\(\int \frac {(c+a^2 c x^2)^2 \arctan (a x)^2}{x^4} \, dx\) [273]

Optimal result

Integrand size = 22, antiderivative size = 216 \[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^2}{x^4} \, dx=-\frac {a^2 c^2}{3 x}-\frac {1}{3} a^3 c^2 \arctan (a x)-\frac {a c^2 \arctan (a x)}{3 x^2}-\frac {2}{3} i a^3 c^2 \arctan (a x)^2-\frac {c^2 \arctan (a x)^2}{3 x^3}-\frac {2 a^2 c^2 \arctan (a x)^2}{x}+a^4 c^2 x \arctan (a x)^2+2 a^3 c^2 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )+\frac {10}{3} a^3 c^2 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {5}{3} i a^3 c^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )+i a^3 c^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right ) \] Output:

-1/3*a^2*c^2/x-1/3*a^3*c^2*arctan(a*x)-1/3*a*c^2*arctan(a*x)/x^2-2/3*I*a^3 
*c^2*arctan(a*x)^2-1/3*c^2*arctan(a*x)^2/x^3-2*a^2*c^2*arctan(a*x)^2/x+a^4 
*c^2*x*arctan(a*x)^2+2*a^3*c^2*arctan(a*x)*ln(2/(1+I*a*x))+10/3*a^3*c^2*ar 
ctan(a*x)*ln(2-2/(1-I*a*x))-5/3*I*a^3*c^2*polylog(2,-1+2/(1-I*a*x))+I*a^3* 
c^2*polylog(2,1-2/(1+I*a*x))
 

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.88 \[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^2}{x^4} \, dx=\frac {c^2 \left (-a^2 x^2-a x \arctan (a x)-a^3 x^3 \arctan (a x)-\arctan (a x)^2-6 a^2 x^2 \arctan (a x)^2-8 i a^3 x^3 \arctan (a x)^2+3 a^4 x^4 \arctan (a x)^2+10 a^3 x^3 \arctan (a x) \log \left (1-e^{2 i \arctan (a x)}\right )+6 a^3 x^3 \arctan (a x) \log \left (1+e^{2 i \arctan (a x)}\right )-3 i a^3 x^3 \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )-5 i a^3 x^3 \operatorname {PolyLog}\left (2,e^{2 i \arctan (a x)}\right )\right )}{3 x^3} \] Input:

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

Output:

(c^2*(-(a^2*x^2) - a*x*ArcTan[a*x] - a^3*x^3*ArcTan[a*x] - ArcTan[a*x]^2 - 
 6*a^2*x^2*ArcTan[a*x]^2 - (8*I)*a^3*x^3*ArcTan[a*x]^2 + 3*a^4*x^4*ArcTan[ 
a*x]^2 + 10*a^3*x^3*ArcTan[a*x]*Log[1 - E^((2*I)*ArcTan[a*x])] + 6*a^3*x^3 
*ArcTan[a*x]*Log[1 + E^((2*I)*ArcTan[a*x])] - (3*I)*a^3*x^3*PolyLog[2, -E^ 
((2*I)*ArcTan[a*x])] - (5*I)*a^3*x^3*PolyLog[2, E^((2*I)*ArcTan[a*x])]))/( 
3*x^3)
 

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 216, 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.

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

\(\Big \downarrow \) 5483

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

\(\Big \downarrow \) 2009

\(\displaystyle a^4 c^2 x \arctan (a x)^2-\frac {2}{3} i a^3 c^2 \arctan (a x)^2-\frac {1}{3} a^3 c^2 \arctan (a x)+2 a^3 c^2 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )+\frac {10}{3} a^3 c^2 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {5}{3} i a^3 c^2 \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )+i a^3 c^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )-\frac {2 a^2 c^2 \arctan (a x)^2}{x}-\frac {a^2 c^2}{3 x}-\frac {c^2 \arctan (a x)^2}{3 x^3}-\frac {a c^2 \arctan (a x)}{3 x^2}\)

Input:

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

Output:

