\(\int \frac {(a+b \arctan (c x))^3}{x^4} \, dx\) [33]

Optimal result

Integrand size = 14, antiderivative size = 213 \[ \int \frac {(a+b \arctan (c x))^3}{x^4} \, dx=-\frac {b^2 c^2 (a+b \arctan (c x))}{x}-\frac {1}{2} b c^3 (a+b \arctan (c x))^2-\frac {b c (a+b \arctan (c x))^2}{2 x^2}+\frac {1}{3} i c^3 (a+b \arctan (c x))^3-\frac {(a+b \arctan (c x))^3}{3 x^3}+b^3 c^3 \log (x)-\frac {1}{2} b^3 c^3 \log \left (1+c^2 x^2\right )-b c^3 (a+b \arctan (c x))^2 \log \left (2-\frac {2}{1-i c x}\right )+i b^2 c^3 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i c x}\right )-\frac {1}{2} b^3 c^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i c x}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.786, Rules used = {4946, 5038, 272, 36, 29, 31, 5004, 5044, 4988, 5112, 6745} \[ \int \frac {(a+b \arctan (c x))^3}{x^4} \, dx=i b^2 c^3 \operatorname {PolyLog}\left (2,\frac {2}{1-i c x}-1\right ) (a+b \arctan (c x))-\frac {b^2 c^2 (a+b \arctan (c x))}{x}+\frac {1}{3} i c^3 (a+b \arctan (c x))^3-\frac {1}{2} b c^3 (a+b \arctan (c x))^2-b c^3 \log \left (2-\frac {2}{1-i c x}\right ) (a+b \arctan (c x))^2-\frac {(a+b \arctan (c x))^3}{3 x^3}-\frac {b c (a+b \arctan (c x))^2}{2 x^2}-\frac {1}{2} b^3 c^3 \operatorname {PolyLog}\left (3,\frac {2}{1-i c x}-1\right )+b^3 c^3 \log (x)-\frac {1}{2} b^3 c^3 \log \left (c^2 x^2+1\right ) \]

[In]

Int[(a + b*ArcTan[c*x])^3/x^4,x]

[Out]

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

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 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 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 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]

Rule 5112

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

Rule 6745

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /
;  !FalseQ[w]] /; FreeQ[n, x]

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

Mathematica [A] (verified)

Time = 1.38 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.43 \[ \int \frac {(a+b \arctan (c x))^3}{x^4} \, dx=\frac {1}{6} \left (-\frac {2 a^3}{x^3}-\frac {3 a^2 b c}{x^2}-\frac {6 a^2 b \arctan (c x)}{x^3}-6 a^2 b c^3 \log (x)+3 a^2 b c^3 \log \left (1+c^2 x^2\right )+\frac {6 i a b^2 \left (i c^2 x^2+\left (i+c^3 x^3\right ) \arctan (c x)^2+i c x \arctan (c x) \left (1+c^2 x^2+2 c^2 x^2 \log \left (1-e^{2 i \arctan (c x)}\right )\right )+c^3 x^3 \operatorname {PolyLog}\left (2,e^{2 i \arctan (c x)}\right )\right )}{x^3}+6 b^3 c^3 \left (-i \arctan (c x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (c x)}\right )+\frac {1}{24} \left (i \pi ^3-\frac {24 \arctan (c x)}{c x}+\left (-8 i-\frac {8}{c^3 x^3}\right ) \arctan (c x)^3+\arctan (c x)^2 \left (-12-\frac {12}{c^2 x^2}-24 \log \left (1-e^{-2 i \arctan (c x)}\right )\right )+24 \log (c x)-12 \log \left (1+c^2 x^2\right )-12 \operatorname {PolyLog}\left (3,e^{-2 i \arctan (c x)}\right )\right )\right )\right ) \]

[In]

Integrate[(a + b*ArcTan[c*x])^3/x^4,x]

[Out]

((-2*a^3)/x^3 - (3*a^2*b*c)/x^2 - (6*a^2*b*ArcTan[c*x])/x^3 - 6*a^2*b*c^3*Log[x] + 3*a^2*b*c^3*Log[1 + c^2*x^2
] + ((6*I)*a*b^2*(I*c^2*x^2 + (I + c^3*x^3)*ArcTan[c*x]^2 + I*c*x*ArcTan[c*x]*(1 + c^2*x^2 + 2*c^2*x^2*Log[1 -
E^((2*I)*ArcTan[c*x])]) + c^3*x^3*PolyLog[2, E^((2*I)*ArcTan[c*x])]))/x^3 + 6*b^3*c^3*((-I)*ArcTan[c*x]*PolyL
og[2, E^((-2*I)*ArcTan[c*x])] + (I*Pi^3 - (24*ArcTan[c*x])/(c*x) + (-8*I - 8/(c^3*x^3))*ArcTan[c*x]^3 + ArcTan
[c*x]^2*(-12 - 12/(c^2*x^2) - 24*Log[1 - E^((-2*I)*ArcTan[c*x])]) + 24*Log[c*x] - 12*Log[1 + c^2*x^2] - 12*Pol
yLog[3, E^((-2*I)*ArcTan[c*x])])/24))/6

