\(\int \frac {(c+a^2 c x^2)^2 \arctan (a x)^3}{x^2} \, dx\) [376]

Optimal result

Integrand size = 22, antiderivative size = 284 \[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^3}{x^2} \, dx=a^2 c^2 x \arctan (a x)-\frac {1}{2} a c^2 \arctan (a x)^2-\frac {1}{2} a^3 c^2 x^2 \arctan (a x)^2+\frac {2}{3} i a c^2 \arctan (a x)^3-\frac {c^2 \arctan (a x)^3}{x}+2 a^2 c^2 x \arctan (a x)^3+\frac {1}{3} a^4 c^2 x^3 \arctan (a x)^3+5 a c^2 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )-\frac {1}{2} a c^2 \log \left (1+a^2 x^2\right )+3 a c^2 \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )-3 i a c^2 \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )+5 i a c^2 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\frac {3}{2} a c^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )+\frac {5}{2} a c^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right ) \] Output:

a^2*c^2*x*arctan(a*x)-1/2*a*c^2*arctan(a*x)^2-1/2*a^3*c^2*x^2*arctan(a*x)^ 
2+2/3*I*a*c^2*arctan(a*x)^3-c^2*arctan(a*x)^3/x+2*a^2*c^2*x*arctan(a*x)^3+ 
1/3*a^4*c^2*x^3*arctan(a*x)^3+5*a*c^2*arctan(a*x)^2*ln(2/(1+I*a*x))-1/2*a* 
c^2*ln(a^2*x^2+1)+3*a*c^2*arctan(a*x)^2*ln(2-2/(1-I*a*x))-3*I*a*c^2*arctan 
(a*x)*polylog(2,-1+2/(1-I*a*x))+5*I*a*c^2*arctan(a*x)*polylog(2,1-2/(1+I*a 
*x))+3/2*a*c^2*polylog(3,-1+2/(1-I*a*x))+5/2*a*c^2*polylog(3,1-2/(1+I*a*x) 
)
 

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 246, normalized size of antiderivative = 0.87 \[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^3}{x^2} \, dx=\frac {c^2 \left (-3 i a \pi ^3 x+24 a^2 x^2 \arctan (a x)-12 a x \arctan (a x)^2-12 a^3 x^3 \arctan (a x)^2-24 \arctan (a x)^3-16 i a x \arctan (a x)^3+48 a^2 x^2 \arctan (a x)^3+8 a^4 x^4 \arctan (a x)^3+72 a x \arctan (a x)^2 \log \left (1-e^{-2 i \arctan (a x)}\right )+120 a x \arctan (a x)^2 \log \left (1+e^{2 i \arctan (a x)}\right )-12 a x \log \left (1+a^2 x^2\right )+72 i a x \arctan (a x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )-120 i a x \arctan (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )+36 a x \operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )+60 a x \operatorname {PolyLog}\left (3,-e^{2 i \arctan (a x)}\right )\right )}{24 x} \] Input:

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

Output:

(c^2*((-3*I)*a*Pi^3*x + 24*a^2*x^2*ArcTan[a*x] - 12*a*x*ArcTan[a*x]^2 - 12 
*a^3*x^3*ArcTan[a*x]^2 - 24*ArcTan[a*x]^3 - (16*I)*a*x*ArcTan[a*x]^3 + 48* 
a^2*x^2*ArcTan[a*x]^3 + 8*a^4*x^4*ArcTan[a*x]^3 + 72*a*x*ArcTan[a*x]^2*Log 
[1 - E^((-2*I)*ArcTan[a*x])] + 120*a*x*ArcTan[a*x]^2*Log[1 + E^((2*I)*ArcT 
an[a*x])] - 12*a*x*Log[1 + a^2*x^2] + (72*I)*a*x*ArcTan[a*x]*PolyLog[2, E^ 
((-2*I)*ArcTan[a*x])] - (120*I)*a*x*ArcTan[a*x]*PolyLog[2, -E^((2*I)*ArcTa 
n[a*x])] + 36*a*x*PolyLog[3, E^((-2*I)*ArcTan[a*x])] + 60*a*x*PolyLog[3, - 
E^((2*I)*ArcTan[a*x])]))/(24*x)
 

Rubi [A] (verified)

Time = 1.06 (sec) , antiderivative size = 284, 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.

