Optimal. Leaf size=185 \[ -\frac {\sqrt {1-\frac {x^2}{a^2}} \text {Int}\left (\frac {x}{\left (1-\frac {x^2}{a^2}\right )^2 \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}},x\right )}{6 a^5 \sqrt {a^2-x^2}}-\frac {\sqrt {1-\frac {x^2}{a^2}} \text {Int}\left (\frac {x}{\left (1-\frac {x^2}{a^2}\right ) \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}},x\right )}{3 a^5 \sqrt {a^2-x^2}}+\frac {x \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}}{3 a^2 \left (a^2-x^2\right )^{3/2}}+\frac {2 x \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}}{3 a^4 \sqrt {a^2-x^2}} \]
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Rubi [A] time = 0.16, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}}{\left (a^2-x^2\right )^{5/2}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}}{\left (a^2-x^2\right )^{5/2}} \, dx &=\frac {x \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}}{3 a^2 \left (a^2-x^2\right )^{3/2}}+\frac {2 \int \frac {\sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}}{\left (a^2-x^2\right )^{3/2}} \, dx}{3 a^2}-\frac {\sqrt {1-\frac {x^2}{a^2}} \int \frac {x}{\left (1-\frac {x^2}{a^2}\right )^2 \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}} \, dx}{6 a^5 \sqrt {a^2-x^2}}\\ &=\frac {x \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}}{3 a^2 \left (a^2-x^2\right )^{3/2}}+\frac {2 x \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}}{3 a^4 \sqrt {a^2-x^2}}-\frac {\sqrt {1-\frac {x^2}{a^2}} \int \frac {x}{\left (1-\frac {x^2}{a^2}\right )^2 \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}} \, dx}{6 a^5 \sqrt {a^2-x^2}}-\frac {\sqrt {1-\frac {x^2}{a^2}} \int \frac {x}{\left (1-\frac {x^2}{a^2}\right ) \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}} \, dx}{3 a^5 \sqrt {a^2-x^2}}\\ \end {align*}
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Mathematica [A] time = 2.10, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}}{\left (a^2-x^2\right )^{5/2}} \, dx \]
Verification is Not applicable to the result.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\arcsin \left (\frac {x}{a}\right )}}{{\left (a^{2} - x^{2}\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.56, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\arcsin \left (\frac {x}{a}\right )}}{\left (a^{2}-x^{2}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {\mathrm {asin}\left (\frac {x}{a}\right )}}{{\left (a^2-x^2\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\operatorname {asin}{\left (\frac {x}{a} \right )}}}{\left (- \left (- a + x\right ) \left (a + x\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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