Optimal. Leaf size=60 \[ \frac{1}{4} \left (\frac{1}{x^2}-1\right )^{5/2} x^4+\frac{5}{8} \left (\frac{1}{x^2}-1\right )^{3/2} x^2-\frac{15}{8} \sqrt{\frac{1}{x^2}-1}+\frac{15}{8} \tan ^{-1}\left (\sqrt{\frac{1}{x^2}-1}\right ) \]
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Rubi [A] time = 0.0199571, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {25, 266, 47, 50, 63, 203} \[ \frac{1}{4} \left (\frac{1}{x^2}-1\right )^{5/2} x^4+\frac{5}{8} \left (\frac{1}{x^2}-1\right )^{3/2} x^2-\frac{15}{8} \sqrt{\frac{1}{x^2}-1}+\frac{15}{8} \tan ^{-1}\left (\sqrt{\frac{1}{x^2}-1}\right ) \]
Antiderivative was successfully verified.
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Rule 25
Rule 266
Rule 47
Rule 50
Rule 63
Rule 203
Rubi steps
\begin{align*} \int \frac{\sqrt{-1+\frac{1}{x^2}} \left (-1+x^2\right )^2}{x} \, dx &=\int \left (-1+\frac{1}{x^2}\right )^{5/2} x^3 \, dx\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{(-1+x)^{5/2}}{x^3} \, dx,x,\frac{1}{x^2}\right )\right )\\ &=\frac{1}{4} \left (-1+\frac{1}{x^2}\right )^{5/2} x^4-\frac{5}{8} \operatorname{Subst}\left (\int \frac{(-1+x)^{3/2}}{x^2} \, dx,x,\frac{1}{x^2}\right )\\ &=\frac{5}{8} \left (-1+\frac{1}{x^2}\right )^{3/2} x^2+\frac{1}{4} \left (-1+\frac{1}{x^2}\right )^{5/2} x^4-\frac{15}{16} \operatorname{Subst}\left (\int \frac{\sqrt{-1+x}}{x} \, dx,x,\frac{1}{x^2}\right )\\ &=-\frac{15}{8} \sqrt{-1+\frac{1}{x^2}}+\frac{5}{8} \left (-1+\frac{1}{x^2}\right )^{3/2} x^2+\frac{1}{4} \left (-1+\frac{1}{x^2}\right )^{5/2} x^4+\frac{15}{16} \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+x} x} \, dx,x,\frac{1}{x^2}\right )\\ &=-\frac{15}{8} \sqrt{-1+\frac{1}{x^2}}+\frac{5}{8} \left (-1+\frac{1}{x^2}\right )^{3/2} x^2+\frac{1}{4} \left (-1+\frac{1}{x^2}\right )^{5/2} x^4+\frac{15}{8} \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{-1+\frac{1}{x^2}}\right )\\ &=-\frac{15}{8} \sqrt{-1+\frac{1}{x^2}}+\frac{5}{8} \left (-1+\frac{1}{x^2}\right )^{3/2} x^2+\frac{1}{4} \left (-1+\frac{1}{x^2}\right )^{5/2} x^4+\frac{15}{8} \tan ^{-1}\left (\sqrt{-1+\frac{1}{x^2}}\right )\\ \end{align*}
Mathematica [C] time = 0.0071944, size = 35, normalized size = 0.58 \[ -\frac{\sqrt{\frac{1}{x^2}-1} \, _2F_1\left (-\frac{5}{2},-\frac{1}{2};\frac{1}{2};x^2\right )}{\sqrt{1-x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 69, normalized size = 1.2 \begin{align*} -{\frac{1}{8}\sqrt{-{\frac{{x}^{2}-1}{{x}^{2}}}} \left ( 2\,{x}^{2} \left ( -{x}^{2}+1 \right ) ^{3/2}+8\, \left ( -{x}^{2}+1 \right ) ^{3/2}+15\,{x}^{2}\sqrt{-{x}^{2}+1}+15\,\arcsin \left ( x \right ) x \right ){\frac{1}{\sqrt{-{x}^{2}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.02796, size = 90, normalized size = 1.5 \begin{align*} -x^{2} \sqrt{\frac{1}{x^{2}} - 1} - \sqrt{\frac{1}{x^{2}} - 1} - \frac{{\left (\frac{1}{x^{2}} - 1\right )}^{\frac{3}{2}} - \sqrt{\frac{1}{x^{2}} - 1}}{8 \,{\left ({\left (\frac{1}{x^{2}} - 1\right )}^{2} + \frac{2}{x^{2}} - 1\right )}} + \frac{15}{8} \, \arctan \left (\sqrt{\frac{1}{x^{2}} - 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54223, size = 124, normalized size = 2.07 \begin{align*} \frac{1}{8} \,{\left (2 \, x^{4} - 9 \, x^{2} - 8\right )} \sqrt{-\frac{x^{2} - 1}{x^{2}}} + \frac{15}{4} \, \arctan \left (\frac{x \sqrt{-\frac{x^{2} - 1}{x^{2}}} - 1}{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 73.4257, size = 60, normalized size = 1. \begin{align*} \frac{x^{4} \sqrt{-1 + \frac{1}{x^{2}}} \left (2 - \frac{1}{x^{2}}\right )}{8} - x^{2} \sqrt{-1 + \frac{1}{x^{2}}} - \sqrt{-1 + \frac{1}{x^{2}}} + \frac{15 \operatorname{atan}{\left (\sqrt{-1 + \frac{1}{x^{2}}} \right )}}{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11504, size = 90, normalized size = 1.5 \begin{align*} \frac{1}{8} \,{\left (2 \, x^{2} \mathrm{sgn}\left (x\right ) - 9 \, \mathrm{sgn}\left (x\right )\right )} \sqrt{-x^{2} + 1} x - \frac{15}{8} \, \arcsin \left (x\right ) \mathrm{sgn}\left (x\right ) + \frac{x \mathrm{sgn}\left (x\right )}{2 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}} - \frac{{\left (\sqrt{-x^{2} + 1} - 1\right )} \mathrm{sgn}\left (x\right )}{2 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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