Optimal. Leaf size=96 \[ \frac {x e^{\sin ^{-1}(a x)}}{3 \left (1-a^2 x^2\right )^{3/2}}-\frac {e^{\sin ^{-1}(a x)}}{6 a \left (1-a^2 x^2\right )}+\frac {\left (\frac {2}{3}-\frac {4 i}{3}\right ) e^{(1+2 i) \sin ^{-1}(a x)} \, _2F_1\left (1-\frac {i}{2},2;2-\frac {i}{2};-e^{2 i \sin ^{-1}(a x)}\right )}{a} \]
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Rubi [A] time = 0.29, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {4836, 6688, 6720, 4448, 4451} \[ \frac {x e^{\sin ^{-1}(a x)}}{3 \left (1-a^2 x^2\right )^{3/2}}-\frac {e^{\sin ^{-1}(a x)}}{6 a \left (1-a^2 x^2\right )}+\frac {\left (\frac {2}{3}-\frac {4 i}{3}\right ) e^{(1+2 i) \sin ^{-1}(a x)} \, _2F_1\left (1-\frac {i}{2},2;2-\frac {i}{2};-e^{2 i \sin ^{-1}(a x)}\right )}{a} \]
Antiderivative was successfully verified.
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Rule 4448
Rule 4451
Rule 4836
Rule 6688
Rule 6720
Rubi steps
\begin {align*} \int \frac {e^{\sin ^{-1}(a x)}}{\left (1-a^2 x^2\right )^{5/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {e^x \cos (x)}{\left (1-\sin ^2(x)\right )^{5/2}} \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac {\operatorname {Subst}\left (\int \frac {e^x \cos (x)}{\cos ^2(x)^{5/2}} \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac {\operatorname {Subst}\left (\int e^x \sec ^4(x) \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac {e^{\sin ^{-1}(a x)} x}{3 \left (1-a^2 x^2\right )^{3/2}}-\frac {e^{\sin ^{-1}(a x)}}{6 a \left (1-a^2 x^2\right )}+\frac {5 \operatorname {Subst}\left (\int e^x \sec ^2(x) \, dx,x,\sin ^{-1}(a x)\right )}{6 a}\\ &=\frac {e^{\sin ^{-1}(a x)} x}{3 \left (1-a^2 x^2\right )^{3/2}}-\frac {e^{\sin ^{-1}(a x)}}{6 a \left (1-a^2 x^2\right )}+\frac {\left (\frac {2}{3}-\frac {4 i}{3}\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 = 0.20, size = 84, normalized size = 0.88 \[ \frac {e^{\sin ^{-1}(a x)} \left (\frac {2 a x}{\sqrt {1-a^2 x^2}}+(1-2 i) \left (1+e^{2 i \sin ^{-1}(a x)}\right )^2 \, _2F_1\left (1-\frac {i}{2},2;2-\frac {i}{2};-e^{2 i \sin ^{-1}(a x)}\right )-1\right )}{6 \left (a-a^3 x^2\right )} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a^{2} x^{2} + 1} e^{\left (\arcsin \left (a x\right )\right )}}{a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1}, x\right ) \]
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 \[ \int \frac {e^{\left (\arcsin \left (a x\right )\right )}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.13, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{\arcsin \left (a x \right )}}{\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}\, dx \]
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 \[ \int \frac {e^{\left (\arcsin \left (a x\right )\right )}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {e}}^{\mathrm {asin}\left (a\,x\right )}}{{\left (1-a^2\,x^2\right )}^{5/2}} \,d x \]
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 \[ \int \frac {e^{\operatorname {asin}{\left (a x \right )}}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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