Optimal. Leaf size=29 \[ -\frac {1}{4} x^2 \sqrt {16-x^4}+4 \sin ^{-1}\left (\frac {x^2}{4}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {281, 327, 222}
\begin {gather*} 4 \text {ArcSin}\left (\frac {x^2}{4}\right )-\frac {1}{4} x^2 \sqrt {16-x^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 222
Rule 281
Rule 327
Rubi steps
\begin {align*} \int \frac {x^5}{\sqrt {16-x^4}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^2}{\sqrt {16-x^2}} \, dx,x,x^2\right )\\ &=-\frac {1}{4} x^2 \sqrt {16-x^4}+4 \text {Subst}\left (\int \frac {1}{\sqrt {16-x^2}} \, dx,x,x^2\right )\\ &=-\frac {1}{4} x^2 \sqrt {16-x^4}+4 \sin ^{-1}\left (\frac {x^2}{4}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.09, size = 37, normalized size = 1.28 \begin {gather*} -\frac {1}{4} x^2 \sqrt {16-x^4}-4 \tan ^{-1}\left (\frac {\sqrt {16-x^4}}{x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.27, size = 24, normalized size = 0.83
method | result | size |
default | \(4 \arcsin \left (\frac {x^{2}}{4}\right )-\frac {x^{2} \sqrt {-x^{4}+16}}{4}\) | \(24\) |
elliptic | \(4 \arcsin \left (\frac {x^{2}}{4}\right )-\frac {x^{2} \sqrt {-x^{4}+16}}{4}\) | \(24\) |
risch | \(\frac {x^{2} \left (x^{4}-16\right )}{4 \sqrt {-x^{4}+16}}+4 \arcsin \left (\frac {x^{2}}{4}\right )\) | \(29\) |
meijerg | \(\frac {4 i \left (\frac {i \sqrt {\pi }\, x^{2} \sqrt {1-\frac {x^{4}}{16}}}{4}-i \sqrt {\pi }\, \arcsin \left (\frac {x^{2}}{4}\right )\right )}{\sqrt {\pi }}\) | \(38\) |
trager | \(-\frac {x^{2} \sqrt {-x^{4}+16}}{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{4}+16}+x^{2}\right )\) | \(45\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.49, size = 44, normalized size = 1.52 \begin {gather*} \frac {4 \, \sqrt {-x^{4} + 16}}{x^{2} {\left (\frac {x^{4} - 16}{x^{4}} - 1\right )}} - 4 \, \arctan \left (\frac {\sqrt {-x^{4} + 16}}{x^{2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.35, size = 33, normalized size = 1.14 \begin {gather*} -\frac {1}{4} \, \sqrt {-x^{4} + 16} x^{2} - 8 \, \arctan \left (\frac {\sqrt {-x^{4} + 16} - 4}{x^{2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] Result contains complex when optimal does not.
time = 0.93, size = 78, normalized size = 2.69 \begin {gather*} \begin {cases} - \frac {i x^{6}}{4 \sqrt {x^{4} - 16}} + \frac {4 i x^{2}}{\sqrt {x^{4} - 16}} - 4 i \operatorname {acosh}{\left (\frac {x^{2}}{4} \right )} & \text {for}\: \left |{x^{4}}\right | > 16 \\\frac {x^{6}}{4 \sqrt {16 - x^{4}}} - \frac {4 x^{2}}{\sqrt {16 - x^{4}}} + 4 \operatorname {asin}{\left (\frac {x^{2}}{4} \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.11, size = 23, normalized size = 0.79 \begin {gather*} -\frac {1}{4} \, \sqrt {-x^{4} + 16} x^{2} + 4 \, \arcsin \left (\frac {1}{4} \, x^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {x^5}{\sqrt {16-x^4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________