3.4.88 \(\int \frac {x^3 \arctan (a x)^3}{c+a^2 c x^2} \, dx\) [388]

3.4.88.1 Optimal result

Integrand size = 22, antiderivative size = 260 \[ \int \frac {x^3 \arctan (a x)^3}{c+a^2 c x^2} \, dx=-\frac {3 i \arctan (a x)^2}{2 a^4 c}-\frac {3 x \arctan (a x)^2}{2 a^3 c}+\frac {\arctan (a x)^3}{2 a^4 c}+\frac {x^2 \arctan (a x)^3}{2 a^2 c}+\frac {i \arctan (a x)^4}{4 a^4 c}-\frac {3 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{a^4 c}+\frac {\arctan (a x)^3 \log \left (\frac {2}{1+i a x}\right )}{a^4 c}-\frac {3 i \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{2 a^4 c}+\frac {3 i \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{2 a^4 c}+\frac {3 \arctan (a x) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{2 a^4 c}-\frac {3 i \operatorname {PolyLog}\left (4,1-\frac {2}{1+i a x}\right )}{4 a^4 c} \]

output

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

3.4.88.2 Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.62 \[ \int \frac {x^3 \arctan (a x)^3}{c+a^2 c x^2} \, dx=\frac {6 i \arctan (a x)^2-6 a x \arctan (a x)^2+2 \left (1+a^2 x^2\right ) \arctan (a x)^3-i \arctan (a x)^4-12 \arctan (a x) \log \left (1+e^{2 i \arctan (a x)}\right )+4 \arctan (a x)^3 \log \left (1+e^{2 i \arctan (a x)}\right )-6 i \left (-1+\arctan (a x)^2\right ) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )+6 \arctan (a x) \operatorname {PolyLog}\left (3,-e^{2 i \arctan (a x)}\right )+3 i \operatorname {PolyLog}\left (4,-e^{2 i \arctan (a x)}\right )}{4 a^4 c} \]

input

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

output

((6*I)*ArcTan[a*x]^2 - 6*a*x*ArcTan[a*x]^2 + 2*(1 + a^2*x^2)*ArcTan[a*x]^3 
 - I*ArcTan[a*x]^4 - 12*ArcTan[a*x]*Log[1 + E^((2*I)*ArcTan[a*x])] + 4*Arc 
Tan[a*x]^3*Log[1 + E^((2*I)*ArcTan[a*x])] - (6*I)*(-1 + ArcTan[a*x]^2)*Pol 
yLog[2, -E^((2*I)*ArcTan[a*x])] + 6*ArcTan[a*x]*PolyLog[3, -E^((2*I)*ArcTa 
n[a*x])] + (3*I)*PolyLog[4, -E^((2*I)*ArcTan[a*x])])/(4*a^4*c)
 

3.4.88.3 Rubi [A] (verified)

Time = 1.69 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.03, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.591, Rules used = {5451, 27, 5361, 5451, 5345, 5419, 5455, 5379, 2849, 2752, 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.

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

\(\Big \downarrow \) 5451

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5361

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

\(\Big \downarrow \) 5451

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

\(\Big \downarrow \) 5345

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

\(\Big \downarrow \) 5419

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

\(\Big \downarrow \) 5455

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

\(\Big \downarrow \) 5379

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

\(\Big \downarrow \) 2849

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

\(\Big \downarrow \) 2752

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

\(\Big \downarrow \) 5529

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

\(\Big \downarrow \) 5533

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

\(\Big \downarrow \) 7164

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

input

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

output

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

3.4.88.3.1 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 2752

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo 
g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
 

rule 2849

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Simp 
[-e/g   Subst[Int[Log[2*d*x]/(1 - 2*d*x), x], x, 1/(d + e*x)], x] /; FreeQ[ 
{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]
 

rule 5345

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a 
+ b*ArcTan[c*x^n])^p, x] - Simp[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 5361

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] - Simp[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 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 5419

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbo 
l] :> 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 5451

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

3.4.88.4 Maple [A] (verified)

Time = 39.60 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.00

method	result	size
derivativedivides	\(\frac {-\frac {i \arctan \left (a x \right )^{4}}{4 c}+\frac {\arctan \left (a x \right )^{2} \left (-i \arctan \left (a x \right )+x \arctan \left (a x \right ) a -3\right ) \left (a x +i\right )}{2 c}+\frac {\arctan \left (a x \right )^{3} \ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )}{c}-\frac {3 i \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2 c}+\frac {3 \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2 c}+\frac {3 i \operatorname {polylog}\left (4, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{4 c}+\frac {3 i \arctan \left (a x \right )^{2}}{c}-\frac {3 \arctan \left (a x \right ) \ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )}{c}+\frac {3 i \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2 c}}{a^{4}}\)	\(259\)
default	\(\frac {-\frac {i \arctan \left (a x \right )^{4}}{4 c}+\frac {\arctan \left (a x \right )^{2} \left (-i \arctan \left (a x \right )+x \arctan \left (a x \right ) a -3\right ) \left (a x +i\right )}{2 c}+\frac {\arctan \left (a x \right )^{3} \ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )}{c}-\frac {3 i \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2 c}+\frac {3 \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2 c}+\frac {3 i \operatorname {polylog}\left (4, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{4 c}+\frac {3 i \arctan \left (a x \right )^{2}}{c}-\frac {3 \arctan \left (a x \right ) \ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )}{c}+\frac {3 i \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2 c}}{a^{4}}\)	\(259\)

input

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

output

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

3.4.88.5 Fricas [F]

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

input

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

output

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

3.4.88.6 Sympy [F]

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

input

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

output

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

3.4.88.7 Maxima [F]

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

input

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

output

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

3.4.88.8 Giac [F]

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

input

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

output

sage0*x
 

3.4.88.9 Mupad [F(-1)]

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

input

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

output

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