Integrand size = 20, antiderivative size = 185 \[ \int e^{n \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right ) \, dx=c \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{n/2} x-\frac {2 c (1-n) \left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{n/2} \operatorname {Hypergeometric2F1}\left (1,\frac {n}{2},\frac {2+n}{2},\frac {a+\frac {1}{x}}{a-\frac {1}{x}}\right )}{a n}-\frac {2^{n/2} c \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (1-\frac {n}{2},1-\frac {n}{2},2-\frac {n}{2},\frac {a-\frac {1}{x}}{2 a}\right )}{a (2-n)} \]
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Time = 0.09 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6314, 130, 71, 98, 133} \[ \int e^{n \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right ) \, dx=-\frac {c 2^{n/2} \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (1-\frac {n}{2},1-\frac {n}{2},2-\frac {n}{2},\frac {a-\frac {1}{x}}{2 a}\right )}{a (2-n)}-\frac {2 c (1-n) \left (\frac {1}{a x}+1\right )^{n/2} \left (1-\frac {1}{a x}\right )^{-n/2} \operatorname {Hypergeometric2F1}\left (1,\frac {n}{2},\frac {n+2}{2},\frac {a+\frac {1}{x}}{a-\frac {1}{x}}\right )}{a n}+c x \left (\frac {1}{a x}+1\right )^{n/2} \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \]
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Rule 71
Rule 98
Rule 130
Rule 133
Rule 6314
Rubi steps \begin{align*} \text {integral}& = -\left (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^2} \, dx,x,\frac {1}{x}\right )\right ) \\ & = -\left (c \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{-n/2} \left (1+\frac {x}{a}\right )^{-1+\frac {n}{2}}}{x^2} \, dx,x,\frac {1}{x}\right )\right )+\frac {c \text {Subst}\left (\int \left (1-\frac {x}{a}\right )^{-n/2} \left (1+\frac {x}{a}\right )^{-1+\frac {n}{2}} \, dx,x,\frac {1}{x}\right )}{a^2} \\ & = c \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{n/2} x-\frac {2^{n/2} c \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (1-\frac {n}{2},1-\frac {n}{2},2-\frac {n}{2},\frac {a-\frac {1}{x}}{2 a}\right )}{a (2-n)}+\frac {(c (1-n)) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{-n/2} \left (1+\frac {x}{a}\right )^{-1+\frac {n}{2}}}{x} \, dx,x,\frac {1}{x}\right )}{a} \\ & = c \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{n/2} x-\frac {2 c (1-n) \left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{n/2} \operatorname {Hypergeometric2F1}\left (1,\frac {n}{2},\frac {2+n}{2},\frac {a+\frac {1}{x}}{a-\frac {1}{x}}\right )}{a n}-\frac {2^{n/2} c \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (1-\frac {n}{2},1-\frac {n}{2},2-\frac {n}{2},\frac {a-\frac {1}{x}}{2 a}\right )}{a (2-n)} \\ \end{align*}
Time = 0.38 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.84 \[ \int e^{n \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right ) \, dx=\frac {c e^{n \coth ^{-1}(a x)} \left (-e^{2 \coth ^{-1}(a x)} n \operatorname {Hypergeometric2F1}\left (1,1+\frac {n}{2},2+\frac {n}{2},-e^{2 \coth ^{-1}(a x)}\right )+e^{2 \coth ^{-1}(a x)} (-1+n) n \operatorname {Hypergeometric2F1}\left (1,1+\frac {n}{2},2+\frac {n}{2},e^{2 \coth ^{-1}(a x)}\right )+(2+n) \left (a n x+\operatorname {Hypergeometric2F1}\left (1,\frac {n}{2},1+\frac {n}{2},-e^{2 \coth ^{-1}(a x)}\right )+(-1+n) \operatorname {Hypergeometric2F1}\left (1,\frac {n}{2},1+\frac {n}{2},e^{2 \coth ^{-1}(a x)}\right )\right )\right )}{a n (2+n)} \]
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\[\int {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} \left (c -\frac {c}{a x}\right )d x\]
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\[ \int e^{n \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right ) \, dx=\int { {\left (c - \frac {c}{a x}\right )} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \]
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\[ \int e^{n \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right ) \, dx=\frac {c \left (\int a e^{n \operatorname {acoth}{\left (a x \right )}}\, dx + \int \left (- \frac {e^{n \operatorname {acoth}{\left (a x \right )}}}{x}\right )\, dx\right )}{a} \]
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\[ \int e^{n \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right ) \, dx=\int { {\left (c - \frac {c}{a x}\right )} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \]
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\[ \int e^{n \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right ) \, dx=\int { {\left (c - \frac {c}{a x}\right )} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \]
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Timed out. \[ \int e^{n \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right ) \, dx=\int {\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}\,\left (c-\frac {c}{a\,x}\right ) \,d x \]
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