Integrand size = 21, antiderivative size = 90 \[ \int e^{-2 \arctan (a x)} \left (c+a^2 c x^2\right )^p \, dx=\frac {i 2^{i+p} (1-i a x)^{(1-i)+p} \left (1+a^2 x^2\right )^{-p} \left (c+a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (-i-p,(1-i)+p,(2-i)+p,\frac {1}{2} (1-i a x)\right )}{a ((1-i)+p)} \]
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Time = 0.05 (sec) , antiderivative size = 90, 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} \[ \int e^{-2 \arctan (a x)} \left (c+a^2 c x^2\right )^p \, dx=\frac {i 2^{p+i} (1-i a x)^{p+(1-i)} \left (a^2 x^2+1\right )^{-p} \left (a^2 c x^2+c\right )^p \operatorname {Hypergeometric2F1}\left (-p-i,p+(1-i),p+(2-i),\frac {1}{2} (1-i a x)\right )}{a (p+(1-i))} \]
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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^{-2 \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)^{-i+p} (1+i a x)^{i+p} \, dx \\ & = \frac {i 2^{i+p} (1-i a x)^{(1-i)+p} \left (1+a^2 x^2\right )^{-p} \left (c+a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (-i-p,(1-i)+p,(2-i)+p,\frac {1}{2} (1-i a x)\right )}{a ((1-i)+p)} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00 \[ \int e^{-2 \arctan (a x)} \left (c+a^2 c x^2\right )^p \, dx=\frac {i 2^{i+p} (1-i a x)^{(1-i)+p} \left (1+a^2 x^2\right )^{-p} \left (c+a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (-i-p,(1-i)+p,(2-i)+p,\frac {1}{2} (1-i a x)\right )}{a ((1-i)+p)} \]
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\[\int \left (a^{2} c \,x^{2}+c \right )^{p} {\mathrm e}^{-2 \arctan \left (a x \right )}d x\]
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\[ \int e^{-2 \arctan (a x)} \left (c+a^2 c x^2\right )^p \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{p} e^{\left (-2 \, \arctan \left (a x\right )\right )} \,d x } \]
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\[ \int e^{-2 \arctan (a x)} \left (c+a^2 c x^2\right )^p \, dx=\int \left (c \left (a^{2} x^{2} + 1\right )\right )^{p} e^{- 2 \operatorname {atan}{\left (a x \right )}}\, dx \]
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\[ \int e^{-2 \arctan (a x)} \left (c+a^2 c x^2\right )^p \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{p} e^{\left (-2 \, \arctan \left (a x\right )\right )} \,d x } \]
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\[ \int e^{-2 \arctan (a x)} \left (c+a^2 c x^2\right )^p \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{p} e^{\left (-2 \, \arctan \left (a x\right )\right )} \,d x } \]
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Timed out. \[ \int e^{-2 \arctan (a x)} \left (c+a^2 c x^2\right )^p \, dx=\int {\mathrm {e}}^{-2\,\mathrm {atan}\left (a\,x\right )}\,{\left (c\,a^2\,x^2+c\right )}^p \,d x \]
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