-1/3*(a^2*c^2)/x - (a^3*c^2*ArcTan[a*x])/3 - (a*c^2*ArcTan[a*x])/(3*x^2) - 
 ((2*I)/3)*a^3*c^2*ArcTan[a*x]^2 - (c^2*ArcTan[a*x]^2)/(3*x^3) - (2*a^2*c^ 
2*ArcTan[a*x]^2)/x + a^4*c^2*x*ArcTan[a*x]^2 + 2*a^3*c^2*ArcTan[a*x]*Log[2 
/(1 + I*a*x)] + (10*a^3*c^2*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)])/3 - ((5*I) 
/3)*a^3*c^2*PolyLog[2, -1 + 2/(1 - I*a*x)] + I*a^3*c^2*PolyLog[2, 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 = 1.09 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.34

method	result	size
derivativedivides	\(a^{3} \left (a \,c^{2} x \arctan \left (a x \right )^{2}-\frac {2 c^{2} \arctan \left (a x \right )^{2}}{a x}-\frac {c^{2} \arctan \left (a x \right )^{2}}{3 a^{3} x^{3}}-\frac {2 c^{2} \left (4 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )+\frac {\arctan \left (a x \right )}{2 a^{2} x^{2}}-5 \arctan \left (a x \right ) \ln \left (a x \right )+\frac {1}{2 a x}+\frac {\arctan \left (a x \right )}{2}-\frac {5 i \ln \left (a x \right ) \ln \left (i a x +1\right )}{2}+\frac {5 i \ln \left (a x \right ) \ln \left (-i a x +1\right )}{2}-\frac {5 i \operatorname {dilog}\left (i a x +1\right )}{2}+\frac {5 i \operatorname {dilog}\left (-i a x +1\right )}{2}+2 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )\right )-2 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )\right )\right )}{3}\right )\)	\(290\)
default	\(a^{3} \left (a \,c^{2} x \arctan \left (a x \right )^{2}-\frac {2 c^{2} \arctan \left (a x \right )^{2}}{a x}-\frac {c^{2} \arctan \left (a x \right )^{2}}{3 a^{3} x^{3}}-\frac {2 c^{2} \left (4 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )+\frac {\arctan \left (a x \right )}{2 a^{2} x^{2}}-5 \arctan \left (a x \right ) \ln \left (a x \right )+\frac {1}{2 a x}+\frac {\arctan \left (a x \right )}{2}-\frac {5 i \ln \left (a x \right ) \ln \left (i a x +1\right )}{2}+\frac {5 i \ln \left (a x \right ) \ln \left (-i a x +1\right )}{2}-\frac {5 i \operatorname {dilog}\left (i a x +1\right )}{2}+\frac {5 i \operatorname {dilog}\left (-i a x +1\right )}{2}+2 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )\right )-2 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )\right )\right )}{3}\right )\)	\(290\)
parts	\(a^{4} c^{2} x \arctan \left (a x \right )^{2}-\frac {c^{2} \arctan \left (a x \right )^{2}}{3 x^{3}}-\frac {2 a^{2} c^{2} \arctan \left (a x \right )^{2}}{x}-\frac {2 c^{2} \left (4 a^{3} \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )+\frac {a \arctan \left (a x \right )}{2 x^{2}}-5 a^{3} \arctan \left (a x \right ) \ln \left (a x \right )-\frac {a^{3} \left (-\frac {1}{a x}-\arctan \left (a x \right )+5 i \ln \left (a x \right ) \ln \left (i a x +1\right )-5 i \ln \left (a x \right ) \ln \left (-i a x +1\right )+5 i \operatorname {dilog}\left (i a x +1\right )-5 i \operatorname {dilog}\left (-i a x +1\right )-4 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )\right )+4 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )\right )\right )}{2}\right )}{3}\)	\(295\)

Input:

int((a^2*c*x^2+c)^2*arctan(a*x)^2/x^4,x,method=_RETURNVERBOSE)
 

