Optimal. Leaf size=79 \[ \frac{x^{m+1} \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};-a^2 x^2\right )}{m+1}-\frac{i a x^{m+2} \, _2F_1\left (\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};-a^2 x^2\right )}{m+2} \]
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Rubi [A] time = 0.0403945, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {5060, 808, 364} \[ \frac{x^{m+1} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+1}{2},\frac{m+3}{2},-a^2 x^2\right )}{m+1}-\frac{i a x^{m+2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+2}{2},\frac{m+4}{2},-a^2 x^2\right )}{m+2} \]
Antiderivative was successfully verified.
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Rule 5060
Rule 808
Rule 364
Rubi steps
\begin{align*} \int e^{-i \tan ^{-1}(a x)} x^m \, dx &=\int \frac{x^m (1-i a x)}{\sqrt{1+a^2 x^2}} \, dx\\ &=-\left ((i a) \int \frac{x^{1+m}}{\sqrt{1+a^2 x^2}} \, dx\right )+\int \frac{x^m}{\sqrt{1+a^2 x^2}} \, dx\\ &=\frac{x^{1+m} \, _2F_1\left (\frac{1}{2},\frac{1+m}{2};\frac{3+m}{2};-a^2 x^2\right )}{1+m}-\frac{i a x^{2+m} \, _2F_1\left (\frac{1}{2},\frac{2+m}{2};\frac{4+m}{2};-a^2 x^2\right )}{2+m}\\ \end{align*}
Mathematica [C] time = 0.0409323, size = 85, normalized size = 1.08 \[ -\frac{i \sqrt{1+i a x} \sqrt{a x+i} x^{m+1} F_1\left (m+1;\frac{1}{2},-\frac{1}{2};m+2;-i a x,i a x\right )}{(m+1) \sqrt{1-i a x} \sqrt{a x-i}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.414, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{m}}{1+iax}\sqrt{{a}^{2}{x}^{2}+1}}\, 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{\sqrt{a^{2} x^{2} + 1} x^{m}}{i \, a x + 1}\,{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 (-\frac{i \, \sqrt{a^{2} x^{2} + 1} x^{m}}{a x - i}, 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 \frac{x^{m} \sqrt{a^{2} x^{2} + 1}}{i a x + 1}\, 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{\sqrt{a^{2} x^{2} + 1} x^{m}}{i \, a x + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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