Optimal. Leaf size=50 \[ \frac{\pi x^2}{8}+\frac{x^{3/2}}{12}-\frac{1}{4} x^2 \tan ^{-1}\left (\sqrt{x}\right )-\frac{\sqrt{x}}{4}+\frac{1}{4} \tan ^{-1}\left (\sqrt{x}\right ) \]
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Rubi [A] time = 0.0176926, antiderivative size = 50, normalized size of antiderivative = 1., 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 5159
Rule 30
Rule 5033
Rule 50
Rule 63
Rule 203
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*}
Mathematica [A] time = 0.0293783, 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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Maple [A] time = 0.054, size = 35, normalized size = 0.7 \begin{align*} -{\frac{{x}^{2}}{2}\arctan \left ( \sqrt{x}-\sqrt{x+1} \right ) }+{\frac{1}{12}{x}^{{\frac{3}{2}}}}-{\frac{1}{4}\sqrt{x}}+{\frac{1}{4}\arctan \left ( \sqrt{x} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.56974, size = 46, normalized size = 0.92 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.87696, size = 92, normalized size = 1.84 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11343, size = 46, normalized size = 0.92 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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