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 = 0.57, 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 \[ \int \frac {{\left (i \, a x + 1\right )}^{4} x^{m}}{{\left (a^{2} x^{2} + 1\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.57, size = 417, normalized size = 8.34 \[ \frac {\left (a^{2}\right )^{-\frac {1}{2}-\frac {m}{2}} \left (\frac {2 x^{1+m} \left (a^{2}\right )^{\frac {1}{2}+\frac {m}{2}}}{2 a^{2} x^{2}+2}+\frac {2 x^{1+m} \left (a^{2}\right )^{\frac {1}{2}+\frac {m}{2}} \left (-\frac {m^{2}}{4}+\frac {1}{4}\right ) \Phi \left (-a^{2} x^{2}, 1, \frac {1}{2}+\frac {m}{2}\right )}{1+m}\right )}{2}+\frac {2 i \left (a^{2}\right )^{-\frac {m}{2}} \left (\frac {x^{m} \left (a^{2}\right )^{\frac {m}{2}} \left (-m -2\right )}{\left (2+m \right ) \left (a^{2} x^{2}+1\right )}+\frac {x^{m} \left (a^{2}\right )^{\frac {m}{2}} m \Phi \left (-a^{2} x^{2}, 1, \frac {m}{2}\right )}{2}\right )}{a}-3 \left (a^{2}\right )^{-\frac {1}{2}-\frac {m}{2}} \left (\frac {x^{1+m} \left (a^{2}\right )^{\frac {3}{2}+\frac {m}{2}} \left (-3-m \right )}{\left (3+m \right ) a^{2} \left (a^{2} x^{2}+1\right )}+\frac {x^{1+m} \left (a^{2}\right )^{\frac {3}{2}+\frac {m}{2}} \left (1+m \right ) \Phi \left (-a^{2} x^{2}, 1, \frac {1}{2}+\frac {m}{2}\right )}{2 a^{2}}\right )-\frac {2 i \left (a^{2}\right )^{-\frac {m}{2}} \left (\frac {x^{m} \left (a^{2}\right )^{\frac {m}{2}} \left (2 a^{2} x^{2}+m +2\right )}{\left (a^{2} x^{2}+1\right ) m}-\frac {x^{m} \left (a^{2}\right )^{\frac {m}{2}} \left (2+m \right ) \Phi \left (-a^{2} x^{2}, 1, \frac {m}{2}\right )}{2}\right )}{a}+\frac {\left (a^{2}\right )^{-\frac {1}{2}-\frac {m}{2}} \left (\frac {x^{1+m} \left (a^{2}\right )^{\frac {5}{2}+\frac {m}{2}} \left (2 a^{2} x^{2}+m +3\right )}{\left (a^{2} x^{2}+1\right ) a^{4} \left (1+m \right )}-\frac {x^{1+m} \left (a^{2}\right )^{\frac {5}{2}+\frac {m}{2}} \left (3+m \right ) \Phi \left (-a^{2} x^{2}, 1, \frac {1}{2}+\frac {m}{2}\right )}{2 a^{4}}\right )}{2} \]
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 \[ \int \frac {{\left (i \, a x + 1\right )}^{4} x^{m}}{{\left (a^{2} x^{2} + 1\right )}^{2}}\,{d x} \]
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 \[ \int \frac {x^m\,{\left (1+a\,x\,1{}\mathrm {i}\right )}^4}{{\left (a^2\,x^2+1\right )}^2} \,d x \]
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 \[ \int \frac {x^{m} \left (a x - i\right )^{4}}{\left (a^{2} x^{2} + 1\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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