Optimal. Leaf size=86 \[ \frac {\sqrt {1-x}}{28 x^{7/2}}+\frac {3 \sqrt {1-x}}{70 x^{5/2}}+\frac {2 \sqrt {1-x}}{35 x^{3/2}}+\frac {4 \sqrt {1-x}}{35 \sqrt {x}}-\frac {\text {ArcCos}\left (\sqrt {x}\right )}{4 x^4} \]
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Rubi [A]
time = 0.02, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4927, 12, 47,
37} \begin {gather*} -\frac {\text {ArcCos}\left (\sqrt {x}\right )}{4 x^4}+\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 {4 \sqrt {1-x}}{35 \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 37
Rule 47
Rule 4927
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*}
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Mathematica [A]
time = 0.03, size = 42, normalized size = 0.49 \begin {gather*} \frac {\sqrt {-((-1+x) x)} \left (5+6 x+8 x^2+16 x^3\right )-35 \text {ArcCos}\left (\sqrt {x}\right )}{140 x^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 59, normalized size = 0.69
method | result | size |
derivativedivides | \(-\frac {\arccos \left (\sqrt {x}\right )}{4 x^{4}}+\frac {\sqrt {1-x}}{28 x^{\frac {7}{2}}}+\frac {3 \sqrt {1-x}}{70 x^{\frac {5}{2}}}+\frac {2 \sqrt {1-x}}{35 x^{\frac {3}{2}}}+\frac {4 \sqrt {1-x}}{35 \sqrt {x}}\) | \(59\) |
default | \(-\frac {\arccos \left (\sqrt {x}\right )}{4 x^{4}}+\frac {\sqrt {1-x}}{28 x^{\frac {7}{2}}}+\frac {3 \sqrt {1-x}}{70 x^{\frac {5}{2}}}+\frac {2 \sqrt {1-x}}{35 x^{\frac {3}{2}}}+\frac {4 \sqrt {1-x}}{35 \sqrt {x}}\) | \(59\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 58, normalized size = 0.67 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.97, size = 38, normalized size = 0.44 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 32.01, size = 80, normalized size = 0.93 \begin {gather*} - \frac {\begin {cases} - \frac {\sqrt {1 - x}}{\sqrt {x}} - \frac {\left (1 - x\right )^{\frac {3}{2}}}{x^{\frac {3}{2}}} - \frac {3 \left (1 - x\right )^{\frac {5}{2}}}{5 x^{\frac {5}{2}}} - \frac {\left (1 - x\right )^{\frac {7}{2}}}{7 x^{\frac {7}{2}}} & \text {for}\: \sqrt {x} > -1 \wedge \sqrt {x} < 1 \end {cases}}{4} - \frac {\operatorname {acos}{\left (\sqrt {x} \right )}}{4 x^{4}} \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. 138 vs.
\(2 (58) = 116\).
time = 0.43, size = 138, normalized size = 1.60 \begin {gather*} \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 {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 {acos}\left (\sqrt {x}\right )}{x^5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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