\(\int \frac {x^2 \arctan (a x)^3}{c+a^2 c x^2} \, dx\) [389]

Optimal result

Integrand size = 22, antiderivative size = 130 \[ \int \frac {x^2 \arctan (a x)^3}{c+a^2 c x^2} \, dx=\frac {i \arctan (a x)^3}{a^3 c}+\frac {x \arctan (a x)^3}{a^2 c}-\frac {\arctan (a x)^4}{4 a^3 c}+\frac {3 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a^3 c}+\frac {3 i \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{a^3 c}+\frac {3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{2 a^3 c} \]

[Out]

I*arctan(a*x)^3/a^3/c+x*arctan(a*x)^3/a^2/c-1/4*arctan(a*x)^4/a^3/c+3*arctan(a*x)^2*ln(2/(1+I*a*x))/a^3/c+3*I*
arctan(a*x)*polylog(2,1-2/(1+I*a*x))/a^3/c+3/2*polylog(3,1-2/(1+I*a*x))/a^3/c

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {5036, 4930, 5040, 4964, 5004, 5114, 6745} \[ \int \frac {x^2 \arctan (a x)^3}{c+a^2 c x^2} \, dx=\frac {3 i \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{a^3 c}-\frac {\arctan (a x)^4}{4 a^3 c}+\frac {i \arctan (a x)^3}{a^3 c}+\frac {3 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a^3 c}+\frac {3 \operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right )}{2 a^3 c}+\frac {x \arctan (a x)^3}{a^2 c} \]

[In]

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

[Out]

(I*ArcTan[a*x]^3)/(a^3*c) + (x*ArcTan[a*x]^3)/(a^2*c) - ArcTan[a*x]^4/(4*a^3*c) + (3*ArcTan[a*x]^2*Log[2/(1 +
I*a*x)])/(a^3*c) + ((3*I)*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)])/(a^3*c) + (3*PolyLog[3, 1 - 2/(1 + I*a*x)
])/(2*a^3*c)

Rule 4930

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcTan[c*x^n])^p, x] - Dist[b*c
*n*p, Int[x^n*((a + b*ArcTan[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0
] && (EqQ[n, 1] || EqQ[p, 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 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 5036

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

Rule 5040

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*e*(p + 1))), x] - Dist[1/(c*d), Int[(a + b*ArcTan[c*x])^p/(I - c*x), x], x] /; FreeQ[{a, b
, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]

Rule 5114

Int[(Log[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-I)*(a + b*Ar
cTan[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 {\int \frac {\arctan (a x)^3}{c+a^2 c x^2} \, dx}{a^2}+\frac {\int \arctan (a x)^3 \, dx}{a^2 c} \\ & = \frac {x \arctan (a x)^3}{a^2 c}-\frac {\arctan (a x)^4}{4 a^3 c}-\frac {3 \int \frac {x \arctan (a x)^2}{1+a^2 x^2} \, dx}{a c} \\ & = \frac {i \arctan (a x)^3}{a^3 c}+\frac {x \arctan (a x)^3}{a^2 c}-\frac {\arctan (a x)^4}{4 a^3 c}+\frac {3 \int \frac {\arctan (a x)^2}{i-a x} \, dx}{a^2 c} \\ & = \frac {i \arctan (a x)^3}{a^3 c}+\frac {x \arctan (a x)^3}{a^2 c}-\frac {\arctan (a x)^4}{4 a^3 c}+\frac {3 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a^3 c}-\frac {6 \int \frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a^2 c} \\ & = \frac {i \arctan (a x)^3}{a^3 c}+\frac {x \arctan (a x)^3}{a^2 c}-\frac {\arctan (a x)^4}{4 a^3 c}+\frac {3 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a^3 c}+\frac {3 i \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{a^3 c}-\frac {(3 i) \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a^2 c} \\ & = \frac {i \arctan (a x)^3}{a^3 c}+\frac {x \arctan (a x)^3}{a^2 c}-\frac {\arctan (a x)^4}{4 a^3 c}+\frac {3 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a^3 c}+\frac {3 i \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{a^3 c}+\frac {3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{2 a^3 c} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.72 \[ \int \frac {x^2 \arctan (a x)^3}{c+a^2 c x^2} \, dx=\frac {-\frac {1}{4} \arctan (a x)^2 \left ((4 i-4 a x) \arctan (a x)+\arctan (a x)^2-12 \log \left (1+e^{2 i \arctan (a x)}\right )\right )-3 i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )+\frac {3}{2} \operatorname {PolyLog}\left (3,-e^{2 i \arctan (a x)}\right )}{a^3 c} \]

