Optimal. Leaf size=58 \[ \frac {1}{4} x^4 \csc ^{-1}\left (\sqrt {x}\right )+\frac {1}{28} (x-1)^{7/2}+\frac {3}{20} (x-1)^{5/2}+\frac {1}{4} (x-1)^{3/2}+\frac {\sqrt {x-1}}{4} \]
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Rubi [A] time = 0.02, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5271, 12, 43} \[ \frac {1}{4} x^4 \csc ^{-1}\left (\sqrt {x}\right )+\frac {1}{28} (x-1)^{7/2}+\frac {3}{20} (x-1)^{5/2}+\frac {1}{4} (x-1)^{3/2}+\frac {\sqrt {x-1}}{4} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 5271
Rubi steps
\begin {align*} \int x^3 \csc ^{-1}\left (\sqrt {x}\right ) \, dx &=\frac {1}{4} x^4 \csc ^{-1}\left (\sqrt {x}\right )+\frac {1}{4} \int \frac {x^3}{2 \sqrt {-1+x}} \, dx\\ &=\frac {1}{4} x^4 \csc ^{-1}\left (\sqrt {x}\right )+\frac {1}{8} \int \frac {x^3}{\sqrt {-1+x}} \, dx\\ &=\frac {1}{4} x^4 \csc ^{-1}\left (\sqrt {x}\right )+\frac {1}{8} \int \left (\frac {1}{\sqrt {-1+x}}+3 \sqrt {-1+x}+3 (-1+x)^{3/2}+(-1+x)^{5/2}\right ) \, dx\\ &=\frac {\sqrt {-1+x}}{4}+\frac {1}{4} (-1+x)^{3/2}+\frac {3}{20} (-1+x)^{5/2}+\frac {1}{28} (-1+x)^{7/2}+\frac {1}{4} x^4 \csc ^{-1}\left (\sqrt {x}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 40, normalized size = 0.69 \[ \frac {1}{4} x^4 \csc ^{-1}\left (\sqrt {x}\right )+\frac {1}{140} \sqrt {x-1} \left (5 x^3+6 x^2+8 x+16\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 32, normalized size = 0.55 \[ \frac {1}{4} \, x^{4} \operatorname {arccsc}\left (\sqrt {x}\right ) + \frac {1}{140} \, {\left (5 \, x^{3} + 6 \, x^{2} + 8 \, x + 16\right )} \sqrt {x - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 152, normalized size = 2.62 \[ \frac {1}{3584} \, x^{\frac {7}{2}} {\left (\sqrt {-\frac {1}{x} + 1} - 1\right )}^{7} + \frac {7}{2560} \, x^{\frac {5}{2}} {\left (\sqrt {-\frac {1}{x} + 1} - 1\right )}^{5} + \frac {1}{4} \, x^{4} \arcsin \left (\frac {1}{\sqrt {x}}\right ) + \frac {7}{512} \, x^{\frac {3}{2}} {\left (\sqrt {-\frac {1}{x} + 1} - 1\right )}^{3} + \frac {35}{512} \, \sqrt {x} {\left (\sqrt {-\frac {1}{x} + 1} - 1\right )} - \frac {1225 \, x^{3} {\left (\sqrt {-\frac {1}{x} + 1} - 1\right )}^{6} + 245 \, x^{2} {\left (\sqrt {-\frac {1}{x} + 1} - 1\right )}^{4} + 49 \, x {\left (\sqrt {-\frac {1}{x} + 1} - 1\right )}^{2} + 5}{17920 \, x^{\frac {7}{2}} {\left (\sqrt {-\frac {1}{x} + 1} - 1\right )}^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 43, normalized size = 0.74 \[ \frac {x^{4} \mathrm {arccsc}\left (\sqrt {x}\right )}{4}+\frac {\left (x -1\right ) \left (5 x^{3}+6 x^{2}+8 x +16\right )}{140 \sqrt {\frac {x -1}{x}}\, \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 66, normalized size = 1.14 \[ \frac {1}{28} \, x^{\frac {7}{2}} {\left (-\frac {1}{x} + 1\right )}^{\frac {7}{2}} + \frac {3}{20} \, x^{\frac {5}{2}} {\left (-\frac {1}{x} + 1\right )}^{\frac {5}{2}} + \frac {1}{4} \, x^{4} \operatorname {arccsc}\left (\sqrt {x}\right ) + \frac {1}{4} \, x^{\frac {3}{2}} {\left (-\frac {1}{x} + 1\right )}^{\frac {3}{2}} + \frac {1}{4} \, \sqrt {x} \sqrt {-\frac {1}{x} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x^3\,\mathrm {asin}\left (\frac {1}{\sqrt {x}}\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 64.53, size = 119, normalized size = 2.05 \[ \frac {x^{4} \operatorname {acsc}{\left (\sqrt {x} \right )}}{4} + \frac {\begin {cases} \frac {2 x^{3} \sqrt {x - 1}}{7} + \frac {12 x^{2} \sqrt {x - 1}}{35} + \frac {16 x \sqrt {x - 1}}{35} + \frac {32 \sqrt {x - 1}}{35} & \text {for}\: \left |{x}\right | > 1 \\\frac {2 i x^{3} \sqrt {1 - x}}{7} + \frac {12 i x^{2} \sqrt {1 - x}}{35} + \frac {16 i x \sqrt {1 - x}}{35} + \frac {32 i \sqrt {1 - x}}{35} & \text {otherwise} \end {cases}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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