Integrand size = 23, antiderivative size = 147 \[ \int \frac {e^{n \cot ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{2/3}} \, dx=-\frac {3 \left (1+\frac {1}{a^2 x^2}\right )^{2/3} \left (\frac {a-\frac {i}{x}}{a+\frac {i}{x}}\right )^{\frac {1}{6} (4-3 i n)} \left (1-\frac {i}{a x}\right )^{\frac {1}{6} (-4+3 i n)} \left (1+\frac {i}{a x}\right )^{\frac {1}{6} (2-3 i n)} x \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{6} (4-3 i n),\frac {4}{3},\frac {2 i}{\left (a+\frac {i}{x}\right ) x}\right )}{\left (c+a^2 c x^2\right )^{2/3}} \]
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Time = 0.14 (sec) , antiderivative size = 147, 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 = {5230, 5234, 134} \[ \int \frac {e^{n \cot ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{2/3}} \, dx=-\frac {3 x \left (\frac {1}{a^2 x^2}+1\right )^{2/3} \left (\frac {a-\frac {i}{x}}{a+\frac {i}{x}}\right )^{\frac {1}{6} (4-3 i n)} \left (1-\frac {i}{a x}\right )^{\frac {1}{6} (-4+3 i n)} \left (1+\frac {i}{a x}\right )^{\frac {1}{6} (2-3 i n)} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{6} (4-3 i n),\frac {4}{3},\frac {2 i}{\left (a+\frac {i}{x}\right ) x}\right )}{\left (a^2 c x^2+c\right )^{2/3}} \]
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Rule 134
Rule 5230
Rule 5234
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\left (1+\frac {1}{a^2 x^2}\right )^{2/3} x^{4/3}\right ) \int \frac {e^{n \cot ^{-1}(a x)}}{\left (1+\frac {1}{a^2 x^2}\right )^{2/3} x^{4/3}} \, dx}{\left (c+a^2 c x^2\right )^{2/3}} \\ & = -\frac {\left (1+\frac {1}{a^2 x^2}\right )^{2/3} \text {Subst}\left (\int \frac {\left (1-\frac {i x}{a}\right )^{-\frac {2}{3}+\frac {i n}{2}} \left (1+\frac {i x}{a}\right )^{-\frac {2}{3}-\frac {i n}{2}}}{x^{2/3}} \, dx,x,\frac {1}{x}\right )}{\left (\frac {1}{x}\right )^{4/3} \left (c+a^2 c x^2\right )^{2/3}} \\ & = -\frac {3 \left (1+\frac {1}{a^2 x^2}\right )^{2/3} \left (\frac {a-\frac {i}{x}}{a+\frac {i}{x}}\right )^{\frac {1}{6} (4-3 i n)} \left (1-\frac {i}{a x}\right )^{\frac {1}{6} (-4+3 i n)} \left (1+\frac {i}{a x}\right )^{\frac {1}{6} (2-3 i n)} x \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{6} (4-3 i n),\frac {4}{3},\frac {2 i}{\left (a+\frac {i}{x}\right ) x}\right )}{\left (c+a^2 c x^2\right )^{2/3}} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.61 \[ \int \frac {e^{n \cot ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{2/3}} \, dx=-\frac {3 e^{(-2 i+n) \cot ^{-1}(a x)} \left (-1+e^{2 i \cot ^{-1}(a x)}\right ) \sqrt [3]{c+a^2 c x^2} \operatorname {Hypergeometric2F1}\left (1,\frac {2}{3}+\frac {i n}{2},\frac {4}{3}+\frac {i n}{2},e^{-2 i \cot ^{-1}(a x)}\right )}{a c (-2 i+3 n)} \]
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\[\int \frac {{\mathrm e}^{n \,\operatorname {arccot}\left (a x \right )}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {2}{3}}}d x\]
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\[ \int \frac {e^{n \cot ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{2/3}} \, dx=\int { \frac {e^{\left (n \operatorname {arccot}\left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {2}{3}}} \,d x } \]
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\[ \int \frac {e^{n \cot ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{2/3}} \, dx=\int \frac {e^{n \operatorname {acot}{\left (a x \right )}}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {2}{3}}}\, dx \]
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\[ \int \frac {e^{n \cot ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{2/3}} \, dx=\int { \frac {e^{\left (n \operatorname {arccot}\left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {2}{3}}} \,d x } \]
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\[ \int \frac {e^{n \cot ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{2/3}} \, dx=\int { \frac {e^{\left (n \operatorname {arccot}\left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {2}{3}}} \,d x } \]
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Timed out. \[ \int \frac {e^{n \cot ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{2/3}} \, dx=\int \frac {{\mathrm {e}}^{n\,\mathrm {acot}\left (a\,x\right )}}{{\left (c\,a^2\,x^2+c\right )}^{2/3}} \,d x \]
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