\(\int \frac {\cot ^{-1}(a x)}{x^3} \, dx\) [9]

Optimal result

Integrand size = 8, antiderivative size = 31 \[ \int \frac {\cot ^{-1}(a x)}{x^3} \, dx=\frac {a}{2 x}-\frac {\cot ^{-1}(a x)}{2 x^2}+\frac {1}{2} a^2 \arctan (a x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4947, 331, 209} \[ \int \frac {\cot ^{-1}(a x)}{x^3} \, dx=\frac {1}{2} a^2 \arctan (a x)-\frac {\cot ^{-1}(a x)}{2 x^2}+\frac {a}{2 x} \]

[In]

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

[Out]

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

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)}{2 x^2}-\frac {1}{2} a \int \frac {1}{x^2 \left (1+a^2 x^2\right )} \, dx \\ & = \frac {a}{2 x}-\frac {\cot ^{-1}(a x)}{2 x^2}+\frac {1}{2} a^3 \int \frac {1}{1+a^2 x^2} \, dx \\ & = \frac {a}{2 x}-\frac {\cot ^{-1}(a x)}{2 x^2}+\frac {1}{2} a^2 \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 = 1.16 \[ \int \frac {\cot ^{-1}(a x)}{x^3} \, dx=-\frac {\cot ^{-1}(a x)}{2 x^2}+\frac {a \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-a^2 x^2\right )}{2 x} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84

method	result	size
parallelrisch	\(-\frac {\operatorname {arccot}\left (a x \right ) a^{2} x^{2}-a x +\operatorname {arccot}\left (a x \right )}{2 x^{2}}\)	\(26\)
parts	\(-\frac {\operatorname {arccot}\left (a x \right )}{2 x^{2}}-\frac {a \left (-\frac {1}{x}-a \arctan \left (a x \right )\right )}{2}\)	\(27\)
derivativedivides	\(a^{2} \left (-\frac {\operatorname {arccot}\left (a x \right )}{2 a^{2} x^{2}}+\frac {\arctan \left (a x \right )}{2}+\frac {1}{2 a x}\right )\)	\(32\)
default	\(a^{2} \left (-\frac {\operatorname {arccot}\left (a x \right )}{2 a^{2} x^{2}}+\frac {\arctan \left (a x \right )}{2}+\frac {1}{2 a x}\right )\)	\(32\)
risch	\(-\frac {i \ln \left (i a x +1\right )}{4 x^{2}}-\frac {-i a^{2} \ln \left (-a x -i\right ) x^{2}+i a^{2} \ln \left (-a x +i\right ) x^{2}-i \ln \left (-i a x +1\right )-2 a x +\pi }{4 x^{2}}\)	\(72\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.77 \[ \int \frac {\cot ^{-1}(a x)}{x^3} \, dx=\frac {a x - {\left (a^{2} x^{2} + 1\right )} \operatorname {arccot}\left (a x\right )}{2 \, x^{2}} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.77 \[ \int \frac {\cot ^{-1}(a x)}{x^3} \, dx=- \frac {a^{2} \operatorname {acot}{\left (a x \right )}}{2} + \frac {a}{2 x} - \frac {\operatorname {acot}{\left (a x \right )}}{2 x^{2}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.74 \[ \int \frac {\cot ^{-1}(a x)}{x^3} \, dx=\frac {1}{2} \, {\left (a \arctan \left (a x\right ) + \frac {1}{x}\right )} a - \frac {\operatorname {arccot}\left (a x\right )}{2 \, x^{2}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.29 \[ \int \frac {\cot ^{-1}(a x)}{x^3} \, dx=\frac {1}{2} \, {\left (a {\left (\frac {1}{a x} - \arctan \left (\frac {1}{a x}\right )\right )} - \frac {\arctan \left (\frac {1}{a x}\right )}{a x^{2}}\right )} a \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.81 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.42 \[ \int \frac {\cot ^{-1}(a x)}{x^3} \, dx=\left \{\begin {array}{cl} -\frac {\pi }{4\,x^2} & \text {\ if\ \ }a=0\\ \frac {a^3\,\mathrm {atan}\left (a\,x\right )+\frac {a^2}{x}}{2\,a}-\frac {\mathrm {acot}\left (a\,x\right )}{2\,x^2} & \text {\ if\ \ }a\neq 0 \end {array}\right . \]

[In]

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

[Out]

piecewise(a == 0, -pi/(4*x^2), a ~= 0, (a^3*atan(a*x) + a^2/x)/(2*a) - acot(a*x)/(2*x^2))