\(\int \frac {\cot ^{-1}(a x)^2}{x^5} \, dx\) [22]

Optimal result

Integrand size = 10, antiderivative size = 89 \[ \int \frac {\cot ^{-1}(a x)^2}{x^5} \, dx=-\frac {a^2}{12 x^2}+\frac {a \cot ^{-1}(a x)}{6 x^3}-\frac {a^3 \cot ^{-1}(a x)}{2 x}+\frac {1}{4} a^4 \cot ^{-1}(a x)^2-\frac {\cot ^{-1}(a x)^2}{4 x^4}-\frac {2}{3} a^4 \log (x)+\frac {1}{3} a^4 \log \left (1+a^2 x^2\right ) \]

[Out]

-1/12*a^2/x^2+1/6*a*arccot(a*x)/x^3-1/2*a^3*arccot(a*x)/x+1/4*a^4*arccot(a*x)^2-1/4*arccot(a*x)^2/x^4-2/3*a^4*
ln(x)+1/3*a^4*ln(a^2*x^2+1)

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {4947, 5039, 272, 46, 36, 29, 31, 5005} \[ \int \frac {\cot ^{-1}(a x)^2}{x^5} \, dx=-\frac {2}{3} a^4 \log (x)+\frac {1}{4} a^4 \cot ^{-1}(a x)^2-\frac {a^3 \cot ^{-1}(a x)}{2 x}-\frac {a^2}{12 x^2}+\frac {1}{3} a^4 \log \left (a^2 x^2+1\right )-\frac {\cot ^{-1}(a x)^2}{4 x^4}+\frac {a \cot ^{-1}(a x)}{6 x^3} \]

[In]

Int[ArcCot[a*x]^2/x^5,x]

[Out]

-1/12*a^2/x^2 + (a*ArcCot[a*x])/(6*x^3) - (a^3*ArcCot[a*x])/(2*x) + (a^4*ArcCot[a*x]^2)/4 - ArcCot[a*x]^2/(4*x
^4) - (2*a^4*Log[x])/3 + (a^4*Log[1 + a^2*x^2])/3

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 46

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x
)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 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}{4 x^4}-\frac {1}{2} a \int \frac {\cot ^{-1}(a x)}{x^4 \left (1+a^2 x^2\right )} \, dx \\ & = -\frac {\cot ^{-1}(a x)^2}{4 x^4}-\frac {1}{2} a \int \frac {\cot ^{-1}(a x)}{x^4} \, dx+\frac {1}{2} a^3 \int \frac {\cot ^{-1}(a x)}{x^2 \left (1+a^2 x^2\right )} \, dx \\ & = \frac {a \cot ^{-1}(a x)}{6 x^3}-\frac {\cot ^{-1}(a x)^2}{4 x^4}+\frac {1}{6} a^2 \int \frac {1}{x^3 \left (1+a^2 x^2\right )} \, dx+\frac {1}{2} a^3 \int \frac {\cot ^{-1}(a x)}{x^2} \, dx-\frac {1}{2} a^5 \int \frac {\cot ^{-1}(a x)}{1+a^2 x^2} \, dx \\ & = \frac {a \cot ^{-1}(a x)}{6 x^3}-\frac {a^3 \cot ^{-1}(a x)}{2 x}+\frac {1}{4} a^4 \cot ^{-1}(a x)^2-\frac {\cot ^{-1}(a x)^2}{4 x^4}+\frac {1}{12} a^2 \text {Subst}\left (\int \frac {1}{x^2 \left (1+a^2 x\right )} \, dx,x,x^2\right )-\frac {1}{2} a^4 \int \frac {1}{x \left (1+a^2 x^2\right )} \, dx \\ & = \frac {a \cot ^{-1}(a x)}{6 x^3}-\frac {a^3 \cot ^{-1}(a x)}{2 x}+\frac {1}{4} a^4 \cot ^{-1}(a x)^2-\frac {\cot ^{-1}(a x)^2}{4 x^4}+\frac {1}{12} a^2 \text {Subst}\left (\int \left (\frac {1}{x^2}-\frac {a^2}{x}+\frac {a^4}{1+a^2 x}\right ) \, dx,x,x^2\right )-\frac {1}{4} a^4 \text {Subst}\left (\int \frac {1}{x \left (1+a^2 x\right )} \, dx,x,x^2\right ) \\ & = -\frac {a^2}{12 x^2}+\frac {a \cot ^{-1}(a x)}{6 x^3}-\frac {a^3 \cot ^{-1}(a x)}{2 x}+\frac {1}{4} a^4 \cot ^{-1}(a x)^2-\frac {\cot ^{-1}(a x)^2}{4 x^4}-\frac {1}{6} a^4 \log (x)+\frac {1}{12} a^4 \log \left (1+a^2 x^2\right )-\frac {1}{4} a^4 \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )+\frac {1}{4} a^6 \text {Subst}\left (\int \frac {1}{1+a^2 x} \, dx,x,x^2\right ) \\ & = -\frac {a^2}{12 x^2}+\frac {a \cot ^{-1}(a x)}{6 x^3}-\frac {a^3 \cot ^{-1}(a x)}{2 x}+\frac {1}{4} a^4 \cot ^{-1}(a x)^2-\frac {\cot ^{-1}(a x)^2}{4 x^4}-\frac {2}{3} a^4 \log (x)+\frac {1}{3} a^4 \log \left (1+a^2 x^2\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.91 \[ \int \frac {\cot ^{-1}(a x)^2}{x^5} \, dx=-\frac {a^2}{12 x^2}-\frac {a \left (-1+3 a^2 x^2\right ) \cot ^{-1}(a x)}{6 x^3}+\frac {\left (-1+a^4 x^4\right ) \cot ^{-1}(a x)^2}{4 x^4}-\frac {2}{3} a^4 \log (x)+\frac {1}{3} a^4 \log \left (1+a^2 x^2\right ) \]

