Integrand size = 16, antiderivative size = 79 \[ \int e^{n \coth ^{-1}(a x)} (c-a c x) \, dx=-\frac {8 c \left (1-\frac {1}{a x}\right )^{2-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-4+n)} \operatorname {Hypergeometric2F1}\left (3,2-\frac {n}{2},3-\frac {n}{2},\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a (4-n)} \]
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Time = 0.06 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {6310, 6315, 133} \[ \int e^{n \coth ^{-1}(a x)} (c-a c x) \, dx=-\frac {8 c \left (1-\frac {1}{a x}\right )^{2-\frac {n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n-4}{2}} \operatorname {Hypergeometric2F1}\left (3,2-\frac {n}{2},3-\frac {n}{2},\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a (4-n)} \]
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Rule 133
Rule 6310
Rule 6315
Rubi steps \begin{align*} \text {integral}& = -\left ((a c) \int e^{n \coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right ) x \, dx\right ) \\ & = (a c) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{1-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{n/2}}{x^3} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {8 c \left (1-\frac {1}{a x}\right )^{2-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-4+n)} \operatorname {Hypergeometric2F1}\left (3,2-\frac {n}{2},3-\frac {n}{2},\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a (4-n)} \\ \end{align*}
Time = 0.59 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.32 \[ \int e^{n \coth ^{-1}(a x)} (c-a c x) \, dx=-\frac {c e^{n \coth ^{-1}(a x)} \left (e^{2 \coth ^{-1}(a x)} (-2+n) n \operatorname {Hypergeometric2F1}\left (1,1+\frac {n}{2},2+\frac {n}{2},e^{2 \coth ^{-1}(a x)}\right )+(2+n) \left (-1+a (-2+n) x+a^2 x^2+(-2+n) \operatorname {Hypergeometric2F1}\left (1,\frac {n}{2},1+\frac {n}{2},e^{2 \coth ^{-1}(a x)}\right )\right )\right )}{2 a (2+n)} \]
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\[\int {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} \left (-a c x +c \right )d x\]
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\[ \int e^{n \coth ^{-1}(a x)} (c-a c x) \, dx=\int { -{\left (a c x - c\right )} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \]
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\[ \int e^{n \coth ^{-1}(a x)} (c-a c x) \, dx=- c \left (\int a x e^{n \operatorname {acoth}{\left (a x \right )}}\, dx + \int \left (- e^{n \operatorname {acoth}{\left (a x \right )}}\right )\, dx\right ) \]
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\[ \int e^{n \coth ^{-1}(a x)} (c-a c x) \, dx=\int { -{\left (a c x - c\right )} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \]
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\[ \int e^{n \coth ^{-1}(a x)} (c-a c x) \, dx=\int { -{\left (a c x - c\right )} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \]
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Timed out. \[ \int e^{n \coth ^{-1}(a x)} (c-a c x) \, dx=\int {\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}\,\left (c-a\,c\,x\right ) \,d x \]
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