Optimal. Leaf size=159 \[ -\frac{3 x^{m+1} \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};-a^2 x^2\right )}{m+1}+\frac{4 x^{m+1} \, _2F_1\left (\frac{3}{2},\frac{m+1}{2};\frac{m+3}{2};-a^2 x^2\right )}{m+1}+\frac{i a x^{m+2} \, _2F_1\left (\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};-a^2 x^2\right )}{m+2}-\frac{4 i a x^{m+2} \, _2F_1\left (\frac{3}{2},\frac{m+2}{2};\frac{m+4}{2};-a^2 x^2\right )}{m+2} \]
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Rubi [A] time = 0.704954, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {5060, 6742, 364, 850, 808} \[ -\frac{3 x^{m+1} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+1}{2},\frac{m+3}{2},-a^2 x^2\right )}{m+1}+\frac{4 x^{m+1} \text{Hypergeometric2F1}\left (\frac{3}{2},\frac{m+1}{2},\frac{m+3}{2},-a^2 x^2\right )}{m+1}+\frac{i a x^{m+2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+2}{2},\frac{m+4}{2},-a^2 x^2\right )}{m+2}-\frac{4 i a x^{m+2} \text{Hypergeometric2F1}\left (\frac{3}{2},\frac{m+2}{2},\frac{m+4}{2},-a^2 x^2\right )}{m+2} \]
Antiderivative was successfully verified.
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Rule 5060
Rule 6742
Rule 364
Rule 850
Rule 808
Rubi steps
\begin{align*} \int e^{-3 i \tan ^{-1}(a x)} x^m \, dx &=\int \frac{x^m (1-i a x)^2}{(1+i a x) \sqrt{1+a^2 x^2}} \, dx\\ &=\int \left (-\frac{3 x^m}{\sqrt{1+a^2 x^2}}+\frac{i a x^{1+m}}{\sqrt{1+a^2 x^2}}+\frac{4 x^m}{(1+i a x) \sqrt{1+a^2 x^2}}\right ) \, dx\\ &=-\left (3 \int \frac{x^m}{\sqrt{1+a^2 x^2}} \, dx\right )+4 \int \frac{x^m}{(1+i a x) \sqrt{1+a^2 x^2}} \, dx+(i a) \int \frac{x^{1+m}}{\sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{3 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}+4 \int \frac{x^m (1-i a x)}{\left (1+a^2 x^2\right )^{3/2}} \, dx\\ &=-\frac{3 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}+4 \int \frac{x^m}{\left (1+a^2 x^2\right )^{3/2}} \, dx-(4 i a) \int \frac{x^{1+m}}{\left (1+a^2 x^2\right )^{3/2}} \, dx\\ &=-\frac{3 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}+\frac{4 x^{1+m} \, _2F_1\left (\frac{3}{2},\frac{1+m}{2};\frac{3+m}{2};-a^2 x^2\right )}{1+m}-\frac{4 i a x^{2+m} \, _2F_1\left (\frac{3}{2},\frac{2+m}{2};\frac{4+m}{2};-a^2 x^2\right )}{2+m}\\ \end{align*}
Mathematica [C] time = 0.0713403, size = 113, normalized size = 0.71 \[ \frac{i \sqrt{1+i a x} \sqrt{a x+i} x^{m+1} \left (F_1\left (m+1;\frac{1}{2},-\frac{1}{2};m+2;-i a x,i a x\right )-2 F_1\left (m+1;\frac{3}{2},-\frac{1}{2};m+2;-i a x,i a x\right )\right )}{(m+1) \sqrt{1-i a x} \sqrt{a x-i}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.41, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{m}}{ \left ( 1+iax \right ) ^{3}} \left ({a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{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 \frac{{\left (a^{2} x^{2} + 1\right )}^{\frac{3}{2}} x^{m}}{{\left (i \, a x + 1\right )}^{3}}\,{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{\sqrt{a^{2} x^{2} + 1}{\left (i \, a x - 1\right )} x^{m}}{a^{2} x^{2} - 2 i \, a x - 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{{\left (a^{2} x^{2} + 1\right )}^{\frac{3}{2}} x^{m}}{{\left (i \, a x + 1\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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