\(\int \cot ^{-1}(a x)^2 \, dx\) [17]

Optimal result

Integrand size = 6, antiderivative size = 67 \[ \int \cot ^{-1}(a x)^2 \, dx=\frac {i \cot ^{-1}(a x)^2}{a}+x \cot ^{-1}(a x)^2-\frac {2 \cot ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{a} \] Output:

I*arccot(a*x)^2/a+x*arccot(a*x)^2-2*arccot(a*x)*ln(2/(1+I*a*x))/a+I*polylo 
g(2,1-2/(1+I*a*x))/a
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.84 \[ \int \cot ^{-1}(a x)^2 \, dx=\frac {\cot ^{-1}(a x) \left ((i+a x) \cot ^{-1}(a x)-2 \log \left (1-e^{2 i \cot ^{-1}(a x)}\right )\right )+i \operatorname {PolyLog}\left (2,e^{2 i \cot ^{-1}(a x)}\right )}{a} \] Input:

Integrate[ArcCot[a*x]^2,x]
 

Output:

(ArcCot[a*x]*((I + a*x)*ArcCot[a*x] - 2*Log[1 - E^((2*I)*ArcCot[a*x])]) + 
I*PolyLog[2, E^((2*I)*ArcCot[a*x])])/a
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.19, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {5346, 5456, 5380, 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.

\(\displaystyle \int \cot ^{-1}(a x)^2 \, dx\)

\(\Big \downarrow \) 5346

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

\(\Big \downarrow \) 5456

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

\(\Big \downarrow \) 5380

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

\(\Big \downarrow \) 2849

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

\(\Big \downarrow \) 2752

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

Input:

Int[ArcCot[a*x]^2,x]
 

Output:

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

Defintions of rubi rules used

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 5346

Int[((a_.) + ArcCot[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a 
+ b*ArcCot[c*x^n])^p, x] + Simp[b*c*n*p   Int[x^n*((a + b*ArcCot[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 5380

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] 
 :> Simp[(-(a + b*ArcCot[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] - Simp[b*c*( 
p/e)   Int[(a + b*ArcCot[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 5456

Int[(((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), 
x_Symbol] :> Simp[I*((a + b*ArcCot[c*x])^(p + 1)/(b*e*(p + 1))), x] - Simp[ 
1/(c*d)   Int[(a + b*ArcCot[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. 129 vs. \(2 (63 ) = 126\).

Time = 0.37 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.94

method	result	size
derivativedivides	\(\frac {\operatorname {arccot}\left (a x \right )^{2} \left (a x -i\right )-2 \,\operatorname {arccot}\left (a x \right ) \ln \left (1+\frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )-2 \,\operatorname {arccot}\left (a x \right ) \ln \left (1-\frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )+2 i \operatorname {arccot}\left (a x \right )^{2}+2 i \operatorname {polylog}\left (2, -\frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )+2 i \operatorname {polylog}\left (2, \frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )}{a}\)	\(130\)
default	\(\frac {\operatorname {arccot}\left (a x \right )^{2} \left (a x -i\right )-2 \,\operatorname {arccot}\left (a x \right ) \ln \left (1+\frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )-2 \,\operatorname {arccot}\left (a x \right ) \ln \left (1-\frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )+2 i \operatorname {arccot}\left (a x \right )^{2}+2 i \operatorname {polylog}\left (2, -\frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )+2 i \operatorname {polylog}\left (2, \frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )}{a}\)	\(130\)
risch	\(\frac {i \pi \ln \left (i a x +1\right ) x}{2}+\frac {i \pi ^{2}}{4 a}+\frac {\pi \ln \left (i a x +1\right )}{2 a}+\frac {i \operatorname {dilog}\left (\frac {1}{2}-\frac {i a x}{2}\right )}{a}+\frac {\ln \left (i a x +1\right ) \ln \left (-i a x +1\right ) x}{2}-\frac {i \pi \ln \left (-i a x +1\right ) x}{2}+\frac {i \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (\frac {1}{2}-\frac {i a x}{2}\right )}{a}+\frac {\pi \ln \left (-i a x +1\right )}{2 a}-\frac {i \ln \left (i a x +1\right )}{a}-\frac {i \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (-i a x +1\right )}{a}+\frac {i \ln \left (i a x +1\right ) \ln \left (-i a x +1\right )}{2 a}+\frac {i}{a}+\frac {i \ln \left (a^{2} x^{2}+1\right )}{2 a}-\frac {i \ln \left (-i a x +1\right )^{2}}{4 a}+\frac {i \ln \left (i a x +1\right )^{2}}{4 a}-\frac {\ln \left (i a x +1\right )^{2} x}{4}-\frac {\ln \left (-i a x +1\right )^{2} x}{4}-\frac {\pi }{a}-\frac {\arctan \left (a x \right )}{a}+\frac {\pi ^{2} x}{4}\)	\(279\)

