Integrand size = 22, antiderivative size = 18 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx=\frac {e^{n \coth ^{-1}(a x)}}{a c n} \]
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Time = 0.03 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {6318} \[ \int \frac {e^{n \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx=\frac {e^{n \coth ^{-1}(a x)}}{a c n} \]
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Rule 6318
Rubi steps \begin{align*} \text {integral}& = \frac {e^{n \coth ^{-1}(a x)}}{a c n} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx=\frac {e^{n \coth ^{-1}(a x)}}{a c n} \]
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Time = 1.08 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00
method | result | size |
gosper | \(\frac {{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )}}{a c n}\) | \(18\) |
parallelrisch | \(\frac {{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )}}{a c n}\) | \(18\) |
risch | \(\frac {\left (a x -1\right )^{-\frac {n}{2}} \left (a x +1\right )^{\frac {n}{2}}}{c a n}\) | \(29\) |
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none
Time = 0.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.50 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx=\frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a c n} \]
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Result contains complex when optimal does not.
Time = 0.51 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.72 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx=\begin {cases} \frac {x}{c} & \text {for}\: a = 0 \wedge n = 0 \\\frac {x e^{\frac {i \pi n}{2}}}{c} & \text {for}\: a = 0 \\- \frac {\log {\left (x - \frac {1}{a} \right )}}{2 a c} + \frac {\log {\left (x + \frac {1}{a} \right )}}{2 a c} & \text {for}\: n = 0 \\\frac {e^{n \operatorname {acoth}{\left (a x \right )}}}{a c n} & \text {otherwise} \end {cases} \]
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none
Time = 0.21 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.67 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx=\frac {e^{\left (\frac {1}{2} \, n \log \left (a x + 1\right ) - \frac {1}{2} \, n \log \left (a x - 1\right )\right )}}{a c n} \]
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\[ \int \frac {e^{n \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx=\int { -\frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a^{2} c x^{2} - c} \,d x } \]
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Time = 4.06 (sec) , antiderivative size = 39, normalized size of antiderivative = 2.17 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx=\frac {{\left (\frac {1}{a\,x}+1\right )}^{n/2}}{a\,c\,n\,{\left (1-\frac {1}{a\,x}\right )}^{n/2}} \]
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