Optimal. Leaf size=39 \[ -\frac {a+b \sin ^{-1}(c x)}{2 x^2}-\frac {b c \sqrt {1-c^2 x^2}}{2 x} \]
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Rubi [A] time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4627, 264} \[ -\frac {a+b \sin ^{-1}(c x)}{2 x^2}-\frac {b c \sqrt {1-c^2 x^2}}{2 x} \]
Antiderivative was successfully verified.
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Rule 264
Rule 4627
Rubi steps
\begin {align*} \int \frac {a+b \sin ^{-1}(c x)}{x^3} \, dx &=-\frac {a+b \sin ^{-1}(c x)}{2 x^2}+\frac {1}{2} (b c) \int \frac {1}{x^2 \sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {b c \sqrt {1-c^2 x^2}}{2 x}-\frac {a+b \sin ^{-1}(c x)}{2 x^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 44, normalized size = 1.13 \[ -\frac {a}{2 x^2}-\frac {b c \sqrt {1-c^2 x^2}}{2 x}-\frac {b \sin ^{-1}(c x)}{2 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 35, normalized size = 0.90 \[ -\frac {\sqrt {-c^{2} x^{2} + 1} b c x - a x^{2} + b \arcsin \left (c x\right ) + a}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.29, size = 163, normalized size = 4.18 \[ -\frac {b c^{4} x^{2} \arcsin \left (c x\right )}{8 \, {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{2}} - \frac {a c^{4} x^{2}}{8 \, {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{2}} + \frac {b c^{3} x}{4 \, {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}} - \frac {1}{4} \, b c^{2} \arcsin \left (c x\right ) - \frac {1}{4} \, a c^{2} - \frac {b c {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}}{4 \, x} - \frac {b {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{2} \arcsin \left (c x\right )}{8 \, x^{2}} - \frac {a {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{2}}{8 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 50, normalized size = 1.28 \[ c^{2} \left (-\frac {a}{2 c^{2} x^{2}}+b \left (-\frac {\arcsin \left (c x \right )}{2 c^{2} x^{2}}-\frac {\sqrt {-c^{2} x^{2}+1}}{2 c x}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 36, normalized size = 0.92 \[ -\frac {1}{2} \, b {\left (\frac {\sqrt {-c^{2} x^{2} + 1} c}{x} + \frac {\arcsin \left (c x\right )}{x^{2}}\right )} - \frac {a}{2 \, x^{2}} \]
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 \[ \int \frac {a+b\,\mathrm {asin}\left (c\,x\right )}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.45, size = 61, normalized size = 1.56 \[ - \frac {a}{2 x^{2}} + \frac {b c \left (\begin {cases} - \frac {i \sqrt {c^{2} x^{2} - 1}}{x} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- \frac {\sqrt {- c^{2} x^{2} + 1}}{x} & \text {otherwise} \end {cases}\right )}{2} - \frac {b \operatorname {asin}{\left (c x \right )}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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