\(\int \frac {\cot ^{-1}(a x)^2}{x^3} \, dx\) [20]

Optimal result

Integrand size = 10, antiderivative size = 59 \[ \int \frac {\cot ^{-1}(a x)^2}{x^3} \, dx=\frac {a \cot ^{-1}(a x)}{x}-\frac {1}{2} a^2 \cot ^{-1}(a x)^2-\frac {\cot ^{-1}(a x)^2}{2 x^2}+a^2 \log (x)-\frac {1}{2} a^2 \log \left (1+a^2 x^2\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {4947, 5039, 272, 36, 29, 31, 5005} \[ \int \frac {\cot ^{-1}(a x)^2}{x^3} \, dx=-\frac {1}{2} a^2 \log \left (a^2 x^2+1\right )+a^2 \log (x)-\frac {1}{2} a^2 \cot ^{-1}(a x)^2-\frac {\cot ^{-1}(a x)^2}{2 x^2}+\frac {a \cot ^{-1}(a x)}{x} \]

[In]

Int[ArcCot[a*x]^2/x^3,x]

[Out]

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

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 4947

Int[((a_.) + ArcCot[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcCot[c*x^
n])^p/(m + 1)), x] + Dist[b*c*n*(p/(m + 1)), Int[x^(m + n)*((a + b*ArcCot[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 5005

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[-(a + b*ArcCot[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 5039

Int[(((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/d,
Int[(f*x)^m*(a + b*ArcCot[c*x])^p, x], x] - Dist[e/(d*f^2), Int[(f*x)^(m + 2)*((a + b*ArcCot[c*x])^p/(d + e*x
^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && LtQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {\cot ^{-1}(a x)^2}{2 x^2}-a \int \frac {\cot ^{-1}(a x)}{x^2 \left (1+a^2 x^2\right )} \, dx \\ & = -\frac {\cot ^{-1}(a x)^2}{2 x^2}-a \int \frac {\cot ^{-1}(a x)}{x^2} \, dx+a^3 \int \frac {\cot ^{-1}(a x)}{1+a^2 x^2} \, dx \\ & = \frac {a \cot ^{-1}(a x)}{x}-\frac {1}{2} a^2 \cot ^{-1}(a x)^2-\frac {\cot ^{-1}(a x)^2}{2 x^2}+a^2 \int \frac {1}{x \left (1+a^2 x^2\right )} \, dx \\ & = \frac {a \cot ^{-1}(a x)}{x}-\frac {1}{2} a^2 \cot ^{-1}(a x)^2-\frac {\cot ^{-1}(a x)^2}{2 x^2}+\frac {1}{2} a^2 \text {Subst}\left (\int \frac {1}{x \left (1+a^2 x\right )} \, dx,x,x^2\right ) \\ & = \frac {a \cot ^{-1}(a x)}{x}-\frac {1}{2} a^2 \cot ^{-1}(a x)^2-\frac {\cot ^{-1}(a x)^2}{2 x^2}+\frac {1}{2} a^2 \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )-\frac {1}{2} a^4 \text {Subst}\left (\int \frac {1}{1+a^2 x} \, dx,x,x^2\right ) \\ & = \frac {a \cot ^{-1}(a x)}{x}-\frac {1}{2} a^2 \cot ^{-1}(a x)^2-\frac {\cot ^{-1}(a x)^2}{2 x^2}+a^2 \log (x)-\frac {1}{2} a^2 \log \left (1+a^2 x^2\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.95 \[ \int \frac {\cot ^{-1}(a x)^2}{x^3} \, dx=\frac {a \cot ^{-1}(a x)}{x}+\frac {\left (-1-a^2 x^2\right ) \cot ^{-1}(a x)^2}{2 x^2}+a^2 \log (x)-\frac {1}{2} a^2 \log \left (1+a^2 x^2\right ) \]

[In]

Integrate[ArcCot[a*x]^2/x^3,x]

[Out]

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

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.08

method	result	size
derivativedivides	\(a^{2} \left (-\frac {\operatorname {arccot}\left (a x \right )^{2}}{2 a^{2} x^{2}}+\frac {\operatorname {arccot}\left (a x \right )}{a x}+\operatorname {arccot}\left (a x \right ) \arctan \left (a x \right )+\ln \left (a x \right )-\frac {\ln \left (a^{2} x^{2}+1\right )}{2}+\frac {\arctan \left (a x \right )^{2}}{2}\right )\)	\(64\)
default	\(a^{2} \left (-\frac {\operatorname {arccot}\left (a x \right )^{2}}{2 a^{2} x^{2}}+\frac {\operatorname {arccot}\left (a x \right )}{a x}+\operatorname {arccot}\left (a x \right ) \arctan \left (a x \right )+\ln \left (a x \right )-\frac {\ln \left (a^{2} x^{2}+1\right )}{2}+\frac {\arctan \left (a x \right )^{2}}{2}\right )\)	\(64\)
parallelrisch	\(\frac {-a^{2} x^{2} \operatorname {arccot}\left (a x \right )^{2}+2 a^{2} \ln \left (x \right ) x^{2}-a^{2} \ln \left (a^{2} x^{2}+1\right ) x^{2}+2 \,\operatorname {arccot}\left (a x \right ) a x -\operatorname {arccot}\left (a x \right )^{2}}{2 x^{2}}\)	\(65\)
parts	\(-\frac {\operatorname {arccot}\left (a x \right )^{2}}{2 x^{2}}-a^{2} \left (-\frac {\operatorname {arccot}\left (a x \right )}{a x}-\operatorname {arccot}\left (a x \right ) \arctan \left (a x \right )-\ln \left (a x \right )+\frac {\ln \left (a^{2} x^{2}+1\right )}{2}-\frac {\arctan \left (a x \right )^{2}}{2}\right )\)	\(67\)
risch	\(\frac {\left (a^{2} x^{2}+1\right ) \ln \left (i a x +1\right )^{2}}{8 x^{2}}-\frac {i \left (-i x^{2} \ln \left (-i a x +1\right ) a^{2}-2 a x +\pi -i \ln \left (-i a x +1\right )\right ) \ln \left (i a x +1\right )}{4 x^{2}}-\frac {2 i a^{2} \ln \left (\left (-\pi a +6 i a \right ) x +6+i \pi \right ) \pi \,x^{2}-2 i a^{2} \ln \left (\left (-\pi a -6 i a \right ) x +6-i \pi \right ) \pi \,x^{2}-a^{2} x^{2} \ln \left (-i a x +1\right )^{2}+4 a^{2} \ln \left (\left (-\pi a +6 i a \right ) x +6+i \pi \right ) x^{2}+4 a^{2} \ln \left (\left (-\pi a -6 i a \right ) x +6-i \pi \right ) x^{2}-8 a^{2} \ln \left (-x \right ) x^{2}+4 i a x \ln \left (-i a x +1\right )-2 i \pi \ln \left (-i a x +1\right )-4 \pi a x +\pi ^{2}-\ln \left (-i a x +1\right )^{2}}{8 x^{2}}\)	\(263\)

