\(\int (c+a^2 c x^2)^3 \arctan (a x)^3 \, dx\) [382]

Optimal result

Integrand size = 19, antiderivative size = 388 \[ \int \left (c+a^2 c x^2\right )^3 \arctan (a x)^3 \, dx=-\frac {13 c^3 \left (1+a^2 x^2\right )}{210 a}-\frac {c^3 \left (1+a^2 x^2\right )^2}{140 a}+\frac {14}{15} c^3 x \arctan (a x)+\frac {13}{105} c^3 x \left (1+a^2 x^2\right ) \arctan (a x)+\frac {1}{35} c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)-\frac {12 c^3 \left (1+a^2 x^2\right ) \arctan (a x)^2}{35 a}-\frac {9 c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)^2}{70 a}-\frac {c^3 \left (1+a^2 x^2\right )^3 \arctan (a x)^2}{14 a}+\frac {16 i c^3 \arctan (a x)^3}{35 a}+\frac {16}{35} c^3 x \arctan (a x)^3+\frac {8}{35} c^3 x \left (1+a^2 x^2\right ) \arctan (a x)^3+\frac {6}{35} c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)^3+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \arctan (a x)^3+\frac {48 c^3 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{35 a}-\frac {7 c^3 \log \left (1+a^2 x^2\right )}{15 a}+\frac {48 i c^3 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{35 a}+\frac {24 c^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{35 a} \]

[Out]

-13/210*c^3*(a^2*x^2+1)/a-1/140*c^3*(a^2*x^2+1)^2/a+14/15*c^3*x*arctan(a*x)+13/105*c^3*x*(a^2*x^2+1)*arctan(a*
x)+1/35*c^3*x*(a^2*x^2+1)^2*arctan(a*x)-12/35*c^3*(a^2*x^2+1)*arctan(a*x)^2/a-9/70*c^3*(a^2*x^2+1)^2*arctan(a*
x)^2/a-1/14*c^3*(a^2*x^2+1)^3*arctan(a*x)^2/a+48/35*I*c^3*arctan(a*x)*polylog(2,1-2/(1+I*a*x))/a+16/35*c^3*x*a
rctan(a*x)^3+8/35*c^3*x*(a^2*x^2+1)*arctan(a*x)^3+6/35*c^3*x*(a^2*x^2+1)^2*arctan(a*x)^3+1/7*c^3*x*(a^2*x^2+1)
^3*arctan(a*x)^3+48/35*c^3*arctan(a*x)^2*ln(2/(1+I*a*x))/a-7/15*c^3*ln(a^2*x^2+1)/a+16/35*I*c^3*arctan(a*x)^3/
a+24/35*c^3*polylog(3,1-2/(1+I*a*x))/a

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {5000, 4930, 5040, 4964, 5004, 5114, 6745, 266, 4998} \[ \int \left (c+a^2 c x^2\right )^3 \arctan (a x)^3 \, dx=\frac {1}{7} c^3 x \left (a^2 x^2+1\right )^3 \arctan (a x)^3+\frac {6}{35} c^3 x \left (a^2 x^2+1\right )^2 \arctan (a x)^3+\frac {8}{35} c^3 x \left (a^2 x^2+1\right ) \arctan (a x)^3-\frac {c^3 \left (a^2 x^2+1\right )^3 \arctan (a x)^2}{14 a}-\frac {9 c^3 \left (a^2 x^2+1\right )^2 \arctan (a x)^2}{70 a}-\frac {12 c^3 \left (a^2 x^2+1\right ) \arctan (a x)^2}{35 a}+\frac {1}{35} c^3 x \left (a^2 x^2+1\right )^2 \arctan (a x)+\frac {13}{105} c^3 x \left (a^2 x^2+1\right ) \arctan (a x)-\frac {c^3 \left (a^2 x^2+1\right )^2}{140 a}-\frac {13 c^3 \left (a^2 x^2+1\right )}{210 a}-\frac {7 c^3 \log \left (a^2 x^2+1\right )}{15 a}+\frac {48 i c^3 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{35 a}+\frac {16}{35} c^3 x \arctan (a x)^3+\frac {16 i c^3 \arctan (a x)^3}{35 a}+\frac {14}{15} c^3 x \arctan (a x)+\frac {48 c^3 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{35 a}+\frac {24 c^3 \operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right )}{35 a} \]