Input:

int(arccot(a*x)^2,x,method=_RETURNVERBOSE)
 

Output:

1/a*(arccot(a*x)^2*(a*x-I)-2*arccot(a*x)*ln(1+(I+a*x)/(a^2*x^2+1)^(1/2))-2 
*arccot(a*x)*ln(1-(I+a*x)/(a^2*x^2+1)^(1/2))+2*I*arccot(a*x)^2+2*I*polylog 
(2,-(I+a*x)/(a^2*x^2+1)^(1/2))+2*I*polylog(2,(I+a*x)/(a^2*x^2+1)^(1/2)))
 

Fricas [F]

\[ \int \cot ^{-1}(a x)^2 \, dx=\int { \operatorname {arccot}\left (a x\right )^{2} \,d x } \] Input:

integrate(arccot(a*x)^2,x, algorithm="fricas")
 

Output:

integral(arccot(a*x)^2, x)
 

Sympy [F]

\[ \int \cot ^{-1}(a x)^2 \, dx=\int \operatorname {acot}^{2}{\left (a x \right )}\, dx \] Input:

integrate(acot(a*x)**2,x)
 

Output:

Integral(acot(a*x)**2, x)
 

Maxima [F]

\[ \int \cot ^{-1}(a x)^2 \, dx=\int { \operatorname {arccot}\left (a x\right )^{2} \,d x } \] Input:

integrate(arccot(a*x)^2,x, algorithm="maxima")
 

Output:

1/4*x*arctan2(1, a*x)^2 + 12*a^2*integrate(1/16*x^2*arctan(1/(a*x))^2/(a^2 
*x^2 + 1), x) + a^2*integrate(1/16*x^2*log(a^2*x^2 + 1)^2/(a^2*x^2 + 1), x 
) + 4*a^2*integrate(1/16*x^2*log(a^2*x^2 + 1)/(a^2*x^2 + 1), x) - 1/16*x*l 
og(a^2*x^2 + 1)^2 + 1/4*arctan(a*x)^3/a + 3/4*arctan(a*x)^2*arctan(1/(a*x) 
)/a + 3/4*arctan(a*x)*arctan(1/(a*x))^2/a + 8*a*integrate(1/16*x*arctan(1/ 
(a*x))/(a^2*x^2 + 1), x) + integrate(1/16*log(a^2*x^2 + 1)^2/(a^2*x^2 + 1) 
, x)
 

Giac [F]

\[ \int \cot ^{-1}(a x)^2 \, dx=\int { \operatorname {arccot}\left (a x\right )^{2} \,d x } \] Input:

integrate(arccot(a*x)^2,x, algorithm="giac")
 

Output:

integrate(arccot(a*x)^2, x)
 

Mupad [B] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.82 \[ \int \cot ^{-1}(a x)^2 \, dx=\frac {-2\,\ln \left (1-{\mathrm {e}}^{\mathrm {acot}\left (a\,x\right )\,2{}\mathrm {i}}\right )\,\mathrm {acot}\left (a\,x\right )+\mathrm {polylog}\left (2,{\mathrm {e}}^{\mathrm {acot}\left (a\,x\right )\,2{}\mathrm {i}}\right )\,1{}\mathrm {i}+{\mathrm {acot}\left (a\,x\right )}^2\,1{}\mathrm {i}}{a}+x\,{\mathrm {acot}\left (a\,x\right )}^2 \] Input:

int(acot(a*x)^2,x)
 

Output:

(polylog(2, exp(acot(a*x)*2i))*1i - 2*log(1 - exp(acot(a*x)*2i))*acot(a*x) 
 + acot(a*x)^2*1i)/a + x*acot(a*x)^2
 

Reduce [F]

\[ \int \cot ^{-1}(a x)^2 \, dx=\int \mathit {acot} \left (a x \right )^{2}d x \] Input:

int(acot(a*x)^2,x)
 

Output:

int(acot(a*x)**2,x)