Integrand size = 21, antiderivative size = 101 \[ \int e^{-\arctan (a x)} \left (c+a^2 c x^2\right )^p \, dx=\frac {2^{\left (1+\frac {i}{2}\right )+p} (1-i a x)^{\left (1-\frac {i}{2}\right )+p} \left (1+a^2 x^2\right )^{-p} \left (c+a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (-\frac {i}{2}-p,\left (1-\frac {i}{2}\right )+p,\left (2-\frac {i}{2}\right )+p,\frac {1}{2} (1-i a x)\right )}{a ((-1-2 i)-2 i p)} \]
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Time = 0.05 (sec) , antiderivative size = 101, 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} \[ \int e^{-\arctan (a x)} \left (c+a^2 c x^2\right )^p \, dx=\frac {2^{p+\left (1+\frac {i}{2}\right )} (1-i a x)^{p+\left (1-\frac {i}{2}\right )} \left (a^2 x^2+1\right )^{-p} \left (a^2 c x^2+c\right )^p \operatorname {Hypergeometric2F1}\left (-p-\frac {i}{2},p+\left (1-\frac {i}{2}\right ),p+\left (2-\frac {i}{2}\right ),\frac {1}{2} (1-i a x)\right )}{a (-2 i p-(1+2 i))} \]
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Rule 71
Rule 5181
Rule 5184
Rubi steps \begin{align*} \text {integral}& = \left (\left (1+a^2 x^2\right )^{-p} \left (c+a^2 c x^2\right )^p\right ) \int e^{-\arctan (a x)} \left (1+a^2 x^2\right )^p \, dx \\ & = \left (\left (1+a^2 x^2\right )^{-p} \left (c+a^2 c x^2\right )^p\right ) \int (1-i a x)^{-\frac {i}{2}+p} (1+i a x)^{\frac {i}{2}+p} \, dx \\ & = \frac {2^{\left (1+\frac {i}{2}\right )+p} (1-i a x)^{\left (1-\frac {i}{2}\right )+p} \left (1+a^2 x^2\right )^{-p} \left (c+a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (-\frac {i}{2}-p,\left (1-\frac {i}{2}\right )+p,\left (2-\frac {i}{2}\right )+p,\frac {1}{2} (1-i a x)\right )}{a ((-1-2 i)-2 i p)} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.01 \[ \int e^{-\arctan (a x)} \left (c+a^2 c x^2\right )^p \, dx=\frac {i 2^{\frac {i}{2}+p} (1-i a x)^{\left (1-\frac {i}{2}\right )+p} \left (1+a^2 x^2\right )^{-p} \left (c+a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (-\frac {i}{2}-p,\left (1-\frac {i}{2}\right )+p,\left (2-\frac {i}{2}\right )+p,\frac {1}{2} (1-i a x)\right )}{a \left (\left (1-\frac {i}{2}\right )+p\right )} \]
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\[\int \left (a^{2} c \,x^{2}+c \right )^{p} {\mathrm e}^{-\arctan \left (a x \right )}d x\]
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\[ \int e^{-\arctan (a x)} \left (c+a^2 c x^2\right )^p \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{p} e^{\left (-\arctan \left (a x\right )\right )} \,d x } \]
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\[ \int e^{-\arctan (a x)} \left (c+a^2 c x^2\right )^p \, dx=\int \left (c \left (a^{2} x^{2} + 1\right )\right )^{p} e^{- \operatorname {atan}{\left (a x \right )}}\, dx \]
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\[ \int e^{-\arctan (a x)} \left (c+a^2 c x^2\right )^p \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{p} e^{\left (-\arctan \left (a x\right )\right )} \,d x } \]
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\[ \int e^{-\arctan (a x)} \left (c+a^2 c x^2\right )^p \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{p} e^{\left (-\arctan \left (a x\right )\right )} \,d x } \]
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Timed out. \[ \int e^{-\arctan (a x)} \left (c+a^2 c x^2\right )^p \, dx=\int {\mathrm {e}}^{-\mathrm {atan}\left (a\,x\right )}\,{\left (c\,a^2\,x^2+c\right )}^p \,d x \]
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