Optimal. Leaf size=45 \[ 2 i e^{\sec ^{-1}(a x)} \, _2F_1\left (-\frac {i}{2},1;1-\frac {i}{2};-e^{2 i \sec ^{-1}(a x)}\right )-i e^{\sec ^{-1}(a x)} \]
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Rubi [A] time = 0.06, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5266, 12, 4442, 2194, 2251} \[ 2 i e^{\sec ^{-1}(a x)} \, _2F_1\left (-\frac {i}{2},1;1-\frac {i}{2};-e^{2 i \sec ^{-1}(a x)}\right )-i e^{\sec ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2194
Rule 2251
Rule 4442
Rule 5266
Rubi steps
\begin {align*} \int \frac {e^{\sec ^{-1}(a x)}}{x} \, dx &=\frac {\operatorname {Subst}\left (\int a e^x \tan (x) \, dx,x,\sec ^{-1}(a x)\right )}{a}\\ &=\operatorname {Subst}\left (\int e^x \tan (x) \, dx,x,\sec ^{-1}(a x)\right )\\ &=i \operatorname {Subst}\left (\int \left (-e^x+\frac {2 e^x}{1+e^{2 i x}}\right ) \, dx,x,\sec ^{-1}(a x)\right )\\ &=-\left (i \operatorname {Subst}\left (\int e^x \, dx,x,\sec ^{-1}(a x)\right )\right )+2 i \operatorname {Subst}\left (\int \frac {e^x}{1+e^{2 i x}} \, dx,x,\sec ^{-1}(a x)\right )\\ &=-i e^{\sec ^{-1}(a x)}+2 i e^{\sec ^{-1}(a x)} \, _2F_1\left (-\frac {i}{2},1;1-\frac {i}{2};-e^{2 i \sec ^{-1}(a x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.05, size = 79, normalized size = 1.76 \[ -i \left (\left (\frac {1}{5}-\frac {2 i}{5}\right ) e^{(1+2 i) \sec ^{-1}(a x)} \, _2F_1\left (1,1-\frac {i}{2};2-\frac {i}{2};-e^{2 i \sec ^{-1}(a x)}\right )-e^{\sec ^{-1}(a x)} \, _2F_1\left (-\frac {i}{2},1;1-\frac {i}{2};-e^{2 i \sec ^{-1}(a x)}\right )\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.99, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {e^{\left (\operatorname {arcsec}\left (a x\right )\right )}}{x}, x\right ) \]
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 \[ \int \frac {e^{\left (\operatorname {arcsec}\left (a x\right )\right )}}{x}\,{d x} \]
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 \[ \int \frac {{\mathrm e}^{\mathrm {arcsec}\left (a x \right )}}{x}\, dx \]
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 \[ \int \frac {e^{\left (\operatorname {arcsec}\left (a x\right )\right )}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\mathrm {e}}^{\mathrm {acos}\left (\frac {1}{a\,x}\right )}}{x} \,d x \]
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 \[ \int \frac {e^{\operatorname {asec}{\left (a x \right )}}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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