Optimal. Leaf size=120 \[ -\frac{2^{\frac{1}{2}-\frac{i n}{2}} \sqrt{a^2 x^2+1} (1-i a x)^{\frac{1}{2} (1+i n)} \, _2F_1\left (\frac{1}{2} (i n+1),\frac{1}{2} (i n+1);\frac{1}{2} (i n+3);\frac{1}{2} (1-i a x)\right )}{a (-n+i) \sqrt{a^2 c x^2+c}} \]
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Rubi [A] time = 0.0833224, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {5076, 5073, 69} \[ -\frac{2^{\frac{1}{2}-\frac{i n}{2}} \sqrt{a^2 x^2+1} (1-i a x)^{\frac{1}{2} (1+i n)} \, _2F_1\left (\frac{1}{2} (i n+1),\frac{1}{2} (i n+1);\frac{1}{2} (i n+3);\frac{1}{2} (1-i a x)\right )}{a (-n+i) \sqrt{a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Rule 5076
Rule 5073
Rule 69
Rubi steps
\begin{align*} \int \frac{e^{n \tan ^{-1}(a x)}}{\sqrt{c+a^2 c x^2}} \, dx &=\frac{\sqrt{1+a^2 x^2} \int \frac{e^{n \tan ^{-1}(a x)}}{\sqrt{1+a^2 x^2}} \, dx}{\sqrt{c+a^2 c x^2}}\\ &=\frac{\sqrt{1+a^2 x^2} \int (1-i a x)^{-\frac{1}{2}+\frac{i n}{2}} (1+i a x)^{-\frac{1}{2}-\frac{i n}{2}} \, dx}{\sqrt{c+a^2 c x^2}}\\ &=-\frac{2^{\frac{1}{2}-\frac{i n}{2}} (1-i a x)^{\frac{1}{2} (1+i n)} \sqrt{1+a^2 x^2} \, _2F_1\left (\frac{1}{2} (1+i n),\frac{1}{2} (1+i n);\frac{1}{2} (3+i n);\frac{1}{2} (1-i a x)\right )}{a (i-n) \sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0249861, size = 117, normalized size = 0.98 \[ \frac{2^{\frac{1}{2}-\frac{i n}{2}} \sqrt{a^2 x^2+1} (1-i a x)^{\frac{1}{2}+\frac{i n}{2}} \, _2F_1\left (\frac{i n}{2}+\frac{1}{2},\frac{i n}{2}+\frac{1}{2};\frac{i n}{2}+\frac{3}{2};\frac{1}{2}-\frac{i a x}{2}\right )}{a (n-i) \sqrt{a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.03, size = 0, normalized size = 0. \begin{align*} \int{{{\rm e}^{n\arctan \left ( ax \right ) }}{\frac{1}{\sqrt{{a}^{2}c{x}^{2}+c}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\left (n \arctan \left (a x\right )\right )}}{\sqrt{a^{2} c x^{2} + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{e^{\left (n \arctan \left (a x\right )\right )}}{\sqrt{a^{2} c x^{2} + c}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{n \operatorname{atan}{\left (a x \right )}}}{\sqrt{c \left (a^{2} x^{2} + 1\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\left (n \arctan \left (a x\right )\right )}}{\sqrt{a^{2} c x^{2} + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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