Optimal. Leaf size=54 \[ -\frac{\sqrt{x-1}}{8 x^2}-\frac{\csc ^{-1}\left (\sqrt{x}\right )}{2 x^2}-\frac{3 \sqrt{x-1}}{16 x}-\frac{3}{16} \tan ^{-1}\left (\sqrt{x-1}\right ) \]
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Rubi [A] time = 0.0201348, antiderivative size = 54, 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 = {5271, 12, 51, 63, 203} \[ -\frac{\sqrt{x-1}}{8 x^2}-\frac{\csc ^{-1}\left (\sqrt{x}\right )}{2 x^2}-\frac{3 \sqrt{x-1}}{16 x}-\frac{3}{16} \tan ^{-1}\left (\sqrt{x-1}\right ) \]
Antiderivative was successfully verified.
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Rule 5271
Rule 12
Rule 51
Rule 63
Rule 203
Rubi steps
\begin{align*} \int \frac{\csc ^{-1}\left (\sqrt{x}\right )}{x^3} \, dx &=-\frac{\csc ^{-1}\left (\sqrt{x}\right )}{2 x^2}-\frac{1}{2} \int \frac{1}{2 \sqrt{-1+x} x^3} \, dx\\ &=-\frac{\csc ^{-1}\left (\sqrt{x}\right )}{2 x^2}-\frac{1}{4} \int \frac{1}{\sqrt{-1+x} x^3} \, dx\\ &=-\frac{\sqrt{-1+x}}{8 x^2}-\frac{\csc ^{-1}\left (\sqrt{x}\right )}{2 x^2}-\frac{3}{16} \int \frac{1}{\sqrt{-1+x} x^2} \, dx\\ &=-\frac{\sqrt{-1+x}}{8 x^2}-\frac{3 \sqrt{-1+x}}{16 x}-\frac{\csc ^{-1}\left (\sqrt{x}\right )}{2 x^2}-\frac{3}{32} \int \frac{1}{\sqrt{-1+x} x} \, dx\\ &=-\frac{\sqrt{-1+x}}{8 x^2}-\frac{3 \sqrt{-1+x}}{16 x}-\frac{\csc ^{-1}\left (\sqrt{x}\right )}{2 x^2}-\frac{3}{16} \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{-1+x}\right )\\ &=-\frac{\sqrt{-1+x}}{8 x^2}-\frac{3 \sqrt{-1+x}}{16 x}-\frac{\csc ^{-1}\left (\sqrt{x}\right )}{2 x^2}-\frac{3}{16} \tan ^{-1}\left (\sqrt{-1+x}\right )\\ \end{align*}
Mathematica [A] time = 0.0472944, size = 55, normalized size = 1.02 \[ \sqrt{\frac{x-1}{x}} \left (-\frac{1}{8 x^{3/2}}-\frac{3}{16 \sqrt{x}}\right )-\frac{\csc ^{-1}\left (\sqrt{x}\right )}{2 x^2}+\frac{3}{16} \sin ^{-1}\left (\frac{1}{\sqrt{x}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.118, size = 57, normalized size = 1.1 \begin{align*} -{\frac{1}{2\,{x}^{2}}{\rm arccsc} \left (\sqrt{x}\right )}-{\frac{1}{16}\sqrt{x-1} \left ( -3\,\arctan \left ({\frac{1}{\sqrt{x-1}}} \right ){x}^{2}+3\,x\sqrt{x-1}+2\,\sqrt{x-1} \right ){\frac{1}{\sqrt{{\frac{x-1}{x}}}}}{x}^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.51452, size = 108, normalized size = 2. \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{arccsc}\left (\sqrt{x}\right )}{2 \, x^{2}} - \frac{3}{16} \, \arctan \left (\sqrt{x} \sqrt{-\frac{1}{x} + 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 3.32783, size = 86, normalized size = 1.59 \begin{align*} \frac{{\left (3 \, x^{2} - 8\right )} \operatorname{arccsc}\left (\sqrt{x}\right ) -{\left (3 \, x + 2\right )} \sqrt{x - 1}}{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 [A] time = 1.13661, size = 51, normalized size = 0.94 \begin{align*} -\frac{3 \,{\left (x - 1\right )}^{\frac{3}{2}} + 5 \, \sqrt{x - 1}}{16 \, x^{2}} - \frac{\arcsin \left (\frac{1}{\sqrt{x}}\right )}{2 \, x^{2}} - \frac{3}{16} \, \arctan \left (\sqrt{x - 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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