Integrand size = 24, antiderivative size = 278 \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{2+\frac {n}{2}} \, dx=-\frac {\left (56+14 n+n^2\right ) \left (1-\frac {1}{a x}\right )^{-2-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} (c-a c x)^{\frac {4+n}{2}}}{a (4+n) (6+n)}+\frac {2 \left (56+14 n+n^2\right ) \left (1-\frac {1}{a x}\right )^{-2-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} (c-a c x)^{\frac {4+n}{2}}}{a^2 (6+n) \left (8+6 n+n^2\right ) x}+\frac {(8+n) \left (1-\frac {1}{a x}\right )^{-2-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x (c-a c x)^{\frac {4+n}{2}}}{6+n}-\frac {\left (a-\frac {1}{x}\right ) \left (1-\frac {1}{a x}\right )^{-2-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x (c-a c x)^{\frac {4+n}{2}}}{a} \]
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Time = 0.25 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6311, 6316, 92, 80, 47, 37} \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{2+\frac {n}{2}} \, dx=\frac {2 \left (n^2+14 n+56\right ) \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-2} (c-a c x)^{\frac {n+4}{2}}}{a^2 (n+6) \left (n^2+6 n+8\right ) x}-\frac {\left (n^2+14 n+56\right ) \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-2} (c-a c x)^{\frac {n+4}{2}}}{a (n+4) (n+6)}+\frac {(n+8) x \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-2} (c-a c x)^{\frac {n+4}{2}}}{n+6}-\frac {x \left (a-\frac {1}{x}\right ) \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-2} (c-a c x)^{\frac {n+4}{2}}}{a} \]
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Rule 37
Rule 47
Rule 80
Rule 92
Rule 6311
Rule 6316
Rubi steps \begin{align*} \text {integral}& = \left (\left (1-\frac {1}{a x}\right )^{-2-\frac {n}{2}} x^{-2-\frac {n}{2}} (c-a c x)^{2+\frac {n}{2}}\right ) \int e^{n \coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right )^{2+\frac {n}{2}} x^{2+\frac {n}{2}} \, dx \\ & = -\left (\left (\left (1-\frac {1}{a x}\right )^{-2-\frac {n}{2}} \left (\frac {1}{x}\right )^{2+\frac {n}{2}} (c-a c x)^{2+\frac {n}{2}}\right ) \text {Subst}\left (\int x^{-4-\frac {n}{2}} \left (1-\frac {x}{a}\right )^2 \left (1+\frac {x}{a}\right )^{n/2} \, dx,x,\frac {1}{x}\right )\right ) \\ & = -\frac {\left (a-\frac {1}{x}\right ) \left (1-\frac {1}{a x}\right )^{-2-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x (c-a c x)^{\frac {4+n}{2}}}{a}+\left (a \left (1-\frac {1}{a x}\right )^{-2-\frac {n}{2}} \left (\frac {1}{x}\right )^{2+\frac {n}{2}} (c-a c x)^{2+\frac {n}{2}}\right ) \text {Subst}\left (\int x^{-4-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{n/2} \left (-\frac {8+n}{2 a}+\frac {(4+n) x}{2 a^2}\right ) \, dx,x,\frac {1}{x}\right ) \\ & = \frac {(8+n) \left (1-\frac {1}{a x}\right )^{-2-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x (c-a c x)^{\frac {4+n}{2}}}{6+n}-\frac {\left (a-\frac {1}{x}\right ) \left (1-\frac {1}{a x}\right )^{-2-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x (c-a c x)^{\frac {4+n}{2}}}{a}+\frac {\left (\left (56+14 n+n^2\right ) \left (1-\frac {1}{a x}\right )^{-2-\frac {n}{2}} \left (\frac {1}{x}\right )^{2+\frac {n}{2}} (c-a c x)^{2+\frac {n}{2}}\right ) \text {Subst}\left (\int x^{-3-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{n/2} \, dx,x,\frac {1}{x}\right )}{2 a (6+n)} \\ & = -\frac {\left (56+14 n+n^2\right ) \left (1-\frac {1}{a x}\right )^{-2-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} (c-a c x)^{\frac {4+n}{2}}}{a (4+n) (6+n)}+\frac {(8+n) \left (1-\frac {1}{a x}\right )^{-2-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x (c-a c x)^{\frac {4+n}{2}}}{6+n}-\frac {\left (a-\frac {1}{x}\right ) \left (1-\frac {1}{a x}\right )^{-2-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x (c-a c x)^{\frac {4+n}{2}}}{a}-\frac {\left (\left (56+14 n+n^2\right ) \left (1-\frac {1}{a x}\right )^{-2-\frac {n}{2}} \left (\frac {1}{x}\right )^{2+\frac {n}{2}} (c-a c x)^{2+\frac {n}{2}}\right ) \text {Subst}\left (\int x^{-2-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{n/2} \, dx,x,\frac {1}{x}\right )}{a^2 (4+n) (6+n)} \\ & = -\frac {\left (56+14 n+n^2\right ) \left (1-\frac {1}{a x}\right )^{-2-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} (c-a