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

\(\Big \downarrow \) 5483

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{3} a^4 c^2 x^3 \arctan (a x)^3-\frac {1}{2} a^3 c^2 x^2 \arctan (a x)^2+2 a^2 c^2 x \arctan (a x)^3+a^2 c^2 x \arctan (a x)-\frac {1}{2} a c^2 \log \left (a^2 x^2+1\right )-3 i a c^2 \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )+5 i a c^2 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )+\frac {2}{3} i a c^2 \arctan (a x)^3-\frac {1}{2} a c^2 \arctan (a x)^2-\frac {c^2 \arctan (a x)^3}{x}+5 a c^2 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )+3 a c^2 \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )+\frac {3}{2} a c^2 \operatorname {PolyLog}\left (3,\frac {2}{1-i a x}-1\right )+\frac {5}{2} a c^2 \operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right )\)

Input:

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

Output:

a^2*c^2*x*ArcTan[a*x] - (a*c^2*ArcTan[a*x]^2)/2 - (a^3*c^2*x^2*ArcTan[a*x] 
^2)/2 + ((2*I)/3)*a*c^2*ArcTan[a*x]^3 - (c^2*ArcTan[a*x]^3)/x + 2*a^2*c^2* 
x*ArcTan[a*x]^3 + (a^4*c^2*x^3*ArcTan[a*x]^3)/3 + 5*a*c^2*ArcTan[a*x]^2*Lo 
g[2/(1 + I*a*x)] - (a*c^2*Log[1 + a^2*x^2])/2 + 3*a*c^2*ArcTan[a*x]^2*Log[ 
2 - 2/(1 - I*a*x)] - (3*I)*a*c^2*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x) 
] + (5*I)*a*c^2*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)] + (3*a*c^2*PolyL 
og[3, -1 + 2/(1 - I*a*x)])/2 + (5*a*c^2*PolyLog[3, 1 - 2/(1 + I*a*x)])/2
 

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 [C] (warning: unable to verify)

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

Time = 307.72 (sec) , antiderivative size = 1851, normalized size of antiderivative = 6.52

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1851\)
default	\(\text {Expression too large to display}\)	\(1851\)
parts	\(\text {Expression too large to display}\)	\(1851\)

Input:

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

Output:

a*(1/3*c^2*arctan(a*x)^3*a^3*x^3+2*c^2*arctan(a*x)^3*a*x-c^2*arctan(a*x)^3 
/a/x-c^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))*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2 
+1)+1))*arctan(a*x)^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)*arctan(a*x)^2+1/2*arctan(a*x)^2+1/2*arctan(a*x)^2*x^2*a^2-3/2 
*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))^3*ar 
ctan(a*x)^2-3/2*I*Pi*csgn(((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^ 
2+1)+1))^3*arctan(a*x)^2+3/2*I*Pi*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+6*I*arctan(a*x)*polylog(2,(1+I*a*x)/ 
(a^2*x^2+1)^(1/2))-6*polylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))-6*polylog(3,-( 
1+I*a*x)/(a^2*x^2+1)^(1/2))+6*I*arctan(a*x)*polylog(2,-(1+I*a*x)/(a^2*x^2+ 
1)^(1/2))-arctan(a*x)*(a*x-I)+5*I*arctan(a*x)*polylog(2,-(1+I*a*x)^2/(a^2* 
x^2+1))-3/2*I*Pi*arctan(a*x)^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*arctan(a*x)^2+2*I*Pi*csgn(I*(1+I*a*x)^2/(a^2*x 
^2+1))^3*arctan(a*x)^2-2*I*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^3*arct 
an(a*x)^2+8/3*I*arctan(a*x)^3-8*arctan(a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/ 
2))-5/2*polylog(3,-(1+I*a*x)^2/(a^2*x^2+1))-ln((1+I*a*x)^2/(a^2*x^2+1)+1)+ 
3/2*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))*c 
sgn(((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...
 

