Optimal. Leaf size=91 \[ -\frac{2^{1-\frac{i n}{2}} (-i a-i b x+1)^{1+\frac{i n}{2}} \, _2F_1\left (\frac{i n}{2}+1,\frac{i n}{2};\frac{i n}{2}+2;\frac{1}{2} (-i a-i b x+1)\right )}{b (-n+2 i)} \]
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Rubi [A] time = 0.013713, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {5093, 69} \[ -\frac{2^{1-\frac{i n}{2}} (-i a-i b x+1)^{1+\frac{i n}{2}} \, _2F_1\left (\frac{i n}{2}+1,\frac{i n}{2};\frac{i n}{2}+2;\frac{1}{2} (-i a-i b x+1)\right )}{b (-n+2 i)} \]
Antiderivative was successfully verified.
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Rule 5093
Rule 69
Rubi steps
\begin{align*} \int e^{n \tan ^{-1}(a+b x)} \, dx &=\int (1-i a-i b x)^{\frac{i n}{2}} (1+i a+i b x)^{-\frac{i n}{2}} \, dx\\ &=-\frac{2^{1-\frac{i n}{2}} (1-i a-i b x)^{1+\frac{i n}{2}} \, _2F_1\left (1+\frac{i n}{2},\frac{i n}{2};2+\frac{i n}{2};\frac{1}{2} (1-i a-i b x)\right )}{b (2 i-n)}\\ \end{align*}
Mathematica [A] time = 0.0283222, size = 60, normalized size = 0.66 \[ \frac{4 e^{(n+2 i) \tan ^{-1}(a+b x)} \, _2F_1\left (2,1-\frac{i n}{2};2-\frac{i n}{2};-e^{2 i \tan ^{-1}(a+b x)}\right )}{b (n+2 i)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.055, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{n\arctan \left ( bx+a \right ) }}\, 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 e^{\left (n \arctan \left (b x + a\right )\right )}\,{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 (e^{\left (n \arctan \left (b x + a\right )\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int e^{n \operatorname{atan}{\left (a + b x \right )}}\, dx \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 e^{\left (n \arctan \left (b x + a\right )\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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