Optimal. Leaf size=68 \[ -\frac {4 \sqrt {1-x}}{45 x^{3/2}}-\frac {\sqrt {1-x}}{15 x^{5/2}}-\frac {\sin ^{-1}\left (\sqrt {x}\right )}{3 x^3}-\frac {8 \sqrt {1-x}}{45 \sqrt {x}} \]
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Rubi [A] time = 0.02, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4842, 12, 45, 37} \[ -\frac {4 \sqrt {1-x}}{45 x^{3/2}}-\frac {\sqrt {1-x}}{15 x^{5/2}}-\frac {\sin ^{-1}\left (\sqrt {x}\right )}{3 x^3}-\frac {8 \sqrt {1-x}}{45 \sqrt {x}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 37
Rule 45
Rule 4842
Rubi steps
\begin {align*} \int \frac {\sin ^{-1}\left (\sqrt {x}\right )}{x^4} \, dx &=-\frac {\sin ^{-1}\left (\sqrt {x}\right )}{3 x^3}+\frac {1}{3} \int \frac {1}{2 \sqrt {1-x} x^{7/2}} \, dx\\ &=-\frac {\sin ^{-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 {\sin ^{-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 {\sin ^{-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 {\sin ^{-1}\left (\sqrt {x}\right )}{3 x^3}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 44, normalized size = 0.65 \[ 2 \left (-\frac {\sin ^{-1}\left (\sqrt {x}\right )}{6 x^3}-\frac {\sqrt {1-x} \left (8 x^2+4 x+3\right )}{90 x^{5/2}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 33, normalized size = 0.49 \[ -\frac {{\left (8 \, x^{2} + 4 \, x + 3\right )} \sqrt {x} \sqrt {-x + 1} + 15 \, \arcsin \left (\sqrt {x}\right )}{45 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.28, size = 106, normalized size = 1.56 \[ -\frac {{\left (\sqrt {-x + 1} - 1\right )}^{5}}{480 \, x^{\frac {5}{2}}} - \frac {5 \, {\left (\sqrt {-x + 1} - 1\right )}^{3}}{288 \, x^{\frac {3}{2}}} - \frac {5 \, {\left (\sqrt {-x + 1} - 1\right )}}{48 \, \sqrt {x}} + \frac {{\left (\frac {150 \, {\left (\sqrt {-x + 1} - 1\right )}^{4}}{x^{2}} + \frac {25 \, {\left (\sqrt {-x + 1} - 1\right )}^{2}}{x} + 3\right )} x^{\frac {5}{2}}}{1440 \, {\left (\sqrt {-x + 1} - 1\right )}^{5}} - \frac {\arcsin \left (\sqrt {x}\right )}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 47, normalized size = 0.69 \[ -\frac {\arcsin \left (\sqrt {x}\right )}{3 x^{3}}-\frac {\sqrt {1-x}}{15 x^{\frac {5}{2}}}-\frac {4 \sqrt {1-x}}{45 x^{\frac {3}{2}}}-\frac {8 \sqrt {1-x}}{45 \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 46, normalized size = 0.68 \[ -\frac {8 \, \sqrt {-x + 1}}{45 \, \sqrt {x}} - \frac {4 \, \sqrt {-x + 1}}{45 \, x^{\frac {3}{2}}} - \frac {\sqrt {-x + 1}}{15 \, x^{\frac {5}{2}}} - \frac {\arcsin \left (\sqrt {x}\right )}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {asin}\left (\sqrt {x}\right )}{x^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 15.69, size = 58, normalized size = 0.85 \[ \frac {\begin {cases} - \frac {\sqrt {1 - x}}{\sqrt {x}} - \frac {2 \left (1 - x\right )^{\frac {3}{2}}}{3 x^{\frac {3}{2}}} - \frac {\left (1 - x\right )^{\frac {5}{2}}}{5 x^{\frac {5}{2}}} & \text {for}\: x \geq 0 \wedge x < 1 \end {cases}}{3} - \frac {\operatorname {asin}{\left (\sqrt {x} \right )}}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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