\(\int \frac {(c+a^2 c x^2)^3 \arctan (a x)^2}{x^2} \, dx\) [279]

Optimal result

Integrand size = 22, antiderivative size = 251 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^2}{x^2} \, dx=\frac {7}{10} a^2 c^3 x+\frac {1}{30} a^4 c^3 x^3-\frac {7}{10} a c^3 \arctan (a x)-\frac {4}{5} a^3 c^3 x^2 \arctan (a x)-\frac {1}{10} a^5 c^3 x^4 \arctan (a x)+\frac {6}{5} i a c^3 \arctan (a x)^2-\frac {c^3 \arctan (a x)^2}{x}+3 a^2 c^3 x \arctan (a x)^2+a^4 c^3 x^3 \arctan (a x)^2+\frac {1}{5} a^6 c^3 x^5 \arctan (a x)^2+\frac {22}{5} a c^3 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )+2 a c^3 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-i a c^3 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )+\frac {11}{5} i a c^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right ) \] Output:

7/10*a^2*c^3*x+1/30*a^4*c^3*x^3-7/10*a*c^3*arctan(a*x)-4/5*a^3*c^3*x^2*arc 
tan(a*x)-1/10*a^5*c^3*x^4*arctan(a*x)+6/5*I*a*c^3*arctan(a*x)^2-c^3*arctan 
(a*x)^2/x+3*a^2*c^3*x*arctan(a*x)^2+a^4*c^3*x^3*arctan(a*x)^2+1/5*a^6*c^3* 
x^5*arctan(a*x)^2+22/5*a*c^3*arctan(a*x)*ln(2/(1+I*a*x))+2*a*c^3*arctan(a* 
x)*ln(2-2/(1-I*a*x))-I*a*c^3*polylog(2,-1+2/(1-I*a*x))+11/5*I*a*c^3*polylo 
g(2,1-2/(1+I*a*x))
 

Mathematica [A] (verified)

Time = 0.51 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.80 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^2}{x^2} \, dx=\frac {c^3 \left (21 a^2 x^2+a^4 x^4-21 a x \arctan (a x)-24 a^3 x^3 \arctan (a x)-3 a^5 x^5 \arctan (a x)-30 \arctan (a x)^2-96 i a x \arctan (a x)^2+90 a^2 x^2 \arctan (a x)^2+30 a^4 x^4 \arctan (a x)^2+6 a^6 x^6 \arctan (a x)^2+60 a x \arctan (a x) \log \left (1-e^{2 i \arctan (a x)}\right )+132 a x \arctan (a x) \log \left (1+e^{2 i \arctan (a x)}\right )-66 i a x \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )-30 i a x \operatorname {PolyLog}\left (2,e^{2 i \arctan (a x)}\right )\right )}{30 x} \] Input:

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

Output:

(c^3*(21*a^2*x^2 + a^4*x^4 - 21*a*x*ArcTan[a*x] - 24*a^3*x^3*ArcTan[a*x] - 
 3*a^5*x^5*ArcTan[a*x] - 30*ArcTan[a*x]^2 - (96*I)*a*x*ArcTan[a*x]^2 + 90* 
a^2*x^2*ArcTan[a*x]^2 + 30*a^4*x^4*ArcTan[a*x]^2 + 6*a^6*x^6*ArcTan[a*x]^2 
 + 60*a*x*ArcTan[a*x]*Log[1 - E^((2*I)*ArcTan[a*x])] + 132*a*x*ArcTan[a*x] 
*Log[1 + E^((2*I)*ArcTan[a*x])] - (66*I)*a*x*PolyLog[2, -E^((2*I)*ArcTan[a 
*x])] - (30*I)*a*x*PolyLog[2, E^((2*I)*ArcTan[a*x])]))/(30*x)
 

Rubi [A] (verified)

Time = 0.93 (sec) , antiderivative size = 251, 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.

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

\(\Big \downarrow \) 5483

\(\displaystyle \int \left (a^6 c^3 x^4 \arctan (a x)^2+3 a^4 c^3 x^2 \arctan (a x)^2+3 a^2 c^3 \arctan (a x)^2+\frac {c^3 \arctan (a x)^2}{x^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{5} a^6 c^3 x^5 \arctan (a x)^2-\frac {1}{10} a^5 c^3 x^4 \arctan (a x)+a^4 c^3 x^3 \arctan (a x)^2+\frac {1}{30} a^4 c^3 x^3-\frac {4}{5} a^3 c^3 x^2 \arctan (a x)+3 a^2 c^3 x \arctan (a x)^2+\frac {7}{10} a^2 c^3 x+\frac {6}{5} i a c^3 \arctan (a x)^2-\frac {7}{10} a c^3 \arctan (a x)-\frac {c^3 \arctan (a x)^2}{x}+\frac {22}{5} a c^3 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )+2 a c^3 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-i a c^3 \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )+\frac {11}{5} i a c^3 \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )\)