[In]

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

[Out]

(-13*c^3*(1 + a^2*x^2))/(210*a) - (c^3*(1 + a^2*x^2)^2)/(140*a) + (14*c^3*x*ArcTan[a*x])/15 + (13*c^3*x*(1 + a
^2*x^2)*ArcTan[a*x])/105 + (c^3*x*(1 + a^2*x^2)^2*ArcTan[a*x])/35 - (12*c^3*(1 + a^2*x^2)*ArcTan[a*x]^2)/(35*a
) - (9*c^3*(1 + a^2*x^2)^2*ArcTan[a*x]^2)/(70*a) - (c^3*(1 + a^2*x^2)^3*ArcTan[a*x]^2)/(14*a) + (((16*I)/35)*c
^3*ArcTan[a*x]^3)/a + (16*c^3*x*ArcTan[a*x]^3)/35 + (8*c^3*x*(1 + a^2*x^2)*ArcTan[a*x]^3)/35 + (6*c^3*x*(1 + a
^2*x^2)^2*ArcTan[a*x]^3)/35 + (c^3*x*(1 + a^2*x^2)^3*ArcTan[a*x]^3)/7 + (48*c^3*ArcTan[a*x]^2*Log[2/(1 + I*a*x
)])/(35*a) - (7*c^3*Log[1 + a^2*x^2])/(15*a) + (((48*I)/35)*c^3*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)])/a +
(24*c^3*PolyLog[3, 1 - 2/(1 + I*a*x)])/(35*a)

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 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 4998

