Optimal. Leaf size=39 \[ -\frac{a}{x}-\frac{b \sqrt{1-\frac{c^2}{x^2}}}{c}-\frac{b \csc ^{-1}\left (\frac{x}{c}\right )}{x} \]
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Rubi [A] time = 0.0359881, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {6715, 4619, 261} \[ -\frac{a}{x}-\frac{b \sqrt{1-\frac{c^2}{x^2}}}{c}-\frac{b \csc ^{-1}\left (\frac{x}{c}\right )}{x} \]
Antiderivative was successfully verified.
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Rule 6715
Rule 4619
Rule 261
Rubi steps
\begin{align*} \int \frac{a+b \sin ^{-1}\left (\frac{c}{x}\right )}{x^2} \, dx &=-\operatorname{Subst}\left (\int \left (a+b \sin ^{-1}(c x)\right ) \, dx,x,\frac{1}{x}\right )\\ &=-\frac{a}{x}-b \operatorname{Subst}\left (\int \sin ^{-1}(c x) \, dx,x,\frac{1}{x}\right )\\ &=-\frac{a}{x}-\frac{b \csc ^{-1}\left (\frac{x}{c}\right )}{x}+(b c) \operatorname{Subst}\left (\int \frac{x}{\sqrt{1-c^2 x^2}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{b \sqrt{1-\frac{c^2}{x^2}}}{c}-\frac{a}{x}-\frac{b \csc ^{-1}\left (\frac{x}{c}\right )}{x}\\ \end{align*}
Mathematica [A] time = 0.0233979, size = 39, normalized size = 1. \[ -\frac{a}{x}-\frac{b \sqrt{1-\frac{c^2}{x^2}}}{c}-\frac{b \sin ^{-1}\left (\frac{c}{x}\right )}{x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 39, normalized size = 1. \begin{align*} -{\frac{1}{c} \left ({\frac{ac}{x}}+b \left ({\frac{c}{x}\arcsin \left ({\frac{c}{x}} \right ) }+\sqrt{1-{\frac{{c}^{2}}{{x}^{2}}}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.40248, size = 50, normalized size = 1.28 \begin{align*} -\frac{b{\left (\frac{c \arcsin \left (\frac{c}{x}\right )}{x} + \sqrt{-\frac{c^{2}}{x^{2}} + 1}\right )}}{c} - \frac{a}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.1986, size = 82, normalized size = 2.1 \begin{align*} -\frac{b c \arcsin \left (\frac{c}{x}\right ) + b x \sqrt{-\frac{c^{2} - x^{2}}{x^{2}}} + a c}{c x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.59582, size = 32, normalized size = 0.82 \begin{align*} \begin{cases} - \frac{a}{x} - \frac{b \operatorname{asin}{\left (\frac{c}{x} \right )}}{x} - \frac{b \sqrt{- \frac{c^{2}}{x^{2}} + 1}}{c} & \text{for}\: c \neq 0 \\- \frac{a}{x} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16384, size = 51, normalized size = 1.31 \begin{align*} -\frac{b{\left (\frac{c \arcsin \left (\frac{c}{x}\right )}{x} + \sqrt{-\frac{c^{2}}{x^{2}} + 1}\right )} + \frac{a c}{x}}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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