Optimal. Leaf size=31 \[ \frac {1}{6} \left (1-x^4\right )^{3/2}-\frac {\sqrt {1-x^4}}{2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac {1}{6} \left (1-x^4\right )^{3/2}-\frac {\sqrt {1-x^4}}{2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^7}{\sqrt {1-x^4}} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {x}{\sqrt {1-x}} \, dx,x,x^4\right )\\ &=\frac {1}{4} \operatorname {Subst}\left (\int \left (\frac {1}{\sqrt {1-x}}-\sqrt {1-x}\right ) \, dx,x,x^4\right )\\ &=-\frac {1}{2} \sqrt {1-x^4}+\frac {1}{6} \left (1-x^4\right )^{3/2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 20, normalized size = 0.65 \[ -\frac {1}{6} \sqrt {1-x^4} \left (x^4+2\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.64, size = 16, normalized size = 0.52 \[ -\frac {1}{6} \, {\left (x^{4} + 2\right )} \sqrt {-x^{4} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.15, size = 23, normalized size = 0.74 \[ \frac {1}{6} \, {\left (-x^{4} + 1\right )}^{\frac {3}{2}} - \frac {1}{2} \, \sqrt {-x^{4} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 28, normalized size = 0.90 \[ \frac {\left (x -1\right ) \left (x +1\right ) \left (x^{2}+1\right ) \left (x^{4}+2\right )}{6 \sqrt {-x^{4}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.19, size = 23, normalized size = 0.74 \[ \frac {1}{6} \, {\left (-x^{4} + 1\right )}^{\frac {3}{2}} - \frac {1}{2} \, \sqrt {-x^{4} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.13, size = 16, normalized size = 0.52 \[ -\frac {\sqrt {1-x^4}\,\left (x^4+2\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.13, size = 24, normalized size = 0.77 \[ - \frac {x^{4} \sqrt {1 - x^{4}}}{6} - \frac {\sqrt {1 - x^{4}}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________