\(\int \frac {(c+a^2 c x^2)^2 \arctan (a x)^3}{x^3} \, dx\) [377]

Optimal result

Integrand size = 22, antiderivative size = 399 \[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^3}{x^3} \, dx=-3 i a^2 c^2 \arctan (a x)^2-\frac {3 a c^2 \arctan (a x)^2}{2 x}-\frac {3}{2} a^3 c^2 x \arctan (a x)^2-\frac {c^2 \arctan (a x)^3}{2 x^2}+\frac {1}{2} a^4 c^2 x^2 \arctan (a x)^3+4 a^2 c^2 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-3 a^2 c^2 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )+3 a^2 c^2 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {3}{2} i a^2 c^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )-\frac {3}{2} i a^2 c^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )-3 i a^2 c^2 \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+3 i a^2 c^2 \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-3 a^2 c^2 \arctan (a x) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+3 a^2 c^2 \arctan (a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right )+\frac {3}{2} i a^2 c^2 \operatorname {PolyLog}\left (4,1-\frac {2}{1+i a x}\right )-\frac {3}{2} i a^2 c^2 \operatorname {PolyLog}\left (4,-1+\frac {2}{1+i a x}\right ) \] Output:

3/2*I*a^2*c^2*polylog(4,1-2/(1+I*a*x))-3/2*a*c^2*arctan(a*x)^2/x-3/2*a^3*c 
^2*x*arctan(a*x)^2-1/2*c^2*arctan(a*x)^3/x^2+1/2*a^4*c^2*x^2*arctan(a*x)^3 
-4*a^2*c^2*arctan(a*x)^3*arctanh(-1+2/(1+I*a*x))-3*a^2*c^2*arctan(a*x)*ln( 
2/(1+I*a*x))+3*a^2*c^2*arctan(a*x)*ln(2-2/(1-I*a*x))-3*I*a^2*c^2*arctan(a* 
x)^2-3*I*a^2*c^2*arctan(a*x)^2*polylog(2,1-2/(1+I*a*x))-3/2*I*a^2*c^2*poly 
log(2,1-2/(1+I*a*x))-3/2*I*a^2*c^2*polylog(4,-1+2/(1+I*a*x))-3*a^2*c^2*arc 
tan(a*x)*polylog(3,1-2/(1+I*a*x))+3*a^2*c^2*arctan(a*x)*polylog(3,-1+2/(1+ 
I*a*x))-3/2*I*a^2*c^2*polylog(2,-1+2/(1-I*a*x))+3*I*a^2*c^2*arctan(a*x)^2* 
polylog(2,-1+2/(1+I*a*x))
 

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 302, normalized size of antiderivative = 0.76 \[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^3}{x^3} \, dx=\frac {1}{32} a^2 c^2 \left (-i \pi ^4-\frac {48 \arctan (a x)^2}{a x}-48 a x \arctan (a x)^2-\frac {16 \arctan (a x)^3}{a^2 x^2}+16 a^2 x^2 \arctan (a x)^3+32 i \arctan (a x)^4+64 \arctan (a x)^3 \log \left (1-e^{-2 i \arctan (a x)}\right )+96 \arctan (a x) \log \left (1-e^{2 i \arctan (a x)}\right )-96 \arctan (a x) \log \left (1+e^{2 i \arctan (a x)}\right )-64 \arctan (a x)^3 \log \left (1+e^{2 i \arctan (a x)}\right )+96 i \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )+48 i \left (1+2 \arctan (a x)^2\right ) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )-48 i \operatorname {PolyLog}\left (2,e^{2 i \arctan (a x)}\right )+96 \arctan (a x) \operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )-96 \arctan (a x) \operatorname {PolyLog}\left (3,-e^{2 i \arctan (a x)}\right )-48 i \operatorname {PolyLog}\left (4,e^{-2 i \arctan (a x)}\right )-48 i \operatorname {PolyLog}\left (4,-e^{2 i \arctan (a x)}\right )\right ) \] Input:

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

Output:

