Optimal. Leaf size=113 \[ -\frac {\sqrt {x}}{2 \left (x^2+1\right )}-\frac {\log \left (x-\sqrt {2} \sqrt {x}+1\right )}{8 \sqrt {2}}+\frac {\log \left (x+\sqrt {2} \sqrt {x}+1\right )}{8 \sqrt {2}}-\frac {\tan ^{-1}\left (1-\sqrt {2} \sqrt {x}\right )}{4 \sqrt {2}}+\frac {\tan ^{-1}\left (\sqrt {2} \sqrt {x}+1\right )}{4 \sqrt {2}} \]
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Rubi [A] time = 0.06, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {288, 329, 211, 1165, 628, 1162, 617, 204} \[ -\frac {\sqrt {x}}{2 \left (x^2+1\right )}-\frac {\log \left (x-\sqrt {2} \sqrt {x}+1\right )}{8 \sqrt {2}}+\frac {\log \left (x+\sqrt {2} \sqrt {x}+1\right )}{8 \sqrt {2}}-\frac {\tan ^{-1}\left (1-\sqrt {2} \sqrt {x}\right )}{4 \sqrt {2}}+\frac {\tan ^{-1}\left (\sqrt {2} \sqrt {x}+1\right )}{4 \sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 211
Rule 288
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {x^{3/2}}{\left (1+x^2\right )^2} \, dx &=-\frac {\sqrt {x}}{2 \left (1+x^2\right )}+\frac {1}{4} \int \frac {1}{\sqrt {x} \left (1+x^2\right )} \, dx\\ &=-\frac {\sqrt {x}}{2 \left (1+x^2\right )}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,\sqrt {x}\right )\\ &=-\frac {\sqrt {x}}{2 \left (1+x^2\right )}+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {x}\right )+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {x}\right )\\ &=-\frac {\sqrt {x}}{2 \left (1+x^2\right )}+\frac {1}{8} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {x}\right )+\frac {1}{8} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {x}\right )-\frac {\operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2}}-\frac {\operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2}}\\ &=-\frac {\sqrt {x}}{2 \left (1+x^2\right )}-\frac {\log \left (1-\sqrt {2} \sqrt {x}+x\right )}{8 \sqrt {2}}+\frac {\log \left (1+\sqrt {2} \sqrt {x}+x\right )}{8 \sqrt {2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {x}\right )}{4 \sqrt {2}}-\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {x}\right )}{4 \sqrt {2}}\\ &=-\frac {\sqrt {x}}{2 \left (1+x^2\right )}-\frac {\tan ^{-1}\left (1-\sqrt {2} \sqrt {x}\right )}{4 \sqrt {2}}+\frac {\tan ^{-1}\left (1+\sqrt {2} \sqrt {x}\right )}{4 \sqrt {2}}-\frac {\log \left (1-\sqrt {2} \sqrt {x}+x\right )}{8 \sqrt {2}}+\frac {\log \left (1+\sqrt {2} \sqrt {x}+x\right )}{8 \sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 106, normalized size = 0.94 \[ \frac {1}{16} \left (-\frac {8 \sqrt {x}}{x^2+1}-\sqrt {2} \log \left (x-\sqrt {2} \sqrt {x}+1\right )+\sqrt {2} \log \left (x+\sqrt {2} \sqrt {x}+1\right )-2 \sqrt {2} \tan ^{-1}\left (1-\sqrt {2} \sqrt {x}\right )+2 \sqrt {2} \tan ^{-1}\left (\sqrt {2} \sqrt {x}+1\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.10, size = 140, normalized size = 1.24 \[ -\frac {4 \, \sqrt {2} {\left (x^{2} + 1\right )} \arctan \left (\sqrt {2} \sqrt {\sqrt {2} \sqrt {x} + x + 1} - \sqrt {2} \sqrt {x} - 1\right ) + 4 \, \sqrt {2} {\left (x^{2} + 1\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {-4 \, \sqrt {2} \sqrt {x} + 4 \, x + 4} - \sqrt {2} \sqrt {x} + 1\right ) - \sqrt {2} {\left (x^{2} + 1\right )} \log \left (4 \, \sqrt {2} \sqrt {x} + 4 \, x + 4\right ) + \sqrt {2} {\left (x^{2} + 1\right )} \log \left (-4 \, \sqrt {2} \sqrt {x} + 4 \, x + 4\right ) + 8 \, \sqrt {x}}{16 \, {\left (x^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.63, size = 86, normalized size = 0.76 \[ \frac {1}{8} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {x}\right )}\right ) + \frac {1}{8} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {x}\right )}\right ) + \frac {1}{16} \, \sqrt {2} \log \left (\sqrt {2} \sqrt {x} + x + 1\right ) - \frac {1}{16} \, \sqrt {2} \log \left (-\sqrt {2} \sqrt {x} + x + 1\right ) - \frac {\sqrt {x}}{2 \, {\left (x^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 74, normalized size = 0.65 \[ \frac {\sqrt {2}\, \arctan \left (\sqrt {2}\, \sqrt {x}-1\right )}{8}+\frac {\sqrt {2}\, \arctan \left (\sqrt {2}\, \sqrt {x}+1\right )}{8}+\frac {\sqrt {2}\, \ln \left (\frac {x +\sqrt {2}\, \sqrt {x}+1}{x -\sqrt {2}\, \sqrt {x}+1}\right )}{16}-\frac {\sqrt {x}}{2 \left (x^{2}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.96, size = 86, normalized size = 0.76 \[ \frac {1}{8} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {x}\right )}\right ) + \frac {1}{8} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {x}\right )}\right ) + \frac {1}{16} \, \sqrt {2} \log \left (\sqrt {2} \sqrt {x} + x + 1\right ) - \frac {1}{16} \, \sqrt {2} \log \left (-\sqrt {2} \sqrt {x} + x + 1\right ) - \frac {\sqrt {x}}{2 \, {\left (x^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.68, size = 51, normalized size = 0.45 \[ -\frac {\sqrt {x}}{2\,\left (x^2+1\right )}+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,\sqrt {x}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {1}{8}+\frac {1}{8}{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,\sqrt {x}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {1}{8}-\frac {1}{8}{}\mathrm {i}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 3.39, size = 257, normalized size = 2.27 \[ - \frac {8 \sqrt {x}}{16 x^{2} + 16} - \frac {\sqrt {2} x^{2} \log {\left (- 4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{16 x^{2} + 16} + \frac {\sqrt {2} x^{2} \log {\left (4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{16 x^{2} + 16} + \frac {2 \sqrt {2} x^{2} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} - 1 \right )}}{16 x^{2} + 16} + \frac {2 \sqrt {2} x^{2} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} + 1 \right )}}{16 x^{2} + 16} - \frac {\sqrt {2} \log {\left (- 4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{16 x^{2} + 16} + \frac {\sqrt {2} \log {\left (4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{16 x^{2} + 16} + \frac {2 \sqrt {2} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} - 1 \right )}}{16 x^{2} + 16} + \frac {2 \sqrt {2} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} + 1 \right )}}{16 x^{2} + 16} \]
Verification of antiderivative is not currently implemented for this CAS.
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