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

Rule 5000

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

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 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 {c^3 \left (1+a^2 x^2\right )^3 \arctan (a x)^2}{14 a}+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \arctan (a x)^3+\frac {1}{7} c \int \left (c+a^2 c x^2\right )^2 \arctan (a x) \, dx+\frac {1}{7} (6 c) \int \left (c+a^2 c x^2\right )^2 \arctan (a x)^3 \, dx \\ & = -\frac {c^3 \left (1+a^2 x^2\right )^2}{140 a}+\frac {1}{35} c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)-\frac {9 c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)^2}{70 a}-\frac {c^3 \left (1+a^2 x^2\right )^3 \arctan (a x)^2}{14 a}+\frac {6}{35} c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)^3+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \arctan (a x)^3+\frac {1}{35} \left (4 c^2\right ) \int \left (c+a^2 c x^2\right ) \arctan (a x) \, dx+\frac {1}{35} \left (9 c^2\right ) \int \left (c+a^2 c x^2\right ) \arctan (a x) \, dx+\frac {1}{35} \left (24 c^2\right ) \int \left (c+a^2 c x^2\right ) \arctan (a x)^3 \, dx \\ & = -\frac {13 c^3 \left (1+a^2 x^2\right )}{210 a}-\frac {c^3 \left (1+a^2 x^2\right )^2}{140 a}+\frac {13}{105} c^3 x \left (1+a^2 x^2\right ) \arctan (a x)+\frac {1}{35} c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)-\frac {12 c^3 \left (1+a^2 x^2\right ) \arctan (a x)^2}{35 a}-\frac {9 c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)^2}{70 a}-\frac {c^3 \left (1+a^2 x^2\right )^3 \arctan (a x)^2}{14 a}+\frac {8}{35} c^3 x \left (1+a^2 x^2\right ) \arctan (a x)^3+\frac {6}{35} c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)^3+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \arctan (a x)^3+\frac {1}{105} \left (8 c^3\right ) \int \arctan (a x) \, dx+\frac {1}{35} \left (6 c^3\right ) \int \arctan (a x) \, dx+\frac {1}{35} \left (16 c^3\right ) \int \arctan (a x)^3 \, dx+\frac {1}{35} \left (24 c^3\right ) \int \arctan (a x) \, dx \\ & = -\frac {13 c^3 \left (1+a^2 x^2\right )}{210 a}-\frac {c^3 \left (1+a^2 x^2\right )^2}{140 a}+\frac {14}{15} c^3 x \arctan (a x)+\frac {13}{105} c^3 x \left (1+a^2 x^2\right ) \arctan (a x)+\frac {1}{35} c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)-\frac {12 c^3 \left (1+a^2 x^2\right ) \arctan (a x)^2}{35 a}-\frac {9 c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)^2}{70 a}-\frac {c^3 \left (1+a^2 x^2\right )^3 \arctan (a x)^2}{14 a}+\frac {16}{35} c^3 x \arctan (a x)^3+\frac {8}{35} c^3 x \left (1+a^2 x^2\right ) \arctan (a x)^3+\frac {6}{35} c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)^3+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \arctan (a x)^3-\frac {1}{105} \left (8 a c^3\right ) \int \frac {x}{1+a^2 x^2} \, dx-\frac {1}{35} \left (6 a c^3\right ) \int \frac {x}{1+a^2 x^2} \, dx-\frac {1}{35} \left (24 a c^3\right ) \int \frac {x}{1+a^2 x^2} \, dx-\frac {1}{35} \left (48 a c^3\right ) \int \frac {x \arctan (a x)^2}{1+a^2 x^2} \, dx \\ & = -\frac {13 c^3 \left (1+a^2 x^2\right )}{210 a}-\frac {c^3 \left (1+a^2 x^2\right )^2}{140 a}+\frac {14}{15} c^3 x \arctan (a x)+\frac {13}{105} c^3 x \left (1+a^2 x^2\right ) \arctan (a x)+\frac {1}{35} c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)-\frac {12 c^3 \left (1+a^2 x^2\right ) \arctan (a x)^2}{35 a}-\frac {9 c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)^2}{70 a}-\frac {c^3 \left (1+a^2 x^2\right )^3 \arctan (a x)^2}{14 a}+\frac {16 i c^3 \arctan (a x)^3}{35 a}+\frac {16}{35} c^3 x \arctan (a x)^3+\frac {8}{35} c^3 x \left (1+a^2 x^2\right ) \arctan (a x)^3+\frac {6}{35} c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)^3+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \arctan (a x)^3-\frac {7 c^3 \log \left (1+a^2 x^2\right )}{15 a}+\frac {1}{35} \left (48 c^3\right ) \int \frac {\arctan (a x)^2}{i-a x} \, dx \\ & = -\frac {13 c^3 \left (1+a^2 x^2\right )}{210 a}-\frac {c^3 \left (1+a^2 x^2\right )^2}{140 a}+\frac {14}{15} c^3 x \arctan (a x)+\frac {13}{105} c^3 x \left (1+a^2 x^2\right ) \arctan (a x)+\frac {1}{35} c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)-\frac {12 c^3 \left (1+a^2 x^2\right ) \arctan (a x)^2}{35 a}-\frac {9 c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)^2}{70 a}-\frac {c^3 \left (1+a^2 x^2\right )^3 \arctan (a x)^2}{14 a}+\frac {16 i c^3 \arctan (a x)^3}{35 a}+\frac {16}{35} c^3 x \arctan (a x)^3+\frac {8}{35} c^3 x \left (1+a^2 x^2\right ) \arctan (a x)^3+\frac {6}{35} c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)^3+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \arctan (a x)^3+\frac {48 c^3 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{35 a}-\frac {7 c^3 \log \left (1+a^2 x^2\right )}{15 a}-\frac {1}{35} \left (96 c^3\right ) \int \frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx \\ & = -\frac {13 c^3 \left (1+a^2 x^2\right )}{210 a}-\frac {c^3 \left (1+a^2 x^2\right )^2}{140 a}+\frac {14}{15} c^3 x \arctan (a x)+\frac {13}{105} c^3 x \left (1+a^2 x^2\right ) \arctan (a x)+\frac {1}{35} c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)-\frac {12 c^3 \left (1+a^2 x^2\right ) \arctan (a x)^2}{35 a}-\frac {9 c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)^2}{70 a}-\frac {c^3 \left (1+a^2 x^2\right )^3 \arctan (a x)^2}{14 a}+\frac {16 i c^3 \arctan (a x)^3}{35 a}+\frac {16}{35} c^3 x \arctan (a x)^3+\frac {8}{35} c^3 x \left (1+a^2 x^2\right ) \arctan (a x)^3+\frac {6}{35} c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)^3+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \arctan (a x)^3+\frac {48 c^3 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{35 a}-\frac {7 c^3 \log \left (1+a^2 x^2\right )}{15 a}+\frac {48 i c^3 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{35 a}-\frac {1}{35} \left (48 i c^3\right ) \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx \\ & = -\frac {13 c^3 \left (1+a^2 x^2\right )}{210 a}-\frac {c^3 \left (1+a^2 x^2\right )^2}{140 a}+\frac {14}{15} c^3 x \arctan (a x)+\frac {13}{105} c^3 x \left (1+a^2 x^2\right ) \arctan (a x)+\frac {1}{35} c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)-\frac {12 c^3 \left (1+a^2 x^2\right ) \arctan (a x)^2}{35 a}-\frac {9 c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)^2}{70 a}-\frac {c^3 \left (1+a^2 x^2\right )^3 \arctan (a x)^2}{14 a}+\frac {16 i c^3 \arctan (a x)^3}{35 a}+\frac {16}{35} c^3 x \arctan (a x)^3+\frac {8}{35} c^3 x \left (1+a^2 x^2\right ) \arctan (a x)^3+\frac {6}{35} c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)^3+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \arctan (a x)^3+\frac {48 c^3 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{35 a}-\frac {7 c^3 \log \left (1+a^2 x^2\right )}{15 a}+\frac {48 i c^3 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{35 a}+\frac {24 c^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{35 a} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.00 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.63 \[ \int \left (c+a^2 c x^2\right )^3 \arctan (a x)^3 \, dx=\frac {c^3 \left (-29-32 a^2 x^2-3 a^4 x^4+456 a x \arctan (a x)+76 a^3 x^3 \arctan (a x)+12 a^5 x^5 \arctan (a x)-228 \arctan (a x)^2-342 a^2 x^2 \arctan (a x)^2-144 a^4 x^4 \arctan (a x)^2-30 a^6 x^6 \arctan (a x)^2-192 i \arctan (a x)^3+420 a x \arctan (a x)^3+420 a^3 x^3 \arctan (a x)^3+252 a^5 x^5 \arctan (a x)^3+60 a^7 x^7 \arctan (a x)^3+576 \arctan (a x)^2 \log \left (1+e^{2 i \arctan (a x)}\right )-196 \log \left (1+a^2 x^2\right )-576 i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )+288 \operatorname {PolyLog}\left (3,-e^{2 i \arctan (a x)}\right )\right )}{420 a} \]

