Optimal. Leaf size=45 \[ \frac{\left (\frac{4}{5}-\frac{8 i}{5}\right ) e^{(1+2 i) \sin ^{-1}(a x)} \text{Hypergeometric2F1}\left (1-\frac{i}{2},2,2-\frac{i}{2},-e^{2 i \sin ^{-1}(a x)}\right )}{a} \]
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Rubi [A] time = 0.258082, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {4836, 6688, 6720, 4451} \[ \frac{\left (\frac{4}{5}-\frac{8 i}{5}\right ) e^{(1+2 i) \sin ^{-1}(a x)} \, _2F_1\left (1-\frac{i}{2},2;2-\frac{i}{2};-e^{2 i \sin ^{-1}(a x)}\right )}{a} \]
Antiderivative was successfully verified.
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Rule 4836
Rule 6688
Rule 6720
Rule 4451
Rubi steps
\begin{align*} \int \frac{e^{\sin ^{-1}(a x)}}{\left (1-a^2 x^2\right )^{3/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{e^x \cos (x)}{\left (1-\sin ^2(x)\right )^{3/2}} \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac{\operatorname{Subst}\left (\int \frac{e^x \cos (x)}{\cos ^2(x)^{3/2}} \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac{\operatorname{Subst}\left (\int e^x \sec ^2(x) \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac{\left (\frac{4}{5}-\frac{8 i}{5}\right ) e^{(1+2 i) \sin ^{-1}(a x)} \, _2F_1\left (1-\frac{i}{2},2;2-\frac{i}{2};-e^{2 i \sin ^{-1}(a x)}\right )}{a}\\ \end{align*}
Mathematica [A] time = 0.0680029, size = 45, normalized size = 1. \[ \frac{\left (\frac{4}{5}-\frac{8 i}{5}\right ) e^{(1+2 i) \sin ^{-1}(a x)} \text{Hypergeometric2F1}\left (1-\frac{i}{2},2,2-\frac{i}{2},-e^{2 i \sin ^{-1}(a x)}\right )}{a} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.08, size = 0, normalized size = 0. \begin{align*} \int{{{\rm e}^{\arcsin \left ( ax \right ) }} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{-{\frac{3}{2}}}}\, 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 )}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}\,{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{\sqrt{-a^{2} x^{2} + 1} e^{\left (\arcsin \left (a x\right )\right )}}{a^{4} x^{4} - 2 \, a^{2} x^{2} + 1}, 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 )}}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}}}\, 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 )}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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