3.3.89 \(\int \frac {\arctan (a x)^2}{x^3 (c+a^2 c x^2)} \, dx\) [289]

3.3.89.1 Optimal result

Integrand size = 22, antiderivative size = 178 \[ \int \frac {\arctan (a x)^2}{x^3 \left (c+a^2 c x^2\right )} \, dx=-\frac {a \arctan (a x)}{c x}-\frac {a^2 \arctan (a x)^2}{2 c}-\frac {\arctan (a x)^2}{2 c x^2}+\frac {i a^2 \arctan (a x)^3}{3 c}+\frac {a^2 \log (x)}{c}-\frac {a^2 \log \left (1+a^2 x^2\right )}{2 c}-\frac {a^2 \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c}+\frac {i a^2 \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{c}-\frac {a^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )}{2 c} \]

output

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

3.3.89.2 Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.80 \[ \int \frac {\arctan (a x)^2}{x^3 \left (c+a^2 c x^2\right )} \, dx=\frac {a^2 \left (\frac {i \pi ^3}{24}-\frac {\arctan (a x)}{a x}-\frac {\left (1+a^2 x^2\right ) \arctan (a x)^2}{2 a^2 x^2}-\frac {1}{3} i \arctan (a x)^3-\arctan (a x)^2 \log \left (1-e^{-2 i \arctan (a x)}\right )+\log (a x)+\log \left (\frac {1}{\sqrt {1+a^2 x^2}}\right )-i \arctan (a x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )\right )}{c} \]

input

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

output

(a^2*((I/24)*Pi^3 - ArcTan[a*x]/(a*x) - ((1 + a^2*x^2)*ArcTan[a*x]^2)/(2*a 
^2*x^2) - (I/3)*ArcTan[a*x]^3 - ArcTan[a*x]^2*Log[1 - E^((-2*I)*ArcTan[a*x 
])] + Log[a*x] + Log[1/Sqrt[1 + a^2*x^2]] - I*ArcTan[a*x]*PolyLog[2, E^((- 
2*I)*ArcTan[a*x])] - PolyLog[3, E^((-2*I)*ArcTan[a*x])]/2))/c
 

3.3.89.3 Rubi [A] (verified)

Time = 1.21 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.97, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {5453, 27, 5361, 5453, 5361, 243, 47, 14, 16, 5419, 5459, 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.

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

\(\Big \downarrow \) 5453

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5361

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

\(\Big \downarrow \) 5453

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

\(\Big \downarrow \) 5361

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 47

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

\(\Big \downarrow \) 14

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 5419

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

\(\Big \downarrow \) 5459

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

\(\Big \downarrow \) 5403

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

\(\Big \downarrow \) 5527

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

\(\Big \downarrow \) 7164

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

input

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

output

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

3.3.89.3.1 Defintions of rubi rules used

rule 14

Int[(a_.)/(x_), x_Symbol] :> Simp[a*Log[x], x] /; FreeQ[a, x]
 

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

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 47

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Simp[b/(b*c 
 - a*d)   Int[1/(a + b*x), x], x] - Simp[d/(b*c - a*d)   Int[1/(c + d*x), x 
], x] /; FreeQ[{a, b, c, d}, x]
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 5361

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

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 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 5453

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

rule 5459

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

3.3.89.4 Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 35.01 (sec) , antiderivative size = 1786, normalized size of antiderivative = 10.03

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1786\)
default	\(\text {Expression too large to display}\)	\(1786\)
parts	\(\text {Expression too large to display}\)	\(2203\)

input

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

output

a^2*(-1/2/c*arctan(a*x)^2/a^2/x^2-1/c*arctan(a*x)^2*ln(a*x)+1/2/c*arctan(a 
*x)^2*ln(a^2*x^2+1)-1/c*(arctan(a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2))-arc 
tan(a*x)^2*ln((1+I*a*x)^2/(a^2*x^2+1)-1)-1/12*arctan(a*x)*(6*I*Pi*arctan(a 
*x)*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))^2*csgn 
(I*((1+I*a*x)^2/(a^2*x^2+1)-1))*a*x-3*I*Pi*arctan(a*x)*csgn(I*((1+I*a*x)^2 
/(a^2*x^2+1)+1)^2)^3*a*x-6*I*Pi*arctan(a*x)*csgn(((1+I*a*x)^2/(a^2*x^2+1)- 
1)/((1+I*a*x)^2/(a^2*x^2+1)+1))^3*a*x-12*I*a*x+3*I*Pi*arctan(a*x)*csgn(I*( 
1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^3*a*x-6*I*Pi*arctan( 
a*x)*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))^3*a*x 
-3*I*Pi*arctan(a*x)*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1))^2*csgn(I*((1+I*a*x 
)^2/(a^2*x^2+1)+1)^2)*a*x+6*I*Pi*arctan(a*x)*csgn(I*((1+I*a*x)^2/(a^2*x^2+ 
1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))^2*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1))*a 
*x-6*I*Pi*arctan(a*x)*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2 
*x^2+1)+1))*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)-1))*csgn(I/((1+I*a*x)^2/(a^2*x 
^2+1)+1))*a*x+3*I*csgn(I*(1+I*a*x)/(a^2*x^2+1)^(1/2))^2*Pi*arctan(a*x)*csg 
n(I*(1+I*a*x)^2/(a^2*x^2+1))*a*x+6*I*Pi*arctan(a*x)*csgn(I*((1+I*a*x)^2/(a 
^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))*csgn(((1+I*a*x)^2/(a^2*x^2+1)-1) 
/((1+I*a*x)^2/(a^2*x^2+1)+1))^2*a*x+4*I*arctan(a*x)^2*a*x-6*I*csgn(I*(1+I* 
a*x)/(a^2*x^2+1)^(1/2))*Pi*arctan(a*x)*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))^2*a 
*x-6*I*Pi*arctan(a*x)*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(...
 

3.3.89.5 Fricas [F]

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

input

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

output

integral(arctan(a*x)^2/(a^2*c*x^5 + c*x^3), x)
 

3.3.89.6 Sympy [F]

\[ \int \frac {\arctan (a x)^2}{x^3 \left (c+a^2 c x^2\right )} \, dx=\frac {\int \frac {\operatorname {atan}^{2}{\left (a x \right )}}{a^{2} x^{5} + x^{3}}\, dx}{c} \]

input

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

output

Integral(atan(a*x)**2/(a**2*x**5 + x**3), x)/c
 

3.3.89.7 Maxima [F]

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

input

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

output

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

3.3.89.8 Giac [F]

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

input

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

output

sage0*x
 

3.3.89.9 Mupad [F(-1)]

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

input

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

output

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