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

3.2.19.1 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
 

3.2.19.2 Mathematica [A] (verified)

Time = 0.23 (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)
 

3.2.19.3 Rubi [A] (verified)

Time = 0.45 (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
 

3.2.19.3.1 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]
 

3.2.19.4 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 = 6.00 (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)))
 

3.2.19.5 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)
 

3.2.19.6 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
 

3.2.19.7 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
 

3.2.19.8 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

sage0*x
 

3.2.19.9 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)