\(\int \frac {x \arctan (a x)^3}{c+a^2 c x^2} \, dx\) [390]

Optimal result

Integrand size = 20, antiderivative size = 138 \[ \int \frac {x \arctan (a x)^3}{c+a^2 c x^2} \, dx=-\frac {i \arctan (a x)^4}{4 a^2 c}-\frac {\arctan (a x)^3 \log \left (\frac {2}{1+i a x}\right )}{a^2 c}-\frac {3 i \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{2 a^2 c}-\frac {3 \arctan (a x) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{2 a^2 c}+\frac {3 i \operatorname {PolyLog}\left (4,1-\frac {2}{1+i a x}\right )}{4 a^2 c} \] Output:

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.86 \[ \int \frac {x \arctan (a x)^3}{c+a^2 c x^2} \, dx=-\frac {i \left (\arctan (a x)^4-4 i \arctan (a x)^3 \log \left (\frac {2 i}{i-a x}\right )+6 \arctan (a x)^2 \operatorname {PolyLog}\left (2,\frac {i+a x}{-i+a x}\right )-6 i \arctan (a x) \operatorname {PolyLog}\left (3,\frac {i+a x}{-i+a x}\right )-3 \operatorname {PolyLog}\left (4,\frac {i+a x}{-i+a x}\right )\right )}{4 a^2 c} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5455, 5379, 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.

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

\(\Big \downarrow \) 5455

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

\(\Big \downarrow \) 5379

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

\(\Big \downarrow \) 5529

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

\(\Big \downarrow \) 5533

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

\(\Big \downarrow \) 7164

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

Input:

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

Output:

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

Defintions of rubi rules used

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 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]
 

Maple [C] (warning: unable to verify)

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

Time = 69.65 (sec) , antiderivative size = 789, normalized size of antiderivative = 5.72

method	result	size
derivativedivides	\(\frac {\frac {\ln \left (a^{2} x^{2}+1\right ) \arctan \left (a x \right )^{3}}{2 c}-\frac {3 \left (\frac {2 \arctan \left (a x \right )^{3} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{3}-\frac {i \arctan \left (a x \right )^{4}}{6}+\frac {\left (-i \pi \,\operatorname {csgn}\left (\frac {i}{\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 (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 )+i \pi \,\operatorname {csgn}\left (\frac {i}{\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}}{\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}+i \pi {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )\right )}^{2} \operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )-2 i \pi \,\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )\right ) {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )}^{2}+i \pi {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )}^{3}-i \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 )+2 i \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}-i \pi \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )^{3}+i \pi \,\operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right ) \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}-i \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}+4 \ln \left (2\right )\right ) \arctan \left (a x \right )^{3}}{6}-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 )+\arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )+\frac {i \operatorname {polylog}\left (4, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}\right )}{2 c}}{a^{2}}\)	\(789\)
default	\(\frac {\frac {\ln \left (a^{2} x^{2}+1\right ) \arctan \left (a x \right )^{3}}{2 c}-\frac {3 \left (\frac {2 \arctan \left (a x \right )^{3} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{3}-\frac {i \arctan \left (a x \right )^{4}}{6}+\frac {\left (-i \pi \,\operatorname {csgn}\left (\frac {i}{\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 (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 )+i \pi \,\operatorname {csgn}\left (\frac {i}{\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}}{\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}+i \pi {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )\right )}^{2} \operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )-2 i \pi \,\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )\right ) {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )}^{2}+i \pi {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )}^{3}-i \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 )+2 i \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}-i \pi \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )^{3}+i \pi \,\operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right ) \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}-i \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}+4 \ln \left (2\right )\right ) \arctan \left (a x \right )^{3}}{6}-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 )+\arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )+\frac {i \operatorname {polylog}\left (4, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}\right )}{2 c}}{a^{2}}\)	\(789\)
parts	\(\frac {\ln \left (a^{2} x^{2}+1\right ) \arctan \left (a x \right )^{3}}{2 c \,a^{2}}-\frac {3 \left (\frac {2 \arctan \left (a x \right )^{3} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{3 a}-\frac {i \arctan \left (a x \right )^{4}}{6 a}+\frac {\left (-i \pi \,\operatorname {csgn}\left (\frac {i}{\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 (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 )+i \pi \,\operatorname {csgn}\left (\frac {i}{\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}}{\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}+i \pi {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )\right )}^{2} \operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )-2 i \pi \,\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )\right ) {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )}^{2}+i \pi {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )}^{3}-i \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 )+2 i \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}-i \pi \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )^{3}+i \pi \,\operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right ) \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}-i \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}+4 \ln \left (2\right )\right ) \arctan \left (a x \right )^{3}}{6 a}-\frac {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 )}{a}+\frac {\arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{a}+\frac {i \operatorname {polylog}\left (4, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2 a}\right )}{2 c a}\)	\(809\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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