Integrand size = 23, antiderivative size = 120 \[ \int \frac {e^{n \arctan (a x)}}{\left (c+a^2 c x^2\right )^{2/3}} \, dx=-\frac {3\ 2^{\frac {1}{3}-\frac {i n}{2}} (1-i a x)^{\frac {1}{6} (2+3 i n)} \left (1+a^2 x^2\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{6} (2+3 i n),\frac {1}{6} (4+3 i n),\frac {1}{6} (8+3 i n),\frac {1}{2} (1-i a x)\right )}{a (2 i-3 n) \left (c+a^2 c x^2\right )^{2/3}} \]
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Time = 0.08 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {5184, 5181, 71} \[ \int \frac {e^{n \arctan (a x)}}{\left (c+a^2 c x^2\right )^{2/3}} \, dx=-\frac {3\ 2^{\frac {1}{3}-\frac {i n}{2}} \left (a^2 x^2+1\right )^{2/3} (1-i a x)^{\frac {1}{6} (2+3 i n)} \operatorname {Hypergeometric2F1}\left (\frac {1}{6} (3 i n+2),\frac {1}{6} (3 i n+4),\frac {1}{6} (3 i n+8),\frac {1}{2} (1-i a x)\right )}{a (-3 n+2 i) \left (a^2 c x^2+c\right )^{2/3}} \]
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Rule 71
Rule 5181
Rule 5184
Rubi steps \begin{align*} \text {integral}& = \frac {\left (1+a^2 x^2\right )^{2/3} \int \frac {e^{n \arctan (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} \operatorname {Hypergeometric2F1}\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*}
Time = 0.04 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00 \[ \int \frac {e^{n \arctan (a x)}}{\left (c+a^2 c x^2\right )^{2/3}} \, dx=\frac {3\ 2^{\frac {1}{3}-\frac {i n}{2}} (1-i a x)^{\frac {1}{3}+\frac {i n}{2}} \left (1+a^2 x^2\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3}+\frac {i n}{2},\frac {2}{3}+\frac {i n}{2},\frac {4}{3}+\frac {i n}{2},\frac {1}{2}-\frac {i a x}{2}\right )}{a (-2 i+3 n) \left (c+a^2 c x^2\right )^{2/3}} \]
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\[\int \frac {{\mathrm e}^{n \arctan \left (a x \right )}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {2}{3}}}d x\]
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\[ \int \frac {e^{n \arctan (a x)}}{\left (c+a^2 c x^2\right )^{2/3}} \, dx=\int { \frac {e^{\left (n \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {2}{3}}} \,d x } \]
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\[ \int \frac {e^{n \arctan (a x)}}{\left (c+a^2 c x^2\right )^{2/3}} \, dx=\int \frac {e^{n \operatorname {atan}{\left (a x \right )}}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {2}{3}}}\, dx \]
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\[ \int \frac {e^{n \arctan (a x)}}{\left (c+a^2 c x^2\right )^{2/3}} \, dx=\int { \frac {e^{\left (n \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {2}{3}}} \,d x } \]
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\[ \int \frac {e^{n \arctan (a x)}}{\left (c+a^2 c x^2\right )^{2/3}} \, dx=\int { \frac {e^{\left (n \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {2}{3}}} \,d x } \]
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Timed out. \[ \int \frac {e^{n \arctan (a x)}}{\left (c+a^2 c x^2\right )^{2/3}} \, dx=\int \frac {{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )}}{{\left (c\,a^2\,x^2+c\right )}^{2/3}} \,d x \]
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