Optimal. Leaf size=28 \[ -a \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-\frac {\sin ^{-1}(a x)}{x} \]
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Rubi [A] time = 0.02, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4627, 266, 63, 208} \[ -a \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-\frac {\sin ^{-1}(a x)}{x} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 4627
Rubi steps
\begin {align*} \int \frac {\sin ^{-1}(a x)}{x^2} \, dx &=-\frac {\sin ^{-1}(a x)}{x}+a \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {\sin ^{-1}(a x)}{x}+\frac {1}{2} a \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac {\sin ^{-1}(a x)}{x}-\frac {\operatorname {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2}} \, dx,x,\sqrt {1-a^2 x^2}\right )}{a}\\ &=-\frac {\sin ^{-1}(a x)}{x}-a \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 28, normalized size = 1.00 \[ -a \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-\frac {\sin ^{-1}(a x)}{x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 49, normalized size = 1.75 \[ -\frac {a x \log \left (\sqrt {-a^{2} x^{2} + 1} + 1\right ) - a x \log \left (\sqrt {-a^{2} x^{2} + 1} - 1\right ) + 2 \, \arcsin \left (a x\right )}{2 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 48, normalized size = 1.71 \[ -\frac {1}{2} \, a {\left (\log \left (\sqrt {-a^{2} x^{2} + 1} + 1\right ) - \log \left (-\sqrt {-a^{2} x^{2} + 1} + 1\right )\right )} - \frac {\arcsin \left (a x\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 31, normalized size = 1.11 \[ a \left (-\frac {\arcsin \left (a x \right )}{a x}-\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 39, normalized size = 1.39 \[ -a \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - \frac {\arcsin \left (a x\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.02, size = 26, normalized size = 0.93 \[ -\frac {\mathrm {asin}\left (a\,x\right )}{x}-a\,\mathrm {atanh}\left (\frac {1}{\sqrt {1-a^2\,x^2}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.44, size = 32, normalized size = 1.14 \[ a \left (\begin {cases} - \operatorname {acosh}{\left (\frac {1}{a x} \right )} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{a x} \right )} & \text {otherwise} \end {cases}\right ) - \frac {\operatorname {asin}{\left (a x \right )}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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