\(\int \frac {x^3 \arctan (a x)^2}{c+a^2 c x^2} \, dx\) [283]

Optimal result

Integrand size = 22, antiderivative size = 169 \[ \int \frac {x^3 \arctan (a x)^2}{c+a^2 c x^2} \, dx=-\frac {x \arctan (a x)}{a^3 c}+\frac {\arctan (a x)^2}{2 a^4 c}+\frac {x^2 \arctan (a x)^2}{2 a^2 c}+\frac {i \arctan (a x)^3}{3 a^4 c}+\frac {\arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a^4 c}+\frac {\log \left (1+a^2 x^2\right )}{2 a^4 c}+\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{a^4 c}+\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{2 a^4 c} \]

[Out]

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

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {5036, 4946, 4930, 266, 5004, 5040, 4964, 5114, 6745} \[ \int \frac {x^3 \arctan (a x)^2}{c+a^2 c x^2} \, dx=\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{a^4 c}+\frac {i \arctan (a x)^3}{3 a^4 c}+\frac {\arctan (a x)^2}{2 a^4 c}+\frac {\arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a^4 c}+\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right )}{2 a^4 c}-\frac {x \arctan (a x)}{a^3 c}+\frac {x^2 \arctan (a x)^2}{2 a^2 c}+\frac {\log \left (a^2 x^2+1\right )}{2 a^4 c} \]

[In]

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

[Out]

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

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

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 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 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 {x \arctan (a x)^2}{c+a^2 c x^2} \, dx}{a^2}+\frac {\int x \arctan (a x)^2 \, dx}{a^2 c} \\ & = \frac {x^2 \arctan (a x)^2}{2 a^2 c}+\frac {i \arctan (a x)^3}{3 a^4 c}+\frac {\int \frac {\arctan (a x)^2}{i-a x} \, dx}{a^3 c}-\frac {\int \frac {x^2 \arctan (a x)}{1+a^2 x^2} \, dx}{a c} \\ & = \frac {x^2 \arctan (a x)^2}{2 a^2 c}+\frac {i \arctan (a x)^3}{3 a^4 c}+\frac {\arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a^4 c}-\frac {\int \arctan (a x) \, dx}{a^3 c}+\frac {\int \frac {\arctan (a x)}{1+a^2 x^2} \, dx}{a^3 c}-\frac {2 \int \frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a^3 c} \\ & = -\frac {x \arctan (a x)}{a^3 c}+\frac {\arctan (a x)^2}{2 a^4 c}+\frac {x^2 \arctan (a x)^2}{2 a^2 c}+\frac {i \arctan (a x)^3}{3 a^4 c}+\frac {\arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a^4 c}+\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{a^4 c}-\frac {i \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a^3 c}+\frac {\int \frac {x}{1+a^2 x^2} \, dx}{a^2 c} \\ & = -\frac {x \arctan (a x)}{a^3 c}+\frac {\arctan (a x)^2}{2 a^4 c}+\frac {x^2 \arctan (a x)^2}{2 a^2 c}+\frac {i \arctan (a x)^3}{3 a^4 c}+\frac {\arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a^4 c}+\frac {\log \left (1+a^2 x^2\right )}{2 a^4 c}+\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{a^4 c}+\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{2 a^4 c} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.73 \[ \int \frac {x^3 \arctan (a x)^2}{c+a^2 c x^2} \, dx=\frac {-a x \arctan (a x)+\frac {1}{2} \left (1+a^2 x^2\right ) \arctan (a x)^2-\frac {1}{3} i \arctan (a x)^3+\arctan (a x)^2 \log \left (1+e^{2 i \arctan (a x)}\right )-\log \left (\frac {1}{\sqrt {1+a^2 x^2}}\right )-i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,-e^{2 i \arctan (a x)}\right )}{a^4 c} \]

[In]

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

[Out]

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

Maple [C] (warning: unable to verify)

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

Time = 19.90 (sec) , antiderivative size = 844, normalized size of antiderivative = 4.99

