3.1.8 \(\int \frac {\cot ^{-1}(a x)}{x^2} \, dx\) [8]

Optimal. Leaf size=30 \[ -\frac {\cot ^{-1}(a x)}{x}-a \log (x)+\frac {1}{2} a \log \left (1+a^2 x^2\right ) \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {4947, 272, 36, 29, 31} \begin {gather*} \frac {1}{2} a \log \left (a^2 x^2+1\right )-a \log (x)-\frac {\cot ^{-1}(a x)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(ArcCot[a*x]/x) - a*Log[x] + (a*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]

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 30, normalized size = 1.00 \begin {gather*} -\frac {\cot ^{-1}(a x)}{x}-a \log (x)+\frac {1}{2} a \log \left (1+a^2 x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.04, size = 34, normalized size = 1.13

method	result	size
derivativedivides	\(a \left (-\frac {\mathrm {arccot}\left (a x \right )}{a x}-\ln \left (a x \right )+\frac {\ln \left (a^{2} x^{2}+1\right )}{2}\right )\)	\(34\)
default	\(a \left (-\frac {\mathrm {arccot}\left (a x \right )}{a x}-\ln \left (a x \right )+\frac {\ln \left (a^{2} x^{2}+1\right )}{2}\right )\)	\(34\)
risch	\(-\frac {i \ln \left (i a x +1\right )}{2 x}-\frac {2 \ln \left (x \right ) a x -a \ln \left (a^{2} x^{2}+1\right ) x -i \ln \left (-i a x +1\right )+\pi }{2 x}\)	\(54\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 30, normalized size = 1.00 \begin {gather*} \frac {1}{2} \, a {\left (\log \left (a^{2} x^{2} + 1\right ) - \log \left (x^{2}\right )\right )} - \frac {\operatorname {arccot}\left (a x\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 3.12, size = 31, normalized size = 1.03 \begin {gather*} \frac {a x \log \left (a^{2} x^{2} + 1\right ) - 2 \, a x \log \left (x\right ) - 2 \, \operatorname {arccot}\left (a x\right )}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.12, size = 24, normalized size = 0.80 \begin {gather*} - a \log {\left (x \right )} + \frac {a \log {\left (a^{2} x^{2} + 1 \right )}}{2} - \frac {\operatorname {acot}{\left (a x \right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.40, size = 32, normalized size = 1.07 \begin {gather*} -\frac {1}{2} \, a {\left (\frac {2 \, \arctan \left (\frac {1}{a x}\right )}{a x} - \log \left (\frac {1}{a^{2} x^{2}} + 1\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.23, size = 28, normalized size = 0.93 \begin {gather*} \frac {a\,\left (\ln \left (a^2\,x^2+1\right )-2\,\ln \left (x\right )\right )}{2}-\frac {\mathrm {acot}\left (a\,x\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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