Optimal. Leaf size=17 \[ \tanh ^{-1}\left (\sqrt{1-\frac{1}{x^2}}\right )+x \csc ^{-1}(x) \]
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Rubi [A] time = 0.0090909, 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., Rules used = {5215, 266, 63, 206} \[ \tanh ^{-1}\left (\sqrt{1-\frac{1}{x^2}}\right )+x \csc ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 5215
Rule 266
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \csc ^{-1}(x) \, dx &=x \csc ^{-1}(x)+\int \frac{1}{\sqrt{1-\frac{1}{x^2}} x} \, dx\\ &=x \csc ^{-1}(x)-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} x} \, dx,x,\frac{1}{x^2}\right )\\ &=x \csc ^{-1}(x)+\operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{1-\frac{1}{x^2}}\right )\\ &=x \csc ^{-1}(x)+\tanh ^{-1}\left (\sqrt{1-\frac{1}{x^2}}\right )\\ \end{align*}
Mathematica [B] time = 0.0386257, 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.
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Maple [A] time = 0.002, size = 20, normalized size = 1.2 \begin{align*} x{\rm arccsc} \left (x\right )+\ln \left ( x+x\sqrt{1-{x}^{-2}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.928191, size = 47, normalized size = 2.76 \begin{align*} 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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.553965, size = 103, normalized size = 6.06 \begin{align*}{\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{acsc}{\left (x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09854, size = 35, normalized size = 2.06 \begin{align*} x \arcsin \left (\frac{1}{x}\right ) - \frac{\log \left ({\left | -x + \sqrt{x^{2} - 1} \right |}\right )}{\mathrm{sgn}\left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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