Optimal. Leaf size=22 \[ -\tanh ^{-1}\left (\sqrt {1-x^2}\right )-\frac {\sin ^{-1}(x)}{x} \]
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Rubi [A] time = 0.02, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {4627, 266, 63, 206} \[ -\tanh ^{-1}\left (\sqrt {1-x^2}\right )-\frac {\sin ^{-1}(x)}{x} \]
Antiderivative was successfully verified.
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Rule 63
Rule 206
Rule 266
Rule 4627
Rubi steps
\begin {align*} \int \frac {\sin ^{-1}(x)}{x^2} \, dx &=-\frac {\sin ^{-1}(x)}{x}+\int \frac {1}{x \sqrt {1-x^2}} \, dx\\ &=-\frac {\sin ^{-1}(x)}{x}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} x} \, dx,x,x^2\right )\\ &=-\frac {\sin ^{-1}(x)}{x}-\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {1-x^2}\right )\\ &=-\frac {\sin ^{-1}(x)}{x}-\tanh ^{-1}\left (\sqrt {1-x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 22, normalized size = 1.00 \[ -\tanh ^{-1}\left (\sqrt {1-x^2}\right )-\frac {\sin ^{-1}(x)}{x} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin ^{-1}(x)}{x^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.00, size = 39, normalized size = 1.77 \[ -\frac {x \log \left (\sqrt {-x^{2} + 1} + 1\right ) - x \log \left (\sqrt {-x^{2} + 1} - 1\right ) + 2 \, \arcsin \relax (x)}{2 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.95, size = 38, normalized size = 1.73 \[ -\frac {\arcsin \relax (x)}{x} - \frac {1}{2} \, \log \left (\sqrt {-x^{2} + 1} + 1\right ) + \frac {1}{2} \, \log \left (-\sqrt {-x^{2} + 1} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 21, normalized size = 0.95
method | result | size |
default | \(-\frac {\arcsin \relax (x )}{x}-\arctanh \left (\frac {1}{\sqrt {-x^{2}+1}}\right )\) | \(21\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 33, normalized size = 1.50 \[ -\frac {\arcsin \relax (x)}{x} - \log \left (\frac {2 \, \sqrt {-x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.02, size = 20, normalized size = 0.91 \[ -\mathrm {atanh}\left (\frac {1}{\sqrt {1-x^2}}\right )-\frac {\mathrm {asin}\relax (x)}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.98, size = 22, normalized size = 1.00 \[ \begin {cases} - \operatorname {acosh}{\left (\frac {1}{x} \right )} & \text {for}\: \frac {1}{\left |{x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{x} \right )} & \text {otherwise} \end {cases} - \frac {\operatorname {asin}{\relax (x )}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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