Optimal. Leaf size=31 \[ \frac {\sqrt {1-x^4}}{2}+\frac {1}{2 \sqrt {1-x^4}} \]
[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} \[ \frac {\sqrt {1-x^4}}{2}+\frac {1}{2 \sqrt {1-x^4}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^7}{\left (1-x^4\right )^{3/2}} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {x}{(1-x)^{3/2}} \, dx,x,x^4\right )\\ &=\frac {1}{4} \operatorname {Subst}\left (\int \left (\frac {1}{(1-x)^{3/2}}-\frac {1}{\sqrt {1-x}}\right ) \, dx,x,x^4\right )\\ &=\frac {1}{2 \sqrt {1-x^4}}+\frac {\sqrt {1-x^4}}{2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 22, normalized size = 0.71 \[ \frac {2-x^4}{2 \sqrt {1-x^4}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.84, size = 23, normalized size = 0.74 \[ \frac {{\left (x^{4} - 2\right )} \sqrt {-x^{4} + 1}}{2 \, {\left (x^{4} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.15, size = 23, normalized size = 0.74 \[ \frac {1}{2} \, \sqrt {-x^{4} + 1} + \frac {1}{2 \, \sqrt {-x^{4} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 28, normalized size = 0.90 \[ \frac {\left (x -1\right ) \left (x +1\right ) \left (x^{2}+1\right ) \left (x^{4}-2\right )}{2 \left (-x^{4}+1\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.30, size = 23, normalized size = 0.74 \[ \frac {1}{2} \, \sqrt {-x^{4} + 1} + \frac {1}{2 \, \sqrt {-x^{4} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.17, size = 16, normalized size = 0.52 \[ -\frac {x^4-2}{2\,\sqrt {1-x^4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.20, size = 22, normalized size = 0.71 \[ - \frac {x^{4}}{2 \sqrt {1 - x^{4}}} + \frac {1}{\sqrt {1 - x^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________