Optimal. Leaf size=39 \[ -\frac {x^{m+1}}{m+1}+\frac {2 x^{m+1} \, _2F_1(1,m+1;m+2;i a x)}{m+1} \]
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Rubi [A] time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {5062, 80, 64} \[ -\frac {x^{m+1}}{m+1}+\frac {2 x^{m+1} \text {Hypergeometric2F1}(1,m+1,m+2,i a x)}{m+1} \]
Antiderivative was successfully verified.
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Rule 64
Rule 80
Rule 5062
Rubi steps
\begin {align*} \int e^{2 i \tan ^{-1}(a x)} x^m \, dx &=\int \frac {x^m (1+i a x)}{1-i a x} \, dx\\ &=-\frac {x^{1+m}}{1+m}+2 \int \frac {x^m}{1-i a x} \, dx\\ &=-\frac {x^{1+m}}{1+m}+\frac {2 x^{1+m} \, _2F_1(1,1+m;2+m;i a x)}{1+m}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 29, normalized size = 0.74 \[ \frac {x^{m+1} (-1+2 \, _2F_1(1,m+1;m+2;i a x))}{m+1} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (a x - i\right )} x^{m}}{a x + i}, x\right ) \]
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 \[ \int \frac {{\left (i \, a x + 1\right )}^{2} x^{m}}{a^{2} x^{2} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.51, size = 175, normalized size = 4.49 \[ \frac {x^{1+m} \left (\frac {1}{2}+\frac {m}{2}\right ) \Phi \left (-a^{2} x^{2}, 1, \frac {1}{2}+\frac {m}{2}\right )}{1+m}+\frac {i \left (a^{2}\right )^{-\frac {m}{2}} \left (\frac {2 x^{m} \left (a^{2}\right )^{\frac {m}{2}}}{m}+\frac {x^{m} \left (a^{2}\right )^{\frac {m}{2}} \left (-m -2\right ) \Phi \left (-a^{2} x^{2}, 1, \frac {m}{2}\right )}{2+m}\right )}{a}-\frac {\left (a^{2}\right )^{-\frac {1}{2}-\frac {m}{2}} \left (\frac {2 x^{1+m} \left (a^{2}\right )^{\frac {3}{2}+\frac {m}{2}}}{\left (1+m \right ) a^{2}}+\frac {x^{1+m} \left (a^{2}\right )^{\frac {3}{2}+\frac {m}{2}} \left (-3-m \right ) \Phi \left (-a^{2} x^{2}, 1, \frac {1}{2}+\frac {m}{2}\right )}{\left (3+m \right ) a^{2}}\right )}{2} \]
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 \[ \int \frac {{\left (i \, a x + 1\right )}^{2} x^{m}}{a^{2} x^{2} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {x^m\,{\left (1+a\,x\,1{}\mathrm {i}\right )}^2}{a^2\,x^2+1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 3.40, size = 136, normalized size = 3.49 \[ \frac {i a m x^{2} x^{m} \Phi \left (a x e^{\frac {5 i \pi }{2}}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{\Gamma \left (m + 3\right )} + \frac {2 i a x^{2} x^{m} \Phi \left (a x e^{\frac {5 i \pi }{2}}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{\Gamma \left (m + 3\right )} + \frac {m x x^{m} \Phi \left (a x e^{\frac {5 i \pi }{2}}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{\Gamma \left (m + 2\right )} + \frac {x x^{m} \Phi \left (a x e^{\frac {5 i \pi }{2}}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{\Gamma \left (m + 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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