(a^2*c^2*((-I)*Pi^4 - (48*ArcTan[a*x]^2)/(a*x) - 48*a*x*ArcTan[a*x]^2 - (1 
6*ArcTan[a*x]^3)/(a^2*x^2) + 16*a^2*x^2*ArcTan[a*x]^3 + (32*I)*ArcTan[a*x] 
^4 + 64*ArcTan[a*x]^3*Log[1 - E^((-2*I)*ArcTan[a*x])] + 96*ArcTan[a*x]*Log 
[1 - E^((2*I)*ArcTan[a*x])] - 96*ArcTan[a*x]*Log[1 + E^((2*I)*ArcTan[a*x]) 
] - 64*ArcTan[a*x]^3*Log[1 + E^((2*I)*ArcTan[a*x])] + (96*I)*ArcTan[a*x]^2 
*PolyLog[2, E^((-2*I)*ArcTan[a*x])] + (48*I)*(1 + 2*ArcTan[a*x]^2)*PolyLog 
[2, -E^((2*I)*ArcTan[a*x])] - (48*I)*PolyLog[2, E^((2*I)*ArcTan[a*x])] + 9 
6*ArcTan[a*x]*PolyLog[3, E^((-2*I)*ArcTan[a*x])] - 96*ArcTan[a*x]*PolyLog[ 
3, -E^((2*I)*ArcTan[a*x])] - (48*I)*PolyLog[4, E^((-2*I)*ArcTan[a*x])] - ( 
48*I)*PolyLog[4, -E^((2*I)*ArcTan[a*x])]))/32
 

Rubi [A] (verified)

Time = 1.13 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {5483, 2009}

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.

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

\(\Big \downarrow \) 5483

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 5483

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_)*((d_) + (e_. 
)*(x_)^2)^(q_), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + 
 b*ArcTan[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2* 
d] && IGtQ[p, 0] && IGtQ[q, 1] && (EqQ[p, 1] || IntegerQ[m])
 

Maple [A] (verified)

Time = 71.29 (sec) , antiderivative size = 622, normalized size of antiderivative = 1.56

method	result	size
derivativedivides	\(a^{2} \left (\frac {c^{2} \arctan \left (a x \right )^{2} \left (x^{2} a^{2} \arctan \left (a x \right )-\arctan \left (a x \right )-3 a x \right ) \left (a x -i\right ) \left (a x +i\right )}{2 a^{2} x^{2}}+2 c^{2} \arctan \left (a x \right )^{3} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+\frac {3 i c^{2} \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}-3 c^{2} \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )+12 i c^{2} \operatorname {polylog}\left (4, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+12 c^{2} \arctan \left (a x \right ) \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-6 i c^{2} \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+3 c^{2} \arctan \left (a x \right ) \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-3 i c^{2} \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+12 c^{2} \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-3 i c^{2} \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+2 c^{2} \arctan \left (a x \right )^{3} \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-\frac {3 i c^{2} \operatorname {polylog}\left (4, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}+3 c^{2} \arctan \left (a x \right ) \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-6 i c^{2} \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-2 c^{2} \arctan \left (a x \right )^{3} \ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )+3 i c^{2} \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )-3 c^{2} \arctan \left (a x \right ) \ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )+12 i c^{2} \operatorname {polylog}\left (4, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right )\)	\(622\)
default	\(a^{2} \left (\frac {c^{2} \arctan \left (a x \right )^{2} \left (x^{2} a^{2} \arctan \left (a x \right )-\arctan \left (a x \right )-3 a x \right ) \left (a x -i\right ) \left (a x +i\right )}{2 a^{2} x^{2}}+2 c^{2} \arctan \left (a x \right )^{3} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+\frac {3 i c^{2} \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}-3 c^{2} \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )+12 i c^{2} \operatorname {polylog}\left (4, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+12 c^{2} \arctan \left (a x \right ) \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-6 i c^{2} \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+3 c^{2} \arctan \left (a x \right ) \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-3 i c^{2} \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+12 c^{2} \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-3 i c^{2} \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+2 c^{2} \arctan \left (a x \right )^{3} \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-\frac {3 i c^{2} \operatorname {polylog}\left (4, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}+3 c^{2} \arctan \left (a x \right ) \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-6 i c^{2} \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-2 c^{2} \arctan \left (a x \right )^{3} \ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )+3 i c^{2} \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )-3 c^{2} \arctan \left (a x \right ) \ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )+12 i c^{2} \operatorname {polylog}\left (4, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right )\)	\(622\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

