\(\int \frac {\cot ^{-1}(a x)}{x^5} \, dx\) [11]

Optimal result

Integrand size = 8, antiderivative size = 41 \[ \int \frac {\cot ^{-1}(a x)}{x^5} \, dx=\frac {a}{12 x^3}-\frac {a^3}{4 x}-\frac {\cot ^{-1}(a x)}{4 x^4}-\frac {1}{4} a^4 \arctan (a x) \]

[Out]

1/12*a/x^3-1/4*a^3/x-1/4*arccot(a*x)/x^4-1/4*a^4*arctan(a*x)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4947, 331, 209} \[ \int \frac {\cot ^{-1}(a x)}{x^5} \, dx=-\frac {1}{4} a^4 \arctan (a x)-\frac {a^3}{4 x}-\frac {\cot ^{-1}(a x)}{4 x^4}+\frac {a}{12 x^3} \]

[In]

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

[Out]

a/(12*x^3) - a^3/(4*x) - ArcCot[a*x]/(4*x^4) - (a^4*ArcTan[a*x])/4

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

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]

Rubi steps \begin{align*} \text {integral}& = -\frac {\cot ^{-1}(a x)}{4 x^4}-\frac {1}{4} a \int \frac {1}{x^4 \left (1+a^2 x^2\right )} \, dx \\ & = \frac {a}{12 x^3}-\frac {\cot ^{-1}(a x)}{4 x^4}+\frac {1}{4} a^3 \int \frac {1}{x^2 \left (1+a^2 x^2\right )} \, dx \\ & = \frac {a}{12 x^3}-\frac {a^3}{4 x}-\frac {\cot ^{-1}(a x)}{4 x^4}-\frac {1}{4} a^5 \int \frac {1}{1+a^2 x^2} \, dx \\ & = \frac {a}{12 x^3}-\frac {a^3}{4 x}-\frac {\cot ^{-1}(a x)}{4 x^4}-\frac {1}{4} a^4 \arctan (a x) \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.00 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.88 \[ \int \frac {\cot ^{-1}(a x)}{x^5} \, dx=-\frac {\cot ^{-1}(a x)}{4 x^4}+\frac {a \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},-a^2 x^2\right )}{12 x^3} \]

[In]

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

[Out]

-1/4*ArcCot[a*x]/x^4 + (a*Hypergeometric2F1[-3/2, 1, -1/2, -(a^2*x^2)])/(12*x^3)

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.85

method	result	size
parts	\(-\frac {\operatorname {arccot}\left (a x \right )}{4 x^{4}}-\frac {a \left (-\frac {1}{3 x^{3}}+\frac {a^{2}}{x}+a^{3} \arctan \left (a x \right )\right )}{4}\)	\(35\)
parallelrisch	\(\frac {3 a^{4} x^{4} \operatorname {arccot}\left (a x \right )-3 a^{3} x^{3}+a x -3 \,\operatorname {arccot}\left (a x \right )}{12 x^{4}}\)	\(36\)
derivativedivides	\(a^{4} \left (-\frac {\operatorname {arccot}\left (a x \right )}{4 a^{4} x^{4}}-\frac {\arctan \left (a x \right )}{4}+\frac {1}{12 a^{3} x^{3}}-\frac {1}{4 a x}\right )\)	\(40\)
default	\(a^{4} \left (-\frac {\operatorname {arccot}\left (a x \right )}{4 a^{4} x^{4}}-\frac {\arctan \left (a x \right )}{4}+\frac {1}{12 a^{3} x^{3}}-\frac {1}{4 a x}\right )\)	\(40\)
risch	\(-\frac {i \ln \left (i a x +1\right )}{8 x^{4}}-\frac {-3 i a^{4} \ln \left (-a x +i\right ) x^{4}+3 i a^{4} \ln \left (-a x -i\right ) x^{4}+6 a^{3} x^{3}-3 i \ln \left (-i a x +1\right )-2 a x +3 \pi }{24 x^{4}}\)	\(82\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.80 \[ \int \frac {\cot ^{-1}(a x)}{x^5} \, dx=-\frac {3 \, a^{3} x^{3} - a x - 3 \, {\left (a^{4} x^{4} - 1\right )} \operatorname {arccot}\left (a x\right )}{12 \, x^{4}} \]

[In]

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

[Out]

-1/12*(3*a^3*x^3 - a*x - 3*(a^4*x^4 - 1)*arccot(a*x))/x^4

Sympy [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.78 \[ \int \frac {\cot ^{-1}(a x)}{x^5} \, dx=\frac {a^{4} \operatorname {acot}{\left (a x \right )}}{4} - \frac {a^{3}}{4 x} + \frac {a}{12 x^{3}} - \frac {\operatorname {acot}{\left (a x \right )}}{4 x^{4}} \]

[In]

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

[Out]

a**4*acot(a*x)/4 - a**3/(4*x) + a/(12*x**3) - acot(a*x)/(4*x**4)

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.90 \[ \int \frac {\cot ^{-1}(a x)}{x^5} \, dx=-\frac {1}{12} \, {\left (3 \, a^{3} \arctan \left (a x\right ) + \frac {3 \, a^{2} x^{2} - 1}{x^{3}}\right )} a - \frac {\operatorname {arccot}\left (a x\right )}{4 \, x^{4}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.24 \[ \int \frac {\cot ^{-1}(a x)}{x^5} \, dx=-\frac {1}{12} \, {\left (a^{3} {\left (\frac {3}{a x} - \frac {1}{a^{3} x^{3}} - 3 \, \arctan \left (\frac {1}{a x}\right )\right )} + \frac {3 \, \arctan \left (\frac {1}{a x}\right )}{a x^{4}}\right )} a \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.84 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.15 \[ \int \frac {\cot ^{-1}(a x)}{x^5} \, dx=\left \{\begin {array}{cl} -\frac {\pi }{8\,x^4} & \text {\ if\ \ }a=0\\ -\frac {a^4\,\mathrm {atan}\left (a\,x\right )}{4}-\frac {\frac {\mathrm {acot}\left (a\,x\right )}{4}-\frac {a\,x}{12}+\frac {a^3\,x^3}{4}}{x^4} & \text {\ if\ \ }a\neq 0 \end {array}\right . \]

[In]

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

[Out]

piecewise(a == 0, -pi/(8*x^4), a ~= 0, - (a^4*atan(a*x))/4 - (acot(a*x)/4 - (a*x)/12 + (a^3*x^3)/4)/x^4)