\(\int \frac {\arctan (a x)^2}{c x-i a c x^2} \, dx\) [119]

Optimal result

Integrand size = 22, antiderivative size = 76 \[ \int \frac {\arctan (a x)^2}{c x-i a c x^2} \, dx=\frac {\arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c}-\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{c}+\frac {\operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )}{2 c} \] Output:

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

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.08 \[ \int \frac {\arctan (a x)^2}{c x-i a c x^2} \, dx=\frac {-i \pi ^3+16 i \arctan (a x)^3+24 \arctan (a x)^2 \log \left (1-e^{-2 i \arctan (a x)}\right )+24 i \arctan (a x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )+12 \operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )}{24 c} \] Input:

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

Output:

((-I)*Pi^3 + (16*I)*ArcTan[a*x]^3 + 24*ArcTan[a*x]^2*Log[1 - E^((-2*I)*Arc 
Tan[a*x])] + (24*I)*ArcTan[a*x]*PolyLog[2, E^((-2*I)*ArcTan[a*x])] + 12*Po 
lyLog[3, E^((-2*I)*ArcTan[a*x])])/(24*c)
 

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.12, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2026, 5403, 5527, 7164}

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.

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

\(\Big \downarrow \) 2026

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

\(\Big \downarrow \) 5403

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

\(\Big \downarrow \) 5527

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

\(\Big \downarrow \) 7164

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2026

Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p 
*r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ 
erQ[p] &&  !MonomialQ[Px, x] && (ILtQ[p, 0] ||  !PolyQ[u, x])
 

rule 5403

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x_ 
Symbol] :> Simp[(a + b*ArcTan[c*x])^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] - Si 
mp[b*c*(p/d)   Int[(a + b*ArcTan[c*x])^(p - 1)*(Log[2 - 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 5527

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

rule 7164

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, 
x]}, Simp[w*PolyLog[n + 1, v], x] /;  !FalseQ[w]] /; FreeQ[n, x]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 192 vs. \(2 (70 ) = 140\).

Time = 2.05 (sec) , antiderivative size = 193, normalized size of antiderivative = 2.54

method	result	size
derivativedivides	\(\frac {\frac {a \arctan \left (a x \right )^{2} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}-\frac {2 i a \arctan \left (a x \right ) \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}+\frac {2 a \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}+\frac {a \arctan \left (a x \right )^{2} \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}-\frac {2 i a \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}+\frac {2 a \operatorname {polylog}\left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}}{a}\)	\(193\)
default	\(\frac {\frac {a \arctan \left (a x \right )^{2} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}-\frac {2 i a \arctan \left (a x \right ) \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}+\frac {2 a \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}+\frac {a \arctan \left (a x \right )^{2} \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}-\frac {2 i a \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}+\frac {2 a \operatorname {polylog}\left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}}{a}\)	\(193\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(-1/4*I*log(-(a*x + I)/(a*x - I))^2/(a*c*x^2 + I*c*x), x)
 

Sympy [F]

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

integrate(atan(a*x)**2/(c*x-I*a*c*x**2),x)
 

Output:

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

Maxima [F]

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

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

Output:

1/96*(8*I*arctan(a*x)^3 - 12*arctan(a*x)^2*log(a^2*x^2 + 1) - 6*I*arctan(a 
*x)*log(a^2*x^2 + 1)^2 + log(a^2*x^2 + 1)^3 + 24*I*(arctan(a*x)^3/c + 4*a* 
integrate(1/16*x*log(a^2*x^2 + 1)^2/(a^2*c*x^3 + c*x), x) - 16*integrate(1 
/16*arctan(a*x)*log(a^2*x^2 + 1)/(a^2*c*x^3 + c*x), x))*c + 96*c*integrate 
(1/16*(4*a*x*arctan(a*x)*log(a^2*x^2 + 1) + 12*arctan(a*x)^2 + log(a^2*x^2 
 + 1)^2)/(a^2*c*x^3 + c*x), x))/c
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\arctan (a x)^2}{c x-i a c x^2} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^2}{c\,x-a\,c\,x^2\,1{}\mathrm {i}} \,d x \] Input:

int(atan(a*x)^2/(c*x - a*c*x^2*1i),x)
 

Output:

int(atan(a*x)^2/(c*x - a*c*x^2*1i), x)
 

Reduce [F]

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

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

Output:

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