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 = 0.70, antiderivative size = 159, normalized size of antiderivative = 1.00, 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 364
Rule 808
Rule 850
Rule 5060
Rule 6742
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*}
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Mathematica [C] time = 0.08, 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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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a^{2} x^{2} + 1} {\left (i \, a x - 1\right )} x^{m}}{a^{2} x^{2} - 2 i \, a x - 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.48, size = 0, normalized size = 0.00 \[ \int \frac {x^{m} \left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}{\left (i a x +1\right )^{3}}\, dx \]
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 \[ \int \frac {{\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{m}}{{\left (i \, a x + 1\right )}^{3}}\,{d x} \]
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 \[ \int \frac {x^m\,{\left (a^2\,x^2+1\right )}^{3/2}}{{\left (1+a\,x\,1{}\mathrm {i}\right )}^3} \,d x \]
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 \[ i \left (\int \frac {x^{m} \sqrt {a^{2} x^{2} + 1}}{a^{3} x^{3} - 3 i a^{2} x^{2} - 3 a x + i}\, dx + \int \frac {a^{2} x^{2} x^{m} \sqrt {a^{2} x^{2} + 1}}{a^{3} x^{3} - 3 i a^{2} x^{2} - 3 a x + i}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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