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 = 0.12, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, 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 12
Rule 2251
Rule 4471
Rule 5267
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*}
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Mathematica [A] time = 0.45, 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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fricas [F] time = 0.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{2} e^{\left (\operatorname {arccsc}\left (a x\right )\right )}, x\right ) \]
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 \[ \int x^{2} e^{\left (\operatorname {arccsc}\left (a x\right )\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{\mathrm {arccsc}\left (a x \right )} x^{2}\, dx \]
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 \[ \int x^{2} e^{\left (\operatorname {arccsc}\left (a x\right )\right )}\,{d x} \]
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 \[ \int x^2\,{\mathrm {e}}^{\mathrm {asin}\left (\frac {1}{a\,x}\right )} \,d x \]
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 \[ \int x^{2} e^{\operatorname {acsc}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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