Fricas [F]

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

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

1/96*(4*(a^4*c^2*x^4 + 6*a^2*c^2*x^2 - 3*c^2)*arctan(a*x)^3 - 3*(a^4*c^2*x 
^4 + 6*a^2*c^2*x^2 - 3*c^2)*arctan(a*x)*log(a^2*x^2 + 1)^2 + 3*(896*a^6*c^ 
2*integrate(1/32*x^6*arctan(a*x)^3/(a^2*x^4 + x^2), x) + 96*a^6*c^2*integr 
ate(1/32*x^6*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^2*x^4 + x^2), x) + 128*a^6* 
c^2*integrate(1/32*x^6*arctan(a*x)*log(a^2*x^2 + 1)/(a^2*x^4 + x^2), x) - 
128*a^5*c^2*integrate(1/32*x^5*arctan(a*x)^2/(a^2*x^4 + x^2), x) + 32*a^5* 
c^2*integrate(1/32*x^5*log(a^2*x^2 + 1)^2/(a^2*x^4 + x^2), x) + 21*a*c^2*a 
rctan(a*x)^4 + 2688*a^4*c^2*integrate(1/32*x^4*arctan(a*x)^3/(a^2*x^4 + x^ 
2), x) + 288*a^4*c^2*integrate(1/32*x^4*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^ 
2*x^4 + x^2), x) + 768*a^4*c^2*integrate(1/32*x^4*arctan(a*x)*log(a^2*x^2 
+ 1)/(a^2*x^4 + x^2), x) - 768*a^3*c^2*integrate(1/32*x^3*arctan(a*x)^2/(a 
^2*x^4 + x^2), x) + a*c^2*log(a^2*x^2 + 1)^3 + 288*a^2*c^2*integrate(1/32* 
x^2*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^2*x^4 + x^2), x) - 384*a^2*c^2*integ 
rate(1/32*x^2*arctan(a*x)*log(a^2*x^2 + 1)/(a^2*x^4 + x^2), x) + 384*a*c^2 
*integrate(1/32*x*arctan(a*x)^2/(a^2*x^4 + x^2), x) - 96*a*c^2*integrate(1 
/32*x*log(a^2*x^2 + 1)^2/(a^2*x^4 + x^2), x) + 896*c^2*integrate(1/32*arct 
an(a*x)^3/(a^2*x^4 + x^2), x) + 96*c^2*integrate(1/32*arctan(a*x)*log(a^2* 
x^2 + 1)^2/(a^2*x^4 + x^2), x))*x)/x
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^3}{x^2} \, dx=\frac {c^{2} \left (2 \mathit {atan} \left (a x \right )^{3} a^{4} x^{4}+12 \mathit {atan} \left (a x \right )^{3} a^{2} x^{2}-6 \mathit {atan} \left (a x \right )^{3}-3 \mathit {atan} \left (a x \right )^{2} a^{3} x^{3}-3 \mathit {atan} \left (a x \right )^{2} a x +6 \mathit {atan} \left (a x \right ) a^{2} x^{2}+18 \left (\int \frac {\mathit {atan} \left (a x \right )^{2}}{a^{2} x^{3}+x}d x \right ) a x -30 \left (\int \frac {\mathit {atan} \left (a x \right )^{2} x}{a^{2} x^{2}+1}d x \right ) a^{3} x -3 \,\mathrm {log}\left (a^{2} x^{2}+1\right ) a x \right )}{6 x} \] Input:

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

Output:

(c**2*(2*atan(a*x)**3*a**4*x**4 + 12*atan(a*x)**3*a**2*x**2 - 6*atan(a*x)* 
*3 - 3*atan(a*x)**2*a**3*x**3 - 3*atan(a*x)**2*a*x + 6*atan(a*x)*a**2*x**2 
 + 18*int(atan(a*x)**2/(a**2*x**3 + x),x)*a*x - 30*int((atan(a*x)**2*x)/(a 
**2*x**2 + 1),x)*a**3*x - 3*log(a**2*x**2 + 1)*a*x))/(6*x)