Optimal. Leaf size=41 \[ \frac{1}{5} a^2 e^{\csc ^{-1}(a x)} \cos \left (2 \csc ^{-1}(a x)\right )-\frac{1}{10} a^2 e^{\csc ^{-1}(a x)} \sin \left (2 \csc ^{-1}(a x)\right ) \]
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Rubi [A] time = 0.0431414, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {5267, 12, 4469, 4432} \[ \frac{1}{5} a^2 e^{\csc ^{-1}(a x)} \cos \left (2 \csc ^{-1}(a x)\right )-\frac{1}{10} a^2 e^{\csc ^{-1}(a x)} \sin \left (2 \csc ^{-1}(a x)\right ) \]
Antiderivative was successfully verified.
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Rule 5267
Rule 12
Rule 4469
Rule 4432
Rubi steps
\begin{align*} \int \frac{e^{\csc ^{-1}(a x)}}{x^3} \, dx &=-\frac{\operatorname{Subst}\left (\int a^3 e^x \cos (x) \sin (x) \, dx,x,\csc ^{-1}(a x)\right )}{a}\\ &=-\left (a^2 \operatorname{Subst}\left (\int e^x \cos (x) \sin (x) \, dx,x,\csc ^{-1}(a x)\right )\right )\\ &=-\left (a^2 \operatorname{Subst}\left (\int \frac{1}{2} e^x \sin (2 x) \, dx,x,\csc ^{-1}(a x)\right )\right )\\ &=-\left (\frac{1}{2} a^2 \operatorname{Subst}\left (\int e^x \sin (2 x) \, dx,x,\csc ^{-1}(a x)\right )\right )\\ &=\frac{1}{5} a^2 e^{\csc ^{-1}(a x)} \cos \left (2 \csc ^{-1}(a x)\right )-\frac{1}{10} a^2 e^{\csc ^{-1}(a x)} \sin \left (2 \csc ^{-1}(a x)\right )\\ \end{align*}
Mathematica [A] time = 0.0459281, size = 30, normalized size = 0.73 \[ -\frac{1}{10} a^2 e^{\csc ^{-1}(a x)} \left (\sin \left (2 \csc ^{-1}(a x)\right )-2 \cos \left (2 \csc ^{-1}(a x)\right )\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.18, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{{\rm arccsc} \left (ax\right )}}}{{x}^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\left (\operatorname{arccsc}\left (a x\right )\right )}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 3.58809, size = 80, normalized size = 1.95 \begin{align*} \frac{{\left (a^{2} x^{2} - \sqrt{a^{2} x^{2} - 1} - 2\right )} e^{\left (\operatorname{arccsc}\left (a x\right )\right )}}{5 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\operatorname{acsc}{\left (a x \right )}}}{x^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\left (\operatorname{arccsc}\left (a x\right )\right )}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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