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 6.80 (sec) , antiderivative size = 2097, normalized size of antiderivative = 9.85

\[\text {Expression too large to display}\]

[In]

int((a+b*arctan(c*x))^3/x^4,x)

[Out]

c^3*(-1/3/c^3/x^3*a^3+b^3*(-1/3/c^3/x^3*arctan(c*x)^3-1/2/c^2/x^2*arctan(c*x)^2-ln(c*x)*arctan(c*x)^2+1/2*arct
an(c*x)^2*ln(c^2*x^2+1)-arctan(c*x)^2*ln((1+I*c*x)/(c^2*x^2+1)^(1/2))+arctan(c*x)^2*ln((1+I*c*x)^2/(c^2*x^2+1)
-1)+1/12*arctan(c*x)*(3*I*arctan(c*x)*csgn(I*(1+I*c*x)^2/(c^2*x^2+1)/(1+(1+I*c*x)^2/(c^2*x^2+1))^2)*csgn(I/(1+
(1+I*c*x)^2/(c^2*x^2+1))^2)*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))*Pi*c*x-6*I*arctan(c*x)*csgn(I*((1+I*c*x)^2/(c^2*x^
2+1)-1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))*csgn(((1+I*c*x)^2/(c^2*x^2+1)-1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))*Pi*c*x-3*I*
arctan(c*x)*csgn(I*(1+(1+I*c*x)^2/(c^2*x^2+1))^2)^3*Pi*c*x-3*I*arctan(c*x)*csgn(I*(1+(1+I*c*x)^2/(c^2*x^2+1))^
2)*csgn(I*(1+(1+I*c*x)^2/(c^2*x^2+1)))^2*Pi*c*x-3*I*arctan(c*x)*csgn(I*(1+I*c*x)^2/(c^2*x^2+1)/(1+(1+I*c*x)^2/
(c^2*x^2+1))^2)^2*csgn(I/(1+(1+I*c*x)^2/(c^2*x^2+1))^2)*Pi*c*x+6*I*arctan(c*x)*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)
-1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))^2*csgn(I/(1+(1+I*c*x)^2/(c^2*x^2+1)))*Pi*c*x+3*I*arctan(c*x)*csgn(I*(1+I*c*x)
^2/(c^2*x^2+1)/(1+(1+I*c*x)^2/(c^2*x^2+1))^2)^3*Pi*c*x-6*I*arctan(c*x)*csgn(((1+I*c*x)^2/(c^2*x^2+1)-1)/(1+(1+
I*c*x)^2/(c^2*x^2+1)))^3*Pi*c*x-6*I*arctan(c*x)*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1)/(1+(1+I*c*x)^2/(c^2*x^2+1))
)^3*Pi*c*x-12*I*c*x-6*I*arctan(c*x)*Pi*c*x+6*I*arctan(c*x)*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1)/(1+(1+I*c*x)^2/(
c^2*x^2+1)))^2*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1))*Pi*c*x+3*I*arctan(c*x)*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))^3*Pi
*c*x-6*I*arctan(c*x)*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))*csgn(I/(1+(1+I*c*x)^2/(c^
2*x^2+1)))*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1))*Pi*c*x-3*I*arctan(c*x)*csgn(I*(1+I*c*x)^2/(c^2*x^2+1)/(1+(1+I*c
*x)^2/(c^2*x^2+1))^2)^2*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))*Pi*c*x+6*I*arctan(c*x)*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)
-1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))*csgn(((1+I*c*x)^2/(c^2*x^2+1)-1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))^2*Pi*c*x+3*I*ar
ctan(c*x)*csgn(I*(1+I*c*x)/(c^2*x^2+1)^(1/2))^2*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))*Pi*c*x+6*I*arctan(c*x)*csgn(((
1+I*c*x)^2/(c^2*x^2+1)-1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))^2*Pi*c*x+6*I*arctan(c*x)*csgn(I*(1+(1+I*c*x)^2/(c^2*x^2
+1))^2)^2*csgn(I*(1+(1+I*c*x)^2/(c^2*x^2+1)))*Pi*c*x+4*I*arctan(c*x)^2*c*x-12*arctan(c*x)*ln(2)*c*x-6*c*x*arct
an(c*x)-12-6*I*arctan(c*x)*csgn(I*(1+I*c*x)/(c^2*x^2+1)^(1/2))*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))^2*Pi*c*x)/c/x+l
n((1+I*c*x)/(c^2*x^2+1)^(1/2)-1)+ln(1+(1+I*c*x)/(c^2*x^2+1)^(1/2))-arctan(c*x)^2*ln(1-(1+I*c*x)/(c^2*x^2+1)^(1
/2))+2*I*arctan(c*x)*polylog(2,(1+I*c*x)/(c^2*x^2+1)^(1/2))-2*polylog(3,(1+I*c*x)/(c^2*x^2+1)^(1/2))-arctan(c*
x)^2*ln(1+(1+I*c*x)/(c^2*x^2+1)^(1/2))+2*I*arctan(c*x)*polylog(2,-(1+I*c*x)/(c^2*x^2+1)^(1/2))-2*polylog(3,-(1
+I*c*x)/(c^2*x^2+1)^(1/2)))+3*a*b^2*(-1/3/c^3/x^3*arctan(c*x)^2+1/3*arctan(c*x)*ln(c^2*x^2+1)-1/3/c^2/x^2*arct
an(c*x)-2/3*ln(c*x)*arctan(c*x)+1/6*I*(ln(c*x-I)*ln(c^2*x^2+1)-1/2*ln(c*x-I)^2-dilog(-1/2*I*(c*x+I))-ln(c*x-I)
*ln(-1/2*I*(c*x+I)))-1/6*I*(ln(c*x+I)*ln(c^2*x^2+1)-1/2*ln(c*x+I)^2-dilog(1/2*I*(c*x-I))-ln(c*x+I)*ln(1/2*I*(c
*x-I)))-1/3/c/x-1/3*arctan(c*x)-1/3*I*ln(c*x)*ln(1+I*c*x)+1/3*I*ln(c*x)*ln(1-I*c*x)-1/3*I*dilog(1+I*c*x)+1/3*I
*dilog(1-I*c*x))+3*a^2*b*(-1/3/c^3/x^3*arctan(c*x)+1/6*ln(c^2*x^2+1)-1/6/c^2/x^2-1/3*ln(c*x)))

