Integrand size = 21, antiderivative size = 93 \[ \int \frac {e^{\arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\frac {(1+i) 2^{-\frac {1}{2}-\frac {i}{2}} (1-i a x)^{\frac {1}{2}+\frac {i}{2}} \sqrt {1+a^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2}+\frac {i}{2},\frac {1}{2}+\frac {i}{2},\frac {3}{2}+\frac {i}{2},\frac {1}{2} (1-i a x)\right )}{a \sqrt {c+a^2 c x^2}} \]
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Time = 0.05 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5184, 5181, 71} \[ \int \frac {e^{\arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\frac {(1+i) 2^{-\frac {1}{2}-\frac {i}{2}} (1-i a x)^{\frac {1}{2}+\frac {i}{2}} \sqrt {a^2 x^2+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2}+\frac {i}{2},\frac {1}{2}+\frac {i}{2},\frac {3}{2}+\frac {i}{2},\frac {1}{2} (1-i a x)\right )}{a \sqrt {a^2 c x^2+c}} \]
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Rule 71
Rule 5181
Rule 5184
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1+a^2 x^2} \int \frac {e^{\arctan (a x)}}{\sqrt {1+a^2 x^2}} \, dx}{\sqrt {c+a^2 c x^2}} \\ & = \frac {\sqrt {1+a^2 x^2} \int (1-i a x)^{-\frac {1}{2}+\frac {i}{2}} (1+i a x)^{-\frac {1}{2}-\frac {i}{2}} \, dx}{\sqrt {c+a^2 c x^2}} \\ & = \frac {(1+i) 2^{-\frac {1}{2}-\frac {i}{2}} (1-i a x)^{\frac {1}{2}+\frac {i}{2}} \sqrt {1+a^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2}+\frac {i}{2},\frac {1}{2}+\frac {i}{2},\frac {3}{2}+\frac {i}{2},\frac {1}{2} (1-i a x)\right )}{a \sqrt {c+a^2 c x^2}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\frac {(1+i) 2^{-\frac {1}{2}-\frac {i}{2}} (1-i a x)^{\frac {1}{2}+\frac {i}{2}} \sqrt {1+a^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2}+\frac {i}{2},\frac {1}{2}+\frac {i}{2},\frac {3}{2}+\frac {i}{2},\frac {1}{2} (1-i a x)\right )}{a \sqrt {c+a^2 c x^2}} \]
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\[\int \frac {{\mathrm e}^{\arctan \left (a x \right )}}{\sqrt {a^{2} c \,x^{2}+c}}d x\]
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\[ \int \frac {e^{\arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\int { \frac {e^{\left (\arctan \left (a x\right )\right )}}{\sqrt {a^{2} c x^{2} + c}} \,d x } \]
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\[ \int \frac {e^{\arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {e^{\operatorname {atan}{\left (a x \right )}}}{\sqrt {c \left (a^{2} x^{2} + 1\right )}}\, dx \]
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\[ \int \frac {e^{\arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\int { \frac {e^{\left (\arctan \left (a x\right )\right )}}{\sqrt {a^{2} c x^{2} + c}} \,d x } \]
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\[ \int \frac {e^{\arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\int { \frac {e^{\left (\arctan \left (a x\right )\right )}}{\sqrt {a^{2} c x^{2} + c}} \,d x } \]
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Timed out. \[ \int \frac {e^{\arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {{\mathrm {e}}^{\mathrm {atan}\left (a\,x\right )}}{\sqrt {c\,a^2\,x^2+c}} \,d x \]
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