Optimal. Leaf size=87 \[ -\frac {(1-i) e^{(1+i) \csc ^{-1}(a x)} \, _2F_1\left (\frac {1}{2}-\frac {i}{2},1;\frac {3}{2}-\frac {i}{2};e^{2 i \csc ^{-1}(a x)}\right )}{a}+\frac {(2-2 i) e^{(1+i) \csc ^{-1}(a x)} \, _2F_1\left (\frac {1}{2}-\frac {i}{2},2;\frac {3}{2}-\frac {i}{2};e^{2 i \csc ^{-1}(a x)}\right )}{a} \]
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Rubi [A]
time = 0.07, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5375, 4559,
2283} \begin {gather*} \frac {(2-2 i) e^{(1+i) \csc ^{-1}(a x)} \, _2F_1\left (\frac {1}{2}-\frac {i}{2},2;\frac {3}{2}-\frac {i}{2};e^{2 i \csc ^{-1}(a x)}\right )}{a}-\frac {(1-i) e^{(1+i) \csc ^{-1}(a x)} \, _2F_1\left (\frac {1}{2}-\frac {i}{2},1;\frac {3}{2}-\frac {i}{2};e^{2 i \csc ^{-1}(a x)}\right )}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 2283
Rule 4559
Rule 5375
Rubi steps
\begin {align*} \int e^{\csc ^{-1}(a x)} \, dx &=-\frac {\text {Subst}\left (\int e^x \cot (x) \csc (x) \, dx,x,\csc ^{-1}(a x)\right )}{a}\\ &=-\frac {\text {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,\csc ^{-1}(a x)\right )}{a}\\ &=-\frac {2 \text {Subst}\left (\int \frac {e^{(1+i) x}}{1-e^{2 i x}} \, dx,x,\csc ^{-1}(a x)\right )}{a}+\frac {4 \text {Subst}\left (\int \frac {e^{(1+i) x}}{\left (-1+e^{2 i x}\right )^2} \, dx,x,\csc ^{-1}(a x)\right )}{a}\\ &=-\frac {(1-i) e^{(1+i) \csc ^{-1}(a x)} \, _2F_1\left (\frac {1}{2}-\frac {i}{2},1;\frac {3}{2}-\frac {i}{2};e^{2 i \csc ^{-1}(a x)}\right )}{a}+\frac {(2-2 i) e^{(1+i) \csc ^{-1}(a x)} \, _2F_1\left (\frac {1}{2}-\frac {i}{2},2;\frac {3}{2}-\frac {i}{2};e^{2 i \csc ^{-1}(a x)}\right )}{a}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 54, normalized size = 0.62 \begin {gather*} \frac {e^{\csc ^{-1}(a x)} \left (a x+(1+i) e^{i \csc ^{-1}(a x)} \, _2F_1\left (\frac {1}{2}-\frac {i}{2},1;\frac {3}{2}-\frac {i}{2};e^{2 i \csc ^{-1}(a x)}\right )\right )}{a} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{\mathrm {arccsc}\left (a x \right )}\, dx\]
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 e^{\operatorname {acsc}{\left (a x \right )}}\, 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 = -0.01 \begin {gather*} \int {\mathrm {e}}^{\mathrm {asin}\left (\frac {1}{a\,x}\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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