Integrand size = 20, antiderivative size = 245 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx=-\frac {a \left (1-\frac {1}{a x}\right )^{\frac {2-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x^2}{(5+n) (c-a c x)^{7/2}}+\frac {3 a^2 \left (1-\frac {1}{a x}\right )^{\frac {4-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x^3}{2 \left (15+8 n+n^2\right ) (c-a c x)^{7/2}}-\frac {3 a^2 \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{\frac {3+n}{2}} \left (1-\frac {1}{a x}\right )^{\frac {4-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x^3 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3+n}{2},\frac {3}{2},\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{2 \left (15+8 n+n^2\right ) (c-a c x)^{7/2}} \]
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Time = 0.20 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6311, 6316, 96, 134} \[ \int \frac {e^{n \coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx=-\frac {3 a^2 x^3 \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{\frac {n+3}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \left (1-\frac {1}{a x}\right )^{\frac {4-n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+3}{2},\frac {3}{2},\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{2 \left (n^2+8 n+15\right ) (c-a c x)^{7/2}}+\frac {3 a^2 x^3 \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \left (1-\frac {1}{a x}\right )^{\frac {4-n}{2}}}{2 \left (n^2+8 n+15\right ) (c-a c x)^{7/2}}-\frac {a x^2 \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \left (1-\frac {1}{a x}\right )^{\frac {2-n}{2}}}{(n+5) (c-a c x)^{7/2}} \]
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Rule 96
Rule 134
Rule 6311
Rule 6316
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\left (1-\frac {1}{a x}\right )^{7/2} x^{7/2}\right ) \int \frac {e^{n \coth ^{-1}(a x)}}{\left (1-\frac {1}{a x}\right )^{7/2} x^{7/2}} \, dx}{(c-a c x)^{7/2}} \\ & = -\frac {\left (1-\frac {1}{a x}\right )^{7/2} \text {Subst}\left (\int x^{3/2} \left (1-\frac {x}{a}\right )^{-\frac {7}{2}-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{n/2} \, dx,x,\frac {1}{x}\right )}{\left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2}} \\ & = -\frac {a \left (1-\frac {1}{a x}\right )^{\frac {2-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x^2}{(5+n) (c-a c x)^{7/2}}+\frac {\left (3 a \left (1-\frac {1}{a x}\right )^{7/2}\right ) \text {Subst}\left (\int \sqrt {x} \left (1-\frac {x}{a}\right )^{-\frac {5}{2}-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{n/2} \, dx,x,\frac {1}{x}\right )}{2 (5+n) \left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2}} \\ & = -\frac {a \left (1-\frac {1}{a x}\right )^{\frac {2-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x^2}{(5+n) (c-a c x)^{7/2}}+\frac {3 a^2 \left (1-\frac {1}{a x}\right )^{\frac {4-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x^3}{2 \left (15+8 n+n^2\right ) (c-a c x)^{7/2}}-\frac {\left (3 a^2 \left (1-\frac {1}{a x}\right )^{7/2}\right ) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{-\frac {3}{2}-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{n/2}}{\sqrt {x}} \, dx,x,\frac {1}{x}\right )}{4 (3+n) (5+n) \left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2}} \\ & = -\frac {a \left (1-\frac {1}{a x}\right )^{\frac {2-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x^2}{(5+n) (c-a c x)^{7/2}}+\frac {3 a^2 \left (1-\frac {1}{a x}\right )^{\frac {4-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x^3}{2 \left (15+8 n+n^2\right ) (c-a c x)^{7/2}}-\frac {3 a^2 \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{\frac {3+n}{2}} \left (1-\frac {1}{a x}\right )^{\frac {4-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x^3 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3+n}{2},\frac {3}{2},\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{2 \left (15+8 n+n^2\right ) (c-a c x)^{7/2}} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.56 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx=\frac {\left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{n/2} \left ((9+2 n-3 a x) (1+a x)+3 (-1+a x)^2 \left (\frac {-1+a x}{1+a x}\right )^{\frac {1+n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3+n}{2},\frac {3}{2},\frac {2}{1+a x}\right )\right )}{2 a c^3 (3+n) (5+n) (-1+a x)^2 \sqrt {c-a c x}} \]
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\[\int \frac {{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )}}{\left (-a c x +c \right )^{\frac {7}{2}}}d x\]
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\[ \int \frac {e^{n \coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx=\int { \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a c x + c\right )}^{\frac {7}{2}}} \,d x } \]
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Exception generated. \[ \int \frac {e^{n \coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx=\text {Exception raised: HeuristicGCDFailed} \]
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\[ \int \frac {e^{n \coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx=\int { \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a c x + c\right )}^{\frac {7}{2}}} \,d x } \]
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\[ \int \frac {e^{n \coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx=\int { \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a c x + c\right )}^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {e^{n \coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx=\int \frac {{\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}}{{\left (c-a\,c\,x\right )}^{7/2}} \,d x \]
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