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 = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 2.000, 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 63
Rule 206
Rule 266
Rule 5215
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*}
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Mathematica [B] time = 0.05, 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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fricas [B] time = 0.44, size = 35, normalized size = 2.06 \[ {\left (x - 2\right )} \operatorname {arccsc}\relax (x) - 4 \, \arctan \left (-x + \sqrt {x^{2} - 1}\right ) - \log \left (-x + \sqrt {x^{2} - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.03, size = 37, normalized size = 2.18 \[ x \arcsin \left (\frac {1}{x}\right ) + \frac {1}{2} \, \log \left (\sqrt {-\frac {1}{x^{2}} + 1} + 1\right ) - \frac {1}{2} \, \log \left (-\sqrt {-\frac {1}{x^{2}} + 1} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 20, normalized size = 1.18 \[ x \,\mathrm {arccsc}\relax (x )+\ln \left (x +\sqrt {-\frac {1}{x^{2}}+1}\, x \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.60, size = 35, normalized size = 2.06 \[ x \operatorname {arccsc}\relax (x) + \frac {1}{2} \, \log \left (\sqrt {-\frac {1}{x^{2}} + 1} + 1\right ) - \frac {1}{2} \, \log \left (-\sqrt {-\frac {1}{x^{2}} + 1} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.21, size = 20, normalized size = 1.18 \[ x\,\mathrm {asin}\left (\frac {1}{x}\right )+\ln \left (x+\sqrt {x^2-1}\right )\,\mathrm {sign}\relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.24, size = 17, normalized size = 1.00 \[ x \operatorname {acsc}{\relax (x )} + \begin {cases} \operatorname {acosh}{\relax (x )} & \text {for}\: \left |{x^{2}}\right | > 1 \\- i \operatorname {asin}{\relax (x )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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