Optimal. Leaf size=42 \[ -\frac{1}{6} \left (1-x^4\right )^{3/2}+\sqrt{1-x^4}+\frac{1}{2 \sqrt{1-x^4}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0184253, antiderivative size = 42, normalized size of antiderivative = 1., 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} \[ -\frac{1}{6} \left (1-x^4\right )^{3/2}+\sqrt{1-x^4}+\frac{1}{2 \sqrt{1-x^4}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^{11}}{\left (1-x^4\right )^{3/2}} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{x^2}{(1-x)^{3/2}} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (\frac{1}{(1-x)^{3/2}}-\frac{2}{\sqrt{1-x}}+\sqrt{1-x}\right ) \, dx,x,x^4\right )\\ &=\frac{1}{2 \sqrt{1-x^4}}+\sqrt{1-x^4}-\frac{1}{6} \left (1-x^4\right )^{3/2}\\ \end{align*}
Mathematica [A] time = 0.0092291, size = 27, normalized size = 0.64 \[ \frac{-x^8-4 x^4+8}{6 \sqrt{1-x^4}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.003, size = 33, normalized size = 0.8 \begin{align*}{\frac{ \left ( -1+x \right ) \left ( 1+x \right ) \left ({x}^{2}+1 \right ) \left ({x}^{8}+4\,{x}^{4}-8 \right ) }{6} \left ( -{x}^{4}+1 \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.997337, size = 43, normalized size = 1.02 \begin{align*} -\frac{1}{6} \,{\left (-x^{4} + 1\right )}^{\frac{3}{2}} + \sqrt{-x^{4} + 1} + \frac{1}{2 \, \sqrt{-x^{4} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.50461, size = 65, normalized size = 1.55 \begin{align*} \frac{{\left (x^{8} + 4 \, x^{4} - 8\right )} \sqrt{-x^{4} + 1}}{6 \,{\left (x^{4} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.86091, size = 39, normalized size = 0.93 \begin{align*} - \frac{x^{8}}{6 \sqrt{1 - x^{4}}} - \frac{2 x^{4}}{3 \sqrt{1 - x^{4}}} + \frac{4}{3 \sqrt{1 - x^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.1948, size = 43, normalized size = 1.02 \begin{align*} -\frac{1}{6} \,{\left (-x^{4} + 1\right )}^{\frac{3}{2}} + \sqrt{-x^{4} + 1} + \frac{1}{2 \, \sqrt{-x^{4} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]