c x)^{\frac {4+n}{2}}}{a (4+n) (6+n)}+\frac {2 \left (56+14 n+n^2\right ) \left (1-\frac {1}{a x}\right )^{-2-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} (c-a c x)^{\frac {4+n}{2}}}{a^2 (2+n) (4+n) (6+n) x}+\frac {(8+n) \left (1-\frac {1}{a x}\right )^{-2-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x (c-a c x)^{\frac {4+n}{2}}}{6+n}-\frac {\left (a-\frac {1}{x}\right ) \left (1-\frac {1}{a x}\right )^{-2-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x (c-a c x)^{\frac {4+n}{2}}}{a} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.42 \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{2+\frac {n}{2}} \, dx=\frac {2 c^2 \left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{n/2} (1+a x) (c-a c x)^{n/2} \left (n^2 (-1+a x)^2+8 \left (7-4 a x+a^2 x^2\right )+2 n \left (7-10 a x+3 a^2 x^2\right )\right )}{a (2+n) (4+n) (6+n)} \]
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Time = 1.57 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.37
method | result | size |
gosper | \(\frac {2 \left (a x +1\right ) \left (a^{2} n^{2} x^{2}+6 n \,x^{2} a^{2}+8 a^{2} x^{2}-2 n^{2} x a -20 a n x -32 a x +n^{2}+14 n +56\right ) {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} \left (-a c x +c \right )^{2+\frac {n}{2}}}{\left (a x -1\right )^{2} a \left (n^{3}+12 n^{2}+44 n +48\right )}\) | \(104\) |
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Time = 0.26 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.67 \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{2+\frac {n}{2}} \, dx=\frac {2 \, {\left ({\left (a^{3} n^{2} + 6 \, a^{3} n + 8 \, a^{3}\right )} x^{3} - {\left (a^{2} n^{2} + 14 \, a^{2} n + 24 \, a^{2}\right )} x^{2} + n^{2} - {\left (a n^{2} + 6 \, a n - 24 \, a\right )} x + 14 \, n + 56\right )} {\left (-a c x + c\right )}^{\frac {1}{2} \, n + 2} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a n^{3} + 12 \, a n^{2} + {\left (a^{3} n^{3} + 12 \, a^{3} n^{2} + 44 \, a^{3} n + 48 \, a^{3}\right )} x^{2} + 44 \, a n - 2 \, {\left (a^{2} n^{3} + 12 \, a^{2} n^{2} + 44 \, a^{2} n + 48 \, a^{2}\right )} x + 48 \, a} \]
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\[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{2+\frac {n}{2}} \, dx=\int \left (- c \left (a x - 1\right )\right )^{\frac {n}{2} + 2} e^{n \operatorname {acoth}{\left (a x \right )}}\, dx \]
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Time = 0.22 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.44 \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{2+\frac {n}{2}} \, dx=\frac {2 \, {\left ({\left (n^{2} + 6 \, n + 8\right )} a^{3} \left (-c\right )^{\frac {1}{2} \, n} c^{2} x^{3} - {\left (n^{2} + 14 \, n + 24\right )} a^{2} \left (-c\right )^{\frac {1}{2} \, n} c^{2} x^{2} - {\left (n^{2} + 6 \, n - 24\right )} a \left (-c\right )^{\frac {1}{2} \, n} c^{2} x + {\left (n^{2} + 14 \, n + 56\right )} \left (-c\right )^{\frac {1}{2} \, n} c^{2}\right )} {\left (a x + 1\right )}^{\frac {1}{2} \, n}}{{\left (n^{3} + 12 \, n^{2} + 44 \, n + 48\right )} a} \]
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\[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{2+\frac {n}{2}} \, dx=\int { {\left (-a c x + c\right )}^{\frac {1}{2} \, n + 2} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \]
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Time = 4.59 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.80 \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{2+\frac {n}{2}} \, dx=\frac {{\left (\frac {a\,x+1}{a\,x}\right )}^{n/2}\,\left (\frac {x^3\,{\left (c-a\,c\,x\right )}^{\frac {n}{2}+2}\,\left (2\,n^2+12\,n+16\right )}{n^3+12\,n^2+44\,n+48}+\frac {{\left (c-a\,c\,x\right )}^{\frac {n}{2}+2}\,\left (2\,n^2+28\,n+112\right )}{a^3\,\left (n^3+12\,n^2+44\,n+48\right )}-\frac {2\,x\,{\left (c-a\,c\,x\right )}^{\frac {n}{2}+2}\,\left (n^2+6\,n-24\right )}{a^2\,\left (n^3+12\,n^2+44\,n+48\right )}-\frac {x^2\,{\left (c-a\,c\,x\right )}^{\frac {n}{2}+2}\,\left (2\,n^2+28\,n+48\right )}{a\,\left (n^3+12\,n^2+44\,n+48\right )}\right )}{{\left (\frac {a\,x-1}{a\,x}\right )}^{n/2}\,\left (\frac {1}{a^2}-\frac {2\,x}{a}+x^2\right )} \]
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