Optimal. Leaf size=10 \[ 2 F\left (\left .\sin ^{-1}\left (\sqrt {x}\right )\right |-1\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.00, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {116}
\begin {gather*} 2 F\left (\left .\text {ArcSin}\left (\sqrt {x}\right )\right |-1\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 116
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1-x} \sqrt {x} \sqrt {1+x}} \, dx &=2 F\left (\left .\sin ^{-1}\left (\sqrt {x}\right )\right |-1\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 0.42, size = 44, normalized size = 4.40 \begin {gather*} \frac {2 x \sqrt {1-x^2} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};x^2\right )}{\sqrt {-((-1+x) x)} \sqrt {1+x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(23\) vs.
\(2(8)=16\).
time = 0.10, size = 24, normalized size = 2.40
method | result | size |
default | \(\frac {\sqrt {2}\, \sqrt {-x}\, \EllipticF \left (\sqrt {1+x}, \frac {\sqrt {2}}{2}\right )}{\sqrt {x}}\) | \(24\) |
elliptic | \(\frac {\sqrt {-x \left (x^{2}-1\right )}\, \sqrt {2-2 x}\, \sqrt {-x}\, \EllipticF \left (\sqrt {1+x}, \frac {\sqrt {2}}{2}\right )}{\sqrt {1-x}\, \sqrt {x}\, \sqrt {-x^{3}+x}}\) | \(54\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 66 vs. \(2 (7) = 14\).
time = 12.04, size = 66, normalized size = 6.60 \begin {gather*} \frac {i {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {1}{2}, 1, 1 & \frac {3}{4}, \frac {3}{4}, \frac {5}{4} \\\frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, \frac {5}{4} & 0 \end {matrix} \middle | {\frac {1}{x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} - \frac {i {G_{6, 6}^{3, 5}\left (\begin {matrix} - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4} & 1 \\0, \frac {1}{2}, 0 & - \frac {1}{4}, \frac {1}{4}, \frac {1}{4} \end {matrix} \middle | {\frac {e^{- 2 i \pi }}{x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.10 \begin {gather*} \int \frac {1}{\sqrt {x}\,\sqrt {1-x}\,\sqrt {x+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________