\(\int \frac {x^2 \arctan (a x)^2}{c+a^2 c x^2} \, dx\) [284]

Optimal result

Integrand size = 22, antiderivative size = 98 \[ \int \frac {x^2 \arctan (a x)^2}{c+a^2 c x^2} \, dx=\frac {i \arctan (a x)^2}{a^3 c}+\frac {x \arctan (a x)^2}{a^2 c}-\frac {\arctan (a x)^3}{3 a^3 c}+\frac {2 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{a^3 c}+\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{a^3 c} \] Output:

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

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.70 \[ \int \frac {x^2 \arctan (a x)^2}{c+a^2 c x^2} \, dx=\frac {-\frac {1}{3} \arctan (a x) \left ((3 i-3 a x) \arctan (a x)+\arctan (a x)^2-6 \log \left (1+e^{2 i \arctan (a x)}\right )\right )-i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )}{a^3 c} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.06, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {5451, 27, 5345, 5419, 5455, 5379, 2849, 2752}

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.

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

\(\Big \downarrow \) 5451

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5345

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

\(\Big \downarrow \) 5419

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

\(\Big \downarrow \) 5455

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

\(\Big \downarrow \) 5379

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

\(\Big \downarrow \) 2849

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

\(\Big \downarrow \) 2752

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

Input:

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

Output:

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

Defintions of rubi rules used

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 2752

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo 
g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
 

rule 2849

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Simp 
[-e/g   Subst[Int[Log[2*d*x]/(1 - 2*d*x), x], x, 1/(d + e*x)], x] /; FreeQ[ 
{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]
 

rule 5345

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a 
+ b*ArcTan[c*x^n])^p, x] - Simp[b*c*n*p   Int[x^n*((a + b*ArcTan[c*x^n])^(p 
 - 1)/(1 + c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0] && 
 (EqQ[n, 1] || EqQ[p, 1])
 

rule 5379

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] 
 :> Simp[(-(a + b*ArcTan[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] + Simp[b*c*( 
p/e)   Int[(a + b*ArcTan[c*x])^(p - 1)*(Log[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 5451

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

rule 5455

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

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 186 vs. \(2 (92 ) = 184\).

Time = 0.91 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.91

method	result	size
derivativedivides	\(\frac {\frac {\arctan \left (a x \right )^{2} a x}{c}-\frac {\arctan \left (a x \right )^{3}}{c}-\frac {2 \left (\frac {\arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{2}+\frac {i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )\right )}{4}-\frac {i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )\right )}{4}-\frac {\arctan \left (a x \right )^{3}}{3}\right )}{c}}{a^{3}}\)	\(187\)
default	\(\frac {\frac {\arctan \left (a x \right )^{2} a x}{c}-\frac {\arctan \left (a x \right )^{3}}{c}-\frac {2 \left (\frac {\arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{2}+\frac {i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )\right )}{4}-\frac {i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )\right )}{4}-\frac {\arctan \left (a x \right )^{3}}{3}\right )}{c}}{a^{3}}\)	\(187\)
parts	\(\frac {x \arctan \left (a x \right )^{2}}{a^{2} c}-\frac {\arctan \left (a x \right )^{3}}{a^{3} c}-\frac {2 \left (\frac {\frac {\arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{2}+\frac {i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )\right )}{4}-\frac {i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )\right )}{4}}{a^{3}}-\frac {\arctan \left (a x \right )^{3}}{3 a^{3}}\right )}{c}\)	\(196\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

\[ \int \frac {x^2 \arctan (a x)^2}{c+a^2 c x^2} \, dx=\frac {\int \frac {x^{2} \operatorname {atan}^{2}{\left (a x \right )}}{a^{2} x^{2} + 1}\, dx}{c} \] Input:

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

Output:

Integral(x**2*atan(a*x)**2/(a**2*x**2 + 1), x)/c
 

Maxima [F]

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

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

Output:

1/48*(4*(144*a^2*integrate(1/16*x^2*arctan(a*x)^2/(a^4*c*x^2 + a^2*c), x) 
+ 12*a^2*integrate(1/16*x^2*log(a^2*x^2 + 1)^2/(a^4*c*x^2 + a^2*c), x) + 4 
8*a^2*integrate(1/16*x^2*log(a^2*x^2 + 1)/(a^4*c*x^2 + a^2*c), x) - 96*a*i 
ntegrate(1/16*x*arctan(a*x)/(a^4*c*x^2 + a^2*c), x) + arctan(a*x)^3/(a^3*c 
) + 12*integrate(1/16*log(a^2*x^2 + 1)^2/(a^4*c*x^2 + a^2*c), x))*a^3*c + 
12*a*x*arctan(a*x)^2 - 3*a*x*log(a^2*x^2 + 1)^2 - 8*arctan(a*x)^3)/(a^3*c)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int((atan(a*x)**2*x**2)/(a**2*x**2 + 1),x)/c