Optimal. Leaf size=122 \[ \frac{i (1-i a x)^{\frac{i n}{2}} (1+i a x)^{-\frac{i n}{2}}}{a^2 c n}-\frac{i 2^{1-\frac{i n}{2}} (1-i a x)^{\frac{i n}{2}} \, _2F_1\left (\frac{i n}{2},\frac{i n}{2};\frac{i n}{2}+1;\frac{1}{2} (1-i a x)\right )}{a^2 c n} \]
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Rubi [A] time = 0.0779773, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {5082, 79, 69} \[ \frac{i (1-i a x)^{\frac{i n}{2}} (1+i a x)^{-\frac{i n}{2}}}{a^2 c n}-\frac{i 2^{1-\frac{i n}{2}} (1-i a x)^{\frac{i n}{2}} \, _2F_1\left (\frac{i n}{2},\frac{i n}{2};\frac{i n}{2}+1;\frac{1}{2} (1-i a x)\right )}{a^2 c n} \]
Antiderivative was successfully verified.
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Rule 5082
Rule 79
Rule 69
Rubi steps
\begin{align*} \int \frac{e^{n \tan ^{-1}(a x)} x}{c+a^2 c x^2} \, dx &=\frac{\int x (1-i a x)^{-1+\frac{i n}{2}} (1+i a x)^{-1-\frac{i n}{2}} \, dx}{c}\\ &=\frac{i (1-i a x)^{\frac{i n}{2}} (1+i a x)^{-\frac{i n}{2}}}{a^2 c n}-\frac{i \int (1-i a x)^{-1+\frac{i n}{2}} (1+i a x)^{-\frac{i n}{2}} \, dx}{a c}\\ &=\frac{i (1-i a x)^{\frac{i n}{2}} (1+i a x)^{-\frac{i n}{2}}}{a^2 c n}-\frac{i 2^{1-\frac{i n}{2}} (1-i a x)^{\frac{i n}{2}} \, _2F_1\left (\frac{i n}{2},\frac{i n}{2};1+\frac{i n}{2};\frac{1}{2} (1-i a x)\right )}{a^2 c n}\\ \end{align*}
Mathematica [A] time = 0.0597226, size = 109, normalized size = 0.89 \[ \frac{i (1-i a x)^{\frac{i n}{2}} (2+2 i a x)^{-\frac{i n}{2}} \left (2^{\frac{i n}{2}}-2 (1+i a x)^{\frac{i n}{2}} \, _2F_1\left (\frac{i n}{2},\frac{i n}{2};\frac{i n}{2}+1;\frac{1}{2} (1-i a x)\right )\right )}{a^2 c n} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.318, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{n\arctan \left ( ax \right ) }}x}{{a}^{2}c{x}^{2}+c}}\, 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 \frac{x e^{\left (n \arctan \left (a x\right )\right )}}{a^{2} c x^{2} + c}\,{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 (\frac{x e^{\left (n \arctan \left (a x\right )\right )}}{a^{2} c x^{2} + c}, 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*} \frac{\int \frac{x e^{n \operatorname{atan}{\left (a x \right )}}}{a^{2} x^{2} + 1}\, dx}{c} \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 \frac{x e^{\left (n \arctan \left (a x\right )\right )}}{a^{2} c x^{2} + c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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