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