3.4.96 \(\int \frac {x^3 \arctan (a x)^3}{(c+a^2 c x^2)^2} \, dx\) [396]

3.4.96.1 Optimal result

Integrand size = 22, antiderivative size = 270 \[ \int \frac {x^3 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {3 x}{8 a^3 c^2 \left (1+a^2 x^2\right )}+\frac {3 \arctan (a x)}{8 a^4 c^2}-\frac {3 \arctan (a x)}{4 a^4 c^2 \left (1+a^2 x^2\right )}-\frac {3 x \arctan (a x)^2}{4 a^3 c^2 \left (1+a^2 x^2\right )}-\frac {\arctan (a x)^3}{4 a^4 c^2}+\frac {\arctan (a x)^3}{2 a^4 c^2 \left (1+a^2 x^2\right )}-\frac {i \arctan (a x)^4}{4 a^4 c^2}-\frac {\arctan (a x)^3 \log \left (\frac {2}{1+i a x}\right )}{a^4 c^2}-\frac {3 i \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{2 a^4 c^2}-\frac {3 \arctan (a x) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{2 a^4 c^2}+\frac {3 i \operatorname {PolyLog}\left (4,1-\frac {2}{1+i a x}\right )}{4 a^4 c^2} \]

output

3/8*x/a^3/c^2/(a^2*x^2+1)+3/8*arctan(a*x)/a^4/c^2-3/4*arctan(a*x)/a^4/c^2/ 
(a^2*x^2+1)-3/4*x*arctan(a*x)^2/a^3/c^2/(a^2*x^2+1)-1/4*arctan(a*x)^3/a^4/ 
c^2+1/2*arctan(a*x)^3/a^4/c^2/(a^2*x^2+1)-1/4*I*arctan(a*x)^4/a^4/c^2-arct 
an(a*x)^3*ln(2/(1+I*a*x))/a^4/c^2-3/2*I*arctan(a*x)^2*polylog(2,1-2/(1+I*a 
*x))/a^4/c^2-3/2*arctan(a*x)*polylog(3,1-2/(1+I*a*x))/a^4/c^2+3/4*I*polylo 
g(4,1-2/(1+I*a*x))/a^4/c^2
 

3.4.96.2 Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.58 \[ \int \frac {x^3 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {4 i \arctan (a x)^4-6 \arctan (a x) \cos (2 \arctan (a x))+4 \arctan (a x)^3 \cos (2 \arctan (a x))-16 \arctan (a x)^3 \log \left (1+e^{2 i \arctan (a x)}\right )+24 i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )-24 \arctan (a x) \operatorname {PolyLog}\left (3,-e^{2 i \arctan (a x)}\right )-12 i \operatorname {PolyLog}\left (4,-e^{2 i \arctan (a x)}\right )+3 \sin (2 \arctan (a x))-6 \arctan (a x)^2 \sin (2 \arctan (a x))}{16 a^4 c^2} \]

input

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

output

((4*I)*ArcTan[a*x]^4 - 6*ArcTan[a*x]*Cos[2*ArcTan[a*x]] + 4*ArcTan[a*x]^3* 
Cos[2*ArcTan[a*x]] - 16*ArcTan[a*x]^3*Log[1 + E^((2*I)*ArcTan[a*x])] + (24 
*I)*ArcTan[a*x]^2*PolyLog[2, -E^((2*I)*ArcTan[a*x])] - 24*ArcTan[a*x]*Poly 
Log[3, -E^((2*I)*ArcTan[a*x])] - (12*I)*PolyLog[4, -E^((2*I)*ArcTan[a*x])] 
 + 3*Sin[2*ArcTan[a*x]] - 6*ArcTan[a*x]^2*Sin[2*ArcTan[a*x]])/(16*a^4*c^2)
 

3.4.96.3 Rubi [A] (verified)

Time = 1.61 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.04, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {5499, 27, 5455, 5379, 5465, 5427, 5465, 215, 216, 5529, 5533, 7164}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {x^3 \arctan (a x)^3}{\left (a^2 c x^2+c\right )^2} \, dx\)

\(\Big \downarrow \) 5499

\(\displaystyle \frac {\int \frac {x \arctan (a x)^3}{c \left (a^2 x^2+1\right )}dx}{a^2 c}-\frac {\int \frac {x \arctan (a x)^3}{c^2 \left (a^2 x^2+1\right )^2}dx}{a^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {x \arctan (a x)^3}{a^2 x^2+1}dx}{a^2 c^2}-\frac {\int \frac {x \arctan (a x)^3}{\left (a^2 x^2+1\right )^2}dx}{a^2 c^2}\)

\(\Big \downarrow \) 5455

\(\displaystyle -\frac {\int \frac {x \arctan (a x)^3}{\left (a^2 x^2+1\right )^2}dx}{a^2 c^2}+\frac {-\frac {\int \frac {\arctan (a x)^3}{i-a x}dx}{a}-\frac {i \arctan (a x)^4}{4 a^2}}{a^2 c^2}\)

