Optimal. Leaf size=120 \[ -\frac{3\ 2^{\frac{1}{3}-\frac{i n}{2}} \left (a^2 x^2+1\right )^{2/3} (1-i a x)^{\frac{1}{6} (2+3 i n)} \, _2F_1\left (\frac{1}{6} (3 i n+2),\frac{1}{6} (3 i n+4);\frac{1}{6} (3 i n+8);\frac{1}{2} (1-i a x)\right )}{a (-3 n+2 i) \left (a^2 c x^2+c\right )^{2/3}} \]
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Rubi [A] time = 0.110613, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {5076, 5073, 69} \[ -\frac{3\ 2^{\frac{1}{3}-\frac{i n}{2}} \left (a^2 x^2+1\right )^{2/3} (1-i a x)^{\frac{1}{6} (2+3 i n)} \, _2F_1\left (\frac{1}{6} (3 i n+2),\frac{1}{6} (3 i n+4);\frac{1}{6} (3 i n+8);\frac{1}{2} (1-i a x)\right )}{a (-3 n+2 i) \left (a^2 c x^2+c\right )^{2/3}} \]
Antiderivative was successfully verified.
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Rule 5076
Rule 5073
Rule 69
Rubi steps
\begin{align*} \int \frac{e^{n \tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{2/3}} \, dx &=\frac{\left (1+a^2 x^2\right )^{2/3} \int \frac{e^{n \tan ^{-1}(a x)}}{\left (1+a^2 x^2\right )^{2/3}} \, dx}{\left (c+a^2 c x^2\right )^{2/3}}\\ &=\frac{\left (1+a^2 x^2\right )^{2/3} \int (1-i a x)^{-\frac{2}{3}+\frac{i n}{2}} (1+i a x)^{-\frac{2}{3}-\frac{i n}{2}} \, dx}{\left (c+a^2 c x^2\right )^{2/3}}\\ &=-\frac{3\ 2^{\frac{1}{3}-\frac{i n}{2}} (1-i a x)^{\frac{1}{6} (2+3 i n)} \left (1+a^2 x^2\right )^{2/3} \, _2F_1\left (\frac{1}{6} (2+3 i n),\frac{1}{6} (4+3 i n);\frac{1}{6} (8+3 i n);\frac{1}{2} (1-i a x)\right )}{a (2 i-3 n) \left (c+a^2 c x^2\right )^{2/3}}\\ \end{align*}
Mathematica [A] time = 0.0433382, size = 120, normalized size = 1. \[ \frac{3\ 2^{\frac{1}{3}-\frac{i n}{2}} \left (a^2 x^2+1\right )^{2/3} (1-i a x)^{\frac{1}{3}+\frac{i n}{2}} \, _2F_1\left (\frac{i n}{2}+\frac{1}{3},\frac{i n}{2}+\frac{2}{3};\frac{i n}{2}+\frac{4}{3};\frac{1}{2}-\frac{i a x}{2}\right )}{a (3 n-2 i) \left (a^2 c x^2+c\right )^{2/3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.341, size = 0, normalized size = 0. \begin{align*} \int{{{\rm e}^{n\arctan \left ( ax \right ) }} \left ({a}^{2}c{x}^{2}+c \right ) ^{-{\frac{2}{3}}}}\, 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{e^{\left (n \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{\frac{2}{3}}}\,{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{e^{\left (n \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{\frac{2}{3}}}, 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 \frac{e^{n \operatorname{atan}{\left (a x \right )}}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac{2}{3}}}\, 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 \frac{e^{\left (n \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{\frac{2}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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