Integrand size = 8, antiderivative size = 87 \[ \int e^{\csc ^{-1}(a x)} x \, dx=\frac {\left (\frac {8}{5}+\frac {4 i}{5}\right ) e^{(1+2 i) \csc ^{-1}(a x)} \operatorname {Hypergeometric2F1}\left (1-\frac {i}{2},2,2-\frac {i}{2},e^{2 i \csc ^{-1}(a x)}\right )}{a^2}-\frac {\left (\frac {16}{5}+\frac {8 i}{5}\right ) e^{(1+2 i) \csc ^{-1}(a x)} \operatorname {Hypergeometric2F1}\left (1-\frac {i}{2},3,2-\frac {i}{2},e^{2 i \csc ^{-1}(a x)}\right )}{a^2} \]
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Time = 0.08 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5375, 12, 4559, 2283} \[ \int e^{\csc ^{-1}(a x)} x \, dx=\frac {\left (\frac {8}{5}+\frac {4 i}{5}\right ) e^{(1+2 i) \csc ^{-1}(a x)} \operatorname {Hypergeometric2F1}\left (1-\frac {i}{2},2,2-\frac {i}{2},e^{2 i \csc ^{-1}(a x)}\right )}{a^2}-\frac {\left (\frac {16}{5}+\frac {8 i}{5}\right ) e^{(1+2 i) \csc ^{-1}(a x)} \operatorname {Hypergeometric2F1}\left (1-\frac {i}{2},3,2-\frac {i}{2},e^{2 i \csc ^{-1}(a x)}\right )}{a^2} \]
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Rule 12
Rule 2283
Rule 4559
Rule 5375
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {e^x \cot (x) \csc ^2(x)}{a} \, dx,x,\csc ^{-1}(a x)\right )}{a} \\ & = -\frac {\text {Subst}\left (\int e^x \cot (x) \csc ^2(x) \, dx,x,\csc ^{-1}(a x)\right )}{a^2} \\ & = -\frac {\text {Subst}\left (\int \left (-\frac {8 i e^{(1+2 i) x}}{\left (-1+e^{2 i x}\right )^3}-\frac {4 i e^{(1+2 i) x}}{\left (-1+e^{2 i x}\right )^2}\right ) \, dx,x,\csc ^{-1}(a x)\right )}{a^2} \\ & = \frac {(4 i) \text {Subst}\left (\int \frac {e^{(1+2 i) x}}{\left (-1+e^{2 i x}\right )^2} \, dx,x,\csc ^{-1}(a x)\right )}{a^2}+\frac {(8 i) \text {Subst}\left (\int \frac {e^{(1+2 i) x}}{\left (-1+e^{2 i x}\right )^3} \, dx,x,\csc ^{-1}(a x)\right )}{a^2} \\ & = \frac {\left (\frac {8}{5}+\frac {4 i}{5}\right ) e^{(1+2 i) \csc ^{-1}(a x)} \operatorname {Hypergeometric2F1}\left (1-\frac {i}{2},2,2-\frac {i}{2},e^{2 i \csc ^{-1}(a x)}\right )}{a^2}-\frac {\left (\frac {16}{5}+\frac {8 i}{5}\right ) e^{(1+2 i) \csc ^{-1}(a x)} \operatorname {Hypergeometric2F1}\left (1-\frac {i}{2},3,2-\frac {i}{2},e^{2 i \csc ^{-1}(a x)}\right )}{a^2} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.16 \[ \int e^{\csc ^{-1}(a x)} x \, dx=\frac {\left (\frac {1}{5}+\frac {i}{10}\right ) e^{\csc ^{-1}(a x)} \left ((2-i) a x \left (\sqrt {1-\frac {1}{a^2 x^2}}+a x\right )+(1+2 i) \operatorname {Hypergeometric2F1}\left (-\frac {i}{2},1,1-\frac {i}{2},e^{2 i \csc ^{-1}(a x)}\right )+e^{2 i \csc ^{-1}(a x)} \operatorname {Hypergeometric2F1}\left (1,1-\frac {i}{2},2-\frac {i}{2},e^{2 i \csc ^{-1}(a x)}\right )\right )}{a^2} \]
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\[\int {\mathrm e}^{\operatorname {arccsc}\left (a x \right )} x d x\]
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\[ \int e^{\csc ^{-1}(a x)} x \, dx=\int { x e^{\left (\operatorname {arccsc}\left (a x\right )\right )} \,d x } \]
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\[ \int e^{\csc ^{-1}(a x)} x \, dx=\int x e^{\operatorname {acsc}{\left (a x \right )}}\, dx \]
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\[ \int e^{\csc ^{-1}(a x)} x \, dx=\int { x e^{\left (\operatorname {arccsc}\left (a x\right )\right )} \,d x } \]
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\[ \int e^{\csc ^{-1}(a x)} x \, dx=\int { x e^{\left (\operatorname {arccsc}\left (a x\right )\right )} \,d x } \]
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Timed out. \[ \int e^{\csc ^{-1}(a x)} x \, dx=\int x\,{\mathrm {e}}^{\mathrm {asin}\left (\frac {1}{a\,x}\right )} \,d x \]
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