method	result	size
derivativedivides	\(\frac {\frac {\arctan \left (a x \right )^{2} a^{2} x^{2}}{2 c}-\frac {\arctan \left (a x \right )^{2} \ln \left (a^{2} x^{2}+1\right )}{2 c}-\frac {-\arctan \left (a x \right )^{2} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+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 {i \arctan \left (a x \right ) \left (3 \arctan \left (a x \right ) \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}-3 \arctan \left (a x \right ) \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 )^{2} \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )-3 \arctan \left (a x \right ) \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 )^{2} \operatorname {csgn}\left (\frac {i}{\left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )+3 \arctan \left (a x \right ) \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 ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {i}{\left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )-3 \arctan \left (a x \right ) \pi {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )}^{3}+6 \arctan \left (a x \right ) \pi {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )}^{2} \operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )\right )-3 \arctan \left (a x \right ) \pi \,\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right ) {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )\right )}^{2}+3 \arctan \left (a x \right ) \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 )-6 \arctan \left (a x \right ) \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}+3 \arctan \left (a x \right ) \pi \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )^{3}+4 \arctan \left (a x \right )^{2}+12 i \arctan \left (a x \right ) \ln \left (2\right )+6 i \arctan \left (a x \right )-12-12 i a x \right )}{12}+\ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )}{c}}{a^{4}}\)	\(844\)
default	\(\frac {\frac {\arctan \left (a x \right )^{2} a^{2} x^{2}}{2 c}-\frac {\arctan \left (a x \right )^{2} \ln \left (a^{2} x^{2}+1\right )}{2 c}-\frac {-\arctan \left (a x \right )^{2} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+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 {i \arctan \left (a x \right ) \left (3 \arctan \left (a x \right ) \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}-3 \arctan \left (a x \right ) \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 )^{2} \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )-3 \arctan \left (a x \right ) \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 )^{2} \operatorname {csgn}\left (\frac {i}{\left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )+3 \arctan \left (a x \right ) \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 ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {i}{\left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )-3 \arctan \left (a x \right ) \pi {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )}^{3}+6 \arctan \left (a x \right ) \pi {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )}^{2} \operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )\right )-3 \arctan \left (a x \right ) \pi \,\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right ) {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )\right )}^{2}+3 \arctan \left (a x \right ) \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 )-6 \arctan \left (a x \right ) \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}+3 \arctan \left (a x \right ) \pi \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )^{3}+4 \arctan \left (a x \right )^{2}+12 i \arctan \left (a x \right ) \ln \left (2\right )+6 i \arctan \left (a x \right )-12-12 i a x \right )}{12}+\ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )}{c}}{a^{4}}\)	\(844\)
parts	\(\frac {x^{2} \arctan \left (a x \right )^{2}}{2 a^{2} c}-\frac {\arctan \left (a x \right )^{2} \ln \left (a^{2} x^{2}+1\right )}{2 c \,a^{4}}-\frac {a \left (-\frac {\arctan \left (a x \right )^{2} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{a^{5}}+\frac {i \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{a^{5}}-\frac {\operatorname {polylog}\left (3, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2 a^{5}}+\frac {i \arctan \left (a x \right ) \left (3 \arctan \left (a x \right ) \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}-3 \arctan \left (a x \right ) \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 )^{2} \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )-3 \arctan \left (a x \right ) \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 )^{2} \operatorname {csgn}\left (\frac {i}{\left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )+3 \arctan \left (a x \right ) \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 ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {i}{\left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )-3 \arctan \left (a x \right ) \pi {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )}^{3}+6 \arctan \left (a x \right ) \pi {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )}^{2} \operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )\right )-3 \arctan \left (a x \right ) \pi \,\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right ) {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )\right )}^{2}+3 \arctan \left (a x \right ) \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 )-6 \arctan \left (a x \right ) \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}+3 \arctan \left (a x \right ) \pi \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )^{3}+4 \arctan \left (a x \right )^{2}+12 i \arctan \left (a x \right ) \ln \left (2\right )+6 i \arctan \left (a x \right )-12-12 i a x \right )}{12 a^{5}}+\frac {\ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )}{a^{5}}\right )}{c}\)	\(860\)

[In]

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

[Out]

1/a^4*(1/2/c*arctan(a*x)^2*a^2*x^2-1/2/c*arctan(a*x)^2*ln(a^2*x^2+1)-1/c*(-arctan(a*x)^2*ln((1+I*a*x)/(a^2*x^2
+1)^(1/2))+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))+1/12*I*ar
ctan(a*x)*(3*arctan(a*x)*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)^3-3*arctan(a*x)*Pi*c
sgn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^2*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))-3*arctan(a*x)*P
i*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*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)+3*arct
an(a*x)*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)*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)^2)-3*arctan(a*x)*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^3+6*arctan(a*x)*Pi*csg
n(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^2*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1))-3*arctan(a*x)*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+3*arctan(a*x)*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))-6*arctan(a*x)*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+3*arctan(a*x)*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))^3+4*arctan(a*x)^2+12*I*arctan(a*x)*ln(2)+6*I*arctan
(a*x)-12-12*I*a*x)+ln((1+I*a*x)^2/(a^2*x^2+1)+1)))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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