Optimal. Leaf size=50 \[ -\frac{\sqrt{1-x}}{6 x^{3/2}}-\frac{\sin ^{-1}\left (\sqrt{x}\right )}{2 x^2}-\frac{\sqrt{1-x}}{3 \sqrt{x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.017673, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4842, 12, 45, 37} \[ -\frac{\sqrt{1-x}}{6 x^{3/2}}-\frac{\sin ^{-1}\left (\sqrt{x}\right )}{2 x^2}-\frac{\sqrt{1-x}}{3 \sqrt{x}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4842
Rule 12
Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{\sin ^{-1}\left (\sqrt{x}\right )}{x^3} \, dx &=-\frac{\sin ^{-1}\left (\sqrt{x}\right )}{2 x^2}+\frac{1}{2} \int \frac{1}{2 \sqrt{1-x} x^{5/2}} \, dx\\ &=-\frac{\sin ^{-1}\left (\sqrt{x}\right )}{2 x^2}+\frac{1}{4} \int \frac{1}{\sqrt{1-x} x^{5/2}} \, dx\\ &=-\frac{\sqrt{1-x}}{6 x^{3/2}}-\frac{\sin ^{-1}\left (\sqrt{x}\right )}{2 x^2}+\frac{1}{6} \int \frac{1}{\sqrt{1-x} x^{3/2}} \, dx\\ &=-\frac{\sqrt{1-x}}{6 x^{3/2}}-\frac{\sqrt{1-x}}{3 \sqrt{x}}-\frac{\sin ^{-1}\left (\sqrt{x}\right )}{2 x^2}\\ \end{align*}
Mathematica [A] time = 0.0289701, size = 32, normalized size = 0.64 \[ -\frac{\sqrt{-(x-1) x} (2 x+1)+3 \sin ^{-1}\left (\sqrt{x}\right )}{6 x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.003, size = 35, normalized size = 0.7 \begin{align*} -{\frac{1}{2\,{x}^{2}}\arcsin \left ( \sqrt{x} \right ) }-{\frac{1}{6}\sqrt{1-x}{x}^{-{\frac{3}{2}}}}-{\frac{1}{3}\sqrt{1-x}{\frac{1}{\sqrt{x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.41861, size = 46, normalized size = 0.92 \begin{align*} -\frac{\sqrt{-x + 1}}{3 \, \sqrt{x}} - \frac{\sqrt{-x + 1}}{6 \, x^{\frac{3}{2}}} - \frac{\arcsin \left (\sqrt{x}\right )}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.29218, size = 85, normalized size = 1.7 \begin{align*} -\frac{{\left (2 \, x + 1\right )} \sqrt{x} \sqrt{-x + 1} + 3 \, \arcsin \left (\sqrt{x}\right )}{6 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 28.9816, size = 42, normalized size = 0.84 \begin{align*} \frac{\begin{cases} - \frac{\sqrt{1 - x}}{\sqrt{x}} - \frac{\left (1 - x\right )^{\frac{3}{2}}}{3 x^{\frac{3}{2}}} & \text{for}\: x \geq 0 \wedge x < 1 \end{cases}}{2} - \frac{\operatorname{asin}{\left (\sqrt{x} \right )}}{2 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.14515, size = 100, normalized size = 2. \begin{align*} -\frac{{\left (\sqrt{-x + 1} - 1\right )}^{3}}{48 \, x^{\frac{3}{2}}} - \frac{3 \,{\left (\sqrt{-x + 1} - 1\right )}}{16 \, \sqrt{x}} + \frac{x^{\frac{3}{2}}{\left (\frac{9 \,{\left (\sqrt{-x + 1} - 1\right )}^{2}}{x} + 1\right )}}{48 \,{\left (\sqrt{-x + 1} - 1\right )}^{3}} - \frac{\arcsin \left (\sqrt{x}\right )}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]