Optimal. Leaf size=16 \[ -\log \left (\sqrt{4-x^2}+1\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0450431, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {2155, 31} \[ -\log \left (\sqrt{4-x^2}+1\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2155
Rule 31
Rubi steps
\begin{align*} \int \frac{x}{4-x^2+\sqrt{4-x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{4+\sqrt{4-x}-x} \, dx,x,x^2\right )\\ &=-\operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,\sqrt{4-x^2}\right )\\ &=-\log \left (1+\sqrt{4-x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0267052, size = 16, normalized size = 1. \[ -\log \left (\sqrt{4-x^2}+1\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.043, size = 266, normalized size = 16.6 \begin{align*} -{\frac{\ln \left ({x}^{2}-3 \right ) }{2}}+{\frac{1}{ \left ( 4+2\,\sqrt{3} \right ) \left ( -2+\sqrt{3} \right ) }{\it Artanh} \left ({\frac{2+2\,\sqrt{3} \left ( x+\sqrt{3} \right ) }{2}{\frac{1}{\sqrt{- \left ( x+\sqrt{3} \right ) ^{2}+2\,\sqrt{3} \left ( x+\sqrt{3} \right ) +1}}}} \right ) }-{\frac{1}{ \left ( 4+2\,\sqrt{3} \right ) \left ( -2+\sqrt{3} \right ) }\sqrt{- \left ( x+\sqrt{3} \right ) ^{2}+2\,\sqrt{3} \left ( x+\sqrt{3} \right ) +1}}+{\frac{1}{ \left ( 4+2\,\sqrt{3} \right ) \left ( -2+\sqrt{3} \right ) }\sqrt{- \left ( 2+x \right ) ^{2}+4\,x+8}}+{\frac{1}{ \left ( 4+2\,\sqrt{3} \right ) \left ( -2+\sqrt{3} \right ) }{\it Artanh} \left ({\frac{2-2\,\sqrt{3} \left ( x-\sqrt{3} \right ) }{2}{\frac{1}{\sqrt{- \left ( x-\sqrt{3} \right ) ^{2}-2\,\sqrt{3} \left ( x-\sqrt{3} \right ) +1}}}} \right ) }-{\frac{1}{ \left ( 4+2\,\sqrt{3} \right ) \left ( -2+\sqrt{3} \right ) }\sqrt{- \left ( x-\sqrt{3} \right ) ^{2}-2\,\sqrt{3} \left ( x-\sqrt{3} \right ) +1}}+{\frac{1}{ \left ( 4+2\,\sqrt{3} \right ) \left ( -2+\sqrt{3} \right ) }\sqrt{- \left ( -2+x \right ) ^{2}-4\,x+8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.01516, size = 19, normalized size = 1.19 \begin{align*} -\log \left (\sqrt{-x^{2} + 4} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 0.824438, size = 144, normalized size = 9. \begin{align*} -\frac{1}{2} \, \log \left (x^{2} - 3\right ) + \frac{1}{2} \, \log \left (-\frac{x^{2} + 3 \, \sqrt{-x^{2} + 4} - 6}{x^{2}}\right ) - \frac{1}{2} \, \log \left (-\frac{x^{2} + \sqrt{-x^{2} + 4} - 2}{x^{2}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 2.64185, size = 44, normalized size = 2.75 \begin{align*} \frac{\log{\left (2 \sqrt{4 - x^{2}} \right )}}{2} - \frac{\log{\left (2 \sqrt{4 - x^{2}} + 2 \right )}}{2} - \frac{\log{\left (x^{2} - \sqrt{4 - x^{2}} - 4 \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.10885, size = 19, normalized size = 1.19 \begin{align*} -\log \left (\sqrt{-x^{2} + 4} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]