[In]

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

[Out]

(-1/4*(ArcTan[a*x]^2*((4*I - 4*a*x)*ArcTan[a*x] + ArcTan[a*x]^2 - 12*Log[1 + E^((2*I)*ArcTan[a*x])])) - (3*I)*
ArcTan[a*x]*PolyLog[2, -E^((2*I)*ArcTan[a*x])] + (3*PolyLog[3, -E^((2*I)*ArcTan[a*x])])/2)/(a^3*c)

Maple [C] (warning: unable to verify)

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

Time = 14.14 (sec) , antiderivative size = 785, normalized size of antiderivative = 6.04

method	result	size
derivativedivides	\(\frac {\frac {\arctan \left (a x \right )^{3} a x}{c}-\frac {\arctan \left (a x \right )^{4}}{c}-\frac {3 \left (\frac {\arctan \left (a x \right )^{2} \ln \left (a^{2} x^{2}+1\right )}{2}-\arctan \left (a x \right )^{2} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+\frac {i \arctan \left (a x \right )^{3}}{3}-\frac {\left (-i \pi \,\operatorname {csgn}\left (\frac {i}{\left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )+i \pi \,\operatorname {csgn}\left (\frac {i}{\left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )^{2}+i \pi {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )\right )}^{2} \operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )-2 i \pi \,\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )\right ) {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )}^{2}+i \pi {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )}^{3}-i \pi {\operatorname {csgn}\left (\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )}^{2} \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )+2 i \pi \,\operatorname {csgn}\left (\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )^{2}-i \pi \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )^{3}+i \pi \,\operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )^{2}-i \pi \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )^{3}+4 \ln \left (2\right )\right ) \arctan \left (a x \right )^{2}}{4}+i \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )-\frac {\operatorname {polylog}\left (3, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}-\frac {\arctan \left (a x \right )^{4}}{4}\right )}{c}}{a^{3}}\)	\(785\)
default	\(\frac {\frac {\arctan \left (a x \right )^{3} a x}{c}-\frac {\arctan \left (a x \right )^{4}}{c}-\frac {3 \left (\frac {\arctan \left (a x \right )^{2} \ln \left (a^{2} x^{2}+1\right )}{2}-\arctan \left (a x \right )^{2} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+\frac {i \arctan \left (a x \right )^{3}}{3}-\frac {\left (-i \pi \,\operatorname {csgn}\left (\frac {i}{\left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )+i \pi \,\operatorname {csgn}\left (\frac {i}{\left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )^{2}+i \pi {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )\right )}^{2} \operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )-2 i \pi \,\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )\right ) {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )}^{2}+i \pi {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )}^{3}-i \pi {\operatorname {csgn}\left (\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )}^{2} \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )+2 i \pi \,\operatorname {csgn}\left (\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )^{2}-i \pi \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )^{3}+i \pi \,\operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )^{2}-i \pi \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )^{3}+4 \ln \left (2\right )\right ) \arctan \left (a x \right )^{2}}{4}+i \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )-\frac {\operatorname {polylog}\left (3, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}-\frac {\arctan \left (a x \right )^{4}}{4}\right )}{c}}{a^{3}}\)	\(785\)
parts	\(\frac {x \arctan \left (a x \right )^{3}}{a^{2} c}-\frac {\arctan \left (a x \right )^{4}}{a^{3} c}-\frac {3 \left (\frac {\frac {\arctan \left (a x \right )^{2} \ln \left (a^{2} x^{2}+1\right )}{2}-\arctan \left (a x \right )^{2} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+\frac {i \arctan \left (a x \right )^{3}}{3}-\frac {\left (-i \pi \,\operatorname {csgn}\left (\frac {i}{\left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )+i \pi \,\operatorname {csgn}\left (\frac {i}{\left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )^{2}+i \pi {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )\right )}^{2} \operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )-2 i \pi \,\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )\right ) {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )}^{2}+i \pi {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )}^{3}-i \pi {\operatorname {csgn}\left (\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )}^{2} \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )+2 i \pi \,\operatorname {csgn}\left (\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )^{2}-i \pi \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )^{3}+i \pi \,\operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )^{2}-i \pi \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )^{3}+4 \ln \left (2\right )\right ) \arctan \left (a x \right )^{2}}{4}+i \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )-\frac {\operatorname {polylog}\left (3, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}}{a^{3}}-\frac {\arctan \left (a x \right )^{4}}{4 a^{3}}\right )}{c}\)	\(794\)

