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

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {4931, 5041, 4965, 2449, 2352} \[ \int \cot ^{-1}(a x)^2 \, dx=\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{a}+x \cot ^{-1}(a x)^2+\frac {i \cot ^{-1}(a x)^2}{a}-\frac {2 \log \left (\frac {2}{1+i a x}\right ) \cot ^{-1}(a x)}{a} \]

[In]

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

[Out]

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

Rule 2352

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

Rule 2449

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Dist[-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 4931

Int[((a_.) + ArcCot[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcCot[c*x^n])^p, x] + Dist[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 4965

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] - Dist[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 5041

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] - Dist[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]

Rubi steps \begin{align*} \text {integral}& = x \cot ^{-1}(a x)^2+(2 a) \int \frac {x \cot ^{-1}(a x)}{1+a^2 x^2} \, dx \\ & = \frac {i \cot ^{-1}(a x)^2}{a}+x \cot ^{-1}(a x)^2-2 \int \frac {\cot ^{-1}(a x)}{i-a x} \, 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}-2 \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 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 {(2 i) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{a} \\ & = \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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.84 \[ \int \cot ^{-1}(a x)^2 \, dx=\frac {\cot ^{-1}(a x) \left ((i+a x) \cot ^{-1}(a x)-2 \log \left (1-e^{2 i \cot ^{-1}(a x)}\right )\right )+i \operatorname {PolyLog}\left (2,e^{2 i \cot ^{-1}(a x)}\right )}{a} \]

[In]

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

[Out]

(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

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 = 0.50 (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	\(\frac {i \pi ^{2}}{4 a}-\frac {i \ln \left (-i a x +1\right ) \pi x}{2}+\frac {i \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (\frac {1}{2}-\frac {i a x}{2}\right )}{a}-\frac {i \ln \left (-i a x +1\right )^{2}}{4 a}+\frac {i \ln \left (i a x +1\right )^{2}}{4 a}+\frac {i \ln \left (a^{2} x^{2}+1\right )}{2 a}+\frac {\pi \ln \left (i a x +1\right )}{2 a}+\frac {\ln \left (i a x +1\right ) \ln \left (-i a x +1\right ) x}{2}+\frac {\ln \left (-i a x +1\right ) \pi }{2 a}+\frac {i \operatorname {dilog}\left (\frac {1}{2}-\frac {i a x}{2}\right )}{a}+\frac {i \pi \ln \left (i a x +1\right ) x}{2}-\frac {i \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (-i a x +1\right )}{a}-\frac {\ln \left (i a x +1\right )^{2} x}{4}-\frac {\ln \left (-i a x +1\right )^{2} x}{4}+\frac {\pi ^{2} x}{4}-\frac {\pi }{a}-\frac {i \ln \left (i a x +1\right )}{a}+\frac {i \ln \left (i a x +1\right ) \ln \left (-i a x +1\right )}{2 a}+\frac {i}{a}-\frac {\arctan \left (a x \right )}{a}\)	\(279\)

[In]

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

[Out]

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]

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

[In]

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

[Out]

integral(arccot(a*x)^2, x)

Sympy [F]

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

[In]

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

[Out]

Integral(acot(a*x)**2, x)

Maxima [F]

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

[In]

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

[Out]

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
*log(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]

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

[In]

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

[Out]

integrate(arccot(a*x)^2, x)

Mupad [B] (verification not implemented)

Time = 0.71 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.82 \[ \int \cot ^{-1}(a x)^2 \, dx=\frac {-2\,\ln \left (1-{\mathrm {e}}^{\mathrm {acot}\left (a\,x\right )\,2{}\mathrm {i}}\right )\,\mathrm {acot}\left (a\,x\right )+\mathrm {polylog}\left (2,{\mathrm {e}}^{\mathrm {acot}\left (a\,x\right )\,2{}\mathrm {i}}\right )\,1{}\mathrm {i}+{\mathrm {acot}\left (a\,x\right )}^2\,1{}\mathrm {i}}{a}+x\,{\mathrm {acot}\left (a\,x\right )}^2 \]

[In]

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

[Out]

(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