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 = 0.04, antiderivative size = 39, normalized size of antiderivative = 1.00, 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 261
Rule 4619
Rule 6715
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*}
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Mathematica [A] time = 0.03, size = 39, normalized size = 1.00 \[ -\frac {a}{x}-\frac {b \sqrt {1-\frac {c^2}{x^2}}}{c}-\frac {b \sin ^{-1}\left (\frac {c}{x}\right )}{x} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.39, size = 40, normalized size = 1.03 \[ -\frac {b c \arcsin \left (\frac {c}{x}\right ) + b x \sqrt {-\frac {c^{2} - x^{2}}{x^{2}}} + a c}{c x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 38, normalized size = 0.97 \[ -\frac {\frac {b c \arcsin \left (\frac {c}{x}\right )}{x} + b \sqrt {-\frac {c^{2}}{x^{2}} + 1} + \frac {a c}{x}}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 39, normalized size = 1.00 \[ -\frac {\frac {a c}{x}+b \left (\frac {c \arcsin \left (\frac {c}{x}\right )}{x}+\sqrt {1-\frac {c^{2}}{x^{2}}}\right )}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 37, normalized size = 0.95 \[ -\frac {b {\left (\frac {c \arcsin \left (\frac {c}{x}\right )}{x} + \sqrt {-\frac {c^{2}}{x^{2}} + 1}\right )}}{c} - \frac {a}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.29, size = 37, normalized size = 0.95 \[ -\frac {a}{x}-\frac {b\,\sqrt {1-\frac {c^2}{x^2}}}{c}-\frac {b\,\mathrm {asin}\left (\frac {c}{x}\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.75, size = 32, normalized size = 0.82 \[ \begin {cases} - \frac {a}{x} - \frac {b \operatorname {asin}{\left (\frac {c}{x} \right )}}{x} - \frac {b \sqrt {- \frac {c^{2}}{x^{2}} + 1}}{c} & \text {for}\: c \neq 0 \\- \frac {a}{x} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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