\(\int \frac {\arctan (a x)^3}{x^4 (c+a^2 c x^2)} \, dx\) [395]

Optimal result

Integrand size = 22, antiderivative size = 227 \[ \int \frac {\arctan (a x)^3}{x^4 \left (c+a^2 c x^2\right )} \, dx=-\frac {a^2 \arctan (a x)}{c x}-\frac {a^3 \arctan (a x)^2}{2 c}-\frac {a \arctan (a x)^2}{2 c x^2}+\frac {4 i a^3 \arctan (a x)^3}{3 c}-\frac {\arctan (a x)^3}{3 c x^3}+\frac {a^2 \arctan (a x)^3}{c x}+\frac {a^3 \arctan (a x)^4}{4 c}+\frac {a^3 \log (x)}{c}-\frac {a^3 \log \left (1+a^2 x^2\right )}{2 c}-\frac {4 a^3 \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c}+\frac {4 i a^3 \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{c}-\frac {2 a^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )}{c} \] Output:

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

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.79 \[ \int \frac {\arctan (a x)^3}{x^4 \left (c+a^2 c x^2\right )} \, dx=\frac {a^3 \left (\frac {i \pi ^3}{6}-\frac {\arctan (a x)}{a x}-\frac {1}{2} \arctan (a x)^2-\frac {\arctan (a x)^2}{2 a^2 x^2}-\frac {4}{3} i \arctan (a x)^3-\frac {\arctan (a x)^3}{3 a^3 x^3}+\frac {\arctan (a x)^3}{a x}+\frac {1}{4} \arctan (a x)^4-4 \arctan (a x)^2 \log \left (1-e^{-2 i \arctan (a x)}\right )+\log \left (\frac {a x}{\sqrt {1+a^2 x^2}}\right )-4 i \arctan (a x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )-2 \operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )\right )}{c} \] Input:

Integrate[ArcTan[a*x]^3/(x^4*(c + a^2*c*x^2)),x]
 

Output:

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

Rubi [A] (verified)

Time = 2.48 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.41, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.773, Rules used = {5453, 27, 5361, 5453, 5361, 5419, 5453, 5361, 243, 47, 14, 16, 5419, 5459, 5403, 5527, 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.

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

\(\Big \downarrow \) 5453

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5361

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

\(\Big \downarrow \) 5453

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

\(\Big \downarrow \) 5361

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

\(\Big \downarrow \) 5419

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

\(\Big \downarrow \) 5453

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

\(\Big \downarrow \) 5361

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 47

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

\(\Big \downarrow \) 14

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 5419

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

\(\Big \downarrow \) 5459

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

\(\Big \downarrow \) 5403

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

\(\Big \downarrow \) 5527

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

\(\Big \downarrow \) 7164

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

Input:

Int[ArcTan[a*x]^3/(x^4*(c + a^2*c*x^2)),x]
 

Output:

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

Defintions of rubi rules used

rule 14

Int[(a_.)/(x_), x_Symbol] :> Simp[a*Log[x], x] /; FreeQ[a, x]
 

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

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 47

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Simp[b/(b*c 
 - a*d)   Int[1/(a + b*x), x], x] - Simp[d/(b*c - a*d)   Int[1/(c + d*x), x 
], x] /; FreeQ[{a, b, c, d}, x]
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

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 5403

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x_ 
Symbol] :> Simp[(a + b*ArcTan[c*x])^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] - Si 
mp[b*c*(p/d)   Int[(a + b*ArcTan[c*x])^(p - 1)*(Log[2 - 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 5453

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e 
_.)*(x_)^2), x_Symbol] :> Simp[1/d   Int[(f*x)^m*(a + b*ArcTan[c*x])^p, x], 
 x] - Simp[e/(d*f^2)   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] && LtQ[m, -1]
 

rule 5459

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

rule 5527

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 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 = 5.87 (sec) , antiderivative size = 1872, normalized size of antiderivative = 8.25

\[\text {Expression too large to display}\]

Input:

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

Output:

a^3*(1/c*arctan(a*x)^4-1/3/c*arctan(a*x)^3/a^3/x^3+1/c*arctan(a*x)^3/a/x-1 
/c*(1/2*arctan(a*x)^2-ln((1+I*a*x)/(a^2*x^2+1)^(1/2)-1)-ln(1+(1+I*a*x)/(a^ 
2*x^2+1)^(1/2))+8*polylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))+8*polylog(3,-(1+I 
*a*x)/(a^2*x^2+1)^(1/2))-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*arctan(a*x)^2-I*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))^3*a 
rctan(a*x)^2+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*arctan(a*x)^2+I*Pi*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)*c 
sgn(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+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)^2-2*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)/((1+I*a*x)^2/(a^2*x^2+1)+1 
))^2*arctan(a*x)^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)*arctan(a*x)^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))*arctan(a*x)^2-2*I*Pi*csgn(I*((1+I 
*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))*csgn(((1+I*a*x)^2/(a^2 
*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))^2*arctan(a*x)^2+4*arctan(a*x)^2*ln 
((1+I*a*x)/(a^2*x^2+1)^(1/2))-4*arctan(a*x)^2*ln((1+I*a*x)^2/(a^2*x^2+1)-1 
)+1/2*arctan(a*x)*(I*a*x-(a^2*x^2+1)^(1/2)+1)/a/x+1/2*arctan(a*x)*(I*a*x+( 
a^2*x^2+1)^(1/2)+1)/a/x-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*arctan(a*x)^2-2*I*Pi*csgn(I/((1+I*a*x)...
 

Fricas [F]

\[ \int \frac {\arctan (a x)^3}{x^4 \left (c+a^2 c x^2\right )} \, dx=\int { \frac {\arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )} x^{4}} \,d x } \] Input:

integrate(arctan(a*x)^3/x^4/(a^2*c*x^2+c),x, algorithm="fricas")
 

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-1)]

Timed out. \[ \int \frac {\arctan (a x)^3}{x^4 \left (c+a^2 c x^2\right )} \, dx=\text {Timed out} \] Input:

integrate(arctan(a*x)^3/x^4/(a^2*c*x^2+c),x, algorithm="maxima")
 

Output:

Timed out
 

Giac [F]

\[ \int \frac {\arctan (a x)^3}{x^4 \left (c+a^2 c x^2\right )} \, dx=\int { \frac {\arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )} x^{4}} \,d x } \] Input:

integrate(arctan(a*x)^3/x^4/(a^2*c*x^2+c),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\arctan (a x)^3}{x^4 \left (c+a^2 c x^2\right )} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^3}{x^4\,\left (c\,a^2\,x^2+c\right )} \,d x \] Input:

int(atan(a*x)^3/(x^4*(c + a^2*c*x^2)),x)
 

Output:

int(atan(a*x)^3/(x^4*(c + a^2*c*x^2)), x)
 

Reduce [F]

\[ \int \frac {\arctan (a x)^3}{x^4 \left (c+a^2 c x^2\right )} \, dx=\frac {3 \mathit {atan} \left (a x \right )^{4} a^{3} x^{3}+12 \mathit {atan} \left (a x \right )^{3} a^{2} x^{2}-4 \mathit {atan} \left (a x \right )^{3}+18 \mathit {atan} \left (a x \right )^{2} a^{3} x^{3}+18 \mathit {atan} \left (a x \right )^{2} a x +36 \mathit {atan} \left (a x \right ) a^{2} x^{2}+48 \left (\int \frac {\mathit {atan} \left (a x \right )^{2}}{a^{2} x^{5}+x^{3}}d x \right ) a \,x^{3}+18 \,\mathrm {log}\left (a^{2} x^{2}+1\right ) a^{3} x^{3}-36 \,\mathrm {log}\left (x \right ) a^{3} x^{3}}{12 c \,x^{3}} \] Input:

int(atan(a*x)^3/x^4/(a^2*c*x^2+c),x)
 

Output:

(3*atan(a*x)**4*a**3*x**3 + 12*atan(a*x)**3*a**2*x**2 - 4*atan(a*x)**3 + 1 
8*atan(a*x)**2*a**3*x**3 + 18*atan(a*x)**2*a*x + 36*atan(a*x)*a**2*x**2 + 
48*int(atan(a*x)**2/(a**2*x**5 + x**3),x)*a*x**3 + 18*log(a**2*x**2 + 1)*a 
**3*x**3 - 36*log(x)*a**3*x**3)/(12*c*x**3)