Optimal. Leaf size=86 \[ \frac{2 \sqrt{1-x}}{35 x^{3/2}}+\frac{3 \sqrt{1-x}}{70 x^{5/2}}+\frac{\sqrt{1-x}}{28 x^{7/2}}-\frac{\cos ^{-1}\left (\sqrt{x}\right )}{4 x^4}+\frac{4 \sqrt{1-x}}{35 \sqrt{x}} \]
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Rubi [A] time = 0.0277524, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4843, 12, 45, 37} \[ \frac{2 \sqrt{1-x}}{35 x^{3/2}}+\frac{3 \sqrt{1-x}}{70 x^{5/2}}+\frac{\sqrt{1-x}}{28 x^{7/2}}-\frac{\cos ^{-1}\left (\sqrt{x}\right )}{4 x^4}+\frac{4 \sqrt{1-x}}{35 \sqrt{x}} \]
Antiderivative was successfully verified.
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Rule 4843
Rule 12
Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{\cos ^{-1}\left (\sqrt{x}\right )}{x^5} \, dx &=-\frac{\cos ^{-1}\left (\sqrt{x}\right )}{4 x^4}-\frac{1}{4} \int \frac{1}{2 \sqrt{1-x} x^{9/2}} \, dx\\ &=-\frac{\cos ^{-1}\left (\sqrt{x}\right )}{4 x^4}-\frac{1}{8} \int \frac{1}{\sqrt{1-x} x^{9/2}} \, dx\\ &=\frac{\sqrt{1-x}}{28 x^{7/2}}-\frac{\cos ^{-1}\left (\sqrt{x}\right )}{4 x^4}-\frac{3}{28} \int \frac{1}{\sqrt{1-x} x^{7/2}} \, dx\\ &=\frac{\sqrt{1-x}}{28 x^{7/2}}+\frac{3 \sqrt{1-x}}{70 x^{5/2}}-\frac{\cos ^{-1}\left (\sqrt{x}\right )}{4 x^4}-\frac{3}{35} \int \frac{1}{\sqrt{1-x} x^{5/2}} \, dx\\ &=\frac{\sqrt{1-x}}{28 x^{7/2}}+\frac{3 \sqrt{1-x}}{70 x^{5/2}}+\frac{2 \sqrt{1-x}}{35 x^{3/2}}-\frac{\cos ^{-1}\left (\sqrt{x}\right )}{4 x^4}-\frac{2}{35} \int \frac{1}{\sqrt{1-x} x^{3/2}} \, dx\\ &=\frac{\sqrt{1-x}}{28 x^{7/2}}+\frac{3 \sqrt{1-x}}{70 x^{5/2}}+\frac{2 \sqrt{1-x}}{35 x^{3/2}}+\frac{4 \sqrt{1-x}}{35 \sqrt{x}}-\frac{\cos ^{-1}\left (\sqrt{x}\right )}{4 x^4}\\ \end{align*}
Mathematica [A] time = 0.0436597, size = 42, normalized size = 0.49 \[ \frac{\sqrt{-(x-1) x} \left (16 x^3+8 x^2+6 x+5\right )-35 \cos ^{-1}\left (\sqrt{x}\right )}{140 x^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 59, normalized size = 0.7 \begin{align*} -{\frac{1}{4\,{x}^{4}}\arccos \left ( \sqrt{x} \right ) }+{\frac{1}{28}\sqrt{1-x}{x}^{-{\frac{7}{2}}}}+{\frac{3}{70}\sqrt{1-x}{x}^{-{\frac{5}{2}}}}+{\frac{2}{35}\sqrt{1-x}{x}^{-{\frac{3}{2}}}}+{\frac{4}{35}\sqrt{1-x}{\frac{1}{\sqrt{x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.49298, size = 78, normalized size = 0.91 \begin{align*} \frac{4 \, \sqrt{-x + 1}}{35 \, \sqrt{x}} + \frac{2 \, \sqrt{-x + 1}}{35 \, x^{\frac{3}{2}}} + \frac{3 \, \sqrt{-x + 1}}{70 \, x^{\frac{5}{2}}} + \frac{\sqrt{-x + 1}}{28 \, x^{\frac{7}{2}}} - \frac{\arccos \left (\sqrt{x}\right )}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.32641, size = 111, normalized size = 1.29 \begin{align*} \frac{{\left (16 \, x^{3} + 8 \, x^{2} + 6 \, x + 5\right )} \sqrt{x} \sqrt{-x + 1} - 35 \, \arccos \left (\sqrt{x}\right )}{140 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.28499, size = 186, normalized size = 2.16 \begin{align*} \frac{{\left (\sqrt{-x + 1} - 1\right )}^{7}}{3584 \, x^{\frac{7}{2}}} + \frac{7 \,{\left (\sqrt{-x + 1} - 1\right )}^{5}}{2560 \, x^{\frac{5}{2}}} + \frac{7 \,{\left (\sqrt{-x + 1} - 1\right )}^{3}}{512 \, x^{\frac{3}{2}}} + \frac{35 \,{\left (\sqrt{-x + 1} - 1\right )}}{512 \, \sqrt{x}} - \frac{{\left (\frac{1225 \,{\left (\sqrt{-x + 1} - 1\right )}^{6}}{x^{3}} + \frac{245 \,{\left (\sqrt{-x + 1} - 1\right )}^{4}}{x^{2}} + \frac{49 \,{\left (\sqrt{-x + 1} - 1\right )}^{2}}{x} + 5\right )} x^{\frac{7}{2}}}{17920 \,{\left (\sqrt{-x + 1} - 1\right )}^{7}} - \frac{\arccos \left (\sqrt{x}\right )}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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