Optimal. Leaf size=83 \[ (1-i) a e^{(1+i) \sin ^{-1}(a x)} \, _2F_1\left (\frac {1}{2}-\frac {i}{2},1;\frac {3}{2}-\frac {i}{2};e^{2 i \sin ^{-1}(a x)}\right )-(2-2 i) a e^{(1+i) \sin ^{-1}(a x)} \, _2F_1\left (\frac {1}{2}-\frac {i}{2},2;\frac {3}{2}-\frac {i}{2};e^{2 i \sin ^{-1}(a x)}\right ) \]
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Rubi [A] time = 0.11, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4836, 12, 4471, 2251} \[ (1-i) a e^{(1+i) \sin ^{-1}(a x)} \, _2F_1\left (\frac {1}{2}-\frac {i}{2},1;\frac {3}{2}-\frac {i}{2};e^{2 i \sin ^{-1}(a x)}\right )-(2-2 i) a e^{(1+i) \sin ^{-1}(a x)} \, _2F_1\left (\frac {1}{2}-\frac {i}{2},2;\frac {3}{2}-\frac {i}{2};e^{2 i \sin ^{-1}(a x)}\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 2251
Rule 4471
Rule 4836
Rubi steps
\begin {align*} \int \frac {e^{\sin ^{-1}(a x)}}{x^2} \, dx &=\frac {\operatorname {Subst}\left (\int a^2 e^x \cot (x) \csc (x) \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=a \operatorname {Subst}\left (\int e^x \cot (x) \csc (x) \, dx,x,\sin ^{-1}(a x)\right )\\ &=a \operatorname {Subst}\left (\int \left (\frac {2 e^{(1+i) x}}{1-e^{2 i x}}-\frac {4 e^{(1+i) x}}{\left (-1+e^{2 i x}\right )^2}\right ) \, dx,x,\sin ^{-1}(a x)\right )\\ &=(2 a) \operatorname {Subst}\left (\int \frac {e^{(1+i) x}}{1-e^{2 i x}} \, dx,x,\sin ^{-1}(a x)\right )-(4 a) \operatorname {Subst}\left (\int \frac {e^{(1+i) x}}{\left (-1+e^{2 i x}\right )^2} \, dx,x,\sin ^{-1}(a x)\right )\\ &=(1-i) a e^{(1+i) \sin ^{-1}(a x)} \, _2F_1\left (\frac {1}{2}-\frac {i}{2},1;\frac {3}{2}-\frac {i}{2};e^{2 i \sin ^{-1}(a x)}\right )-(2-2 i) a e^{(1+i) \sin ^{-1}(a x)} \, _2F_1\left (\frac {1}{2}-\frac {i}{2},2;\frac {3}{2}-\frac {i}{2};e^{2 i \sin ^{-1}(a x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.11, size = 54, normalized size = 0.65 \[ -\frac {e^{\sin ^{-1}(a x)}+(1+i) a x e^{(1+i) \sin ^{-1}(a x)} \, _2F_1\left (\frac {1}{2}-\frac {i}{2},1;\frac {3}{2}-\frac {i}{2};e^{2 i \sin ^{-1}(a x)}\right )}{x} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {e^{\left (\arcsin \left (a x\right )\right )}}{x^{2}}, x\right ) \]
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 \[ \int \frac {e^{\left (\arcsin \left (a x\right )\right )}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{\arcsin \left (a x \right )}}{x^{2}}\, dx \]
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 \[ \int \frac {e^{\left (\arcsin \left (a x\right )\right )}}{x^{2}}\,{d x} \]
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 \[ \int \frac {{\mathrm {e}}^{\mathrm {asin}\left (a\,x\right )}}{x^2} \,d x \]
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 \[ \int \frac {e^{\operatorname {asin}{\left (a x \right )}}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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