Optimal. Leaf size=8 \[ 2 \cosh ^{-1}\left (\sqrt {x}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {336, 54}
\begin {gather*} 2 \cosh ^{-1}\left (\sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 54
Rule 336
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}} \, dx &=2 \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,\sqrt {x}\right )\\ &=2 \cosh ^{-1}\left (\sqrt {x}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(38\) vs. \(2(8)=16\).
time = 0.82, size = 38, normalized size = 4.75 \begin {gather*} -8 \tanh ^{-1}\left (\frac {-1+\sqrt {-1+\sqrt {x}}}{\sqrt {3}-\sqrt {1+\sqrt {x}}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(39\) vs.
\(2(6)=12\).
time = 0.39, size = 40, normalized size = 5.00
method | result | size |
derivativedivides | \(\frac {2 \sqrt {\left (\sqrt {x}+1\right ) \left (-1+\sqrt {x}\right )}\, \ln \left (\sqrt {x}+\sqrt {x -1}\right )}{\sqrt {\sqrt {x}+1}\, \sqrt {-1+\sqrt {x}}}\) | \(40\) |
default | \(\frac {2 \sqrt {\left (\sqrt {x}+1\right ) \left (-1+\sqrt {x}\right )}\, \ln \left (\sqrt {x}+\sqrt {x -1}\right )}{\sqrt {\sqrt {x}+1}\, \sqrt {-1+\sqrt {x}}}\) | \(40\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 16 vs.
\(2 (6) = 12\).
time = 0.29, size = 16, normalized size = 2.00 \begin {gather*} 2 \, \log \left (2 \, \sqrt {x - 1} + 2 \, \sqrt {x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 27 vs.
\(2 (6) = 12\).
time = 1.99, size = 27, normalized size = 3.38 \begin {gather*} -\log \left (2 \, \sqrt {x} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} - 2 \, x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {x} \sqrt {\sqrt {x} - 1} \sqrt {\sqrt {x} + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 20 vs.
\(2 (6) = 12\).
time = 1.61, size = 20, normalized size = 2.50 \begin {gather*} -4 \, \log \left (\sqrt {\sqrt {x} + 1} - \sqrt {\sqrt {x} - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 5.29, size = 6, normalized size = 0.75 \begin {gather*} 2\,\mathrm {acosh}\left (\sqrt {x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________