[In]

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

[Out]

1/a^3*(1/c*arctan(a*x)^3*a*x-1/c*arctan(a*x)^4-3/c*(1/2*arctan(a*x)^2*ln(a^2*x^2+1)-arctan(a*x)^2*ln((1+I*a*x)
/(a^2*x^2+1)^(1/2))+1/3*I*arctan(a*x)^3-1/4*(-I*Pi*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)*csgn(I*(1+I*a*x)^2/(a
^2*x^2+1))*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)+I*Pi*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+
1)^2)*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^2+I*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1))
^2*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)-2*I*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1))*csgn(I*((1+I*a*x)^2/(a^2*x
^2+1)+1)^2)^2+I*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^3-I*Pi*csgn(I*(1+I*a*x)/(a^2*x^2+1)^(1/2))^2*csgn(I*(
1+I*a*x)^2/(a^2*x^2+1))+2*I*Pi*csgn(I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))^2-I*Pi*csgn
(I*(1+I*a*x)^2/(a^2*x^2+1))^3+I*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2
/(a^2*x^2+1)+1)^2)^2-I*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^3+4*ln(2))*arctan(a*x)
^2+I*arctan(a*x)*polylog(2,-(1+I*a*x)^2/(a^2*x^2+1))-1/2*polylog(3,-(1+I*a*x)^2/(a^2*x^2+1))-1/4*arctan(a*x)^4
))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {x^2 \arctan (a x)^3}{c+a^2 c x^2} \, dx=\frac {\int \frac {x^{2} \operatorname {atan}^{3}{\left (a x \right )}}{a^{2} x^{2} + 1}\, dx}{c} \]

[In]

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

[Out]

Integral(x**2*atan(a*x)**3/(a**2*x**2 + 1), x)/c

Maxima [F]

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

[In]

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

[Out]

1/1024*(16*(7168*a^2*integrate(1/128*x^2*arctan(a*x)^3/(a^4*c*x^2 + a^2*c), x) + 768*a^2*integrate(1/128*x^2*a
rctan(a*x)*log(a^2*x^2 + 1)^2/(a^4*c*x^2 + a^2*c), x) + 3072*a^2*integrate(1/128*x^2*arctan(a*x)*log(a^2*x^2 +
1)/(a^4*c*x^2 + a^2*c), x) - 768*a*integrate(1/128*x*arctan(a*x)^2*log(a^2*x^2 + 1)/(a^4*c*x^2 + a^2*c), x) -
192*a*integrate(1/128*x*log(a^2*x^2 + 1)^3/(a^4*c*x^2 + a^2*c), x) - 3072*a*integrate(1/128*x*arctan(a*x)^2/(
a^4*c*x^2 + a^2*c), x) + 768*a*integrate(1/128*x*log(a^2*x^2 + 1)^2/(a^4*c*x^2 + a^2*c), x) + 3*arctan(a*x)^4/
(a^3*c) + 384*integrate(1/128*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^4*c*x^2 + a^2*c), x))*a^3*c + 128*a*x*arctan(a
*x)^3 - 80*arctan(a*x)^4 + 3*log(a^2*x^2 + 1)^4 - 24*(4*a*x*arctan(a*x) - arctan(a*x)^2)*log(a^2*x^2 + 1)^2)/(
a^3*c)

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

int((x^2*atan(a*x)^3)/(c + a^2*c*x^2),x)

[Out]

int((x^2*atan(a*x)^3)/(c + a^2*c*x^2), x)