[In]

int(arccot(a*x)^2/x^3,x,method=_RETURNVERBOSE)

[Out]

a^2*(-1/2/a^2/x^2*arccot(a*x)^2+1/a/x*arccot(a*x)+arccot(a*x)*arctan(a*x)+ln(a*x)-1/2*ln(a^2*x^2+1)+1/2*arctan
(a*x)^2)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.97 \[ \int \frac {\cot ^{-1}(a x)^2}{x^3} \, dx=-\frac {a^{2} x^{2} \log \left (a^{2} x^{2} + 1\right ) - 2 \, a^{2} x^{2} \log \left (x\right ) - 2 \, a x \operatorname {arccot}\left (a x\right ) + {\left (a^{2} x^{2} + 1\right )} \operatorname {arccot}\left (a x\right )^{2}}{2 \, x^{2}} \]

[In]

integrate(arccot(a*x)^2/x^3,x, algorithm="fricas")

[Out]

-1/2*(a^2*x^2*log(a^2*x^2 + 1) - 2*a^2*x^2*log(x) - 2*a*x*arccot(a*x) + (a^2*x^2 + 1)*arccot(a*x)^2)/x^2

Sympy [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.90 \[ \int \frac {\cot ^{-1}(a x)^2}{x^3} \, dx=a^{2} \log {\left (x \right )} - \frac {a^{2} \log {\left (a^{2} x^{2} + 1 \right )}}{2} - \frac {a^{2} \operatorname {acot}^{2}{\left (a x \right )}}{2} + \frac {a \operatorname {acot}{\left (a x \right )}}{x} - \frac {\operatorname {acot}^{2}{\left (a x \right )}}{2 x^{2}} \]

[In]

integrate(acot(a*x)**2/x**3,x)

[Out]

a**2*log(x) - a**2*log(a**2*x**2 + 1)/2 - a**2*acot(a*x)**2/2 + a*acot(a*x)/x - acot(a*x)**2/(2*x**2)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.95 \[ \int \frac {\cot ^{-1}(a x)^2}{x^3} \, dx=\frac {1}{2} \, {\left (\arctan \left (a x\right )^{2} - \log \left (a^{2} x^{2} + 1\right ) + 2 \, \log \left (x\right )\right )} a^{2} + {\left (a \arctan \left (a x\right ) + \frac {1}{x}\right )} a \operatorname {arccot}\left (a x\right ) - \frac {\operatorname {arccot}\left (a x\right )^{2}}{2 \, x^{2}} \]

[In]

integrate(arccot(a*x)^2/x^3,x, algorithm="maxima")

[Out]

1/2*(arctan(a*x)^2 - log(a^2*x^2 + 1) + 2*log(x))*a^2 + (a*arctan(a*x) + 1/x)*a*arccot(a*x) - 1/2*arccot(a*x)^
2/x^2

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.02 \[ \int \frac {\cot ^{-1}(a x)^2}{x^3} \, dx=-\frac {1}{2} \, {\left ({\left (\arctan \left (\frac {1}{a x}\right )^{2} - \frac {2 \, \arctan \left (\frac {1}{a x}\right )}{a x} + \log \left (\frac {1}{a^{2} x^{2}} + 1\right )\right )} a + \frac {\arctan \left (\frac {1}{a x}\right )^{2}}{a x^{2}}\right )} a \]

[In]

integrate(arccot(a*x)^2/x^3,x, algorithm="giac")

[Out]

-1/2*((arctan(1/(a*x))^2 - 2*arctan(1/(a*x))/(a*x) + log(1/(a^2*x^2) + 1))*a + arctan(1/(a*x))^2/(a*x^2))*a

Mupad [B] (verification not implemented)

Time = 0.75 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.85 \[ \int \frac {\cot ^{-1}(a x)^2}{x^3} \, dx=a^2\,\ln \left (x\right )-{\mathrm {acot}\left (a\,x\right )}^2\,\left (\frac {a^2}{2}+\frac {1}{2\,x^2}\right )-\frac {a^2\,\ln \left (a^2\,x^2+1\right )}{2}+\frac {a\,\mathrm {acot}\left (a\,x\right )}{x} \]

[In]

int(acot(a*x)^2/x^3,x)

[Out]

a^2*log(x) - acot(a*x)^2*(a^2/2 + 1/(2*x^2)) - (a^2*log(a^2*x^2 + 1))/2 + (a*acot(a*x))/x