Optimal. Leaf size=90 \[ \frac{\sqrt{-x-1}}{8 \sqrt{-x} x^{3/2}}-\frac{\text{csch}^{-1}\left (\sqrt{x}\right )}{2 x^2}-\frac{3 \sqrt{-x-1}}{16 \sqrt{-x} \sqrt{x}}+\frac{3 \sqrt{x} \tan ^{-1}\left (\sqrt{-x-1}\right )}{16 \sqrt{-x}} \]
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Rubi [A] time = 0.0281949, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ \frac{\sqrt{-x-1}}{8 \sqrt{-x} x^{3/2}}-\frac{\text{csch}^{-1}\left (\sqrt{x}\right )}{2 x^2}-\frac{3 \sqrt{-x-1}}{16 \sqrt{-x} \sqrt{x}}+\frac{3 \sqrt{x} \tan ^{-1}\left (\sqrt{-x-1}\right )}{16 \sqrt{-x}} \]
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^3} \, dx &=-\frac{\text{csch}^{-1}\left (\sqrt{x}\right )}{2 x^2}+\frac{\sqrt{x} \int \frac{1}{2 \sqrt{-1-x} x^3} \, dx}{2 \sqrt{-x}}\\ &=-\frac{\text{csch}^{-1}\left (\sqrt{x}\right )}{2 x^2}+\frac{\sqrt{x} \int \frac{1}{\sqrt{-1-x} x^3} \, dx}{4 \sqrt{-x}}\\ &=\frac{\sqrt{-1-x}}{8 \sqrt{-x} x^{3/2}}-\frac{\text{csch}^{-1}\left (\sqrt{x}\right )}{2 x^2}-\frac{\left (3 \sqrt{x}\right ) \int \frac{1}{\sqrt{-1-x} x^2} \, dx}{16 \sqrt{-x}}\\ &=\frac{\sqrt{-1-x}}{8 \sqrt{-x} x^{3/2}}-\frac{3 \sqrt{-1-x}}{16 \sqrt{-x} \sqrt{x}}-\frac{\text{csch}^{-1}\left (\sqrt{x}\right )}{2 x^2}+\frac{\left (3 \sqrt{x}\right ) \int \frac{1}{\sqrt{-1-x} x} \, dx}{32 \sqrt{-x}}\\ &=\frac{\sqrt{-1-x}}{8 \sqrt{-x} x^{3/2}}-\frac{3 \sqrt{-1-x}}{16 \sqrt{-x} \sqrt{x}}-\frac{\text{csch}^{-1}\left (\sqrt{x}\right )}{2 x^2}-\frac{\left (3 \sqrt{x}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,\sqrt{-1-x}\right )}{16 \sqrt{-x}}\\ &=\frac{\sqrt{-1-x}}{8 \sqrt{-x} x^{3/2}}-\frac{3 \sqrt{-1-x}}{16 \sqrt{-x} \sqrt{x}}-\frac{\text{csch}^{-1}\left (\sqrt{x}\right )}{2 x^2}+\frac{3 \sqrt{x} \tan ^{-1}\left (\sqrt{-1-x}\right )}{16 \sqrt{-x}}\\ \end{align*}
Mathematica [A] time = 0.0336742, size = 47, normalized size = 0.52 \[ \frac{3 x^2 \sinh ^{-1}\left (\frac{1}{\sqrt{x}}\right )+\sqrt{\frac{1}{x}+1} (2-3 x) \sqrt{x}-8 \text{csch}^{-1}\left (\sqrt{x}\right )}{16 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.112, size = 57, normalized size = 0.6 \begin{align*} -{\frac{1}{2\,{x}^{2}}{\rm arccsch} \left (\sqrt{x}\right )}+{\frac{1}{16}\sqrt{1+x} \left ( 3\,{\it Artanh} \left ({\frac{1}{\sqrt{1+x}}} \right ){x}^{2}-3\,x\sqrt{1+x}+2\,\sqrt{1+x} \right ){\frac{1}{\sqrt{{\frac{1+x}{x}}}}}{x}^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0186, size = 124, normalized size = 1.38 \begin{align*} -\frac{3 \, x^{\frac{3}{2}}{\left (\frac{1}{x} + 1\right )}^{\frac{3}{2}} - 5 \, \sqrt{x} \sqrt{\frac{1}{x} + 1}}{16 \,{\left (x^{2}{\left (\frac{1}{x} + 1\right )}^{2} - 2 \, x{\left (\frac{1}{x} + 1\right )} + 1\right )}} - \frac{\operatorname{arcsch}\left (\sqrt{x}\right )}{2 \, x^{2}} + \frac{3}{32} \, \log \left (\sqrt{x} \sqrt{\frac{1}{x} + 1} + 1\right ) - \frac{3}{32} \, \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 = 2.50649, size = 132, normalized size = 1.47 \begin{align*} -\frac{{\left (3 \, x - 2\right )} \sqrt{x} \sqrt{\frac{x + 1}{x}} -{\left (3 \, x^{2} - 8\right )} \log \left (\frac{x \sqrt{\frac{x + 1}{x}} + \sqrt{x}}{x}\right )}{16 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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