\(\Big \downarrow \) 5379

\(\displaystyle -\frac {\int \frac {x \arctan (a x)^3}{\left (a^2 x^2+1\right )^2}dx}{a^2 c^2}+\frac {-\frac {\frac {\arctan (a x)^3 \log \left (\frac {2}{1+i a x}\right )}{a}-3 \int \frac {\arctan (a x)^2 \log \left (\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx}{a}-\frac {i \arctan (a x)^4}{4 a^2}}{a^2 c^2}\)

\(\Big \downarrow \) 5465

\(\displaystyle -\frac {\frac {3 \int \frac {\arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx}{2 a}-\frac {\arctan (a x)^3}{2 a^2 \left (a^2 x^2+1\right )}}{a^2 c^2}+\frac {-\frac {\frac {\arctan (a x)^3 \log \left (\frac {2}{1+i a x}\right )}{a}-3 \int \frac {\arctan (a x)^2 \log \left (\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx}{a}-\frac {i \arctan (a x)^4}{4 a^2}}{a^2 c^2}\)

\(\Big \downarrow \) 5427

\(\displaystyle -\frac {\frac {3 \left (-a \int \frac {x \arctan (a x)}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^3}{6 a}\right )}{2 a}-\frac {\arctan (a x)^3}{2 a^2 \left (a^2 x^2+1\right )}}{a^2 c^2}+\frac {-\frac {\frac {\arctan (a x)^3 \log \left (\frac {2}{1+i a x}\right )}{a}-3 \int \frac {\arctan (a x)^2 \log \left (\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx}{a}-\frac {i \arctan (a x)^4}{4 a^2}}{a^2 c^2}\)

\(\Big \downarrow \) 5465

\(\displaystyle -\frac {\frac {3 \left (-a \left (\frac {\int \frac {1}{\left (a^2 x^2+1\right )^2}dx}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^3}{6 a}\right )}{2 a}-\frac {\arctan (a x)^3}{2 a^2 \left (a^2 x^2+1\right )}}{a^2 c^2}+\frac {-\frac {\frac {\arctan (a x)^3 \log \left (\frac {2}{1+i a x}\right )}{a}-3 \int \frac {\arctan (a x)^2 \log \left (\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx}{a}-\frac {i \arctan (a x)^4}{4 a^2}}{a^2 c^2}\)

\(\Big \downarrow \) 215

\(\displaystyle -\frac {\frac {3 \left (-a \left (\frac {\frac {1}{2} \int \frac {1}{a^2 x^2+1}dx+\frac {x}{2 \left (a^2 x^2+1\right )}}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^3}{6 a}\right )}{2 a}-\frac {\arctan (a x)^3}{2 a^2 \left (a^2 x^2+1\right )}}{a^2 c^2}+\frac {-\frac {\frac {\arctan (a x)^3 \log \left (\frac {2}{1+i a x}\right )}{a}-3 \int \frac {\arctan (a x)^2 \log \left (\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx}{a}-\frac {i \arctan (a x)^4}{4 a^2}}{a^2 c^2}\)

\(\Big \downarrow \) 216

\(\displaystyle -\frac {\frac {3 \left (\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}-a \left (\frac {\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^3}{6 a}\right )}{2 a}-\frac {\arctan (a x)^3}{2 a^2 \left (a^2 x^2+1\right )}}{a^2 c^2}+\frac {-\frac {\frac {\arctan (a x)^3 \log \left (\frac {2}{1+i a x}\right )}{a}-3 \int \frac {\arctan (a x)^2 \log \left (\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx}{a}-\frac {i \arctan (a x)^4}{4 a^2}}{a^2 c^2}\)

\(\Big \downarrow \) 5529

\(\displaystyle -\frac {\frac {3 \left (\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}-a \left (\frac {\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^3}{6 a}\right )}{2 a}-\frac {\arctan (a x)^3}{2 a^2 \left (a^2 x^2+1\right )}}{a^2 c^2}+\frac {-\frac {\frac {\arctan (a x)^3 \log \left (\frac {2}{1+i a x}\right )}{a}-3 \left (i \int \frac {\arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx-\frac {i \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{2 a}\right )}{a}-\frac {i \arctan (a x)^4}{4 a^2}}{a^2 c^2}\)

\(\Big \downarrow \) 5533

\(\displaystyle -\frac {\frac {3 \left (\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}-a \left (\frac {\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^3}{6 a}\right )}{2 a}-\frac {\arctan (a x)^3}{2 a^2 \left (a^2 x^2+1\right )}}{a^2 c^2}+\frac {-\frac {\frac {\arctan (a x)^3 \log \left (\frac {2}{1+i a x}\right )}{a}-3 \left (i \left (\frac {i \arctan (a x) \operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right )}{2 a}-\frac {1}{2} i \int \frac {\operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx\right )-\frac {i \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{2 a}\right )}{a}-\frac {i \arctan (a x)^4}{4 a^2}}{a^2 c^2}\)

