Optimal. Leaf size=117 \[ \frac {2}{3} x^{3/2} \tan ^{-1}(x)-\frac {4 \sqrt {x}}{3}-\frac {\log \left (x-\sqrt {2} \sqrt {x}+1\right )}{3 \sqrt {2}}+\frac {\log \left (x+\sqrt {2} \sqrt {x}+1\right )}{3 \sqrt {2}}-\frac {1}{3} \sqrt {2} \tan ^{-1}\left (1-\sqrt {2} \sqrt {x}\right )+\frac {1}{3} \sqrt {2} \tan ^{-1}\left (\sqrt {2} \sqrt {x}+1\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.125, Rules used = {4852, 321, 329, 211, 1165, 628, 1162, 617, 204} \[ \frac {2}{3} x^{3/2} \tan ^{-1}(x)-\frac {4 \sqrt {x}}{3}-\frac {\log \left (x-\sqrt {2} \sqrt {x}+1\right )}{3 \sqrt {2}}+\frac {\log \left (x+\sqrt {2} \sqrt {x}+1\right )}{3 \sqrt {2}}-\frac {1}{3} \sqrt {2} \tan ^{-1}\left (1-\sqrt {2} \sqrt {x}\right )+\frac {1}{3} \sqrt {2} \tan ^{-1}\left (\sqrt {2} \sqrt {x}+1\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 204
Rule 211
Rule 321
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 4852
Rubi steps
\begin {align*} \int \sqrt {x} \tan ^{-1}(x) \, dx &=\frac {2}{3} x^{3/2} \tan ^{-1}(x)-\frac {2}{3} \int \frac {x^{3/2}}{1+x^2} \, dx\\ &=-\frac {4 \sqrt {x}}{3}+\frac {2}{3} x^{3/2} \tan ^{-1}(x)+\frac {2}{3} \int \frac {1}{\sqrt {x} \left (1+x^2\right )} \, dx\\ &=-\frac {4 \sqrt {x}}{3}+\frac {2}{3} x^{3/2} \tan ^{-1}(x)+\frac {4}{3} \operatorname {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,\sqrt {x}\right )\\ &=-\frac {4 \sqrt {x}}{3}+\frac {2}{3} x^{3/2} \tan ^{-1}(x)+\frac {2}{3} \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {x}\right )+\frac {2}{3} \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {x}\right )\\ &=-\frac {4 \sqrt {x}}{3}+\frac {2}{3} x^{3/2} \tan ^{-1}(x)+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {x}\right )+\frac {1}{3} \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 )}{3 \sqrt {2}}-\frac {\operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {x}\right )}{3 \sqrt {2}}\\ &=-\frac {4 \sqrt {x}}{3}+\frac {2}{3} x^{3/2} \tan ^{-1}(x)-\frac {\log \left (1-\sqrt {2} \sqrt {x}+x\right )}{3 \sqrt {2}}+\frac {\log \left (1+\sqrt {2} \sqrt {x}+x\right )}{3 \sqrt {2}}+\frac {1}{3} \sqrt {2} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {x}\right )-\frac {1}{3} \sqrt {2} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {x}\right )\\ &=-\frac {4 \sqrt {x}}{3}-\frac {1}{3} \sqrt {2} \tan ^{-1}\left (1-\sqrt {2} \sqrt {x}\right )+\frac {1}{3} \sqrt {2} \tan ^{-1}\left (1+\sqrt {2} \sqrt {x}\right )+\frac {2}{3} x^{3/2} \tan ^{-1}(x)-\frac {\log \left (1-\sqrt {2} \sqrt {x}+x\right )}{3 \sqrt {2}}+\frac {\log \left (1+\sqrt {2} \sqrt {x}+x\right )}{3 \sqrt {2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 108, normalized size = 0.92 \[ \frac {1}{6} \left (4 x^{3/2} \tan ^{-1}(x)-8 \sqrt {x}-\sqrt {2} \log \left (x-\sqrt {2} \sqrt {x}+1\right )+\sqrt {2} \log \left (x+\sqrt {2} \sqrt {x}+1\right )-2 \sqrt {2} \tan ^{-1}\left (1-\sqrt {2} \sqrt {x}\right )+2 \sqrt {2} \tan ^{-1}\left (\sqrt {2} \sqrt {x}+1\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.48, size = 118, normalized size = 1.01 \[ \frac {2}{3} \, {\left (x \arctan \relax (x) - 2\right )} \sqrt {x} - \frac {2}{3} \, \sqrt {2} \arctan \left (\sqrt {2} \sqrt {\sqrt {2} \sqrt {x} + x + 1} - \sqrt {2} \sqrt {x} - 1\right ) - \frac {2}{3} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {-4 \, \sqrt {2} \sqrt {x} + 4 \, x + 4} - \sqrt {2} \sqrt {x} + 1\right ) + \frac {1}{6} \, \sqrt {2} \log \left (4 \, \sqrt {2} \sqrt {x} + 4 \, x + 4\right ) - \frac {1}{6} \, \sqrt {2} \log \left (-4 \, \sqrt {2} \sqrt {x} + 4 \, x + 4\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.00, size = 86, normalized size = 0.74 \[ \frac {2}{3} \, x^{\frac {3}{2}} \arctan \relax (x) + \frac {1}{3} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {x}\right )}\right ) + \frac {1}{3} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {x}\right )}\right ) + \frac {1}{6} \, \sqrt {2} \log \left (\sqrt {2} \sqrt {x} + x + 1\right ) - \frac {1}{6} \, \sqrt {2} \log \left (-\sqrt {2} \sqrt {x} + x + 1\right ) - \frac {4}{3} \, \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 74, normalized size = 0.63 \[ \frac {2 x^{\frac {3}{2}} \arctan \relax (x )}{3}-\frac {4 \sqrt {x}}{3}+\frac {\arctan \left (1+\sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{3}+\frac {\arctan \left (-1+\sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{3}+\frac {\sqrt {2}\, \ln \left (\frac {1+x +\sqrt {2}\, \sqrt {x}}{1+x -\sqrt {2}\, \sqrt {x}}\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.41, size = 86, normalized size = 0.74 \[ \frac {2}{3} \, x^{\frac {3}{2}} \arctan \relax (x) + \frac {1}{3} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {x}\right )}\right ) + \frac {1}{3} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {x}\right )}\right ) + \frac {1}{6} \, \sqrt {2} \log \left (\sqrt {2} \sqrt {x} + x + 1\right ) - \frac {1}{6} \, \sqrt {2} \log \left (-\sqrt {2} \sqrt {x} + x + 1\right ) - \frac {4}{3} \, \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.32, size = 49, normalized size = 0.42 \[ \frac {2\,x^{3/2}\,\mathrm {atan}\relax (x)}{3}-\frac {4\,\sqrt {x}}{3}+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,\sqrt {x}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {1}{3}+\frac {1}{3}{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,\sqrt {x}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {1}{3}-\frac {1}{3}{}\mathrm {i}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 3.22, size = 104, normalized size = 0.89 \[ \frac {2 x^{\frac {3}{2}} \operatorname {atan}{\relax (x )}}{3} - \frac {4 \sqrt {x}}{3} - \frac {\sqrt {2} \log {\left (- \sqrt {2} \sqrt {x} + x + 1 \right )}}{6} + \frac {\sqrt {2} \log {\left (\sqrt {2} \sqrt {x} + x + 1 \right )}}{6} + \frac {\sqrt {2} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} - 1 \right )}}{3} + \frac {\sqrt {2} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} + 1 \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________