[In]

Integrate[ArcCot[a*x]^2/x^5,x]

[Out]

-1/12*a^2/x^2 - (a*(-1 + 3*a^2*x^2)*ArcCot[a*x])/(6*x^3) + ((-1 + a^4*x^4)*ArcCot[a*x]^2)/(4*x^4) - (2*a^4*Log
[x])/3 + (a^4*Log[1 + a^2*x^2])/3

Maple [A] (verified)

Time = 0.29 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.96

method	result	size
parts	\(-\frac {\operatorname {arccot}\left (a x \right )^{2}}{4 x^{4}}-\frac {a^{4} \left (-\frac {\operatorname {arccot}\left (a x \right )}{3 a^{3} x^{3}}+\frac {\operatorname {arccot}\left (a x \right )}{a x}+\operatorname {arccot}\left (a x \right ) \arctan \left (a x \right )+\frac {1}{6 a^{2} x^{2}}+\frac {4 \ln \left (a x \right )}{3}-\frac {2 \ln \left (a^{2} x^{2}+1\right )}{3}+\frac {\arctan \left (a x \right )^{2}}{2}\right )}{2}\)	\(85\)
derivativedivides	\(a^{4} \left (-\frac {\operatorname {arccot}\left (a x \right )^{2}}{4 a^{4} x^{4}}+\frac {\operatorname {arccot}\left (a x \right )}{6 a^{3} x^{3}}-\frac {\operatorname {arccot}\left (a x \right )}{2 a x}-\frac {\operatorname {arccot}\left (a x \right ) \arctan \left (a x \right )}{2}-\frac {1}{12 a^{2} x^{2}}-\frac {2 \ln \left (a x \right )}{3}+\frac {\ln \left (a^{2} x^{2}+1\right )}{3}-\frac {\arctan \left (a x \right )^{2}}{4}\right )\)	\(88\)
default	\(a^{4} \left (-\frac {\operatorname {arccot}\left (a x \right )^{2}}{4 a^{4} x^{4}}+\frac {\operatorname {arccot}\left (a x \right )}{6 a^{3} x^{3}}-\frac {\operatorname {arccot}\left (a x \right )}{2 a x}-\frac {\operatorname {arccot}\left (a x \right ) \arctan \left (a x \right )}{2}-\frac {1}{12 a^{2} x^{2}}-\frac {2 \ln \left (a x \right )}{3}+\frac {\ln \left (a^{2} x^{2}+1\right )}{3}-\frac {\arctan \left (a x \right )^{2}}{4}\right )\)	\(88\)
parallelrisch	\(-\frac {-3 a^{4} x^{4} \operatorname {arccot}\left (a x \right )^{2}+8 a^{4} \ln \left (x \right ) x^{4}-4 a^{4} \ln \left (a^{2} x^{2}+1\right ) x^{4}-a^{4} x^{4}+6 a^{3} x^{3} \operatorname {arccot}\left (a x \right )+a^{2} x^{2}-2 \,\operatorname {arccot}\left (a x \right ) a x +3 \operatorname {arccot}\left (a x \right )^{2}}{12 x^{4}}\)	\(92\)
