Optimal. Leaf size=36 \[ -\frac {\sqrt {1-a^2 x^2}}{a \sin ^{-1}(a x)}-\frac {\text {Si}\left (\sin ^{-1}(a x)\right )}{a} \]
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Rubi [A] time = 0.08, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4621, 4723, 3299} \[ -\frac {\sqrt {1-a^2 x^2}}{a \sin ^{-1}(a x)}-\frac {\text {Si}\left (\sin ^{-1}(a x)\right )}{a} \]
Antiderivative was successfully verified.
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Rule 3299
Rule 4621
Rule 4723
Rubi steps
\begin {align*} \int \frac {1}{\sin ^{-1}(a x)^2} \, dx &=-\frac {\sqrt {1-a^2 x^2}}{a \sin ^{-1}(a x)}-a \int \frac {x}{\sqrt {1-a^2 x^2} \sin ^{-1}(a x)} \, dx\\ &=-\frac {\sqrt {1-a^2 x^2}}{a \sin ^{-1}(a x)}-\frac {\operatorname {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=-\frac {\sqrt {1-a^2 x^2}}{a \sin ^{-1}(a x)}-\frac {\text {Si}\left (\sin ^{-1}(a x)\right )}{a}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 32, normalized size = 0.89 \[ -\frac {\frac {\sqrt {1-a^2 x^2}}{\sin ^{-1}(a x)}+\text {Si}\left (\sin ^{-1}(a x)\right )}{a} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\arcsin \left (a x\right )^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 34, normalized size = 0.94 \[ -\frac {\operatorname {Si}\left (\arcsin \left (a x\right )\right )}{a} - \frac {\sqrt {-a^{2} x^{2} + 1}}{a \arcsin \left (a x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 33, normalized size = 0.92 \[ \frac {-\frac {\sqrt {-a^{2} x^{2}+1}}{\arcsin \left (a x \right )}-\Si \left (\arcsin \left (a x \right )\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {a^{2} \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right ) \int \frac {\sqrt {-a x + 1} x}{\sqrt {a x + 1} {\left (a x - 1\right )} \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right )}\,{d x} - \sqrt {a x + 1} \sqrt {-a x + 1}}{a \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{{\mathrm {asin}\left (a\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\operatorname {asin}^{2}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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