Optimal. Leaf size=113 \[ \frac {\sqrt {x}}{2 \left (x^2+1\right )}-\frac {3 \log \left (x-\sqrt {2} \sqrt {x}+1\right )}{8 \sqrt {2}}+\frac {3 \log \left (x+\sqrt {2} \sqrt {x}+1\right )}{8 \sqrt {2}}-\frac {3 \tan ^{-1}\left (1-\sqrt {2} \sqrt {x}\right )}{4 \sqrt {2}}+\frac {3 \tan ^{-1}\left (\sqrt {2} \sqrt {x}+1\right )}{4 \sqrt {2}} \]
[Out]
________________________________________________________________________________________
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 = {290, 329, 211, 1165, 628, 1162, 617, 204} \[ \frac {\sqrt {x}}{2 \left (x^2+1\right )}-\frac {3 \log \left (x-\sqrt {2} \sqrt {x}+1\right )}{8 \sqrt {2}}+\frac {3 \log \left (x+\sqrt {2} \sqrt {x}+1\right )}{8 \sqrt {2}}-\frac {3 \tan ^{-1}\left (1-\sqrt {2} \sqrt {x}\right )}{4 \sqrt {2}}+\frac {3 \tan ^{-1}\left (\sqrt {2} \sqrt {x}+1\right )}{4 \sqrt {2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 204
Rule 211
Rule 290
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {x} \left (1+x^2\right )^2} \, dx &=\frac {\sqrt {x}}{2 \left (1+x^2\right )}+\frac {3}{4} \int \frac {1}{\sqrt {x} \left (1+x^2\right )} \, dx\\ &=\frac {\sqrt {x}}{2 \left (1+x^2\right )}+\frac {3}{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 {3}{4} \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {x}\right )+\frac {3}{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 {3}{8} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {x}\right )+\frac {3}{8} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {x}\right )-\frac {3 \operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2}}-\frac {3 \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 {3 \log \left (1-\sqrt {2} \sqrt {x}+x\right )}{8 \sqrt {2}}+\frac {3 \log \left (1+\sqrt {2} \sqrt {x}+x\right )}{8 \sqrt {2}}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {x}\right )}{4 \sqrt {2}}-\frac {3 \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 {3 \tan ^{-1}\left (1-\sqrt {2} \sqrt {x}\right )}{4 \sqrt {2}}+\frac {3 \tan ^{-1}\left (1+\sqrt {2} \sqrt {x}\right )}{4 \sqrt {2}}-\frac {3 \log \left (1-\sqrt {2} \sqrt {x}+x\right )}{8 \sqrt {2}}+\frac {3 \log \left (1+\sqrt {2} \sqrt {x}+x\right )}{8 \sqrt {2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 107, normalized size = 0.95 \[ \frac {1}{16} \left (\frac {8 \sqrt {x}}{x^2+1}-3 \sqrt {2} \log \left (x-\sqrt {2} \sqrt {x}+1\right )+3 \sqrt {2} \log \left (x+\sqrt {2} \sqrt {x}+1\right )-6 \sqrt {2} \tan ^{-1}\left (1-\sqrt {2} \sqrt {x}\right )+6 \sqrt {2} \tan ^{-1}\left (\sqrt {2} \sqrt {x}+1\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.13, size = 141, normalized size = 1.25 \[ -\frac {12 \, \sqrt {2} {\left (x^{2} + 1\right )} \arctan \left (\sqrt {2} \sqrt {\sqrt {2} \sqrt {x} + x + 1} - \sqrt {2} \sqrt {x} - 1\right ) + 12 \, \sqrt {2} {\left (x^{2} + 1\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {-4 \, \sqrt {2} \sqrt {x} + 4 \, x + 4} - \sqrt {2} \sqrt {x} + 1\right ) - 3 \, \sqrt {2} {\left (x^{2} + 1\right )} \log \left (4 \, \sqrt {2} \sqrt {x} + 4 \, x + 4\right ) + 3 \, \sqrt {2} {\left (x^{2} + 1\right )} \log \left (-4 \, \sqrt {2} \sqrt {x} + 4 \, x + 4\right ) - 8 \, \sqrt {x}}{16 \, {\left (x^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.60, size = 86, normalized size = 0.76 \[ \frac {3}{8} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {x}\right )}\right ) + \frac {3}{8} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {x}\right )}\right ) + \frac {3}{16} \, \sqrt {2} \log \left (\sqrt {2} \sqrt {x} + x + 1\right ) - \frac {3}{16} \, \sqrt {2} \log \left (-\sqrt {2} \sqrt {x} + x + 1\right ) + \frac {\sqrt {x}}{2 \, {\left (x^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 74, normalized size = 0.65 \[ \frac {3 \sqrt {2}\, \arctan \left (\sqrt {2}\, \sqrt {x}-1\right )}{8}+\frac {3 \sqrt {2}\, \arctan \left (\sqrt {2}\, \sqrt {x}+1\right )}{8}+\frac {3 \sqrt {2}\, \ln \left (\frac {x +\sqrt {2}\, \sqrt {x}+1}{x -\sqrt {2}\, \sqrt {x}+1}\right )}{16}+\frac {\sqrt {x}}{2 x^{2}+2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 2.84, size = 86, normalized size = 0.76 \[ \frac {3}{8} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {x}\right )}\right ) + \frac {3}{8} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {x}\right )}\right ) + \frac {3}{16} \, \sqrt {2} \log \left (\sqrt {2} \sqrt {x} + x + 1\right ) - \frac {3}{16} \, \sqrt {2} \log \left (-\sqrt {2} \sqrt {x} + x + 1\right ) + \frac {\sqrt {x}}{2 \, {\left (x^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.66, size = 50, normalized size = 0.44 \[ \frac {\sqrt {x}}{2\,\left (x^2+1\right )}+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,\sqrt {x}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {3}{8}+\frac {3}{8}{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,\sqrt {x}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {3}{8}-\frac {3}{8}{}\mathrm {i}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 2.99, size = 264, normalized size = 2.34 \[ \frac {8 \sqrt {x}}{16 x^{2} + 16} - \frac {3 \sqrt {2} x^{2} \log {\left (- 4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{16 x^{2} + 16} + \frac {3 \sqrt {2} x^{2} \log {\left (4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{16 x^{2} + 16} + \frac {6 \sqrt {2} x^{2} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} - 1 \right )}}{16 x^{2} + 16} + \frac {6 \sqrt {2} x^{2} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} + 1 \right )}}{16 x^{2} + 16} - \frac {3 \sqrt {2} \log {\left (- 4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{16 x^{2} + 16} + \frac {3 \sqrt {2} \log {\left (4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{16 x^{2} + 16} + \frac {6 \sqrt {2} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} - 1 \right )}}{16 x^{2} + 16} + \frac {6 \sqrt {2} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} + 1 \right )}}{16 x^{2} + 16} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________