Fricas [F]

\[ \int \frac {(a+b \arctan (c x))^3}{x^4} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{3}}{x^{4}} \,d x } \]

[In]

integrate((a+b*arctan(c*x))^3/x^4,x, algorithm="fricas")

[Out]

integral((b^3*arctan(c*x)^3 + 3*a*b^2*arctan(c*x)^2 + 3*a^2*b*arctan(c*x) + a^3)/x^4, x)

Sympy [F]

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

[In]

integrate((a+b*atan(c*x))**3/x**4,x)

[Out]

Integral((a + b*atan(c*x))**3/x**4, x)

Maxima [F]

\[ \int \frac {(a+b \arctan (c x))^3}{x^4} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{3}}{x^{4}} \,d x } \]

[In]

integrate((a+b*arctan(c*x))^3/x^4,x, algorithm="maxima")

[Out]

1/2*((c^2*log(c^2*x^2 + 1) - c^2*log(x^2) - 1/x^2)*c - 2*arctan(c*x)/x^3)*a^2*b - 1/3*a^3/x^3 - 1/96*(4*b^3*ar
ctan(c*x)^3 - 3*b^3*arctan(c*x)*log(c^2*x^2 + 1)^2 - 96*x^3*integrate(-1/32*(4*b^3*c^2*x^2*arctan(c*x)*log(c^2
*x^2 + 1) - 28*(b^3*c^2*x^2 + b^3)*arctan(c*x)^3 - 4*(24*a*b^2*c^2*x^2 + b^3*c*x + 24*a*b^2)*arctan(c*x)^2 + (
b^3*c*x - 3*(b^3*c^2*x^2 + b^3)*arctan(c*x))*log(c^2*x^2 + 1)^2)/(c^2*x^6 + x^4), x))/x^3

Giac [F(-1)]

Timed out. \[ \int \frac {(a+b \arctan (c x))^3}{x^4} \, dx=\text {Timed out} \]

[In]

integrate((a+b*arctan(c*x))^3/x^4,x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \arctan (c x))^3}{x^4} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^3}{x^4} \,d x \]

[In]

int((a + b*atan(c*x))^3/x^4,x)

[Out]

int((a + b*atan(c*x))^3/x^4, x)