Optimal. Leaf size=99 \[ \frac{\left (\frac{24}{5}+\frac{8 i}{5}\right ) e^{(1+3 i) \sec ^{-1}(a x)} \, _2F_1\left (\frac{3}{2}-\frac{i}{2},4;\frac{5}{2}-\frac{i}{2};-e^{2 i \sec ^{-1}(a x)}\right )}{a^3}-\frac{\left (\frac{12}{5}+\frac{4 i}{5}\right ) e^{(1+3 i) \sec ^{-1}(a x)} \, _2F_1\left (\frac{3}{2}-\frac{i}{2},3;\frac{5}{2}-\frac{i}{2};-e^{2 i \sec ^{-1}(a x)}\right )}{a^3} \]
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Rubi [A] time = 0.117391, antiderivative size = 99, 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 = {5266, 12, 4471, 2251} \[ \frac{\left (\frac{24}{5}+\frac{8 i}{5}\right ) e^{(1+3 i) \sec ^{-1}(a x)} \, _2F_1\left (\frac{3}{2}-\frac{i}{2},4;\frac{5}{2}-\frac{i}{2};-e^{2 i \sec ^{-1}(a x)}\right )}{a^3}-\frac{\left (\frac{12}{5}+\frac{4 i}{5}\right ) e^{(1+3 i) \sec ^{-1}(a x)} \, _2F_1\left (\frac{3}{2}-\frac{i}{2},3;\frac{5}{2}-\frac{i}{2};-e^{2 i \sec ^{-1}(a x)}\right )}{a^3} \]
Antiderivative was successfully verified.
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Rule 5266
Rule 12
Rule 4471
Rule 2251
Rubi steps
\begin{align*} \int e^{\sec ^{-1}(a x)} x^2 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{e^x \sec ^3(x) \tan (x)}{a^2} \, dx,x,\sec ^{-1}(a x)\right )}{a}\\ &=\frac{\operatorname{Subst}\left (\int e^x \sec ^3(x) \tan (x) \, dx,x,\sec ^{-1}(a x)\right )}{a^3}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{16 i e^{(1+3 i) x}}{\left (1+e^{2 i x}\right )^4}-\frac{8 i e^{(1+3 i) x}}{\left (1+e^{2 i x}\right )^3}\right ) \, dx,x,\sec ^{-1}(a x)\right )}{a^3}\\ &=-\frac{(8 i) \operatorname{Subst}\left (\int \frac{e^{(1+3 i) x}}{\left (1+e^{2 i x}\right )^3} \, dx,x,\sec ^{-1}(a x)\right )}{a^3}+\frac{(16 i) \operatorname{Subst}\left (\int \frac{e^{(1+3 i) x}}{\left (1+e^{2 i x}\right )^4} \, dx,x,\sec ^{-1}(a x)\right )}{a^3}\\ &=-\frac{\left (\frac{12}{5}+\frac{4 i}{5}\right ) e^{(1+3 i) \sec ^{-1}(a x)} \, _2F_1\left (\frac{3}{2}-\frac{i}{2},3;\frac{5}{2}-\frac{i}{2};-e^{2 i \sec ^{-1}(a x)}\right )}{a^3}+\frac{\left (\frac{24}{5}+\frac{8 i}{5}\right ) e^{(1+3 i) \sec ^{-1}(a x)} \, _2F_1\left (\frac{3}{2}-\frac{i}{2},4;\frac{5}{2}-\frac{i}{2};-e^{2 i \sec ^{-1}(a x)}\right )}{a^3}\\ \end{align*}
Mathematica [A] time = 0.301252, size = 95, normalized size = 0.96 \[ \frac{e^{\sec ^{-1}(a x)} \left (a^4 x^4 \left (\cos \left (2 \sec ^{-1}(a x)\right )-\sin \left (2 \sec ^{-1}(a x)\right )+5\right )-(4+4 i) \left (a x \sqrt{1-\frac{1}{a^2 x^2}}-i\right ) \, _2F_1\left (\frac{1}{2}-\frac{i}{2},1;\frac{3}{2}-\frac{i}{2};-e^{2 i \sec ^{-1}(a x)}\right )\right )}{12 a^4 x} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.181, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{{\rm arcsec} \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{arcsec}\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{arcsec}\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{asec}{\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{arcsec}\left (a x\right )\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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