[In]

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

[Out]

(c^3*(-29 - 32*a^2*x^2 - 3*a^4*x^4 + 456*a*x*ArcTan[a*x] + 76*a^3*x^3*ArcTan[a*x] + 12*a^5*x^5*ArcTan[a*x] - 2
28*ArcTan[a*x]^2 - 342*a^2*x^2*ArcTan[a*x]^2 - 144*a^4*x^4*ArcTan[a*x]^2 - 30*a^6*x^6*ArcTan[a*x]^2 - (192*I)*
ArcTan[a*x]^3 + 420*a*x*ArcTan[a*x]^3 + 420*a^3*x^3*ArcTan[a*x]^3 + 252*a^5*x^5*ArcTan[a*x]^3 + 60*a^7*x^7*Arc
Tan[a*x]^3 + 576*ArcTan[a*x]^2*Log[1 + E^((2*I)*ArcTan[a*x])] - 196*Log[1 + a^2*x^2] - (576*I)*ArcTan[a*x]*Pol
yLog[2, -E^((2*I)*ArcTan[a*x])] + 288*PolyLog[3, -E^((2*I)*ArcTan[a*x])]))/(420*a)

Maple [C] (warning: unable to verify)

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

Time = 86.79 (sec) , antiderivative size = 1267, normalized size of antiderivative = 3.27

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1267\)
default	\(\text {Expression too large to display}\)	\(1267\)
parts	\(\text {Expression too large to display}\)	\(1268\)

[In]

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

[Out]

1/a*(1/7*c^3*arctan(a*x)^3*a^7*x^7+3/5*c^3*arctan(a*x)^3*a^5*x^5+c^3*arctan(a*x)^3*a^3*x^3+c^3*arctan(a*x)^3*a
*x-3/35*c^3*(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)*arctan(a*x)^2+19/3*arctan(a*x)^2+19/2*x^2*arctan(a*x)^2*a^2-16*arcta
n(a*x)^2*ln(2)+1/12*(I+a*x)^4+5/6*a^6*x^6*arctan(a*x)^2-11/3*arctan(a*x)*(a*x-I)^2*(I+a*x)+5/3*arctan(a*x)*(a*
x-I)^4*(I+a*x)+10/3*arctan(a*x)*(a*x-I)^2*(I+a*x)^3+11/3*arctan(a*x)*(a*x-I)*(I+a*x)^2-10/3*arctan(a*x)*(a*x-I
)^3*(I+a*x)^2-5/3*arctan(a*x)*(a*x-I)*(I+a*x)^4+4*a^4*arctan(a*x)^2*x^4+11/9*arctan(a*x)*(a*x-I)^3-8*arctan(a*
x)*(a*x-I)-1/3*arctan(a*x)*(a*x-I)^5-8*polylog(3,-(1+I*a*x)^2/(a^2*x^2+1))-98/9*ln((1+I*a*x)^2/(a^2*x^2+1)+1)+
8*arctan(a*x)^2*ln(a^2*x^2+1)+7/18*(I+a*x)^2-5/3*I*arctan(a*x)*(a*x-I)^4-4*I*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1
)+1)^2)^3*arctan(a*x)^2-10*I*arctan(a*x)*(a*x-I)^2*(I+a*x)^2+20/3*I*arctan(a*x)*(a*x-I)^3*(I+a*x)+20/3*I*arcta
n(a*x)*(a*x-I)*(I+a*x)^3+6*I*arctan(a*x)*(a*x-I)*(I+a*x)+4*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*arctan(a*x)^2+4*I*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))^3*arctan(a*x)^2-8*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*arctan(a*x)^2+8*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*arctan(a*x)^2-4*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)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^2*arctan(a*x)^2+4*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))*arctan(a*x)^2-4*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*arctan(a*x)^2-4*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)+1)^2)*arctan(a*x)^2-16*arctan(a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2))-3*I*arctan(a*x)*(a
*x-I)^2+16*I*arctan(a*x)*polylog(2,-(1+I*a*x)^2/(a^2*x^2+1))-13/9*I*(I+a*x)+16/3*I*arctan(a*x)^3-1/3*I*(I+a*x)
^3))

Fricas [F]

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

[In]

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

[Out]

integral((a^6*c^3*x^6 + 3*a^4*c^3*x^4 + 3*a^2*c^3*x^2 + c^3)*arctan(a*x)^3, x)

Sympy [F]

\[ \int \left (c+a^2 c x^2\right )^3 \arctan (a x)^3 \, dx=c^{3} \left (\int 3 a^{2} x^{2} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int 3 a^{4} x^{4} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int a^{6} x^{6} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int \operatorname {atan}^{3}{\left (a x \right )}\, dx\right ) \]

