Optimal. Leaf size=202 \[ \frac{\sqrt{a^2 x^2+1} (1-i a x)^{\frac{1}{2} (1+i n)} (1+i a x)^{\frac{1}{2} (1-i n)}}{a^2 (1-i n) \sqrt{a^2 c x^2+c}}-\frac{i 2^{\frac{3}{2}-\frac{i n}{2}} n \sqrt{a^2 x^2+1} (1-i a x)^{\frac{1}{2} (1+i n)} \, _2F_1\left (\frac{1}{2} (i n-1),\frac{1}{2} (i n+1);\frac{1}{2} (i n+3);\frac{1}{2} (1-i a x)\right )}{a^2 \left (n^2+1\right ) \sqrt{a^2 c x^2+c}} \]
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Rubi [A] time = 0.186635, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {5085, 5082, 79, 69} \[ \frac{\sqrt{a^2 x^2+1} (1-i a x)^{\frac{1}{2} (1+i n)} (1+i a x)^{\frac{1}{2} (1-i n)}}{a^2 (1-i n) \sqrt{a^2 c x^2+c}}-\frac{i 2^{\frac{3}{2}-\frac{i n}{2}} n \sqrt{a^2 x^2+1} (1-i a x)^{\frac{1}{2} (1+i n)} \, _2F_1\left (\frac{1}{2} (i n-1),\frac{1}{2} (i n+1);\frac{1}{2} (i n+3);\frac{1}{2} (1-i a x)\right )}{a^2 \left (n^2+1\right ) \sqrt{a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Rule 5085
Rule 5082
Rule 79
Rule 69
Rubi steps
\begin{align*} \int \frac{e^{n \tan ^{-1}(a x)} x}{\sqrt{c+a^2 c x^2}} \, dx &=\frac{\sqrt{1+a^2 x^2} \int \frac{e^{n \tan ^{-1}(a x)} x}{\sqrt{1+a^2 x^2}} \, dx}{\sqrt{c+a^2 c x^2}}\\ &=\frac{\sqrt{1+a^2 x^2} \int x (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{(1-i a x)^{\frac{1}{2} (1+i n)} (1+i a x)^{\frac{1}{2} (1-i n)} \sqrt{1+a^2 x^2}}{a^2 (1-i n) \sqrt{c+a^2 c x^2}}-\frac{\left (n \sqrt{1+a^2 x^2}\right ) \int (1-i a x)^{-\frac{1}{2}+\frac{i n}{2}} (1+i a x)^{-\frac{1}{2} i (i+n)} \, dx}{a (1-i n) \sqrt{c+a^2 c x^2}}\\ &=\frac{(1-i a x)^{\frac{1}{2} (1+i n)} (1+i a x)^{\frac{1}{2} (1-i n)} \sqrt{1+a^2 x^2}}{a^2 (1-i n) \sqrt{c+a^2 c x^2}}-\frac{i 2^{\frac{3}{2}-\frac{i n}{2}} n (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^2 \left (1+n^2\right ) \sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.137188, size = 187, normalized size = 0.93 \[ \frac{i 2^{-\frac{1}{2}-\frac{i n}{2}} \sqrt{a^2 x^2+1} (1-i a x)^{\frac{1}{2}+\frac{i n}{2}} (1+i a x)^{-\frac{i n}{2}} \left (2^{\frac{1}{2}+\frac{i n}{2}} (n-i) \sqrt{1+i a x}-4 n (1+i a x)^{\frac{i n}{2}} \, _2F_1\left (\frac{1}{2} (i n+1),\frac{1}{2} i (n+i);\frac{1}{2} (i n+3);\frac{1}{2} (1-i a x)\right )\right )}{a^2 \left (n^2+1\right ) \sqrt{a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.289, size = 0, normalized size = 0. \begin{align*} \int{{{\rm e}^{n\arctan \left ( ax \right ) }}x{\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{x 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{x 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{x 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{x 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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