Optimal. Leaf size=41 \[ -\frac {1}{5} a^2 e^{\sec ^{-1}(a x)} \cos \left (2 \sec ^{-1}(a x)\right )+\frac {1}{10} a^2 e^{\sec ^{-1}(a x)} \sin \left (2 \sec ^{-1}(a x)\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5374, 12, 4557,
4517} \begin {gather*} \frac {1}{10} a^2 e^{\sec ^{-1}(a x)} \sin \left (2 \sec ^{-1}(a x)\right )-\frac {1}{5} a^2 e^{\sec ^{-1}(a x)} \cos \left (2 \sec ^{-1}(a x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 4517
Rule 4557
Rule 5374
Rubi steps
\begin {align*} \int \frac {e^{\sec ^{-1}(a x)}}{x^3} \, dx &=\frac {\text {Subst}\left (\int a^3 e^x \cos (x) \sin (x) \, dx,x,\sec ^{-1}(a x)\right )}{a}\\ &=a^2 \text {Subst}\left (\int e^x \cos (x) \sin (x) \, dx,x,\sec ^{-1}(a x)\right )\\ &=a^2 \text {Subst}\left (\int \frac {1}{2} e^x \sin (2 x) \, dx,x,\sec ^{-1}(a x)\right )\\ &=\frac {1}{2} a^2 \text {Subst}\left (\int e^x \sin (2 x) \, dx,x,\sec ^{-1}(a x)\right )\\ &=-\frac {1}{5} a^2 e^{\sec ^{-1}(a x)} \cos \left (2 \sec ^{-1}(a x)\right )+\frac {1}{10} a^2 e^{\sec ^{-1}(a x)} \sin \left (2 \sec ^{-1}(a x)\right )\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 30, normalized size = 0.73 \begin {gather*} \frac {1}{10} a^2 e^{\sec ^{-1}(a x)} \left (-2 \cos \left (2 \sec ^{-1}(a x)\right )+\sin \left (2 \sec ^{-1}(a x)\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{\mathrm {arcsec}\left (a x \right )}}{x^{3}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.91, size = 30, normalized size = 0.73 \begin {gather*} \frac {{\left (a^{2} x^{2} + \sqrt {a^{2} x^{2} - 1} - 2\right )} e^{\left (\operatorname {arcsec}\left (a x\right )\right )}}{5 \, x^{2}} \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 {e^{\operatorname {asec}{\left (a x \right )}}}{x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\mathrm {e}}^{\mathrm {acos}\left (\frac {1}{a\,x}\right )}}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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