Input:

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

Output:

(7*a^2*c^3*x)/10 + (a^4*c^3*x^3)/30 - (7*a*c^3*ArcTan[a*x])/10 - (4*a^3*c^ 
3*x^2*ArcTan[a*x])/5 - (a^5*c^3*x^4*ArcTan[a*x])/10 + ((6*I)/5)*a*c^3*ArcT 
an[a*x]^2 - (c^3*ArcTan[a*x]^2)/x + 3*a^2*c^3*x*ArcTan[a*x]^2 + a^4*c^3*x^ 
3*ArcTan[a*x]^2 + (a^6*c^3*x^5*ArcTan[a*x]^2)/5 + (22*a*c^3*ArcTan[a*x]*Lo 
g[2/(1 + I*a*x)])/5 + 2*a*c^3*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)] - I*a*c^3 
*PolyLog[2, -1 + 2/(1 - I*a*x)] + ((11*I)/5)*a*c^3*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 = 1.32 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.28

method	result	size
derivativedivides	\(a \left (\frac {a^{5} c^{3} x^{5} \arctan \left (a x \right )^{2}}{5}+a^{3} c^{3} x^{3} \arctan \left (a x \right )^{2}+3 a \,c^{3} x \arctan \left (a x \right )^{2}-\frac {c^{3} \arctan \left (a x \right )^{2}}{a x}-\frac {2 c^{3} \left (\frac {x^{4} \arctan \left (a x \right ) a^{4}}{4}+2 x^{2} a^{2} \arctan \left (a x \right )-5 \arctan \left (a x \right ) \ln \left (a x \right )+8 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )-\frac {a^{3} x^{3}}{12}-\frac {7 a x}{4}+\frac {7 \arctan \left (a x \right )}{4}-\frac {5 i \ln \left (a x \right ) \ln \left (i a x +1\right )}{2}+\frac {5 i \ln \left (a x \right ) \ln \left (-i a x +1\right )}{2}-\frac {5 i \operatorname {dilog}\left (i a x +1\right )}{2}+\frac {5 i \operatorname {dilog}\left (-i a x +1\right )}{2}+4 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )\right )-4 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )\right )\right )}{5}\right )\)	\(321\)
default	\(a \left (\frac {a^{5} c^{3} x^{5} \arctan \left (a x \right )^{2}}{5}+a^{3} c^{3} x^{3} \arctan \left (a x \right )^{2}+3 a \,c^{3} x \arctan \left (a x \right )^{2}-\frac {c^{3} \arctan \left (a x \right )^{2}}{a x}-\frac {2 c^{3} \left (\frac {x^{4} \arctan \left (a x \right ) a^{4}}{4}+2 x^{2} a^{2} \arctan \left (a x \right )-5 \arctan \left (a x \right ) \ln \left (a x \right )+8 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )-\frac {a^{3} x^{3}}{12}-\frac {7 a x}{4}+\frac {7 \arctan \left (a x \right )}{4}-\frac {5 i \ln \left (a x \right ) \ln \left (i a x +1\right )}{2}+\frac {5 i \ln \left (a x \right ) \ln \left (-i a x +1\right )}{2}-\frac {5 i \operatorname {dilog}\left (i a x +1\right )}{2}+\frac {5 i \operatorname {dilog}\left (-i a x +1\right )}{2}+4 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )\right )-4 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )\right )\right )}{5}\right )\)	\(321\)
parts	\(\frac {a^{6} c^{3} x^{5} \arctan \left (a x \right )^{2}}{5}+a^{4} c^{3} x^{3} \arctan \left (a x \right )^{2}+3 a^{2} c^{3} x \arctan \left (a x \right )^{2}-\frac {c^{3} \arctan \left (a x \right )^{2}}{x}-\frac {2 c^{3} \left (\frac {\arctan \left (a x \right ) a^{5} x^{4}}{4}+2 \arctan \left (a x \right ) x^{2} a^{3}-5 a \arctan \left (a x \right ) \ln \left (a x \right )+8 a \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )-\frac {a \left (\frac {a^{3} x^{3}}{3}+7 a x -7 \arctan \left (a x \right )+10 i \ln \left (a x \right ) \ln \left (i a x +1\right )-10 i \ln \left (a x \right ) \ln \left (-i a x +1\right )+10 i \operatorname {dilog}\left (i a x +1\right )-10 i \operatorname {dilog}\left (-i a x +1\right )-16 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )\right )+16 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )\right )\right )}{4}\right )}{5}\)	\(324\)

Input:

