Optimal. Leaf size=45 \[ \frac {\left (\frac {4}{5}-\frac {8 i}{5}\right ) e^{(1+2 i) \text {ArcSin}(a x)} \, _2F_1\left (1-\frac {i}{2},2;2-\frac {i}{2};-e^{2 i \text {ArcSin}(a x)}\right )}{a} \]
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Rubi [A]
time = 0.18, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {4920, 6820,
6852, 4536} \begin {gather*} \frac {\left (\frac {4}{5}-\frac {8 i}{5}\right ) e^{(1+2 i) \text {ArcSin}(a x)} \, _2F_1\left (1-\frac {i}{2},2;2-\frac {i}{2};-e^{2 i \text {ArcSin}(a x)}\right )}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 4536
Rule 4920
Rule 6820
Rule 6852
Rubi steps
\begin {align*} \int \frac {e^{\sin ^{-1}(a x)}}{\left (1-a^2 x^2\right )^{3/2}} \, dx &=\frac {\text {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 {\text {Subst}\left (\int \frac {e^x \cos (x)}{\cos ^2(x)^{3/2}} \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac {\text {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*}
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Mathematica [A]
time = 0.05, size = 45, normalized size = 1.00 \begin {gather*} \frac {\left (\frac {4}{5}-\frac {8 i}{5}\right ) e^{(1+2 i) \text {ArcSin}(a x)} \, _2F_1\left (1-\frac {i}{2},2;2-\frac {i}{2};-e^{2 i \text {ArcSin}(a x)}\right )}{a} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.08, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{\arcsin \left (a x \right )}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}\, dx\]
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 [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {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 {asin}\left (a\,x\right )}}{{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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