1/64*(12*a^4*c^2*x^2*integrate(4*x*arctan(a*x)^3 + x*arctan(a*x)*log(a^2*x 
^2 + 1)^2, x) + 8*a^3*c^2*x^2*integrate(-1/8*(24*(a^2*x^2 + 1)*a*x*arctan( 
a*x)^3 - 18*(a^2*x^2 + 1)*a*x*arctan(a*x)*log(a^2*x^2 + 1)^2 + 36*(a^2*x^2 
 + 1)*arctan(a*x)^2*log(a^2*x^2 + 1) - 3*(a^2*x^2 + 1)*log(a^2*x^2 + 1)^3 
- sqrt(a^2*x^2 + 1)*(12*sqrt(a^2*x^2 + 1)*arctan(a*x)^2*log(a^2*x^2 + 1) - 
 sqrt(a^2*x^2 + 1)*log(a^2*x^2 + 1)^3 - (12*(a^2*x^2 + 1)^2*arctan(a*x)^2* 
log(a^2*x^2 + 1) - (a^2*x^2 + 1)^2*log(a^2*x^2 + 1)^3)*cos(3*arctan(a*x)) 
+ 3*(12*(a^2*x^2 + 1)^(3/2)*arctan(a*x)^2*log(a^2*x^2 + 1) - (a^2*x^2 + 1) 
^(3/2)*log(a^2*x^2 + 1)^3)*cos(2*arctan(a*x)) - 2*(4*(a^2*x^2 + 1)^2*arcta 
n(a*x)^3 - 3*(a^2*x^2 + 1)^2*arctan(a*x)*log(a^2*x^2 + 1)^2)*sin(3*arctan( 
a*x)) + 6*(4*(a^2*x^2 + 1)^(3/2)*arctan(a*x)^3 - 3*(a^2*x^2 + 1)^(3/2)*arc 
tan(a*x)*log(a^2*x^2 + 1)^2)*sin(2*arctan(a*x))))/((a^2*x^2 + 1)^4*cos(3*a 
rctan(a*x))^2 + (a^2*x^2 + 1)^4*sin(3*arctan(a*x))^2 - 6*(a^2*x^2 + 1)^(7/ 
2)*sin(3*arctan(a*x))*sin(2*arctan(a*x)) + 9*(a^2*x^2 + 1)^3*cos(2*arctan( 
a*x))^2 + 9*(a^2*x^2 + 1)^3*sin(2*arctan(a*x))^2 + a^2*x^2 + 6*(a^2*x^2 + 
1)^2*cos(2*arctan(a*x)) + 9*(a^2*x^2 + 1)^2 - 2*(3*(a^2*x^2 + 1)^(7/2)*cos 
(2*arctan(a*x)) + (a^2*x^2 + 1)^(5/2))*cos(3*arctan(a*x)) + 6*((a^2*x^2 + 
1)^2*a*x*sin(3*arctan(a*x)) - 3*(a^2*x^2 + 1)^(3/2)*a*x*sin(2*arctan(a*x)) 
 + (a^2*x^2 + 1)^2*cos(3*arctan(a*x)) - 3*(a^2*x^2 + 1)^(3/2)*cos(2*arctan 
(a*x)) - sqrt(a^2*x^2 + 1))*sqrt(a^2*x^2 + 1) + 1), x) - 12*a^3*c^2*x^2...
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^3}{x^3} \, dx=\frac {c^{2} \left (\mathit {atan} \left (a x \right )^{3} a^{4} x^{4}-\mathit {atan} \left (a x \right )^{3}-3 \mathit {atan} \left (a x \right )^{2} a^{3} x^{3}-3 \mathit {atan} \left (a x \right )^{2} a x +3 \mathit {atan} \left (a x \right ) a^{4} x^{4}-3 \mathit {atan} \left (a x \right )-6 \left (\int \frac {\mathit {atan} \left (a x \right )}{a^{2} x^{5}+x^{3}}d x \right ) x^{2}+4 \left (\int \frac {\mathit {atan} \left (a x \right )^{3}}{x}d x \right ) a^{2} x^{2}-6 \left (\int \frac {\mathit {atan} \left (a x \right ) x^{3}}{a^{2} x^{2}+1}d x \right ) a^{6} x^{2}-3 a^{3} x^{3}-3 a x \right )}{2 x^{2}} \] Input:

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

Output:

(c**2*(atan(a*x)**3*a**4*x**4 - atan(a*x)**3 - 3*atan(a*x)**2*a**3*x**3 - 
3*atan(a*x)**2*a*x + 3*atan(a*x)*a**4*x**4 - 3*atan(a*x) - 6*int(atan(a*x) 
/(a**2*x**5 + x**3),x)*x**2 + 4*int(atan(a*x)**3/x,x)*a**2*x**2 - 6*int((a 
tan(a*x)*x**3)/(a**2*x**2 + 1),x)*a**6*x**2 - 3*a**3*x**3 - 3*a*x))/(2*x** 
2)