Integrand size = 26, antiderivative size = 121 \[ \int \frac {e^{n \arctan (a x)}}{x \sqrt {c+a^2 c x^2}} \, dx=-\frac {2 (1-i a x)^{\frac {1}{2} (1+i n)} (1+i a x)^{\frac {1}{2} (-1-i n)} \sqrt {1+a^2 x^2} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (1+i n),\frac {1}{2} (3+i n),\frac {1-i a x}{1+i a x}\right )}{(1+i n) \sqrt {c+a^2 c x^2}} \]
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Time = 0.14 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {5193, 5190, 133} \[ \int \frac {e^{n \arctan (a x)}}{x \sqrt {c+a^2 c x^2}} \, dx=-\frac {2 \sqrt {a^2 x^2+1} (1-i a x)^{\frac {1}{2} (1+i n)} (1+i a x)^{\frac {1}{2} (-1-i n)} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (i n+1),\frac {1}{2} (i n+3),\frac {1-i a x}{i a x+1}\right )}{(1+i n) \sqrt {a^2 c x^2+c}} \]
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Rule 133
Rule 5190
Rule 5193
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1+a^2 x^2} \int \frac {e^{n \arctan (a x)}}{x \sqrt {1+a^2 x^2}} \, dx}{\sqrt {c+a^2 c x^2}} \\ & = \frac {\sqrt {1+a^2 x^2} \int \frac {(1-i a x)^{-\frac {1}{2}+\frac {i n}{2}} (1+i a x)^{-\frac {1}{2}-\frac {i n}{2}}}{x} \, dx}{\sqrt {c+a^2 c x^2}} \\ & = -\frac {2 (1-i a x)^{\frac {1}{2} (1+i n)} (1+i a x)^{\frac {1}{2} (-1-i n)} \sqrt {1+a^2 x^2} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (1+i n),\frac {1}{2} (3+i n),\frac {1-i a x}{1+i a x}\right )}{(1+i n) \sqrt {c+a^2 c x^2}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.99 \[ \int \frac {e^{n \arctan (a x)}}{x \sqrt {c+a^2 c x^2}} \, dx=\frac {2 (1-i a x)^{\frac {1}{2}+\frac {i n}{2}} (1+i a x)^{-\frac {1}{2}-\frac {i n}{2}} \sqrt {1+a^2 x^2} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2}+\frac {i n}{2},\frac {3}{2}+\frac {i n}{2},\frac {i+a x}{i-a x}\right )}{(-1-i n) \sqrt {c+a^2 c x^2}} \]
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\[\int \frac {{\mathrm e}^{n \arctan \left (a x \right )}}{x \sqrt {a^{2} c \,x^{2}+c}}d x\]
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\[ \int \frac {e^{n \arctan (a x)}}{x \sqrt {c+a^2 c x^2}} \, dx=\int { \frac {e^{\left (n \arctan \left (a x\right )\right )}}{\sqrt {a^{2} c x^{2} + c} x} \,d x } \]
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\[ \int \frac {e^{n \arctan (a x)}}{x \sqrt {c+a^2 c x^2}} \, dx=\int \frac {e^{n \operatorname {atan}{\left (a x \right )}}}{x \sqrt {c \left (a^{2} x^{2} + 1\right )}}\, dx \]
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\[ \int \frac {e^{n \arctan (a x)}}{x \sqrt {c+a^2 c x^2}} \, dx=\int { \frac {e^{\left (n \arctan \left (a x\right )\right )}}{\sqrt {a^{2} c x^{2} + c} x} \,d x } \]
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\[ \int \frac {e^{n \arctan (a x)}}{x \sqrt {c+a^2 c x^2}} \, dx=\int { \frac {e^{\left (n \arctan \left (a x\right )\right )}}{\sqrt {a^{2} c x^{2} + c} x} \,d x } \]
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Timed out. \[ \int \frac {e^{n \arctan (a x)}}{x \sqrt {c+a^2 c x^2}} \, dx=\int \frac {{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )}}{x\,\sqrt {c\,a^2\,x^2+c}} \,d x \]
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