Integrand size = 10, antiderivative size = 84 \[ \int \frac {e^{\csc ^{-1}(a x)}}{x^4} \, dx=-\frac {1}{8} a^3 e^{\csc ^{-1}(a x)} \sqrt {1-\frac {1}{a^2 x^2}}-\frac {a^2 e^{\csc ^{-1}(a x)}}{8 x}+\frac {1}{40} a^3 e^{\csc ^{-1}(a x)} \cos \left (3 \csc ^{-1}(a x)\right )+\frac {3}{40} a^3 e^{\csc ^{-1}(a x)} \sin \left (3 \csc ^{-1}(a x)\right ) \]
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Time = 0.05 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5375, 12, 4557, 4518} \[ \int \frac {e^{\csc ^{-1}(a x)}}{x^4} \, dx=\frac {1}{40} a^3 e^{\csc ^{-1}(a x)} \cos \left (3 \csc ^{-1}(a x)\right )+\frac {3}{40} a^3 e^{\csc ^{-1}(a x)} \sin \left (3 \csc ^{-1}(a x)\right )-\frac {a^2 e^{\csc ^{-1}(a x)}}{8 x}-\frac {1}{8} a^3 \sqrt {1-\frac {1}{a^2 x^2}} e^{\csc ^{-1}(a x)} \]
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Rule 12
Rule 4518
Rule 4557
Rule 5375
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int a^4 e^x \cos (x) \sin ^2(x) \, dx,x,\csc ^{-1}(a x)\right )}{a} \\ & = -\left (a^3 \text {Subst}\left (\int e^x \cos (x) \sin ^2(x) \, dx,x,\csc ^{-1}(a x)\right )\right ) \\ & = -\left (a^3 \text {Subst}\left (\int \left (\frac {1}{4} e^x \cos (x)-\frac {1}{4} e^x \cos (3 x)\right ) \, dx,x,\csc ^{-1}(a x)\right )\right ) \\ & = -\left (\frac {1}{4} a^3 \text {Subst}\left (\int e^x \cos (x) \, dx,x,\csc ^{-1}(a x)\right )\right )+\frac {1}{4} a^3 \text {Subst}\left (\int e^x \cos (3 x) \, dx,x,\csc ^{-1}(a x)\right ) \\ & = -\frac {1}{8} a^3 e^{\csc ^{-1}(a x)} \sqrt {1-\frac {1}{a^2 x^2}}-\frac {a^2 e^{\csc ^{-1}(a x)}}{8 x}+\frac {1}{40} a^3 e^{\csc ^{-1}(a x)} \cos \left (3 \csc ^{-1}(a x)\right )+\frac {3}{40} a^3 e^{\csc ^{-1}(a x)} \sin \left (3 \csc ^{-1}(a x)\right ) \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.64 \[ \int \frac {e^{\csc ^{-1}(a x)}}{x^4} \, dx=\frac {1}{40} a^3 e^{\csc ^{-1}(a x)} \left (-5 \sqrt {1-\frac {1}{a^2 x^2}}-\frac {5}{a x}+\cos \left (3 \csc ^{-1}(a x)\right )+3 \sin \left (3 \csc ^{-1}(a x)\right )\right ) \]
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\[\int \frac {{\mathrm e}^{\operatorname {arccsc}\left (a x \right )}}{x^{4}}d x\]
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none
Time = 0.30 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.49 \[ \int \frac {e^{\csc ^{-1}(a x)}}{x^4} \, dx=\frac {{\left (a^{2} x^{2} - {\left (a^{2} x^{2} + 1\right )} \sqrt {a^{2} x^{2} - 1} - 3\right )} e^{\left (\operatorname {arccsc}\left (a x\right )\right )}}{10 \, x^{3}} \]
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\[ \int \frac {e^{\csc ^{-1}(a x)}}{x^4} \, dx=\int \frac {e^{\operatorname {acsc}{\left (a x \right )}}}{x^{4}}\, dx \]
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\[ \int \frac {e^{\csc ^{-1}(a x)}}{x^4} \, dx=\int { \frac {e^{\left (\operatorname {arccsc}\left (a x\right )\right )}}{x^{4}} \,d x } \]
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\[ \int \frac {e^{\csc ^{-1}(a x)}}{x^4} \, dx=\int { \frac {e^{\left (\operatorname {arccsc}\left (a x\right )\right )}}{x^{4}} \,d x } \]
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Timed out. \[ \int \frac {e^{\csc ^{-1}(a x)}}{x^4} \, dx=\int \frac {{\mathrm {e}}^{\mathrm {asin}\left (\frac {1}{a\,x}\right )}}{x^4} \,d x \]
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