Optimal. Leaf size=32 \[ \sqrt {-\frac {x}{x+1}} (x+1)-\tan ^{-1}\left (\sqrt {-\frac {x}{x+1}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1959, 288, 204} \[ \sqrt {-\frac {x}{x+1}} (x+1)-\tan ^{-1}\left (\sqrt {-\frac {x}{x+1}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 204
Rule 288
Rule 1959
Rubi steps
\begin {align*} \int \sqrt {-\frac {x}{1+x}} \, dx &=-\left (2 \operatorname {Subst}\left (\int \frac {x^2}{\left (-1-x^2\right )^2} \, dx,x,\sqrt {-\frac {x}{1+x}}\right )\right )\\ &=\sqrt {-\frac {x}{1+x}} (1+x)+\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,\sqrt {-\frac {x}{1+x}}\right )\\ &=\sqrt {-\frac {x}{1+x}} (1+x)-\tan ^{-1}\left (\sqrt {-\frac {x}{1+x}}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 43, normalized size = 1.34 \[ \frac {\sqrt {-\frac {x}{x+1}} \left (\sqrt {x} (x+1)-\sqrt {x+1} \sinh ^{-1}\left (\sqrt {x}\right )\right )}{\sqrt {x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.41, size = 28, normalized size = 0.88 \[ {\left (x + 1\right )} \sqrt {-\frac {x}{x + 1}} - \arctan \left (\sqrt {-\frac {x}{x + 1}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.35, size = 36, normalized size = 1.12 \[ \frac {1}{4} \, \pi \mathrm {sgn}\left (x + 1\right ) + \frac {1}{2} \, \arcsin \left (2 \, x + 1\right ) \mathrm {sgn}\left (x + 1\right ) + \sqrt {-x^{2} - x} \mathrm {sgn}\left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 46, normalized size = 1.44 \[ \frac {\sqrt {-\frac {x}{x +1}}\, \left (x +1\right ) \left (-\ln \left (x +\frac {1}{2}+\sqrt {x^{2}+x}\right )+2 \sqrt {x^{2}+x}\right )}{2 \sqrt {\left (x +1\right ) x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.80, size = 37, normalized size = 1.16 \[ -\frac {\sqrt {-\frac {x}{x + 1}}}{\frac {x}{x + 1} - 1} - \arctan \left (\sqrt {-\frac {x}{x + 1}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.13, size = 37, normalized size = 1.16 \[ -\mathrm {atan}\left (\sqrt {-\frac {x}{x+1}}\right )-\frac {\sqrt {-\frac {x}{x+1}}}{\frac {x}{x+1}-1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {- \frac {x}{x + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________