[In]

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

[Out]

c**3*(Integral(3*a**2*x**2*atan(a*x)**3, x) + Integral(3*a**4*x**4*atan(a*x)**3, x) + Integral(a**6*x**6*atan(
a*x)**3, x) + Integral(atan(a*x)**3, x))

Maxima [F]

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

[In]

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

[Out]

980*a^8*c^3*integrate(1/1120*x^8*arctan(a*x)^3/(a^2*x^2 + 1), x) + 105*a^8*c^3*integrate(1/1120*x^8*arctan(a*x
)*log(a^2*x^2 + 1)^2/(a^2*x^2 + 1), x) + 60*a^8*c^3*integrate(1/1120*x^8*arctan(a*x)*log(a^2*x^2 + 1)/(a^2*x^2
+ 1), x) - 60*a^7*c^3*integrate(1/1120*x^7*arctan(a*x)^2/(a^2*x^2 + 1), x) + 15*a^7*c^3*integrate(1/1120*x^7*
log(a^2*x^2 + 1)^2/(a^2*x^2 + 1), x) + 3920*a^6*c^3*integrate(1/1120*x^6*arctan(a*x)^3/(a^2*x^2 + 1), x) + 420
*a^6*c^3*integrate(1/1120*x^6*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^2*x^2 + 1), x) + 252*a^6*c^3*integrate(1/1120*
x^6*arctan(a*x)*log(a^2*x^2 + 1)/(a^2*x^2 + 1), x) - 252*a^5*c^3*integrate(1/1120*x^5*arctan(a*x)^2/(a^2*x^2 +
1), x) + 63*a^5*c^3*integrate(1/1120*x^5*log(a^2*x^2 + 1)^2/(a^2*x^2 + 1), x) + 5880*a^4*c^3*integrate(1/1120
*x^4*arctan(a*x)^3/(a^2*x^2 + 1), x) + 630*a^4*c^3*integrate(1/1120*x^4*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^2*x^
2 + 1), x) + 420*a^4*c^3*integrate(1/1120*x^4*arctan(a*x)*log(a^2*x^2 + 1)/(a^2*x^2 + 1), x) - 420*a^3*c^3*int
egrate(1/1120*x^3*arctan(a*x)^2/(a^2*x^2 + 1), x) + 105*a^3*c^3*integrate(1/1120*x^3*log(a^2*x^2 + 1)^2/(a^2*x
^2 + 1), x) + 7/32*c^3*arctan(a*x)^4/a + 3920*a^2*c^3*integrate(1/1120*x^2*arctan(a*x)^3/(a^2*x^2 + 1), x) + 4
20*a^2*c^3*integrate(1/1120*x^2*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^2*x^2 + 1), x) + 420*a^2*c^3*integrate(1/112
0*x^2*arctan(a*x)*log(a^2*x^2 + 1)/(a^2*x^2 + 1), x) - 420*a*c^3*integrate(1/1120*x*arctan(a*x)^2/(a^2*x^2 + 1
), x) + 105*a*c^3*integrate(1/1120*x*log(a^2*x^2 + 1)^2/(a^2*x^2 + 1), x) + 1/280*(5*a^6*c^3*x^7 + 21*a^4*c^3*
x^5 + 35*a^2*c^3*x^3 + 35*c^3*x)*arctan(a*x)^3 + 105*c^3*integrate(1/1120*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^2*
x^2 + 1), x) - 3/1120*(5*a^6*c^3*x^7 + 21*a^4*c^3*x^5 + 35*a^2*c^3*x^3 + 35*c^3*x)*arctan(a*x)*log(a^2*x^2 + 1
)^2

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

int(atan(a*x)^3*(c + a^2*c*x^2)^3,x)

[Out]

int(atan(a*x)^3*(c + a^2*c*x^2)^3, x)