Optimal. Leaf size=39 \[ \frac {1}{2} a \sqrt {1-\frac {1}{a^2 x^2}} e^{\sec ^{-1}(a x)}-\frac {e^{\sec ^{-1}(a x)}}{2 x} \]
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Rubi [A] time = 0.03, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5266, 12, 4432} \[ \frac {1}{2} a \sqrt {1-\frac {1}{a^2 x^2}} e^{\sec ^{-1}(a x)}-\frac {e^{\sec ^{-1}(a x)}}{2 x} \]
Antiderivative was successfully verified.
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Rule 12
Rule 4432
Rule 5266
Rubi steps
\begin {align*} \int \frac {e^{\sec ^{-1}(a x)}}{x^2} \, dx &=\frac {\operatorname {Subst}\left (\int a^2 e^x \sin (x) \, dx,x,\sec ^{-1}(a x)\right )}{a}\\ &=a \operatorname {Subst}\left (\int e^x \sin (x) \, dx,x,\sec ^{-1}(a x)\right )\\ &=\frac {1}{2} a e^{\sec ^{-1}(a x)} \sqrt {1-\frac {1}{a^2 x^2}}-\frac {e^{\sec ^{-1}(a x)}}{2 x}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 34, normalized size = 0.87 \[ \frac {1}{2} a \left (\sqrt {1-\frac {1}{a^2 x^2}}-\frac {1}{a x}\right ) e^{\sec ^{-1}(a x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.42, size = 23, normalized size = 0.59 \[ \frac {{\left (\sqrt {a^{2} x^{2} - 1} - 1\right )} e^{\left (\operatorname {arcsec}\left (a x\right )\right )}}{2 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 51, normalized size = 1.31 \[ \frac {1}{2} \, a \sqrt {-\frac {1}{a^{2} x^{2}} + 1} e^{\left (\frac {1}{2} \, \pi - \arcsin \left (\frac {1}{a x}\right )\right )} - \frac {e^{\left (\frac {1}{2} \, \pi - \arcsin \left (\frac {1}{a x}\right )\right )}}{2 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{\mathrm {arcsec}\left (a x \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{\left (\operatorname {arcsec}\left (a x\right )\right )}}{x^{2}}\,{d x} \]
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 \[ \int \frac {{\mathrm {e}}^{\mathrm {acos}\left (\frac {1}{a\,x}\right )}}{x^2} \,d x \]
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 \[ \int \frac {e^{\operatorname {asec}{\left (a x \right )}}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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