Optimal. Leaf size=50 \[ \frac {2 b^2}{x}+2 b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )-\frac {\left (a+b \sec ^{-1}(c x)\right )^2}{x} \]
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Rubi [A]
time = 0.04, antiderivative size = 50, 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 = {5330, 3377,
2718} \begin {gather*} 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} \end {gather*}
Antiderivative was successfully verified.
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Rule 2718
Rule 3377
Rule 5330
Rubi steps
\begin {align*} \int \frac {\left (a+b \sec ^{-1}(c x)\right )^2}{x^2} \, dx &=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*}
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Mathematica [A]
time = 0.08, size = 75, normalized size = 1.50 \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(116\) vs.
\(2(48)=96\).
time = 0.16, size = 117, normalized size = 2.34
method | result | size |
derivativedivides | \(c \left (-\frac {a^{2}}{c x}+b^{2} \left (-\frac {\mathrm {arcsec}\left (c x \right )^{2}}{c x}+\frac {2}{c x}+2 \,\mathrm {arcsec}\left (c x \right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\right )+2 a b \left (-\frac {\mathrm {arcsec}\left (c x \right )}{c x}+\frac {c^{2} x^{2}-1}{\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{2} x^{2}}\right )\right )\) | \(117\) |
default | \(c \left (-\frac {a^{2}}{c x}+b^{2} \left (-\frac {\mathrm {arcsec}\left (c x \right )^{2}}{c x}+\frac {2}{c x}+2 \,\mathrm {arcsec}\left (c x \right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\right )+2 a b \left (-\frac {\mathrm {arcsec}\left (c x \right )}{c x}+\frac {c^{2} x^{2}-1}{\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{2} x^{2}}\right )\right )\) | \(117\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 78, normalized size = 1.56 \begin {gather*} 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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.94, size = 57, normalized size = 1.14 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \operatorname {asec}{\left (c x \right )}\right )^{2}}{x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 105 vs.
\(2 (48) = 96\).
time = 0.40, size = 105, normalized size = 2.10 \begin {gather*} {\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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.82, size = 89, normalized size = 1.78 \begin {gather*} 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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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