Integrand size = 19, antiderivative size = 51 \[ \int e^{2 \arctan (a x)} \left (c+a^2 c x^2\right ) \, dx=\frac {\left (\frac {1}{5}+\frac {2 i}{5}\right ) 2^{1-i} c (1-i a x)^{2+i} \operatorname {Hypergeometric2F1}\left (-1+i,2+i,3+i,\frac {1}{2} (1-i a x)\right )}{a} \]
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Time = 0.02 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {5181, 71} \[ \int e^{2 \arctan (a x)} \left (c+a^2 c x^2\right ) \, dx=\frac {\left (\frac {1}{5}+\frac {2 i}{5}\right ) 2^{1-i} c (1-i a x)^{2+i} \operatorname {Hypergeometric2F1}\left (-1+i,2+i,3+i,\frac {1}{2} (1-i a x)\right )}{a} \]
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Rule 71
Rule 5181
Rubi steps \begin{align*} \text {integral}& = c \int (1-i a x)^{1+i} (1+i a x)^{1-i} \, dx \\ & = \frac {\left (\frac {1}{5}+\frac {2 i}{5}\right ) 2^{1-i} c (1-i a x)^{2+i} \operatorname {Hypergeometric2F1}\left (-1+i,2+i,3+i,\frac {1}{2} (1-i a x)\right )}{a} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00 \[ \int e^{2 \arctan (a x)} \left (c+a^2 c x^2\right ) \, dx=\frac {\left (\frac {1}{5}+\frac {2 i}{5}\right ) 2^{1-i} c (1-i a x)^{2+i} \operatorname {Hypergeometric2F1}\left (-1+i,2+i,3+i,\frac {1}{2} (1-i a x)\right )}{a} \]
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\[\int {\mathrm e}^{2 \arctan \left (a x \right )} \left (a^{2} c \,x^{2}+c \right )d x\]
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\[ \int e^{2 \arctan (a x)} \left (c+a^2 c x^2\right ) \, dx=\int { {\left (a^{2} c x^{2} + c\right )} e^{\left (2 \, \arctan \left (a x\right )\right )} \,d x } \]
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\[ \int e^{2 \arctan (a x)} \left (c+a^2 c x^2\right ) \, dx=c \left (\int a^{2} x^{2} e^{2 \operatorname {atan}{\left (a x \right )}}\, dx + \int e^{2 \operatorname {atan}{\left (a x \right )}}\, dx\right ) \]
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\[ \int e^{2 \arctan (a x)} \left (c+a^2 c x^2\right ) \, dx=\int { {\left (a^{2} c x^{2} + c\right )} e^{\left (2 \, \arctan \left (a x\right )\right )} \,d x } \]
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\[ \int e^{2 \arctan (a x)} \left (c+a^2 c x^2\right ) \, dx=\int { {\left (a^{2} c x^{2} + c\right )} e^{\left (2 \, \arctan \left (a x\right )\right )} \,d x } \]
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Timed out. \[ \int e^{2 \arctan (a x)} \left (c+a^2 c x^2\right ) \, dx=\int {\mathrm {e}}^{2\,\mathrm {atan}\left (a\,x\right )}\,\left (c\,a^2\,x^2+c\right ) \,d x \]
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