Optimal. Leaf size=35 \[ \frac{x^3}{3 \left (1-x^2\right )^{3/2}}-\frac{x}{\sqrt{1-x^2}}+\sin ^{-1}(x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0076016, antiderivative size = 35, 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 = {288, 216} \[ \frac{x^3}{3 \left (1-x^2\right )^{3/2}}-\frac{x}{\sqrt{1-x^2}}+\sin ^{-1}(x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 288
Rule 216
Rubi steps
\begin{align*} \int \frac{x^4}{\left (1-x^2\right )^{5/2}} \, dx &=\frac{x^3}{3 \left (1-x^2\right )^{3/2}}-\int \frac{x^2}{\left (1-x^2\right )^{3/2}} \, dx\\ &=\frac{x^3}{3 \left (1-x^2\right )^{3/2}}-\frac{x}{\sqrt{1-x^2}}+\int \frac{1}{\sqrt{1-x^2}} \, dx\\ &=\frac{x^3}{3 \left (1-x^2\right )^{3/2}}-\frac{x}{\sqrt{1-x^2}}+\sin ^{-1}(x)\\ \end{align*}
Mathematica [A] time = 0.004046, size = 26, normalized size = 0.74 \[ \frac{x \left (4 x^2-3\right )}{3 \left (1-x^2\right )^{3/2}}+\sin ^{-1}(x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0., size = 30, normalized size = 0.9 \begin{align*}{\frac{{x}^{3}}{3} \left ( -{x}^{2}+1 \right ) ^{-{\frac{3}{2}}}}+\arcsin \left ( x \right ) -{x{\frac{1}{\sqrt{-{x}^{2}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.41915, size = 59, normalized size = 1.69 \begin{align*} \frac{1}{3} \, x{\left (\frac{3 \, x^{2}}{{\left (-x^{2} + 1\right )}^{\frac{3}{2}}} - \frac{2}{{\left (-x^{2} + 1\right )}^{\frac{3}{2}}}\right )} - \frac{x}{3 \, \sqrt{-x^{2} + 1}} + \arcsin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.86975, size = 146, normalized size = 4.17 \begin{align*} -\frac{6 \,{\left (x^{4} - 2 \, x^{2} + 1\right )} \arctan \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right ) -{\left (4 \, x^{3} - 3 \, x\right )} \sqrt{-x^{2} + 1}}{3 \,{\left (x^{4} - 2 \, x^{2} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 2.56822, size = 105, normalized size = 3. \begin{align*} \frac{3 x^{4} \operatorname{asin}{\left (x \right )}}{3 x^{4} - 6 x^{2} + 3} + \frac{4 x^{3} \sqrt{1 - x^{2}}}{3 x^{4} - 6 x^{2} + 3} - \frac{6 x^{2} \operatorname{asin}{\left (x \right )}}{3 x^{4} - 6 x^{2} + 3} - \frac{3 x \sqrt{1 - x^{2}}}{3 x^{4} - 6 x^{2} + 3} + \frac{3 \operatorname{asin}{\left (x \right )}}{3 x^{4} - 6 x^{2} + 3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.09872, size = 39, normalized size = 1.11 \begin{align*} \frac{{\left (4 \, x^{2} - 3\right )} \sqrt{-x^{2} + 1} x}{3 \,{\left (x^{2} - 1\right )}^{2}} + \arcsin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]