\(\Big \downarrow \) 7164

\(\displaystyle -\frac {\frac {3 \left (\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}-a \left (\frac {\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^3}{6 a}\right )}{2 a}-\frac {\arctan (a x)^3}{2 a^2 \left (a^2 x^2+1\right )}}{a^2 c^2}+\frac {-\frac {i \arctan (a x)^4}{4 a^2}-\frac {\frac {\arctan (a x)^3 \log \left (\frac {2}{1+i a x}\right )}{a}-3 \left (i \left (\frac {i \arctan (a x) \operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right )}{2 a}+\frac {\operatorname {PolyLog}\left (4,1-\frac {2}{i a x+1}\right )}{4 a}\right )-\frac {i \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{2 a}\right )}{a}}{a^2 c^2}\)

input

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

output

-((-1/2*ArcTan[a*x]^3/(a^2*(1 + a^2*x^2)) + (3*((x*ArcTan[a*x]^2)/(2*(1 + 
a^2*x^2)) + ArcTan[a*x]^3/(6*a) - a*(-1/2*ArcTan[a*x]/(a^2*(1 + a^2*x^2)) 
+ (x/(2*(1 + a^2*x^2)) + ArcTan[a*x]/(2*a))/(2*a))))/(2*a))/(a^2*c^2)) + ( 
((-1/4*I)*ArcTan[a*x]^4)/a^2 - ((ArcTan[a*x]^3*Log[2/(1 + I*a*x)])/a - 3*( 
((-1/2*I)*ArcTan[a*x]^2*PolyLog[2, 1 - 2/(1 + I*a*x)])/a + I*(((I/2)*ArcTa 
n[a*x]*PolyLog[3, 1 - 2/(1 + I*a*x)])/a + PolyLog[4, 1 - 2/(1 + I*a*x)]/(4 
*a))))/a)/(a^2*c^2)
 

3.4.96.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 215

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) 
/(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1))   Int[(a + b*x^2)^(p + 1 
), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 
*p])
 

rule 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 5379

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] + Simp[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 5427

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

rule 5455

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), 
x_Symbol] :> Simp[(-I)*((a + b*ArcTan[c*x])^(p + 1)/(b*e*(p + 1))), x] - Si 
mp[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 5465

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

rule 5499

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_)*((d_) + (e_.)*(x_)^2 
)^(q_), x_Symbol] :> Simp[1/e   Int[x^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*Ar 
cTan[c*x])^p, x], x] - Simp[d/e   Int[x^(m - 2)*(d + e*x^2)^q*(a + b*ArcTan 
[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IntegersQ 
[p, 2*q] && LtQ[q, -1] && IGtQ[m, 1] && NeQ[p, -1]
 

rule 5529

Int[(Log[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2 
), x_Symbol] :> Simp[(-I)*(a + b*ArcTan[c*x])^p*(PolyLog[2, 1 - u]/(2*c*d)) 
, x] + Simp[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 5533

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

rule 7164

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]
 

3.4.96.4 Maple [C] (warning: unable to verify)

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

Time = 24.99 (sec) , antiderivative size = 936, normalized size of antiderivative = 3.47

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(936\)
default	\(\text {Expression too large to display}\)	\(936\)
parts	\(\text {Expression too large to display}\)	\(971\)

input

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

output

1/a^4*(1/2/c^2*arctan(a*x)^3*ln(a^2*x^2+1)+1/2*arctan(a*x)^3/c^2/(a^2*x^2+ 
1)-3/2/c^2*(2/3*arctan(a*x)^3*ln((1+I*a*x)/(a^2*x^2+1)^(1/2))-1/6*I*arctan 
(a*x)^4-I*arctan(a*x)^2*(I+a*x)/(8*a*x-8*I)-1/8*arctan(a*x)*(I+a*x)/(a*x-I 
)+I*(I+a*x)/(16*a*x-16*I)+I*arctan(a*x)^2*(a*x-I)/(8*a*x+8*I)-1/8*arctan(a 
*x)*(a*x-I)/(I+a*x)-I*(a*x-I)/(16*a*x+16*I)-I*arctan(a*x)^2*polylog(2,-(1+ 
I*a*x)^2/(a^2*x^2+1))+arctan(a*x)*polylog(3,-(1+I*a*x)^2/(a^2*x^2+1))+1/2* 
I*polylog(4,-(1+I*a*x)^2/(a^2*x^2+1))+1/6*(I*Pi*csgn(I*((1+I*a*x)^2/(a^2*x 
^2+1)+1)^2)^3-2*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*csgn(I*( 
(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))^3+2*I*P 
i*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))*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))*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)/(a^2*x^2+1)^(1/2))^2*csgn(I*(1+I*a*x)^ 
2/(a^2*x^2+1))-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+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+4*ln(2)+1)*arctan(a*x)^3))
 

3.4.96.5 Fricas [F]

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

input

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

output

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

3.4.96.6 Sympy [F]

\[ \int \frac {x^3 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {\int \frac {x^{3} \operatorname {atan}^{3}{\left (a x \right )}}{a^{4} x^{4} + 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \]

input

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

output

Integral(x**3*atan(a*x)**3/(a**4*x**4 + 2*a**2*x**2 + 1), x)/c**2
 

3.4.96.7 Maxima [F]

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

input

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

output

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

3.4.96.8 Giac [F]

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

input

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

output

sage0*x
 

3.4.96.9 Mupad [F(-1)]

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

input

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

output

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