Optimal. Leaf size=37 \[ \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x}-\cosh ^{-1}\left (\sqrt{x}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0208203, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107, Rules used = {280, 330, 52} \[ \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x}-\cosh ^{-1}\left (\sqrt{x}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 280
Rule 330
Rule 52
Rubi steps
\begin{align*} \int \frac{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}}}{\sqrt{x}} \, dx &=\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} \sqrt{x}-\frac{1}{2} \int \frac{1}{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} \sqrt{x}} \, dx\\ &=\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} \sqrt{x}-\operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,\sqrt{x}\right )\\ &=\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} \sqrt{x}-\cosh ^{-1}\left (\sqrt{x}\right )\\ \end{align*}
Mathematica [A] time = 0.0210303, size = 72, normalized size = 1.95 \[ \frac{\sqrt{\sqrt{x}+1} \sqrt{x} \left (\sqrt{x}-1\right )+2 \sqrt{1-\sqrt{x}} \sin ^{-1}\left (\frac{\sqrt{1-\sqrt{x}}}{\sqrt{2}}\right )}{\sqrt{\sqrt{x}-1}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.007, size = 72, normalized size = 2. \begin{align*} \sqrt{-1+\sqrt{x}} \left ( 1+\sqrt{x} \right ) ^{{\frac{3}{2}}}-\sqrt{-1+\sqrt{x}}\sqrt{1+\sqrt{x}}-{\sqrt{ \left ( 1+\sqrt{x} \right ) \left ( -1+\sqrt{x} \right ) }\ln \left ( \sqrt{x}+\sqrt{-1+x} \right ){\frac{1}{\sqrt{-1+\sqrt{x}}}}{\frac{1}{\sqrt{1+\sqrt{x}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.937178, size = 35, normalized size = 0.95 \begin{align*} \sqrt{x - 1} \sqrt{x} - \log \left (2 \, \sqrt{x - 1} + 2 \, \sqrt{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.971627, size = 151, normalized size = 4.08 \begin{align*} \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} + \frac{1}{2} \, \log \left (2 \, \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} - 2 \, x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\sqrt{x} - 1} \sqrt{\sqrt{x} + 1}}{\sqrt{x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]