Optimal. Leaf size=43 \[ i e^{\sin ^{-1}(a x)}-2 i e^{\sin ^{-1}(a x)} \text{Hypergeometric2F1}\left (-\frac{i}{2},1,1-\frac{i}{2},e^{2 i \sin ^{-1}(a x)}\right ) \]
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Rubi [A] time = 0.0569787, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4836, 12, 4443, 2194, 2251} \[ i e^{\sin ^{-1}(a x)}-2 i e^{\sin ^{-1}(a x)} \, _2F_1\left (-\frac{i}{2},1;1-\frac{i}{2};e^{2 i \sin ^{-1}(a x)}\right ) \]
Antiderivative was successfully verified.
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Rule 4836
Rule 12
Rule 4443
Rule 2194
Rule 2251
Rubi steps
\begin{align*} \int \frac{e^{\sin ^{-1}(a x)}}{x} \, dx &=\frac{\operatorname{Subst}\left (\int a e^x \cot (x) \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\operatorname{Subst}\left (\int e^x \cot (x) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-\left (i \operatorname{Subst}\left (\int \left (-e^x-\frac{2 e^x}{-1+e^{2 i x}}\right ) \, dx,x,\sin ^{-1}(a x)\right )\right )\\ &=i \operatorname{Subst}\left (\int e^x \, dx,x,\sin ^{-1}(a x)\right )+2 i \operatorname{Subst}\left (\int \frac{e^x}{-1+e^{2 i x}} \, dx,x,\sin ^{-1}(a x)\right )\\ &=i e^{\sin ^{-1}(a x)}-2 i e^{\sin ^{-1}(a x)} \, _2F_1\left (-\frac{i}{2},1;1-\frac{i}{2};e^{2 i \sin ^{-1}(a x)}\right )\\ \end{align*}
Mathematica [A] time = 0.0500069, size = 75, normalized size = 1.74 \[ i \left (-e^{\sin ^{-1}(a x)} \text{Hypergeometric2F1}\left (-\frac{i}{2},1,1-\frac{i}{2},e^{2 i \sin ^{-1}(a x)}\right )-\left (\frac{1}{5}-\frac{2 i}{5}\right ) e^{(1+2 i) \sin ^{-1}(a x)} \text{Hypergeometric2F1}\left (1,1-\frac{i}{2},2-\frac{i}{2},e^{2 i \sin ^{-1}(a x)}\right )\right ) \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.008, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{\arcsin \left ( ax \right ) }}}{x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\left (\arcsin \left (a x\right )\right )}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{e^{\left (\arcsin \left (a x\right )\right )}}{x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\operatorname{asin}{\left (a x \right )}}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\left (\arcsin \left (a x\right )\right )}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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