Optimal. Leaf size=38 \[ -\frac {x^4}{2}-\frac {2}{5} \left (1-x^2\right )^{5/2}+\frac {2}{3} \left (1-x^2\right )^{3/2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.32, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {6688, 6742, 266, 43} \[ -\frac {x^4}{2}-\frac {2}{5} \left (1-x^2\right )^{5/2}+\frac {2}{3} \left (1-x^2\right )^{3/2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 266
Rule 6688
Rule 6742
Rubi steps
\begin {align*} \int x^3 \left (-\sqrt {1-x}-\sqrt {1+x}\right ) \left (\sqrt {1-x}+\sqrt {1+x}\right ) \, dx &=-\int x^3 \left (\sqrt {1-x}+\sqrt {1+x}\right )^2 \, dx\\ &=-\int \left (2 x^3+2 x^3 \sqrt {1-x^2}\right ) \, dx\\ &=-\frac {x^4}{2}-2 \int x^3 \sqrt {1-x^2} \, dx\\ &=-\frac {x^4}{2}-\operatorname {Subst}\left (\int \sqrt {1-x} x \, dx,x,x^2\right )\\ &=-\frac {x^4}{2}-\operatorname {Subst}\left (\int \left (\sqrt {1-x}-(1-x)^{3/2}\right ) \, dx,x,x^2\right )\\ &=-\frac {x^4}{2}+\frac {2}{3} \left (1-x^2\right )^{3/2}-\frac {2}{5} \left (1-x^2\right )^{5/2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 38, normalized size = 1.00 \[ -\frac {x^4}{2}-\frac {2}{5} \left (1-x^2\right )^{5/2}+\frac {2}{3} \left (1-x^2\right )^{3/2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.43, size = 32, normalized size = 0.84 \[ -\frac {1}{2} \, x^{4} - \frac {2}{15} \, {\left (3 \, x^{4} - x^{2} - 2\right )} \sqrt {x + 1} \sqrt {-x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.54, size = 77, normalized size = 2.03 \[ -\frac {1}{2} \, x^{4} - \frac {1}{60} \, {\left ({\left (2 \, {\left (3 \, {\left (4 \, x - 17\right )} {\left (x + 1\right )} + 133\right )} {\left (x + 1\right )} - 295\right )} {\left (x + 1\right )} + 195\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {1}{12} \, {\left ({\left (2 \, {\left (3 \, x - 10\right )} {\left (x + 1\right )} + 43\right )} {\left (x + 1\right )} - 39\right )} \sqrt {x + 1} \sqrt {-x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 33, normalized size = 0.87 \[ -\frac {x^{4}}{2}-\frac {2 \sqrt {x +1}\, \sqrt {-x +1}\, \left (x^{2}-1\right ) \left (3 x^{2}+2\right )}{15} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.38, size = 31, normalized size = 0.82 \[ -\frac {1}{2} \, x^{4} + \frac {2}{5} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} x^{2} + \frac {4}{15} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.06, size = 42, normalized size = 1.11 \[ \sqrt {1-x}\,\left (\frac {4\,\sqrt {x+1}}{15}+\frac {2\,x^2\,\sqrt {x+1}}{15}-\frac {2\,x^4\,\sqrt {x+1}}{5}\right )-\frac {x^4}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________