Integrand size = 23, antiderivative size = 87 \[ \int e^{2 \arctan (a x)} \sqrt {c+a^2 c x^2} \, dx=\frac {\left (\frac {2}{13}+\frac {3 i}{13}\right ) 2^{\frac {3}{2}-i} (1-i a x)^{\frac {3}{2}+i} \sqrt {c+a^2 c x^2} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2}+i,\frac {3}{2}+i,\frac {5}{2}+i,\frac {1}{2} (1-i a x)\right )}{a \sqrt {1+a^2 x^2}} \]
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Time = 0.06 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {5184, 5181, 71} \[ \int e^{2 \arctan (a x)} \sqrt {c+a^2 c x^2} \, dx=\frac {\left (\frac {2}{13}+\frac {3 i}{13}\right ) 2^{\frac {3}{2}-i} (1-i a x)^{\frac {3}{2}+i} \sqrt {a^2 c x^2+c} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2}+i,\frac {3}{2}+i,\frac {5}{2}+i,\frac {1}{2} (1-i a x)\right )}{a \sqrt {a^2 x^2+1}} \]
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Rule 71
Rule 5181
Rule 5184
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {c+a^2 c x^2} \int e^{2 \arctan (a x)} \sqrt {1+a^2 x^2} \, dx}{\sqrt {1+a^2 x^2}} \\ & = \frac {\sqrt {c+a^2 c x^2} \int (1-i a x)^{\frac {1}{2}+i} (1+i a x)^{\frac {1}{2}-i} \, dx}{\sqrt {1+a^2 x^2}} \\ & = \frac {\left (\frac {2}{13}+\frac {3 i}{13}\right ) 2^{\frac {3}{2}-i} (1-i a x)^{\frac {3}{2}+i} \sqrt {c+a^2 c x^2} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2}+i,\frac {3}{2}+i,\frac {5}{2}+i,\frac {1}{2} (1-i a x)\right )}{a \sqrt {1+a^2 x^2}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00 \[ \int e^{2 \arctan (a x)} \sqrt {c+a^2 c x^2} \, dx=\frac {\left (\frac {2}{13}+\frac {3 i}{13}\right ) 2^{\frac {3}{2}-i} (1-i a x)^{\frac {3}{2}+i} \sqrt {c+a^2 c x^2} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2}+i,\frac {3}{2}+i,\frac {5}{2}+i,\frac {1}{2} (1-i a x)\right )}{a \sqrt {1+a^2 x^2}} \]
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\[\int {\mathrm e}^{2 \arctan \left (a x \right )} \sqrt {a^{2} c \,x^{2}+c}d x\]
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\[ \int e^{2 \arctan (a x)} \sqrt {c+a^2 c x^2} \, dx=\int { \sqrt {a^{2} c x^{2} + c} e^{\left (2 \, \arctan \left (a x\right )\right )} \,d x } \]
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\[ \int e^{2 \arctan (a x)} \sqrt {c+a^2 c x^2} \, dx=\int \sqrt {c \left (a^{2} x^{2} + 1\right )} e^{2 \operatorname {atan}{\left (a x \right )}}\, dx \]
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\[ \int e^{2 \arctan (a x)} \sqrt {c+a^2 c x^2} \, dx=\int { \sqrt {a^{2} c x^{2} + c} e^{\left (2 \, \arctan \left (a x\right )\right )} \,d x } \]
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Exception generated. \[ \int e^{2 \arctan (a x)} \sqrt {c+a^2 c x^2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int e^{2 \arctan (a x)} \sqrt {c+a^2 c x^2} \, dx=\int {\mathrm {e}}^{2\,\mathrm {atan}\left (a\,x\right )}\,\sqrt {c\,a^2\,x^2+c} \,d x \]
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