Integrand size = 27, antiderivative size = 102 \[ \int \frac {e^{n \coth ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=-\frac {e^{n \coth ^{-1}(a x)} (n-3 a x)}{a^3 c \left (9-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}+\frac {e^{n \coth ^{-1}(a x)} \left (3-n^2\right ) (n-a x)}{a^3 c^2 \left (9-10 n^2+n^4\right ) \sqrt {c-a^2 c x^2}} \]
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Time = 0.14 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {6324, 6319} \[ \int \frac {e^{n \coth ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\frac {\left (3-n^2\right ) (n-a x) e^{n \coth ^{-1}(a x)}}{a^3 c^2 \left (n^4-10 n^2+9\right ) \sqrt {c-a^2 c x^2}}-\frac {(n-3 a x) e^{n \coth ^{-1}(a x)}}{a^3 c \left (9-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}} \]
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Rule 6319
Rule 6324
Rubi steps \begin{align*} \text {integral}& = -\frac {e^{n \coth ^{-1}(a x)} (n-3 a x)}{a^3 c \left (9-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}-\frac {\left (3-n^2\right ) \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx}{a^2 c \left (9-n^2\right )} \\ & = -\frac {e^{n \coth ^{-1}(a x)} (n-3 a x)}{a^3 c \left (9-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}+\frac {e^{n \coth ^{-1}(a x)} \left (3-n^2\right ) (n-a x)}{a^3 c^2 \left (9-10 n^2+n^4\right ) \sqrt {c-a^2 c x^2}} \\ \end{align*}
Time = 0.96 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.07 \[ \int \frac {e^{n \coth ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\frac {e^{n \coth ^{-1}(a x)} \left (10 n-2 n^3-9 a x+a n^2 x-2 n \left (-1+n^2\right ) \cosh \left (2 \coth ^{-1}(a x)\right )+3 a \left (-1+n^2\right ) \sqrt {1-\frac {1}{a^2 x^2}} x \cosh \left (3 \coth ^{-1}(a x)\right )\right )}{4 a^3 c^2 \left (9-10 n^2+n^4\right ) \sqrt {c-a^2 c x^2}} \]
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Time = 0.58 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.94
method | result | size |
gosper | \(\frac {\left (a x -1\right ) \left (a x +1\right ) \left (a^{3} n^{2} x^{3}-a^{2} n^{3} x^{2}-3 a^{3} x^{3}+3 n \,x^{2} a^{2}+2 n^{2} x a -2 n \right ) {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )}}{\left (n^{4}-10 n^{2}+9\right ) a^{3} \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}\) | \(96\) |
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Time = 0.27 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.76 \[ \int \frac {e^{n \coth ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=-\frac {\sqrt {-a^{2} c x^{2} + c} {\left (2 \, a n^{2} x + {\left (a^{3} n^{2} - 3 \, a^{3}\right )} x^{3} - {\left (a^{2} n^{3} - 3 \, a^{2} n\right )} x^{2} - 2 \, n\right )} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a^{3} c^{3} n^{4} - 10 \, a^{3} c^{3} n^{2} + 9 \, a^{3} c^{3} + {\left (a^{7} c^{3} n^{4} - 10 \, a^{7} c^{3} n^{2} + 9 \, a^{7} c^{3}\right )} x^{4} - 2 \, {\left (a^{5} c^{3} n^{4} - 10 \, a^{5} c^{3} n^{2} + 9 \, a^{5} c^{3}\right )} x^{2}} \]
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\[ \int \frac {e^{n \coth ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int \frac {x^{2} e^{n \operatorname {acoth}{\left (a x \right )}}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {e^{n \coth ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {x^{2} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {e^{n \coth ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {x^{2} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]
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Time = 4.72 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.72 \[ \int \frac {e^{n \coth ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\frac {{\left (\frac {a\,x+1}{a\,x}\right )}^{n/2}\,\left (\frac {2\,n}{a^5\,c^2\,\left (n^4-10\,n^2+9\right )}-\frac {x^3\,\left (n^2-3\right )}{a^2\,c^2\,\left (n^4-10\,n^2+9\right )}-\frac {2\,n^2\,x}{a^4\,c^2\,\left (n^4-10\,n^2+9\right )}+\frac {n\,x^2\,\left (n^2-3\right )}{a^3\,c^2\,\left (n^4-10\,n^2+9\right )}\right )}{\left (\frac {\sqrt {c-a^2\,c\,x^2}}{a^2}-x^2\,\sqrt {c-a^2\,c\,x^2}\right )\,{\left (\frac {a\,x-1}{a\,x}\right )}^{n/2}} \]
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