\(\int (c+a^2 c x^2) \arctan (a x)^3 \, dx\) [366]

Optimal result

Integrand size = 17, antiderivative size = 172 \[ \int \left (c+a^2 c x^2\right ) \arctan (a x)^3 \, dx=c x \arctan (a x)-\frac {c \left (1+a^2 x^2\right ) \arctan (a x)^2}{2 a}+\frac {2 i c \arctan (a x)^3}{3 a}+\frac {2}{3} c x \arctan (a x)^3+\frac {1}{3} c x \left (1+a^2 x^2\right ) \arctan (a x)^3+\frac {2 c \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a}-\frac {c \log \left (1+a^2 x^2\right )}{2 a}+\frac {2 i c \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{a}+\frac {c \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{a} \]

[Out]

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

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used = {5000, 4930, 5040, 4964, 5004, 5114, 6745, 266} \[ \int \left (c+a^2 c x^2\right ) \arctan (a x)^3 \, dx=\frac {1}{3} c x \left (a^2 x^2+1\right ) \arctan (a x)^3-\frac {c \left (a^2 x^2+1\right ) \arctan (a x)^2}{2 a}-\frac {c \log \left (a^2 x^2+1\right )}{2 a}+\frac {2 i c \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{a}+\frac {2 i c \arctan (a x)^3}{3 a}+\frac {2}{3} c x \arctan (a x)^3+c x \arctan (a x)+\frac {2 c \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a}+\frac {c \operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right )}{a} \]

[In]

Int[(c + a^2*c*x^2)*ArcTan[a*x]^3,x]

[Out]

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.84 \[ \int \left (c+a^2 c x^2\right ) \arctan (a x)^3 \, dx=\frac {c \left (6 a x \arctan (a x)-3 \arctan (a x)^2-3 a^2 x^2 \arctan (a x)^2-4 i \arctan (a x)^3+6 a x \arctan (a x)^3+2 a^3 x^3 \arctan (a x)^3+12 \arctan (a x)^2 \log \left (1+e^{2 i \arctan (a x)}\right )-3 \log \left (1+a^2 x^2\right )-12 i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )+6 \operatorname {PolyLog}\left (3,-e^{2 i \arctan (a x)}\right )\right )}{6 a} \]

[In]

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

[Out]

(c*(6*a*x*ArcTan[a*x] - 3*ArcTan[a*x]^2 - 3*a^2*x^2*ArcTan[a*x]^2 - (4*I)*ArcTan[a*x]^3 + 6*a*x*ArcTan[a*x]^3
+ 2*a^3*x^3*ArcTan[a*x]^3 + 12*ArcTan[a*x]^2*Log[1 + E^((2*I)*ArcTan[a*x])] - 3*Log[1 + a^2*x^2] - (12*I)*ArcT
an[a*x]*PolyLog[2, -E^((2*I)*ArcTan[a*x])] + 6*PolyLog[3, -E^((2*I)*ArcTan[a*x])]))/(6*a)

Maple [C] (warning: unable to verify)

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

Time = 12.82 (sec) , antiderivative size = 860, normalized size of antiderivative = 5.00

