Optimal. Leaf size=95 \[ \frac{\left (\frac{4}{5}-\frac{12 i}{5}\right ) e^{(1+3 i) \csc ^{-1}(a x)} \, _2F_1\left (\frac{3}{2}-\frac{i}{2},3;\frac{5}{2}-\frac{i}{2};e^{2 i \csc ^{-1}(a x)}\right )}{a^3}-\frac{\left (\frac{8}{5}-\frac{24 i}{5}\right ) e^{(1+3 i) \csc ^{-1}(a x)} \, _2F_1\left (\frac{3}{2}-\frac{i}{2},4;\frac{5}{2}-\frac{i}{2};e^{2 i \csc ^{-1}(a x)}\right )}{a^3} \]
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Rubi [A] time = 0.120934, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {5267, 12, 4471, 2251} \[ \frac{\left (\frac{4}{5}-\frac{12 i}{5}\right ) e^{(1+3 i) \csc ^{-1}(a x)} \, _2F_1\left (\frac{3}{2}-\frac{i}{2},3;\frac{5}{2}-\frac{i}{2};e^{2 i \csc ^{-1}(a x)}\right )}{a^3}-\frac{\left (\frac{8}{5}-\frac{24 i}{5}\right ) e^{(1+3 i) \csc ^{-1}(a x)} \, _2F_1\left (\frac{3}{2}-\frac{i}{2},4;\frac{5}{2}-\frac{i}{2};e^{2 i \csc ^{-1}(a x)}\right )}{a^3} \]
Antiderivative was successfully verified.
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Rule 5267
Rule 12
Rule 4471
Rule 2251
Rubi steps
\begin{align*} \int e^{\csc ^{-1}(a x)} x^2 \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{e^x \cot (x) \csc ^3(x)}{a^2} \, dx,x,\csc ^{-1}(a x)\right )}{a}\\ &=-\frac{\operatorname{Subst}\left (\int e^x \cot (x) \csc ^3(x) \, dx,x,\csc ^{-1}(a x)\right )}{a^3}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{16 e^{(1+3 i) x}}{\left (-1+e^{2 i x}\right )^4}+\frac{8 e^{(1+3 i) x}}{\left (-1+e^{2 i x}\right )^3}\right ) \, dx,x,\csc ^{-1}(a x)\right )}{a^3}\\ &=-\frac{8 \operatorname{Subst}\left (\int \frac{e^{(1+3 i) x}}{\left (-1+e^{2 i x}\right )^3} \, dx,x,\csc ^{-1}(a x)\right )}{a^3}-\frac{16 \operatorname{Subst}\left (\int \frac{e^{(1+3 i) x}}{\left (-1+e^{2 i x}\right )^4} \, dx,x,\csc ^{-1}(a x)\right )}{a^3}\\ &=\frac{\left (\frac{4}{5}-\frac{12 i}{5}\right ) e^{(1+3 i) \csc ^{-1}(a x)} \, _2F_1\left (\frac{3}{2}-\frac{i}{2},3;\frac{5}{2}-\frac{i}{2};e^{2 i \csc ^{-1}(a x)}\right )}{a^3}-\frac{\left (\frac{8}{5}-\frac{24 i}{5}\right ) e^{(1+3 i) \csc ^{-1}(a x)} \, _2F_1\left (\frac{3}{2}-\frac{i}{2},4;\frac{5}{2}-\frac{i}{2};e^{2 i \csc ^{-1}(a x)}\right )}{a^3}\\ \end{align*}
Mathematica [A] time = 0.451027, size = 79, normalized size = 0.83 \[ \frac{e^{\csc ^{-1}(a x)} \left (a^3 x^3 \left (-\cos \left (2 \csc ^{-1}(a x)\right )+\sin \left (2 \csc ^{-1}(a x)\right )+5\right )+(4+4 i) e^{i \csc ^{-1}(a x)} \, _2F_1\left (\frac{1}{2}-\frac{i}{2},1;\frac{3}{2}-\frac{i}{2};e^{2 i \csc ^{-1}(a x)}\right )\right )}{12 a^3} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.198, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{{\rm arccsc} \left (ax\right )}}{x}^{2}\, 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 x^{2} e^{\left (\operatorname{arccsc}\left (a x\right )\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{2} e^{\left (\operatorname{arccsc}\left (a x\right )\right )}, x\right ) \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 x^{2} e^{\operatorname{acsc}{\left (a x \right )}}\, 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 x^{2} e^{\left (\operatorname{arccsc}\left (a x\right )\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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