Optimal. Leaf size=68 \[ \frac{4 \sqrt{1-x}}{45 x^{3/2}}+\frac{\sqrt{1-x}}{15 x^{5/2}}-\frac{\cos ^{-1}\left (\sqrt{x}\right )}{3 x^3}+\frac{8 \sqrt{1-x}}{45 \sqrt{x}} \]
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Rubi [A] time = 0.0218177, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4843, 12, 45, 37} \[ \frac{4 \sqrt{1-x}}{45 x^{3/2}}+\frac{\sqrt{1-x}}{15 x^{5/2}}-\frac{\cos ^{-1}\left (\sqrt{x}\right )}{3 x^3}+\frac{8 \sqrt{1-x}}{45 \sqrt{x}} \]
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^4} \, dx &=-\frac{\cos ^{-1}\left (\sqrt{x}\right )}{3 x^3}-\frac{1}{3} \int \frac{1}{2 \sqrt{1-x} x^{7/2}} \, dx\\ &=-\frac{\cos ^{-1}\left (\sqrt{x}\right )}{3 x^3}-\frac{1}{6} \int \frac{1}{\sqrt{1-x} x^{7/2}} \, dx\\ &=\frac{\sqrt{1-x}}{15 x^{5/2}}-\frac{\cos ^{-1}\left (\sqrt{x}\right )}{3 x^3}-\frac{2}{15} \int \frac{1}{\sqrt{1-x} x^{5/2}} \, dx\\ &=\frac{\sqrt{1-x}}{15 x^{5/2}}+\frac{4 \sqrt{1-x}}{45 x^{3/2}}-\frac{\cos ^{-1}\left (\sqrt{x}\right )}{3 x^3}-\frac{4}{45} \int \frac{1}{\sqrt{1-x} x^{3/2}} \, dx\\ &=\frac{\sqrt{1-x}}{15 x^{5/2}}+\frac{4 \sqrt{1-x}}{45 x^{3/2}}+\frac{8 \sqrt{1-x}}{45 \sqrt{x}}-\frac{\cos ^{-1}\left (\sqrt{x}\right )}{3 x^3}\\ \end{align*}
Mathematica [A] time = 0.0376777, size = 37, normalized size = 0.54 \[ \frac{\sqrt{-(x-1) x} \left (8 x^2+4 x+3\right )-15 \cos ^{-1}\left (\sqrt{x}\right )}{45 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 47, normalized size = 0.7 \begin{align*} -{\frac{1}{3\,{x}^{3}}\arccos \left ( \sqrt{x} \right ) }+{\frac{1}{15}\sqrt{1-x}{x}^{-{\frac{5}{2}}}}+{\frac{4}{45}\sqrt{1-x}{x}^{-{\frac{3}{2}}}}+{\frac{8}{45}\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 = 1.4762, size = 62, normalized size = 0.91 \begin{align*} \frac{8 \, \sqrt{-x + 1}}{45 \, \sqrt{x}} + \frac{4 \, \sqrt{-x + 1}}{45 \, x^{\frac{3}{2}}} + \frac{\sqrt{-x + 1}}{15 \, x^{\frac{5}{2}}} - \frac{\arccos \left (\sqrt{x}\right )}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.63136, size = 97, normalized size = 1.43 \begin{align*} \frac{{\left (8 \, x^{2} + 4 \, x + 3\right )} \sqrt{x} \sqrt{-x + 1} - 15 \, \arccos \left (\sqrt{x}\right )}{45 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 150.038, size = 60, normalized size = 0.88 \begin{align*} - \frac{\begin{cases} - \frac{\sqrt{1 - x}}{\sqrt{x}} - \frac{2 \left (1 - x\right )^{\frac{3}{2}}}{3 x^{\frac{3}{2}}} - \frac{\left (1 - x\right )^{\frac{5}{2}}}{5 x^{\frac{5}{2}}} & \text{for}\: x \geq 0 \wedge x < 1 \end{cases}}{3} - \frac{\operatorname{acos}{\left (\sqrt{x} \right )}}{3 x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.31196, size = 143, normalized size = 2.1 \begin{align*} \frac{{\left (\sqrt{-x + 1} - 1\right )}^{5}}{480 \, x^{\frac{5}{2}}} + \frac{5 \,{\left (\sqrt{-x + 1} - 1\right )}^{3}}{288 \, x^{\frac{3}{2}}} + \frac{5 \,{\left (\sqrt{-x + 1} - 1\right )}}{48 \, \sqrt{x}} - \frac{{\left (\frac{150 \,{\left (\sqrt{-x + 1} - 1\right )}^{4}}{x^{2}} + \frac{25 \,{\left (\sqrt{-x + 1} - 1\right )}^{2}}{x} + 3\right )} x^{\frac{5}{2}}}{1440 \,{\left (\sqrt{-x + 1} - 1\right )}^{5}} - \frac{\arccos \left (\sqrt{x}\right )}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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