Integrand size = 12, antiderivative size = 32 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^2} \, dx=-b c \sqrt {1-\frac {1}{c^2 x^2}}-\frac {a+b \csc ^{-1}(c x)}{x} \]
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Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5329, 267} \[ \int \frac {a+b \csc ^{-1}(c x)}{x^2} \, dx=-\frac {a+b \csc ^{-1}(c x)}{x}-b c \sqrt {1-\frac {1}{c^2 x^2}} \]
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Rule 267
Rule 5329
Rubi steps \begin{align*} \text {integral}& = -\frac {a+b \csc ^{-1}(c x)}{x}-\frac {b \int \frac {1}{\sqrt {1-\frac {1}{c^2 x^2}} x^3} \, dx}{c} \\ & = -b c \sqrt {1-\frac {1}{c^2 x^2}}-\frac {a+b \csc ^{-1}(c x)}{x} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.28 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^2} \, dx=-\frac {a}{x}-b c \sqrt {\frac {-1+c^2 x^2}{c^2 x^2}}-\frac {b \csc ^{-1}(c x)}{x} \]
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Time = 0.35 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.84
method | result | size |
parts | \(-\frac {a}{x}+b c \left (-\frac {\operatorname {arccsc}\left (c x \right )}{c x}-\frac {c^{2} x^{2}-1}{\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{2} x^{2}}\right )\) | \(59\) |
derivativedivides | \(c \left (-\frac {a}{c x}+b \left (-\frac {\operatorname {arccsc}\left (c x \right )}{c x}-\frac {c^{2} x^{2}-1}{\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{2} x^{2}}\right )\right )\) | \(63\) |
default | \(c \left (-\frac {a}{c x}+b \left (-\frac {\operatorname {arccsc}\left (c x \right )}{c x}-\frac {c^{2} x^{2}-1}{\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{2} x^{2}}\right )\right )\) | \(63\) |
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Time = 0.28 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^2} \, dx=-\frac {b \operatorname {arccsc}\left (c x\right ) + \sqrt {c^{2} x^{2} - 1} b + a}{x} \]
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Time = 0.37 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.16 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^2} \, dx=\begin {cases} - \frac {a}{x} - b c \sqrt {1 - \frac {1}{c^{2} x^{2}}} - \frac {b \operatorname {acsc}{\left (c x \right )}}{x} & \text {for}\: c \neq 0 \\- \frac {a + \tilde {\infty } b}{x} & \text {otherwise} \end {cases} \]
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Time = 0.18 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.03 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^2} \, dx=-{\left (c \sqrt {-\frac {1}{c^{2} x^{2}} + 1} + \frac {\operatorname {arccsc}\left (c x\right )}{x}\right )} b - \frac {a}{x} \]
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Time = 0.30 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.31 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^2} \, dx=-{\left (b \sqrt {-\frac {1}{c^{2} x^{2}} + 1} + \frac {b \arcsin \left (\frac {1}{c x}\right )}{c x} + \frac {a}{c x}\right )} c \]
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Time = 0.79 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.16 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^2} \, dx=-\frac {a}{x}-b\,c\,\sqrt {1-\frac {1}{c^2\,x^2}}-\frac {b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )}{x} \]
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