Optimal. Leaf size=50 \[ -4 x^{m+1} \, _2F_1(1,m+1;m+2;-i a x)+\frac {4 x^{m+1}}{1+i a x}+\frac {x^{m+1}}{m+1} \]
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Rubi [A] time = 0.04, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5062, 89, 80, 64} \[ -4 x^{m+1} \text {Hypergeometric2F1}(1,m+1,m+2,-i a x)+\frac {4 x^{m+1}}{1+i a x}+\frac {x^{m+1}}{m+1} \]
Antiderivative was successfully verified.
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Rule 64
Rule 80
Rule 89
Rule 5062
Rubi steps
\begin {align*} \int e^{-4 i \tan ^{-1}(a x)} x^m \, dx &=\int \frac {x^m (1-i a x)^2}{(1+i a x)^2} \, dx\\ &=\frac {4 x^{1+m}}{1+i a x}+\frac {\int \frac {x^m \left (-a^2 (3+4 m)+i a^3 x\right )}{1+i a x} \, dx}{a^2}\\ &=\frac {x^{1+m}}{1+m}+\frac {4 x^{1+m}}{1+i a x}-(4 (1+m)) \int \frac {x^m}{1+i a x} \, dx\\ &=\frac {x^{1+m}}{1+m}+\frac {4 x^{1+m}}{1+i a x}-4 x^{1+m} \, _2F_1(1,1+m;2+m;-i a x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 58, normalized size = 1.16 \[ \frac {x^{m+1} (-4 (m+1) (a x-i) \, _2F_1(1,m+1;m+2;-i a x)+a x-4 i m-5 i)}{(m+1) (a x-i)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{2} x^{2} + 2 i \, a x - 1\right )} x^{m}}{a^{2} x^{2} - 2 i \, a x - 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a^{2} x^{2} + 1\right )}^{2} x^{m}}{{\left (i \, a x + 1\right )}^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.56, size = 428, normalized size = 8.56 \[ -\frac {i \left (i a \right )^{-m} \left (\frac {x^{m} \left (i a \right )^{m} \left (6 a^{4} x^{4} m +6 i a^{3} x^{3} m +a^{2} x^{2} m^{4}+24 i a^{3} x^{3}+11 a^{2} x^{2} m^{3}-2 i a x \,m^{4}+46 a^{2} m^{2} x^{2}-21 i a x \,m^{3}+90 a^{2} m \,x^{2}-79 i a x \,m^{2}+72 a^{2} x^{2}-126 i a m x -m^{4}-72 i a x -10 m^{3}-35 m^{2}-50 m -24\right )}{\left (1+m \right ) m \left (i a x +1\right )^{3}}+x^{m} \left (i a \right )^{m} \left (m^{3}+9 m^{2}+26 m +24\right ) \Phi \left (-i a x , 1, m\right )\right )}{6 a}+\frac {i \left (i a \right )^{-m} \left (-\frac {x^{m} \left (i a \right )^{m} \left (-a^{2} m^{2} x^{2}-4 a^{2} m \,x^{2}+2 i a x \,m^{2}-6 a^{2} x^{2}+7 i a m x +6 i a x +m^{2}+3 m +2\right )}{\left (i a x +1\right )^{3}}+x^{m} \left (i a \right )^{m} m \left (m^{2}+3 m +2\right ) \Phi \left (-i a x , 1, m\right )\right )}{3 a}-\frac {i \left (i a \right )^{-m} \left (-\frac {x^{m} \left (i a \right )^{m} \left (-a^{2} m^{2} x^{2}+2 a^{2} m \,x^{2}+2 i a x \,m^{2}-5 i a m x +m^{2}-3 m +2\right )}{\left (i a x +1\right )^{3}}+x^{m} \left (i a \right )^{m} \left (m^{2}-3 m +2\right ) m \Phi \left (-i a x , 1, m\right )\right )}{6 a} \]
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 \[ \int \frac {{\left (a^{2} x^{2} + 1\right )}^{2} x^{m}}{{\left (i \, a x + 1\right )}^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^m\,{\left (a^2\,x^2+1\right )}^2}{{\left (1+a\,x\,1{}\mathrm {i}\right )}^4} \,d x \]
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 \[ \int \frac {x^{m} \left (a^{2} x^{2} + 1\right )^{2}}{\left (a x - i\right )^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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