3.84 \(\int \csc ^{-1}(x) \, dx\)

Optimal. Leaf size=17 \[ \tanh ^{-1}\left (\sqrt{1-\frac{1}{x^2}}\right )+x \csc ^{-1}(x) \]

[Out]

x*ArcCsc[x] + ArcTanh[Sqrt[1 - x^(-2)]]

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Rubi [A]  time = 0.0275982, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 2, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 2. \[ \tanh ^{-1}\left (\sqrt{1-\frac{1}{x^2}}\right )+x \csc ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[ArcCsc[x],x]

[Out]

x*ArcCsc[x] + ArcTanh[Sqrt[1 - x^(-2)]]

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Rubi in Sympy [A]  time = 1.74025, size = 15, normalized size = 0.88 \[ x \operatorname{acsc}{\left (x \right )} + \operatorname{atanh}{\left (\sqrt{1 - \frac{1}{x^{2}}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(acsc(x),x)

[Out]

x*acsc(x) + atanh(sqrt(1 - 1/x**2))

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Mathematica [B]  time = 0.0713409, size = 64, normalized size = 3.76 \[ \frac{\sqrt{x^2-1} \left (\log \left (\frac{x}{\sqrt{x^2-1}}+1\right )-\log \left (1-\frac{x}{\sqrt{x^2-1}}\right )\right )}{2 \sqrt{1-\frac{1}{x^2}} x}+x \csc ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Integrate[ArcCsc[x],x]

[Out]

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

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Maple [A]  time = 0.004, size = 20, normalized size = 1.2 \[ x{\rm arccsc} \left (x\right )+\ln \left ( x+x\sqrt{1-{x}^{-2}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(arccsc(x),x)

[Out]

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

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Maxima [A]  time = 1.49562, size = 47, normalized size = 2.76 \[ x \operatorname{arccsc}\left (x\right ) + \frac{1}{2} \, \log \left (\sqrt{-\frac{1}{x^{2}} + 1} + 1\right ) - \frac{1}{2} \, \log \left (-\sqrt{-\frac{1}{x^{2}} + 1} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arccsc(x),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.229933, size = 47, normalized size = 2.76 \[{\left (x - 2\right )} \operatorname{arccsc}\left (x\right ) - 4 \, \arctan \left (-x + \sqrt{x^{2} - 1}\right ) - \log \left (-x + \sqrt{x^{2} - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arccsc(x),x, algorithm="fricas")

[Out]

(x - 2)*arccsc(x) - 4*arctan(-x + sqrt(x^2 - 1)) - log(-x + sqrt(x^2 - 1))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \operatorname{acsc}{\left (x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(acsc(x),x)

[Out]

Integral(acsc(x), x)

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GIAC/XCAS [A]  time = 0.229477, size = 35, normalized size = 2.06 \[ x \arcsin \left (\frac{1}{x}\right ) - \frac{{\rm ln}\left ({\left | -x + \sqrt{x^{2} - 1} \right |}\right )}{{\rm sign}\left (x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arccsc(x),x, algorithm="giac")

[Out]

x*arcsin(1/x) - ln(abs(-x + sqrt(x^2 - 1)))/sign(x)