3.1.13 \(\int \frac {\csc ^{-1}(\frac {a}{x})}{x^2} \, dx\) [13]

Optimal. Leaf size=32 \[ -\frac {\text {ArcSin}\left (\frac {x}{a}\right )}{x}-\frac {\tanh ^{-1}\left (\sqrt {1-\frac {x^2}{a^2}}\right )}{a} \]

[Out]

-arcsin(x/a)/x-arctanh((1-x^2/a^2)^(1/2))/a

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Rubi [A]
time = 0.02, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5373, 4723, 272, 65, 214} \begin {gather*} -\frac {\tanh ^{-1}\left (\sqrt {1-\frac {x^2}{a^2}}\right )}{a}-\frac {\text {ArcSin}\left (\frac {x}{a}\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcCsc[a/x]/x^2,x]

[Out]

-(ArcSin[x/a]/x) - ArcTanh[Sqrt[1 - x^2/a^2]]/a

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

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

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 4723

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcSi
n[c*x])^n/(d*(m + 1))), x] - Dist[b*c*(n/(d*(m + 1))), Int[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 -
c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 5373

Int[ArcCsc[(c_.)/((a_.) + (b_.)*(x_)^(n_.))]^(m_.)*(u_.), x_Symbol] :> Int[u*ArcSin[a/c + b*(x^n/c)]^m, x] /;
FreeQ[{a, b, c, n, m}, x]

Rubi steps

\begin {align*} \int \frac {\csc ^{-1}\left (\frac {a}{x}\right )}{x^2} \, dx &=\int \frac {\sin ^{-1}\left (\frac {x}{a}\right )}{x^2} \, dx\\ &=-\frac {\sin ^{-1}\left (\frac {x}{a}\right )}{x}+\frac {\int \frac {1}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx}{a}\\ &=-\frac {\sin ^{-1}\left (\frac {x}{a}\right )}{x}+\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,x^2\right )}{2 a}\\ &=-\frac {\sin ^{-1}\left (\frac {x}{a}\right )}{x}-a \text {Subst}\left (\int \frac {1}{a^2-a^2 x^2} \, dx,x,\sqrt {1-\frac {x^2}{a^2}}\right )\\ &=-\frac {\sin ^{-1}\left (\frac {x}{a}\right )}{x}-\frac {\tanh ^{-1}\left (\sqrt {1-\frac {x^2}{a^2}}\right )}{a}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(93\) vs. \(2(32)=64\).
time = 0.10, size = 93, normalized size = 2.91 \begin {gather*} -\frac {\csc ^{-1}\left (\frac {a}{x}\right )}{x}-\frac {\sqrt {-1+\frac {a^2}{x^2}} x \left (-\log \left (1-\frac {a}{\sqrt {-1+\frac {a^2}{x^2}} x}\right )+\log \left (1+\frac {a}{\sqrt {-1+\frac {a^2}{x^2}} x}\right )\right )}{2 a^2 \sqrt {1-\frac {x^2}{a^2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[ArcCsc[a/x]/x^2,x]

[Out]

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

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Maple [A]
time = 0.07, size = 42, normalized size = 1.31

method	result	size
derivativedivides	\(-\frac {\frac {\mathrm {arccsc}\left (\frac {a}{x}\right ) a}{x}+\ln \left (\frac {a}{x}+\frac {a \sqrt {1-\frac {x^{2}}{a^{2}}}}{x}\right )}{a}\)	\(42\)
default	\(-\frac {\frac {\mathrm {arccsc}\left (\frac {a}{x}\right ) a}{x}+\ln \left (\frac {a}{x}+\frac {a \sqrt {1-\frac {x^{2}}{a^{2}}}}{x}\right )}{a}\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.26, size = 52, normalized size = 1.62 \begin {gather*} -\frac {\frac {2 \, a \operatorname {arccsc}\left (\frac {a}{x}\right )}{x} + \log \left (\sqrt {-\frac {x^{2}}{a^{2}} + 1} + 1\right ) - \log \left (-\sqrt {-\frac {x^{2}}{a^{2}} + 1} + 1\right )}{2 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (30) = 60\).
time = 0.38, size = 65, normalized size = 2.03 \begin {gather*} -\frac {2 \, a \operatorname {arccsc}\left (\frac {a}{x}\right ) + x \log \left (x \sqrt {\frac {a^{2} - x^{2}}{x^{2}}} + a\right ) - x \log \left (x \sqrt {\frac {a^{2} - x^{2}}{x^{2}}} - a\right )}{2 \, a x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C] Result contains complex when optimal does not.
time = 1.39, size = 27, normalized size = 0.84 \begin {gather*} - \frac {\operatorname {acsc}{\left (\frac {a}{x} \right )}}{x} + \frac {\begin {cases} - \operatorname {acosh}{\left (\frac {a}{x} \right )} & \text {for}\: \left |{\frac {a^{2}}{x^{2}}}\right | > 1 \\i \operatorname {asin}{\left (\frac {a}{x} \right )} & \text {otherwise} \end {cases}}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-acsc(a/x)/x + Piecewise((-acosh(a/x), Abs(a**2/x**2) > 1), (I*asin(a/x), True))/a

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (30) = 60\).
time = 0.45, size = 61, normalized size = 1.91 \begin {gather*} -\frac {a {\left (\frac {\log \left ({\left | a + \sqrt {a^{2} - x^{2}} \right |}\right )}{a} - \frac {\log \left ({\left | -a + \sqrt {a^{2} - x^{2}} \right |}\right )}{a}\right )}}{2 \, {\left | a \right |}} - \frac {\arcsin \left (\frac {x}{a}\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*a*(log(abs(a + sqrt(a^2 - x^2)))/a - log(abs(-a + sqrt(a^2 - x^2)))/a)/abs(a) - arcsin(x/a)/x

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Mupad [B]
time = 0.59, size = 30, normalized size = 0.94 \begin {gather*} -\frac {\mathrm {asin}\left (\frac {x}{a}\right )}{x}-\frac {\mathrm {atanh}\left (\frac {1}{\sqrt {1-\frac {x^2}{a^2}}}\right )}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asin(x/a)/x^2,x)

[Out]

- asin(x/a)/x - atanh(1/(1 - x^2/a^2)^(1/2))/a

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