Output:

a^3*(a*c^2*x*arctan(a*x)^2-2*c^2*arctan(a*x)^2/a/x-1/3*c^2*arctan(a*x)^2/a 
^3/x^3-2/3*c^2*(4*arctan(a*x)*ln(a^2*x^2+1)+1/2*arctan(a*x)/a^2/x^2-5*arct 
an(a*x)*ln(a*x)+1/2/a/x+1/2*arctan(a*x)-5/2*I*ln(a*x)*ln(1+I*a*x)+5/2*I*ln 
(a*x)*ln(1-I*a*x)-5/2*I*dilog(1+I*a*x)+5/2*I*dilog(1-I*a*x)+2*I*(ln(a*x-I) 
*ln(a^2*x^2+1)-1/2*ln(a*x-I)^2-dilog(-1/2*I*(a*x+I))-ln(a*x-I)*ln(-1/2*I*( 
a*x+I)))-2*I*(ln(a*x+I)*ln(a^2*x^2+1)-1/2*ln(a*x+I)^2-dilog(1/2*I*(a*x-I)) 
-ln(a*x+I)*ln(1/2*I*(a*x-I)))))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

1/48*(12*(144*a^6*c^2*integrate(1/48*x^6*arctan(a*x)^2/(a^2*x^6 + x^4), x) 
 + 12*a^6*c^2*integrate(1/48*x^6*log(a^2*x^2 + 1)^2/(a^2*x^6 + x^4), x) + 
48*a^6*c^2*integrate(1/48*x^6*log(a^2*x^2 + 1)/(a^2*x^6 + x^4), x) + 3*a^3 
*c^2*arctan(a*x)^3 - 96*a^5*c^2*integrate(1/48*x^5*arctan(a*x)/(a^2*x^6 + 
x^4), x) + 36*a^4*c^2*integrate(1/48*x^4*log(a^2*x^2 + 1)^2/(a^2*x^6 + x^4 
), x) - 96*a^4*c^2*integrate(1/48*x^4*log(a^2*x^2 + 1)/(a^2*x^6 + x^4), x) 
 + 192*a^3*c^2*integrate(1/48*x^3*arctan(a*x)/(a^2*x^6 + x^4), x) + 432*a^ 
2*c^2*integrate(1/48*x^2*arctan(a*x)^2/(a^2*x^6 + x^4), x) + 36*a^2*c^2*in 
tegrate(1/48*x^2*log(a^2*x^2 + 1)^2/(a^2*x^6 + x^4), x) - 16*a^2*c^2*integ 
rate(1/48*x^2*log(a^2*x^2 + 1)/(a^2*x^6 + x^4), x) + 32*a*c^2*integrate(1/ 
48*x*arctan(a*x)/(a^2*x^6 + x^4), x) + 144*c^2*integrate(1/48*arctan(a*x)^ 
2/(a^2*x^6 + x^4), x) + 12*c^2*integrate(1/48*log(a^2*x^2 + 1)^2/(a^2*x^6 
+ x^4), x))*x^3 + 4*(3*a^4*c^2*x^4 - 6*a^2*c^2*x^2 - c^2)*arctan(a*x)^2 - 
(3*a^4*c^2*x^4 - 6*a^2*c^2*x^2 - c^2)*log(a^2*x^2 + 1)^2)/x^3
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^2}{x^4} \, dx=\frac {c^{2} \left (-6 \mathit {atan} \left (a x \right )^{2} a^{2} x^{2}-\mathit {atan} \left (a x \right )^{2}-\mathit {atan} \left (a x \right ) a^{3} x^{3}-\mathit {atan} \left (a x \right ) a x +3 \left (\int \mathit {atan} \left (a x \right )^{2}d x \right ) a^{4} x^{3}+10 \left (\int \frac {\mathit {atan} \left (a x \right )}{a^{2} x^{3}+x}d x \right ) a^{3} x^{3}-a^{2} x^{2}\right )}{3 x^{3}} \] Input:

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

Output:

(c**2*( - 6*atan(a*x)**2*a**2*x**2 - atan(a*x)**2 - atan(a*x)*a**3*x**3 - 
atan(a*x)*a*x + 3*int(atan(a*x)**2,x)*a**4*x**3 + 10*int(atan(a*x)/(a**2*x 
**3 + x),x)*a**3*x**3 - a**2*x**2))/(3*x**3)