method	result	size
parts	\(\frac {c \arctan \left (a x \right )^{3} a^{2} x^{3}}{3}+c x \arctan \left (a x \right )^{3}-c \left (\frac {a \arctan \left (a x \right )^{2} x^{2}}{2}+\frac {\arctan \left (a x \right )^{2} \ln \left (a^{2} x^{2}+1\right )}{a}-\frac {2 \arctan \left (a x \right )^{2} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-2 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )+\operatorname {polylog}\left (3, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )-\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 )^{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 (\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}}{a^{2} x^{2}+1}\right )^{3}-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 )}{\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 )-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}+4 \arctan \left (a x \right )^{2}+12 i \arctan \left (a x \right ) \ln \left (2\right )-3 i \arctan \left (a x \right )+6+6 i a x \right )}{6}+\ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )}{a}\right )\)	\(860\)
derivativedivides	\(\frac {\frac {c \arctan \left (a x \right )^{3} a^{3} x^{3}}{3}+c \arctan \left (a x \right )^{3} a x -c \left (\frac {x^{2} \arctan \left (a x \right )^{2} a^{2}}{2}+\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 )+2 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )-\operatorname {polylog}\left (3, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )+\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 )^{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 (\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}}{a^{2} x^{2}+1}\right )^{3}-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 )}{\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 )-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}+4 \arctan \left (a x \right )^{2}+12 i \arctan \left (a x \right ) \ln \left (2\right )-3 i \arctan \left (a x \right )+6+6 i a x \right )}{6}-\ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )\right )}{a}\)	\(862\)
default	\(\frac {\frac {c \arctan \left (a x \right )^{3} a^{3} x^{3}}{3}+c \arctan \left (a x \right )^{3} a x -c \left (\frac {x^{2} \arctan \left (a x \right )^{2} a^{2}}{2}+\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 )+2 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )-\operatorname {polylog}\left (3, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )+\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 )^{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 (\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}}{a^{2} x^{2}+1}\right )^{3}-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 )}{\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 )-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}+4 \arctan \left (a x \right )^{2}+12 i \arctan \left (a x \right ) \ln \left (2\right )-3 i \arctan \left (a x \right )+6+6 i a x \right )}{6}-\ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )\right )}{a}\)	\(862\)

[In]

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

[Out]

1/3*c*arctan(a*x)^3*a^2*x^3+c*x*arctan(a*x)^3-c*(1/2*a*arctan(a*x)^2*x^2+1/a*arctan(a*x)^2*ln(a^2*x^2+1)-1/a*(
2*arctan(a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2))-2*I*arctan(a*x)*polylog(2,-(1+I*a*x)^2/(a^2*x^2+1))+polylog(3,
-(1+I*a*x)^2/(a^2*x^2+1))-1/6*I*arctan(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)^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)^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+I*a*x)^2/(a^2*x^2+1)+1)^2)^3-3*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)^2*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))+3*arctan(a*x)*Pi*csgn(I*(1+I*a*x)^2/(a^2*x
^2+1))^3-6*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)/(a^2*x^2+1)^(1/2))^2*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))-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*csgn(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+4*arctan(a*x)^
2+12*I*arctan(a*x)*ln(2)-3*I*arctan(a*x)+6+6*I*a*x)+ln((1+I*a*x)^2/(a^2*x^2+1)+1)))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

28*a^4*c*integrate(1/32*x^4*arctan(a*x)^3/(a^2*x^2 + 1), x) + 3*a^4*c*integrate(1/32*x^4*arctan(a*x)*log(a^2*x
^2 + 1)^2/(a^2*x^2 + 1), x) + 4*a^4*c*integrate(1/32*x^4*arctan(a*x)*log(a^2*x^2 + 1)/(a^2*x^2 + 1), x) - 4*a^
3*c*integrate(1/32*x^3*arctan(a*x)^2/(a^2*x^2 + 1), x) + a^3*c*integrate(1/32*x^3*log(a^2*x^2 + 1)^2/(a^2*x^2
+ 1), x) + 1/24*(a^2*c*x^3 + 3*c*x)*arctan(a*x)^3 + 7/32*c*arctan(a*x)^4/a + 56*a^2*c*integrate(1/32*x^2*arcta
n(a*x)^3/(a^2*x^2 + 1), x) + 6*a^2*c*integrate(1/32*x^2*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^2*x^2 + 1), x) + 12*
a^2*c*integrate(1/32*x^2*arctan(a*x)*log(a^2*x^2 + 1)/(a^2*x^2 + 1), x) - 1/32*(a^2*c*x^3 + 3*c*x)*arctan(a*x)
*log(a^2*x^2 + 1)^2 - 12*a*c*integrate(1/32*x*arctan(a*x)^2/(a^2*x^2 + 1), x) + 3*a*c*integrate(1/32*x*log(a^2
*x^2 + 1)^2/(a^2*x^2 + 1), x) + 3*c*integrate(1/32*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^2*x^2 + 1), x)

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

int(atan(a*x)^3*(c + a^2*c*x^2),x)

[Out]

int(atan(a*x)^3*(c + a^2*c*x^2), x)