Optimal. Leaf size=63 \[ \frac{\sqrt{-x-1}}{2 \sqrt{-x} \sqrt{x}}-\frac{\sqrt{x} \tan ^{-1}\left (\sqrt{-x-1}\right )}{2 \sqrt{-x}}-\frac{\text{csch}^{-1}\left (\sqrt{x}\right )}{x} \]
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Rubi [A] time = 0.0238083, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6346, 12, 51, 63, 204} \[ \frac{\sqrt{-x-1}}{2 \sqrt{-x} \sqrt{x}}-\frac{\sqrt{x} \tan ^{-1}\left (\sqrt{-x-1}\right )}{2 \sqrt{-x}}-\frac{\text{csch}^{-1}\left (\sqrt{x}\right )}{x} \]
Antiderivative was successfully verified.
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Rule 6346
Rule 12
Rule 51
Rule 63
Rule 204
Rubi steps
\begin{align*} \int \frac{\text{csch}^{-1}\left (\sqrt{x}\right )}{x^2} \, dx &=-\frac{\text{csch}^{-1}\left (\sqrt{x}\right )}{x}+\frac{\sqrt{x} \int \frac{1}{2 \sqrt{-1-x} x^2} \, dx}{\sqrt{-x}}\\ &=-\frac{\text{csch}^{-1}\left (\sqrt{x}\right )}{x}+\frac{\sqrt{x} \int \frac{1}{\sqrt{-1-x} x^2} \, dx}{2 \sqrt{-x}}\\ &=\frac{\sqrt{-1-x}}{2 \sqrt{-x} \sqrt{x}}-\frac{\text{csch}^{-1}\left (\sqrt{x}\right )}{x}-\frac{\sqrt{x} \int \frac{1}{\sqrt{-1-x} x} \, dx}{4 \sqrt{-x}}\\ &=\frac{\sqrt{-1-x}}{2 \sqrt{-x} \sqrt{x}}-\frac{\text{csch}^{-1}\left (\sqrt{x}\right )}{x}+\frac{\sqrt{x} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,\sqrt{-1-x}\right )}{2 \sqrt{-x}}\\ &=\frac{\sqrt{-1-x}}{2 \sqrt{-x} \sqrt{x}}-\frac{\text{csch}^{-1}\left (\sqrt{x}\right )}{x}-\frac{\sqrt{x} \tan ^{-1}\left (\sqrt{-1-x}\right )}{2 \sqrt{-x}}\\ \end{align*}
Mathematica [A] time = 0.0230363, size = 42, normalized size = 0.67 \[ \frac{\sqrt{\frac{x+1}{x}}}{2 \sqrt{x}}-\frac{1}{2} \sinh ^{-1}\left (\frac{1}{\sqrt{x}}\right )-\frac{\text{csch}^{-1}\left (\sqrt{x}\right )}{x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.112, size = 46, normalized size = 0.7 \begin{align*} -{\frac{1}{x}{\rm arccsch} \left (\sqrt{x}\right )}-{\frac{1}{2}\sqrt{1+x} \left ({\it Artanh} \left ({\frac{1}{\sqrt{1+x}}} \right ) x-\sqrt{1+x} \right ){\frac{1}{\sqrt{{\frac{1+x}{x}}}}}{x}^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.997511, size = 88, normalized size = 1.4 \begin{align*} \frac{\sqrt{x} \sqrt{\frac{1}{x} + 1}}{2 \,{\left (x{\left (\frac{1}{x} + 1\right )} - 1\right )}} - \frac{\operatorname{arcsch}\left (\sqrt{x}\right )}{x} - \frac{1}{4} \, \log \left (\sqrt{x} \sqrt{\frac{1}{x} + 1} + 1\right ) + \frac{1}{4} \, \log \left (\sqrt{x} \sqrt{\frac{1}{x} + 1} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.69195, size = 109, normalized size = 1.73 \begin{align*} -\frac{{\left (x + 2\right )} \log \left (\frac{x \sqrt{\frac{x + 1}{x}} + \sqrt{x}}{x}\right ) - \sqrt{x} \sqrt{\frac{x + 1}{x}}}{2 \, x} \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 \frac{\operatorname{acsch}{\left (\sqrt{x} \right )}}{x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arcsch}\left (\sqrt{x}\right )}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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