Integrand size = 21, antiderivative size = 45 \[ \int \frac {e^{\arcsin (a x)}}{\left (1-a^2 x^2\right )^{3/2}} \, dx=\frac {\left (\frac {4}{5}-\frac {8 i}{5}\right ) e^{(1+2 i) \arcsin (a x)} \operatorname {Hypergeometric2F1}\left (1-\frac {i}{2},2,2-\frac {i}{2},-e^{2 i \arcsin (a x)}\right )}{a} \]
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Time = 0.18 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {4920, 6820, 6852, 4536} \[ \int \frac {e^{\arcsin (a x)}}{\left (1-a^2 x^2\right )^{3/2}} \, dx=\frac {\left (\frac {4}{5}-\frac {8 i}{5}\right ) e^{(1+2 i) \arcsin (a x)} \operatorname {Hypergeometric2F1}\left (1-\frac {i}{2},2,2-\frac {i}{2},-e^{2 i \arcsin (a x)}\right )}{a} \]
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Rule 4536
Rule 4920
Rule 6820
Rule 6852
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {e^x \cos (x)}{\left (1-\sin ^2(x)\right )^{3/2}} \, dx,x,\arcsin (a x)\right )}{a} \\ & = \frac {\text {Subst}\left (\int \frac {e^x \cos (x)}{\cos ^2(x)^{3/2}} \, dx,x,\arcsin (a x)\right )}{a} \\ & = \frac {\text {Subst}\left (\int e^x \sec ^2(x) \, dx,x,\arcsin (a x)\right )}{a} \\ & = \frac {\left (\frac {4}{5}-\frac {8 i}{5}\right ) e^{(1+2 i) \arcsin (a x)} \operatorname {Hypergeometric2F1}\left (1-\frac {i}{2},2,2-\frac {i}{2},-e^{2 i \arcsin (a x)}\right )}{a} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\arcsin (a x)}}{\left (1-a^2 x^2\right )^{3/2}} \, dx=\frac {\left (\frac {4}{5}-\frac {8 i}{5}\right ) e^{(1+2 i) \arcsin (a x)} \operatorname {Hypergeometric2F1}\left (1-\frac {i}{2},2,2-\frac {i}{2},-e^{2 i \arcsin (a x)}\right )}{a} \]
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\[\int \frac {{\mathrm e}^{\arcsin \left (a x \right )}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}d x\]
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\[ \int \frac {e^{\arcsin (a x)}}{\left (1-a^2 x^2\right )^{3/2}} \, dx=\int { \frac {e^{\left (\arcsin \left (a x\right )\right )}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {e^{\arcsin (a x)}}{\left (1-a^2 x^2\right )^{3/2}} \, dx=\int \frac {e^{\operatorname {asin}{\left (a x \right )}}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {e^{\arcsin (a x)}}{\left (1-a^2 x^2\right )^{3/2}} \, dx=\int { \frac {e^{\left (\arcsin \left (a x\right )\right )}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {e^{\arcsin (a x)}}{\left (1-a^2 x^2\right )^{3/2}} \, dx=\int { \frac {e^{\left (\arcsin \left (a x\right )\right )}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {e^{\arcsin (a x)}}{\left (1-a^2 x^2\right )^{3/2}} \, dx=\int \frac {{\mathrm {e}}^{\mathrm {asin}\left (a\,x\right )}}{{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \]
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