risch	\(-\frac {\left (a^{4} x^{4}-1\right ) \ln \left (i a x +1\right )^{2}}{16 x^{4}}-\frac {i \left (3 i a^{4} \ln \left (-i a x +1\right ) x^{4}+6 a^{3} x^{3}-3 i \ln \left (-i a x +1\right )-2 a x +3 \pi \right ) \ln \left (i a x +1\right )}{24 x^{4}}-\frac {-6 i a^{4} \ln \left (\left (-\pi a +8 i a \right ) x +8+i \pi \right ) \pi \,x^{4}+6 i a^{4} \ln \left (\left (-\pi a -8 i a \right ) x +8-i \pi \right ) \pi \,x^{4}+3 a^{4} x^{4} \ln \left (-i a x +1\right )^{2}-16 a^{4} \ln \left (\left (-\pi a +8 i a \right ) x +8+i \pi \right ) x^{4}-16 a^{4} \ln \left (\left (-\pi a -8 i a \right ) x +8-i \pi \right ) x^{4}+32 a^{4} \ln \left (-x \right ) x^{4}-12 i a^{3} x^{3} \ln \left (-i a x +1\right )+12 \pi \,a^{3} x^{3}+4 i a x \ln \left (-i a x +1\right )+4 a^{2} x^{2}-6 i \pi \ln \left (-i a x +1\right )-4 \pi a x +3 \pi ^{2}-3 \ln \left (-i a x +1\right )^{2}}{48 x^{4}}\)	\(309\)

[In]

int(arccot(a*x)^2/x^5,x,method=_RETURNVERBOSE)

[Out]

-1/4*arccot(a*x)^2/x^4-1/2*a^4*(-1/3/a^3/x^3*arccot(a*x)+1/a/x*arccot(a*x)+arccot(a*x)*arctan(a*x)+1/6/a^2/x^2
+4/3*ln(a*x)-2/3*ln(a^2*x^2+1)+1/2*arctan(a*x)^2)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.88 \[ \int \frac {\cot ^{-1}(a x)^2}{x^5} \, dx=\frac {4 \, a^{4} x^{4} \log \left (a^{2} x^{2} + 1\right ) - 8 \, a^{4} x^{4} \log \left (x\right ) - a^{2} x^{2} + 3 \, {\left (a^{4} x^{4} - 1\right )} \operatorname {arccot}\left (a x\right )^{2} - 2 \, {\left (3 \, a^{3} x^{3} - a x\right )} \operatorname {arccot}\left (a x\right )}{12 \, x^{4}} \]

[In]

integrate(arccot(a*x)^2/x^5,x, algorithm="fricas")

[Out]

1/12*(4*a^4*x^4*log(a^2*x^2 + 1) - 8*a^4*x^4*log(x) - a^2*x^2 + 3*(a^4*x^4 - 1)*arccot(a*x)^2 - 2*(3*a^3*x^3 -
a*x)*arccot(a*x))/x^4

Sympy [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.90 \[ \int \frac {\cot ^{-1}(a x)^2}{x^5} \, dx=- \frac {2 a^{4} \log {\left (x \right )}}{3} + \frac {a^{4} \log {\left (a^{2} x^{2} + 1 \right )}}{3} + \frac {a^{4} \operatorname {acot}^{2}{\left (a x \right )}}{4} - \frac {a^{3} \operatorname {acot}{\left (a x \right )}}{2 x} - \frac {a^{2}}{12 x^{2}} + \frac {a \operatorname {acot}{\left (a x \right )}}{6 x^{3}} - \frac {\operatorname {acot}^{2}{\left (a x \right )}}{4 x^{4}} \]

