Integrand size = 10, antiderivative size = 91 \[ \int e^{n \arctan (a+b x)} \, dx=-\frac {2^{1-\frac {i n}{2}} (1-i a-i b x)^{1+\frac {i n}{2}} \operatorname {Hypergeometric2F1}\left (1+\frac {i n}{2},\frac {i n}{2},2+\frac {i n}{2},\frac {1}{2} (1-i a-i b x)\right )}{b (2 i-n)} \]
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Time = 0.01 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5201, 71} \[ \int e^{n \arctan (a+b x)} \, dx=-\frac {2^{1-\frac {i n}{2}} (-i a-i b x+1)^{1+\frac {i n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {i n}{2}+1,\frac {i n}{2},\frac {i n}{2}+2,\frac {1}{2} (-i a-i b x+1)\right )}{b (-n+2 i)} \]
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Rule 71
Rule 5201
Rubi steps \begin{align*} \text {integral}& = \int (1-i a-i b x)^{\frac {i n}{2}} (1+i a+i b x)^{-\frac {i n}{2}} \, dx \\ & = -\frac {2^{1-\frac {i n}{2}} (1-i a-i b x)^{1+\frac {i n}{2}} \operatorname {Hypergeometric2F1}\left (1+\frac {i n}{2},\frac {i n}{2},2+\frac {i n}{2},\frac {1}{2} (1-i a-i b x)\right )}{b (2 i-n)} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.66 \[ \int e^{n \arctan (a+b x)} \, dx=\frac {4 e^{(2 i+n) \arctan (a+b x)} \operatorname {Hypergeometric2F1}\left (2,1-\frac {i n}{2},2-\frac {i n}{2},-e^{2 i \arctan (a+b x)}\right )}{b (2 i+n)} \]
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\[\int {\mathrm e}^{n \arctan \left (b x +a \right )}d x\]
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\[ \int e^{n \arctan (a+b x)} \, dx=\int { e^{\left (n \arctan \left (b x + a\right )\right )} \,d x } \]
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\[ \int e^{n \arctan (a+b x)} \, dx=\int e^{n \operatorname {atan}{\left (a + b x \right )}}\, dx \]
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\[ \int e^{n \arctan (a+b x)} \, dx=\int { e^{\left (n \arctan \left (b x + a\right )\right )} \,d x } \]
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\[ \int e^{n \arctan (a+b x)} \, dx=\int { e^{\left (n \arctan \left (b x + a\right )\right )} \,d x } \]
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Timed out. \[ \int e^{n \arctan (a+b x)} \, dx=\int {\mathrm {e}}^{n\,\mathrm {atan}\left (a+b\,x\right )} \,d x \]
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