x \right )^{2}}{2}+\frac {i \operatorname {dilog}\left (-i c x +1\right )}{3}\right )+2 a b d \left (-\frac {\arctan \left (c x \right )}{3 c^{3} x^{3}}-\frac {i \arctan \left (c x \right )}{2 c^{2} x^{2}}-\frac {i}{2 c x}-\frac {1}{6 c^{2} x^{2}}-\frac {\ln \left (c x \right )}{3}+\frac {\ln \left (c^{2} x^{2}+1\right )}{6}-\frac {i \arctan \left (c x \right )}{2}\right )\right )\)	\(411\)

[In]

int((d+I*c*d*x)*(a+b*arctan(c*x))^2/x^4,x,method=_RETURNVERBOSE)

[Out]

a^2*d*(-1/2*I*c/x^2-1/3/x^3)+d*b^2*c^3*(-1/3*arctan(c*x)^2/c^3/x^3-1/2*I*ln(c^2*x^2+1)+I*ln(c*x)-1/3/c^2/x^2*a
rctan(c*x)-2/3*arctan(c*x)*ln(c*x)+1/3*arctan(c*x)*ln(c^2*x^2+1)+1/3*I*ln(c*x)*ln(1-I*c*x)-I*arctan(c*x)/c/x-1
/3*I*ln(c*x)*ln(1+I*c*x)-1/3/c/x-1/2*I*arctan(c*x)^2/c^2/x^2-1/3*arctan(c*x)+1/6*I*(ln(c*x-I)*ln(c^2*x^2+1)-di
log(-1/2*I*(c*x+I))-ln(c*x-I)*ln(-1/2*I*(c*x+I))-1/2*ln(c*x-I)^2)-1/3*I*dilog(1+I*c*x)-1/6*I*(ln(c*x+I)*ln(c^2
*x^2+1)-dilog(1/2*I*(c*x-I))-ln(c*x+I)*ln(1/2*I*(c*x-I))-1/2*ln(c*x+I)^2)-1/2*I*arctan(c*x)^2+1/3*I*dilog(1-I*
c*x))+2*a*b*d*c^3*(-1/3*arctan(c*x)/c^3/x^3-1/2*I*arctan(c*x)/c^2/x^2-1/2*I/c/x-1/6/c^2/x^2-1/3*ln(c*x)+1/6*ln
(c^2*x^2+1)-1/2*I*arctan(c*x))

Fricas [F]

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

[In]

integrate((d+I*c*d*x)*(a+b*arctan(c*x))^2/x^4,x, algorithm="fricas")

[Out]

1/24*(24*x^3*integral(1/6*(6*I*a^2*c^3*d*x^3 + 6*a^2*c^2*d*x^2 + 6*I*a^2*c*d*x + 6*a^2*d - (6*a*b*c^3*d*x^3 +
3*(-2*I*a*b + b^2)*c^2*d*x^2 + 2*(3*a*b - I*b^2)*c*d*x - 6*I*a*b*d)*log(-(c*x + I)/(c*x - I)))/(c^2*x^6 + x^4)
, x) + (3*I*b^2*c*d*x + 2*b^2*d)*log(-(c*x + I)/(c*x - I))^2)/x^3

Sympy [F(-1)]

Timed out. \[ \int \frac {(d+i c d x) (a+b \arctan (c x))^2}{x^4} \, dx=\text {Timed out} \]

[In]

integrate((d+I*c*d*x)*(a+b*atan(c*x))**2/x**4,x)

[Out]

Timed out

Maxima [F]

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

[In]

integrate((d+I*c*d*x)*(a+b*arctan(c*x))^2/x^4,x, algorithm="maxima")

[Out]

-I*((c*arctan(c*x) + 1/x)*c + arctan(c*x)/x^2)*a*b*c*d + 1/3*((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*d - 1/2*I*a^2*c*d/x^2 - 1/3*a^2*d/x^3 + 1/96*(96*I*x^3*integrate(1/48*(20*b^2*c^2*d*x
^2*arctan(c*x) + 36*(b^2*c^3*d*x^3 + b^2*c*d*x)*arctan(c*x)^2 + 3*(b^2*c^3*d*x^3 + b^2*c*d*x)*log(c^2*x^2 + 1)
^2 - 2*(3*b^2*c^3*d*x^3 - 2*b^2*c*d*x + 6*(b^2*c^2*d*x^2 + b^2*d)*arctan(c*x))*log(c^2*x^2 + 1))/(c^2*x^6 + x^
4), x) + 96*x^3*integrate(1/48*(36*(b^2*c^2*d*x^2 + b^2*d)*arctan(c*x)^2 + 3*(b^2*c^2*d*x^2 + b^2*d)*log(c^2*x
^2 + 1)^2 - 4*(3*b^2*c^3*d*x^3 - 2*b^2*c*d*x)*arctan(c*x) - 2*(5*b^2*c^2*d*x^2 - 6*(b^2*c^3*d*x^3 + b^2*c*d*x)
*arctan(c*x))*log(c^2*x^2 + 1))/(c^2*x^6 + x^4), x) - 4*(3*I*b^2*c*d*x + 2*b^2*d)*arctan(c*x)^2 + 4*(3*b^2*c*d
*x - 2*I*b^2*d)*arctan(c*x)*log(c^2*x^2 + 1) + (3*I*b^2*c*d*x + 2*b^2*d)*log(c^2*x^2 + 1)^2)/x^3

Giac [F(-1)]

Timed out. \[ \int \frac {(d+i c d x) (a+b \arctan (c x))^2}{x^4} \, dx=\text {Timed out} \]

[In]

integrate((d+I*c*d*x)*(a+b*arctan(c*x))^2/x^4,x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+i c d x) (a+b \arctan (c x))^2}{x^4} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2\,\left (d+c\,d\,x\,1{}\mathrm {i}\right )}{x^4} \,d x \]

[In]

int(((a + b*atan(c*x))^2*(d + c*d*x*1i))/x^4,x)

[Out]

int(((a + b*atan(c*x))^2*(d + c*d*x*1i))/x^4, x)