[In]

integrate(acot(a*x)**2/x**5,x)

[Out]

-2*a**4*log(x)/3 + a**4*log(a**2*x**2 + 1)/3 + a**4*acot(a*x)**2/4 - a**3*acot(a*x)/(2*x) - a**2/(12*x**2) + a
*acot(a*x)/(6*x**3) - acot(a*x)**2/(4*x**4)

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.07 \[ \int \frac {\cot ^{-1}(a x)^2}{x^5} \, dx=-\frac {1}{6} \, {\left (3 \, a^{3} \arctan \left (a x\right ) + \frac {3 \, a^{2} x^{2} - 1}{x^{3}}\right )} a \operatorname {arccot}\left (a x\right ) - \frac {{\left (3 \, a^{2} x^{2} \arctan \left (a x\right )^{2} - 4 \, a^{2} x^{2} \log \left (a^{2} x^{2} + 1\right ) + 8 \, a^{2} x^{2} \log \left (x\right ) + 1\right )} a^{2}}{12 \, x^{2}} - \frac {\operatorname {arccot}\left (a x\right )^{2}}{4 \, x^{4}} \]

[In]

integrate(arccot(a*x)^2/x^5,x, algorithm="maxima")

[Out]

-1/6*(3*a^3*arctan(a*x) + (3*a^2*x^2 - 1)/x^3)*a*arccot(a*x) - 1/12*(3*a^2*x^2*arctan(a*x)^2 - 4*a^2*x^2*log(a
^2*x^2 + 1) + 8*a^2*x^2*log(x) + 1)*a^2/x^2 - 1/4*arccot(a*x)^2/x^4

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.02 \[ \int \frac {\cot ^{-1}(a x)^2}{x^5} \, dx=\frac {1}{12} \, {\left ({\left (3 \, \arctan \left (\frac {1}{a x}\right )^{2} - \frac {6 \, \arctan \left (\frac {1}{a x}\right )}{a x} - \frac {1}{a^{2} x^{2}} + \frac {2 \, \arctan \left (\frac {1}{a x}\right )}{a^{3} x^{3}} + 4 \, \log \left (\frac {1}{a^{2} x^{2}} + 1\right )\right )} a^{3} - \frac {3 \, \arctan \left (\frac {1}{a x}\right )^{2}}{a x^{4}}\right )} a \]

[In]

integrate(arccot(a*x)^2/x^5,x, algorithm="giac")

[Out]

1/12*((3*arctan(1/(a*x))^2 - 6*arctan(1/(a*x))/(a*x) - 1/(a^2*x^2) + 2*arctan(1/(a*x))/(a^3*x^3) + 4*log(1/(a^
2*x^2) + 1))*a^3 - 3*arctan(1/(a*x))^2/(a*x^4))*a

Mupad [B] (verification not implemented)

Time = 0.81 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.82 \[ \int \frac {\cot ^{-1}(a x)^2}{x^5} \, dx={\mathrm {acot}\left (a\,x\right )}^2\,\left (\frac {a^4}{4}-\frac {1}{4\,x^4}\right )-\frac {2\,a^4\,\ln \left (x\right )}{3}+\frac {a^4\,\ln \left (a^2\,x^2+1\right )}{3}-\frac {a^2}{12\,x^2}-\frac {a^2\,\mathrm {acot}\left (a\,x\right )\,\left (\frac {a\,x^2}{2}-\frac {1}{6\,a}\right )}{x^3} \]

[In]

int(acot(a*x)^2/x^5,x)

[Out]

acot(a*x)^2*(a^4/4 - 1/(4*x^4)) - (2*a^4*log(x))/3 + (a^4*log(a^2*x^2 + 1))/3 - a^2/(12*x^2) - (a^2*acot(a*x)*
((a*x^2)/2 - 1/(6*a)))/x^3