Integrand size = 23, antiderivative size = 52 \[ \int e^{-2 p \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\frac {\left (1-\frac {1}{a^2 x^2}\right )^{-p} \left (1-\frac {1}{a x}\right )^{1+2 p} x \left (c-a^2 c x^2\right )^p}{1+2 p} \]
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Time = 0.09 (sec) , antiderivative size = 52, 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 = {6327, 6331, 37} \[ \int e^{-2 p \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\frac {x \left (1-\frac {1}{a^2 x^2}\right )^{-p} \left (1-\frac {1}{a x}\right )^{2 p+1} \left (c-a^2 c x^2\right )^p}{2 p+1} \]
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Rule 37
Rule 6327
Rule 6331
Rubi steps \begin{align*} \text {integral}& = \left (\left (1-\frac {1}{a^2 x^2}\right )^{-p} x^{-2 p} \left (c-a^2 c x^2\right )^p\right ) \int e^{-2 p \coth ^{-1}(a x)} \left (1-\frac {1}{a^2 x^2}\right )^p x^{2 p} \, dx \\ & = -\left (\left (\left (1-\frac {1}{a^2 x^2}\right )^{-p} \left (\frac {1}{x}\right )^{2 p} \left (c-a^2 c x^2\right )^p\right ) \text {Subst}\left (\int x^{-2-2 p} \left (1-\frac {x}{a}\right )^{2 p} \, dx,x,\frac {1}{x}\right )\right ) \\ & = \frac {\left (1-\frac {1}{a^2 x^2}\right )^{-p} \left (1-\frac {1}{a x}\right )^{1+2 p} x \left (c-a^2 c x^2\right )^p}{1+2 p} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.69 \[ \int e^{-2 p \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\frac {e^{-2 p \coth ^{-1}(a x)} (-1+a x) \left (c-a^2 c x^2\right )^p}{a+2 a p} \]
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Time = 0.87 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.77
method | result | size |
gosper | \(\frac {\left (a x -1\right ) \left (-a^{2} c \,x^{2}+c \right )^{p} {\mathrm e}^{-2 p \,\operatorname {arccoth}\left (a x \right )}}{a \left (1+2 p \right )}\) | \(40\) |
parallelrisch | \(\frac {\left (x \left (-a^{2} c \,x^{2}+c \right )^{p} a -\left (-a^{2} c \,x^{2}+c \right )^{p}\right ) {\mathrm e}^{-2 p \,\operatorname {arccoth}\left (a x \right )}}{a \left (1+2 p \right )}\) | \(54\) |
risch | \(\frac {\left (a x -1\right ) \left (\left (a x -1\right )^{p}\right )^{2} c^{p} {\mathrm e}^{-\frac {i p \pi \left (\operatorname {csgn}\left (i \left (a x -1\right ) \left (a x +1\right )\right )^{3}-\operatorname {csgn}\left (i \left (a x -1\right )\right ) \operatorname {csgn}\left (i \left (a x -1\right ) \left (a x +1\right )\right )^{2}-\operatorname {csgn}\left (i \left (a x -1\right ) \left (a x +1\right )\right )^{2} \operatorname {csgn}\left (i \left (a x +1\right )\right )+\operatorname {csgn}\left (i \left (a x -1\right )\right ) \operatorname {csgn}\left (i \left (a x -1\right ) \left (a x +1\right )\right ) \operatorname {csgn}\left (i \left (a x +1\right )\right )-\operatorname {csgn}\left (i \left (a x -1\right ) \left (a x +1\right )\right ) \operatorname {csgn}\left (i c \left (a x -1\right ) \left (a x +1\right )\right )^{2}+\operatorname {csgn}\left (i \left (a x -1\right ) \left (a x +1\right )\right ) \operatorname {csgn}\left (i c \left (a x -1\right ) \left (a x +1\right )\right ) \operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \left (a x -1\right ) \left (a x +1\right )\right )^{3}-\operatorname {csgn}\left (i c \left (a x -1\right ) \left (a x +1\right )\right )^{2} \operatorname {csgn}\left (i c \right )+2 \operatorname {csgn}\left (i c \left (a x -1\right ) \left (a x +1\right )\right )^{2}-2\right )}{2}}}{a \left (1+2 p \right )}\) | \(279\) |
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Time = 0.25 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.85 \[ \int e^{-2 p \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\frac {{\left (a x - 1\right )} {\left (-a^{2} c x^{2} + c\right )}^{p}}{{\left (2 \, a p + a\right )} \left (\frac {a x + 1}{a x - 1}\right )^{p}} \]
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\[ \int e^{-2 p \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\begin {cases} \frac {i x}{\sqrt {c}} & \text {for}\: a = 0 \wedge p = - \frac {1}{2} \\c^{p} x e^{- i \pi p} & \text {for}\: a = 0 \\\int \frac {e^{\operatorname {acoth}{\left (a x \right )}}}{\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )}}\, dx & \text {for}\: p = - \frac {1}{2} \\\frac {a x \left (- a^{2} c x^{2} + c\right )^{p}}{2 a p e^{2 p \operatorname {acoth}{\left (a x \right )}} + a e^{2 p \operatorname {acoth}{\left (a x \right )}}} - \frac {\left (- a^{2} c x^{2} + c\right )^{p}}{2 a p e^{2 p \operatorname {acoth}{\left (a x \right )}} + a e^{2 p \operatorname {acoth}{\left (a x \right )}}} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.69 \[ \int e^{-2 p \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\frac {{\left (a \left (-c\right )^{p} x - \left (-c\right )^{p}\right )} {\left (a x - 1\right )}^{2 \, p}}{a {\left (2 \, p + 1\right )}} \]
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\[ \int e^{-2 p \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\int { \frac {{\left (-a^{2} c x^{2} + c\right )}^{p}}{\left (\frac {a x + 1}{a x - 1}\right )^{p}} \,d x } \]
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Time = 4.01 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.13 \[ \int e^{-2 p \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\frac {{\left (c-a^2\,c\,x^2\right )}^p\,\left (a\,x-1\right )\,{\left (\frac {a\,x-1}{a\,x}\right )}^p}{a\,\left (2\,p+1\right )\,{\left (\frac {a\,x+1}{a\,x}\right )}^p} \]
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