3.10 \(\int \frac {\cot ^{-1}(a x)}{x^4} \, dx\)

Optimal. Leaf size=46 \[ \frac {1}{3} a^3 \log (x)-\frac {1}{6} a^3 \log \left (a^2 x^2+1\right )-\frac {\cot ^{-1}(a x)}{3 x^3}+\frac {a}{6 x^2} \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4853, 266, 44} \[ -\frac {1}{6} a^3 \log \left (a^2 x^2+1\right )+\frac {1}{3} a^3 \log (x)+\frac {a}{6 x^2}-\frac {\cot ^{-1}(a x)}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[ArcCot[a*x]/x^4,x]

[Out]

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

Rule 44

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] && L
tQ[m + n + 2, 0])

Rule 266

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 4853

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcCo
t[c*x])^p)/(d*(m + 1)), x] + Dist[(b*c*p)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcCot[c*x])^(p - 1))/(1 + c^
2*x^2), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && (EqQ[p, 1] || IntegerQ[m]) && NeQ[m, -1]

Rubi steps

\begin {align*} \int \frac {\cot ^{-1}(a x)}{x^4} \, dx &=-\frac {\cot ^{-1}(a x)}{3 x^3}-\frac {1}{3} a \int \frac {1}{x^3 \left (1+a^2 x^2\right )} \, dx\\ &=-\frac {\cot ^{-1}(a x)}{3 x^3}-\frac {1}{6} a \operatorname {Subst}\left (\int \frac {1}{x^2 \left (1+a^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac {\cot ^{-1}(a x)}{3 x^3}-\frac {1}{6} a \operatorname {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 {a}{6 x^2}-\frac {\cot ^{-1}(a x)}{3 x^3}+\frac {1}{3} a^3 \log (x)-\frac {1}{6} a^3 \log \left (1+a^2 x^2\right )\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 44, normalized size = 0.96 \[ -\frac {1}{6} a \left (a^2 \log \left (a^2 x^2+1\right )-2 a^2 \log (x)-\frac {1}{x^2}\right )-\frac {\cot ^{-1}(a x)}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCot[a*x]/x^4,x]

[Out]

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

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fricas [A]  time = 0.62, size = 43, normalized size = 0.93 \[ -\frac {a^{3} x^{3} \log \left (a^{2} x^{2} + 1\right ) - 2 \, a^{3} x^{3} \log \relax (x) - a x + 2 \, \operatorname {arccot}\left (a x\right )}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(a*x)/x^4,x, algorithm="fricas")

[Out]

-1/6*(a^3*x^3*log(a^2*x^2 + 1) - 2*a^3*x^3*log(x) - a*x + 2*arccot(a*x))/x^3

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giac [A]  time = 0.13, size = 44, normalized size = 0.96 \[ \frac {1}{6} \, {\left (a^{2} {\left (\frac {1}{a^{2} x^{2}} - \log \left (\frac {1}{a^{2} x^{2}} + 1\right )\right )} - \frac {2 \, \arctan \left (\frac {1}{a x}\right )}{a x^{3}}\right )} a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(a*x)/x^4,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.04, size = 41, normalized size = 0.89 \[ -\frac {\mathrm {arccot}\left (a x \right )}{3 x^{3}}+\frac {a}{6 x^{2}}+\frac {a^{3} \ln \left (a x \right )}{3}-\frac {a^{3} \ln \left (a^{2} x^{2}+1\right )}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccot(a*x)/x^4,x)

[Out]

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

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maxima [A]  time = 0.31, size = 42, normalized size = 0.91 \[ -\frac {1}{6} \, {\left (a^{2} \log \left (a^{2} x^{2} + 1\right ) - a^{2} \log \left (x^{2}\right ) - \frac {1}{x^{2}}\right )} a - \frac {\operatorname {arccot}\left (a x\right )}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(a*x)/x^4,x, algorithm="maxima")

[Out]

-1/6*(a^2*log(a^2*x^2 + 1) - a^2*log(x^2) - 1/x^2)*a - 1/3*arccot(a*x)/x^3

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mupad [B]  time = 0.85, size = 58, normalized size = 1.26 \[ \left \{\begin {array}{cl} -\frac {\pi }{6\,x^3} & \text {\ if\ \ }a=0\\ \frac {a^4\,\ln \relax (x)-\frac {a^4\,\ln \left (a^2\,x^2+1\right )}{2}+\frac {a^2}{2\,x^2}}{3\,a}-\frac {\mathrm {acot}\left (a\,x\right )}{3\,x^3} & \text {\ if\ \ }a\neq 0 \end {array}\right . \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acot(a*x)/x^4,x)

[Out]

piecewise(a == 0, -pi/(6*x^3), a ~= 0, (a^4*log(x) - (a^4*log(a^2*x^2 + 1))/2 + a^2/(2*x^2))/(3*a) - acot(a*x)
/(3*x^3))

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sympy [A]  time = 0.59, size = 39, normalized size = 0.85 \[ \frac {a^{3} \log {\relax (x )}}{3} - \frac {a^{3} \log {\left (a^{2} x^{2} + 1 \right )}}{6} + \frac {a}{6 x^{2}} - \frac {\operatorname {acot}{\left (a x \right )}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acot(a*x)/x**4,x)

[Out]

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

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