Integrand size = 14, antiderivative size = 50 \[ \int \frac {\left (a+b \sec ^{-1}(c x)\right )^2}{x^2} \, dx=\frac {2 b^2}{x}+2 b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )-\frac {\left (a+b \sec ^{-1}(c x)\right )^2}{x} \]
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Time = 0.05 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {5330, 3377, 2718} \[ \int \frac {\left (a+b \sec ^{-1}(c x)\right )^2}{x^2} \, dx=2 b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )-\frac {\left (a+b \sec ^{-1}(c x)\right )^2}{x}+\frac {2 b^2}{x} \]
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Rule 2718
Rule 3377
Rule 5330
Rubi steps \begin{align*} \text {integral}& = c \text {Subst}\left (\int (a+b x)^2 \sin (x) \, dx,x,\sec ^{-1}(c x)\right ) \\ & = -\frac {\left (a+b \sec ^{-1}(c x)\right )^2}{x}+(2 b c) \text {Subst}\left (\int (a+b x) \cos (x) \, dx,x,\sec ^{-1}(c x)\right ) \\ & = 2 b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )-\frac {\left (a+b \sec ^{-1}(c x)\right )^2}{x}-\left (2 b^2 c\right ) \text {Subst}\left (\int \sin (x) \, dx,x,\sec ^{-1}(c x)\right ) \\ & = \frac {2 b^2}{x}+2 b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )-\frac {\left (a+b \sec ^{-1}(c x)\right )^2}{x} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.50 \[ \int \frac {\left (a+b \sec ^{-1}(c x)\right )^2}{x^2} \, dx=\frac {-a^2+2 b^2+2 a b c \sqrt {1-\frac {1}{c^2 x^2}} x+2 b \left (-a+b c \sqrt {1-\frac {1}{c^2 x^2}} x\right ) \sec ^{-1}(c x)-b^2 \sec ^{-1}(c x)^2}{x} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(113\) vs. \(2(48)=96\).
Time = 0.56 (sec) , antiderivative size = 114, normalized size of antiderivative = 2.28
method | result | size |
parts | \(-\frac {a^{2}}{x}+b^{2} c \left (-\frac {\operatorname {arcsec}\left (c x \right )^{2}}{c x}+\frac {2}{c x}+2 \,\operatorname {arcsec}\left (c x \right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\right )+2 a b c \left (-\frac {\operatorname {arcsec}\left (c x \right )}{c x}+\frac {c^{2} x^{2}-1}{\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{2} x^{2}}\right )\) | \(114\) |
derivativedivides | \(c \left (-\frac {a^{2}}{c x}+b^{2} \left (-\frac {\operatorname {arcsec}\left (c x \right )^{2}}{c x}+\frac {2}{c x}+2 \,\operatorname {arcsec}\left (c x \right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\right )+2 a b \left (-\frac {\operatorname {arcsec}\left (c x \right )}{c x}+\frac {c^{2} x^{2}-1}{\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{2} x^{2}}\right )\right )\) | \(117\) |
default | \(c \left (-\frac {a^{2}}{c x}+b^{2} \left (-\frac {\operatorname {arcsec}\left (c x \right )^{2}}{c x}+\frac {2}{c x}+2 \,\operatorname {arcsec}\left (c x \right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\right )+2 a b \left (-\frac {\operatorname {arcsec}\left (c x \right )}{c x}+\frac {c^{2} x^{2}-1}{\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{2} x^{2}}\right )\right )\) | \(117\) |
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Time = 0.27 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.14 \[ \int \frac {\left (a+b \sec ^{-1}(c x)\right )^2}{x^2} \, dx=-\frac {b^{2} \operatorname {arcsec}\left (c x\right )^{2} + 2 \, a b \operatorname {arcsec}\left (c x\right ) + a^{2} - 2 \, b^{2} - 2 \, \sqrt {c^{2} x^{2} - 1} {\left (b^{2} \operatorname {arcsec}\left (c x\right ) + a b\right )}}{x} \]
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\[ \int \frac {\left (a+b \sec ^{-1}(c x)\right )^2}{x^2} \, dx=\int \frac {\left (a + b \operatorname {asec}{\left (c x \right )}\right )^{2}}{x^{2}}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.56 \[ \int \frac {\left (a+b \sec ^{-1}(c x)\right )^2}{x^2} \, dx=2 \, {\left (c \sqrt {-\frac {1}{c^{2} x^{2}} + 1} - \frac {\operatorname {arcsec}\left (c x\right )}{x}\right )} a b + 2 \, {\left (c \sqrt {-\frac {1}{c^{2} x^{2}} + 1} \operatorname {arcsec}\left (c x\right ) + \frac {1}{x}\right )} b^{2} - \frac {b^{2} \operatorname {arcsec}\left (c x\right )^{2}}{x} - \frac {a^{2}}{x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (48) = 96\).
Time = 0.29 (sec) , antiderivative size = 105, normalized size of antiderivative = 2.10 \[ \int \frac {\left (a+b \sec ^{-1}(c x)\right )^2}{x^2} \, dx={\left (2 \, b^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} \arccos \left (\frac {1}{c x}\right ) + 2 \, a b \sqrt {-\frac {1}{c^{2} x^{2}} + 1} - \frac {b^{2} \arccos \left (\frac {1}{c x}\right )^{2}}{c x} - \frac {2 \, a b \arccos \left (\frac {1}{c x}\right )}{c x} - \frac {a^{2}}{c x} + \frac {2 \, b^{2}}{c x}\right )} c \]
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Time = 0.92 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.78 \[ \int \frac {\left (a+b \sec ^{-1}(c x)\right )^2}{x^2} \, dx=2\,b^2\,c\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\,\sqrt {1-\frac {1}{c^2\,x^2}}-\frac {b^2\,\left ({\mathrm {acos}\left (\frac {1}{c\,x}\right )}^2-2\right )}{x}-\frac {a^2}{x}+2\,a\,b\,c\,\left (\sqrt {1-\frac {1}{c^2\,x^2}}-\frac {\mathrm {acos}\left (\frac {1}{c\,x}\right )}{c\,x}\right ) \]
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