Optimal. Leaf size=68 \[ -\frac {\sqrt {1-x}}{15 x^{5/2}}-\frac {4 \sqrt {1-x}}{45 x^{3/2}}-\frac {8 \sqrt {1-x}}{45 \sqrt {x}}-\frac {\text {ArcSin}\left (\sqrt {x}\right )}{3 x^3} \]
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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 = {4926, 12, 47,
37} \begin {gather*} -\frac {\text {ArcSin}\left (\sqrt {x}\right )}{3 x^3}-\frac {4 \sqrt {1-x}}{45 x^{3/2}}-\frac {\sqrt {1-x}}{15 x^{5/2}}-\frac {8 \sqrt {1-x}}{45 \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 37
Rule 47
Rule 4926
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 = 0.02, size = 44, normalized size = 0.65 \begin {gather*} 2 \left (-\frac {\sqrt {1-x} \left (3+4 x+8 x^2\right )}{90 x^{5/2}}-\frac {\text {ArcSin}\left (\sqrt {x}\right )}{6 x^3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 47, normalized size = 0.69
method | result | size |
derivativedivides | \(-\frac {\arcsin \left (\sqrt {x}\right )}{3 x^{3}}-\frac {\sqrt {1-x}}{15 x^{\frac {5}{2}}}-\frac {4 \sqrt {1-x}}{45 x^{\frac {3}{2}}}-\frac {8 \sqrt {1-x}}{45 \sqrt {x}}\) | \(47\) |
default | \(-\frac {\arcsin \left (\sqrt {x}\right )}{3 x^{3}}-\frac {\sqrt {1-x}}{15 x^{\frac {5}{2}}}-\frac {4 \sqrt {1-x}}{45 x^{\frac {3}{2}}}-\frac {8 \sqrt {1-x}}{45 \sqrt {x}}\) | \(47\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 46, normalized size = 0.68 \begin {gather*} -\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.01, size = 33, normalized size = 0.49 \begin {gather*} -\frac {{\left (8 \, x^{2} + 4 \, x + 3\right )} \sqrt {x} \sqrt {-x + 1} + 15 \, \arcsin \left (\sqrt {x}\right )}{45 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 11.90, size = 66, normalized size = 0.97 \begin {gather*} \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}\: \sqrt {x} > -1 \wedge \sqrt {x} < 1 \end {cases}}{3} - \frac {\operatorname {asin}{\left (\sqrt {x} \right )}}{3 x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 106 vs.
\(2 (46) = 92\).
time = 0.41, size = 106, normalized size = 1.56 \begin {gather*} -\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}} \end {gather*}
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 \begin {gather*} \int \frac {\mathrm {asin}\left (\sqrt {x}\right )}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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