Optimal. Leaf size=50 \[ \frac {x^{3/2}}{12}+\frac {\pi x^2}{8}-\frac {1}{4} x^2 \tan ^{-1}\left (\sqrt {x}\right )-\frac {\sqrt {x}}{4}+\frac {1}{4} \tan ^{-1}\left (\sqrt {x}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {5159, 30, 5033, 50, 63, 203} \[ \frac {\pi x^2}{8}+\frac {x^{3/2}}{12}-\frac {1}{4} x^2 \tan ^{-1}\left (\sqrt {x}\right )-\frac {\sqrt {x}}{4}+\frac {1}{4} \tan ^{-1}\left (\sqrt {x}\right ) \]
Antiderivative was successfully verified.
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Rule 30
Rule 50
Rule 63
Rule 203
Rule 5033
Rule 5159
Rubi steps
\begin {align*} \int -x \tan ^{-1}\left (\sqrt {x}-\sqrt {1+x}\right ) \, dx &=-\left (\frac {1}{2} \int x \tan ^{-1}\left (\sqrt {x}\right ) \, dx\right )+\frac {1}{4} \pi \int x \, dx\\ &=\frac {\pi x^2}{8}-\frac {1}{4} x^2 \tan ^{-1}\left (\sqrt {x}\right )+\frac {1}{8} \int \frac {x^{3/2}}{1+x} \, dx\\ &=\frac {x^{3/2}}{12}+\frac {\pi x^2}{8}-\frac {1}{4} x^2 \tan ^{-1}\left (\sqrt {x}\right )-\frac {1}{8} \int \frac {\sqrt {x}}{1+x} \, dx\\ &=-\frac {\sqrt {x}}{4}+\frac {x^{3/2}}{12}+\frac {\pi x^2}{8}-\frac {1}{4} x^2 \tan ^{-1}\left (\sqrt {x}\right )+\frac {1}{8} \int \frac {1}{\sqrt {x} (1+x)} \, dx\\ &=-\frac {\sqrt {x}}{4}+\frac {x^{3/2}}{12}+\frac {\pi x^2}{8}-\frac {1}{4} x^2 \tan ^{-1}\left (\sqrt {x}\right )+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {x}\right )\\ &=-\frac {\sqrt {x}}{4}+\frac {x^{3/2}}{12}+\frac {\pi x^2}{8}+\frac {1}{4} \tan ^{-1}\left (\sqrt {x}\right )-\frac {1}{4} x^2 \tan ^{-1}\left (\sqrt {x}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 48, normalized size = 0.96 \[ \frac {1}{12} \left (3 \tan ^{-1}\left (\sqrt {x}\right )-\sqrt {x} \left (6 x^{3/2} \tan ^{-1}\left (\sqrt {x}-\sqrt {x+1}\right )-x+3\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.99, size = 28, normalized size = 0.56 \[ \frac {1}{2} \, {\left (x^{2} - 1\right )} \arctan \left (\sqrt {x + 1} - \sqrt {x}\right ) + \frac {1}{12} \, {\left (x - 3\right )} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 34, normalized size = 0.68 \[ -\frac {1}{2} \, x^{2} \arctan \left (-\sqrt {x + 1} + \sqrt {x}\right ) + \frac {1}{12} \, x^{\frac {3}{2}} - \frac {1}{4} \, \sqrt {x} + \frac {1}{4} \, \arctan \left (\sqrt {x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 35, normalized size = 0.70 \[ -\frac {x^{2} \arctan \left (\sqrt {x}-\sqrt {x +1}\right )}{2}+\frac {x^{\frac {3}{2}}}{12}-\frac {\sqrt {x}}{4}+\frac {\arctan \left (\sqrt {x}\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 34, normalized size = 0.68 \[ \frac {1}{2} \, x^{2} \arctan \left (\sqrt {x + 1} - \sqrt {x}\right ) + \frac {1}{12} \, x^{\frac {3}{2}} - \frac {1}{4} \, \sqrt {x} + \frac {1}{4} \, \arctan \left (\sqrt {x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.85, size = 58, normalized size = 1.16 \[ \frac {x^{3/2}}{12}-\frac {\sqrt {x}}{4}+\frac {\mathrm {atan}\left (\sqrt {x+1}-\sqrt {x}\right )\,\left (x^3+x^2\right )}{2\,x+2}+\frac {\ln \left (\frac {{\left (-1+\sqrt {x}\,1{}\mathrm {i}\right )}^2}{x+1}\right )\,1{}\mathrm {i}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 161.34, size = 39, normalized size = 0.78 \[ \frac {x^{\frac {3}{2}}}{12} - \frac {\sqrt {x}}{4} - \frac {x^{2} \operatorname {atan}{\left (\sqrt {x} - \sqrt {x + 1} \right )}}{2} + \frac {\operatorname {atan}{\left (\sqrt {x} \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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