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 = 0.04, antiderivative size = 79, normalized size of antiderivative = 1.00, 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 364
Rule 808
Rule 5060
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*}
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Mathematica [C] time = 0.04, 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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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {i \, \sqrt {a^{2} x^{2} + 1} x^{m}}{a x - i}, x\right ) \]
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 \[ \int \frac {\sqrt {a^{2} x^{2} + 1} x^{m}}{i \, a x + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.45, size = 0, normalized size = 0.00 \[ \int \frac {x^{m} \sqrt {a^{2} x^{2}+1}}{i a x +1}\, dx \]
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 \[ \int \frac {\sqrt {a^{2} x^{2} + 1} x^{m}}{i \, a x + 1}\,{d x} \]
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 \[ \int \frac {x^m\,\sqrt {a^2\,x^2+1}}{1+a\,x\,1{}\mathrm {i}} \,d x \]
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 \[ - i \int \frac {x^{m} \sqrt {a^{2} x^{2} + 1}}{a x - i}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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