Optimal. Leaf size=37 \[ \frac{\pi x}{4}+\frac{\sqrt{x}}{2}-\frac{1}{2} x \tan ^{-1}\left (\sqrt{x}\right )-\frac{1}{2} \tan ^{-1}\left (\sqrt{x}\right ) \]
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Rubi [A] time = 0.0100762, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5159, 8, 5027, 50, 63, 203} \[ \frac{\pi x}{4}+\frac{\sqrt{x}}{2}-\frac{1}{2} x \tan ^{-1}\left (\sqrt{x}\right )-\frac{1}{2} \tan ^{-1}\left (\sqrt{x}\right ) \]
Antiderivative was successfully verified.
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Rule 5159
Rule 8
Rule 5027
Rule 50
Rule 63
Rule 203
Rubi steps
\begin{align*} \int -\tan ^{-1}\left (\sqrt{x}-\sqrt{1+x}\right ) \, dx &=-\left (\frac{1}{2} \int \tan ^{-1}\left (\sqrt{x}\right ) \, dx\right )+\frac{1}{4} \pi \int 1 \, dx\\ &=\frac{\pi x}{4}-\frac{1}{2} x \tan ^{-1}\left (\sqrt{x}\right )+\frac{1}{4} \int \frac{\sqrt{x}}{1+x} \, dx\\ &=\frac{\sqrt{x}}{2}+\frac{\pi x}{4}-\frac{1}{2} x \tan ^{-1}\left (\sqrt{x}\right )-\frac{1}{4} \int \frac{1}{\sqrt{x} (1+x)} \, dx\\ &=\frac{\sqrt{x}}{2}+\frac{\pi x}{4}-\frac{1}{2} x \tan ^{-1}\left (\sqrt{x}\right )-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{x}\right )\\ &=\frac{\sqrt{x}}{2}+\frac{\pi x}{4}-\frac{1}{2} \tan ^{-1}\left (\sqrt{x}\right )-\frac{1}{2} x \tan ^{-1}\left (\sqrt{x}\right )\\ \end{align*}
Mathematica [A] time = 0.39669, size = 39, normalized size = 1.05 \[ \frac{\sqrt{x}}{2}-\frac{1}{2} \tan ^{-1}\left (\sqrt{x}\right )-x \tan ^{-1}\left (\sqrt{x}-\sqrt{x+1}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.054, size = 28, normalized size = 0.8 \begin{align*} -x\arctan \left ( \sqrt{x}-\sqrt{x+1} \right ) +{\frac{1}{2}\sqrt{x}}-{\frac{1}{2}\arctan \left ( \sqrt{x} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.59613, size = 35, normalized size = 0.95 \begin{align*} x \arctan \left (\sqrt{x + 1} - \sqrt{x}\right ) + \frac{1}{2} \, \sqrt{x} - \frac{1}{2} \, \arctan \left (\sqrt{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.91506, size = 72, normalized size = 1.95 \begin{align*}{\left (x + 1\right )} \arctan \left (\sqrt{x + 1} - \sqrt{x}\right ) + \frac{1}{2} \, \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 77.464, size = 29, normalized size = 0.78 \begin{align*} \frac{\sqrt{x}}{2} - x \operatorname{atan}{\left (\sqrt{x} - \sqrt{x + 1} \right )} - \frac{\operatorname{atan}{\left (\sqrt{x} \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11131, size = 36, normalized size = 0.97 \begin{align*} -x \arctan \left (-\sqrt{x + 1} + \sqrt{x}\right ) + \frac{1}{2} \, \sqrt{x} - \frac{1}{2} \, \arctan \left (\sqrt{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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