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

Output:

a*(1/5*a^5*c^3*x^5*arctan(a*x)^2+a^3*c^3*x^3*arctan(a*x)^2+3*a*c^3*x*arcta 
n(a*x)^2-c^3*arctan(a*x)^2/a/x-2/5*c^3*(1/4*x^4*arctan(a*x)*a^4+2*x^2*a^2* 
arctan(a*x)-5*arctan(a*x)*ln(a*x)+8*arctan(a*x)*ln(a^2*x^2+1)-1/12*a^3*x^3 
-7/4*a*x+7/4*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)+4*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)))-4*I*(l 
n(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]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

1/80*(4*(a^6*c^3*x^6 + 5*a^4*c^3*x^4 + 15*a^2*c^3*x^2 - 5*c^3)*arctan(a*x) 
^2 - (a^6*c^3*x^6 + 5*a^4*c^3*x^4 + 15*a^2*c^3*x^2 - 5*c^3)*log(a^2*x^2 + 
1)^2 + 80*(60*a^8*c^3*integrate(1/80*x^8*arctan(a*x)^2/(a^2*x^4 + x^2), x) 
 + 5*a^8*c^3*integrate(1/80*x^8*log(a^2*x^2 + 1)^2/(a^2*x^4 + x^2), x) + 4 
*a^8*c^3*integrate(1/80*x^8*log(a^2*x^2 + 1)/(a^2*x^4 + x^2), x) - 8*a^7*c 
^3*integrate(1/80*x^7*arctan(a*x)/(a^2*x^4 + x^2), x) + 240*a^6*c^3*integr 
ate(1/80*x^6*arctan(a*x)^2/(a^2*x^4 + x^2), x) + 20*a^6*c^3*integrate(1/80 
*x^6*log(a^2*x^2 + 1)^2/(a^2*x^4 + x^2), x) + 20*a^6*c^3*integrate(1/80*x^ 
6*log(a^2*x^2 + 1)/(a^2*x^4 + x^2), x) - 40*a^5*c^3*integrate(1/80*x^5*arc 
tan(a*x)/(a^2*x^4 + x^2), x) + 360*a^4*c^3*integrate(1/80*x^4*arctan(a*x)^ 
2/(a^2*x^4 + x^2), x) + 30*a^4*c^3*integrate(1/80*x^4*log(a^2*x^2 + 1)^2/( 
a^2*x^4 + x^2), x) + 60*a^4*c^3*integrate(1/80*x^4*log(a^2*x^2 + 1)/(a^2*x 
^4 + x^2), x) + a*c^3*arctan(a*x)^3 - 120*a^3*c^3*integrate(1/80*x^3*arcta 
n(a*x)/(a^2*x^4 + x^2), x) + 20*a^2*c^3*integrate(1/80*x^2*log(a^2*x^2 + 1 
)^2/(a^2*x^4 + x^2), x) - 20*a^2*c^3*integrate(1/80*x^2*log(a^2*x^2 + 1)/( 
a^2*x^4 + x^2), x) + 40*a*c^3*integrate(1/80*x*arctan(a*x)/(a^2*x^4 + x^2) 
, x) + 60*c^3*integrate(1/80*arctan(a*x)^2/(a^2*x^4 + x^2), x) + 5*c^3*int 
egrate(1/80*log(a^2*x^2 + 1)^2/(a^2*x^4 + x^2), x))*x)/x
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^2}{x^2} \, dx=\frac {c^{3} \left (6 \mathit {atan} \left (a x \right )^{2} a^{6} x^{6}+30 \mathit {atan} \left (a x \right )^{2} a^{4} x^{4}+90 \mathit {atan} \left (a x \right )^{2} a^{2} x^{2}-30 \mathit {atan} \left (a x \right )^{2}-3 \mathit {atan} \left (a x \right ) a^{5} x^{5}-24 \mathit {atan} \left (a x \right ) a^{3} x^{3}-21 \mathit {atan} \left (a x \right ) a x +60 \left (\int \frac {\mathit {atan} \left (a x \right )}{a^{2} x^{3}+x}d x \right ) a x -132 \left (\int \frac {\mathit {atan} \left (a x \right ) x}{a^{2} x^{2}+1}d x \right ) a^{3} x +a^{4} x^{4}+21 a^{2} x^{2}\right )}{30 x} \] Input:

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

Output:

(c**3*(6*atan(a*x)**2*a**6*x**6 + 30*atan(a*x)**2*a**4*x**4 + 90*atan(a*x) 
**2*a**2*x**2 - 30*atan(a*x)**2 - 3*atan(a*x)*a**5*x**5 - 24*atan(a*x)*a** 
3*x**3 - 21*atan(a*x)*a*x + 60*int(atan(a*x)/(a**2*x**3 + x),x)*a*x - 132* 
int((atan(a*x)*x)/(a**2*x**2 + 1),x)*a**3*x + a**4*x**4 + 21*a**2*x**2))/( 
30*x)