Integrand size = 24, antiderivative size = 74 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=-\frac {2 (1+a x)}{5 a \left (c-a^2 c x^2\right )^{5/2}}-\frac {x}{5 c \left (c-a^2 c x^2\right )^{3/2}}-\frac {2 x}{5 c^2 \sqrt {c-a^2 c x^2}} \]
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Time = 0.11 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {6302, 6276, 667, 198, 197} \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=-\frac {2 x}{5 c^2 \sqrt {c-a^2 c x^2}}-\frac {x}{5 c \left (c-a^2 c x^2\right )^{3/2}}-\frac {2 (a x+1)}{5 a \left (c-a^2 c x^2\right )^{5/2}} \]
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Rule 197
Rule 198
Rule 667
Rule 6276
Rule 6302
Rubi steps \begin{align*} \text {integral}& = -\int \frac {e^{2 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx \\ & = -\left (c \int \frac {(1+a x)^2}{\left (c-a^2 c x^2\right )^{7/2}} \, dx\right ) \\ & = -\frac {2 (1+a x)}{5 a \left (c-a^2 c x^2\right )^{5/2}}-\frac {3}{5} \int \frac {1}{\left (c-a^2 c x^2\right )^{5/2}} \, dx \\ & = -\frac {2 (1+a x)}{5 a \left (c-a^2 c x^2\right )^{5/2}}-\frac {x}{5 c \left (c-a^2 c x^2\right )^{3/2}}-\frac {2 \int \frac {1}{\left (c-a^2 c x^2\right )^{3/2}} \, dx}{5 c} \\ & = -\frac {2 (1+a x)}{5 a \left (c-a^2 c x^2\right )^{5/2}}-\frac {x}{5 c \left (c-a^2 c x^2\right )^{3/2}}-\frac {2 x}{5 c^2 \sqrt {c-a^2 c x^2}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.72 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=-\frac {2+a x-4 a^2 x^2+2 a^3 x^3}{5 a c^2 (-1+a x)^2 \sqrt {c-a^2 c x^2}} \]
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Time = 0.61 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.64
method | result | size |
gosper | \(-\frac {\left (a x +1\right )^{2} \left (2 a^{3} x^{3}-4 a^{2} x^{2}+a x +2\right )}{5 a \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}\) | \(47\) |
trager | \(\frac {\left (2 a^{3} x^{3}-4 a^{2} x^{2}+a x +2\right ) \sqrt {-a^{2} c \,x^{2}+c}}{5 c^{3} \left (a x -1\right )^{3} a \left (a x +1\right )}\) | \(57\) |
default | \(\frac {x}{3 c \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}+\frac {2 x}{3 c^{2} \sqrt {-a^{2} c \,x^{2}+c}}+\frac {\frac {2}{5 a c \left (x -\frac {1}{a}\right ) \left (-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c \right )^{\frac {3}{2}}}-\frac {8 a \left (-\frac {-2 a^{2} c \left (x -\frac {1}{a}\right )-2 a c}{6 a^{2} c^{2} \left (-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c \right )^{\frac {3}{2}}}-\frac {-2 a^{2} c \left (x -\frac {1}{a}\right )-2 a c}{3 a^{2} c^{3} \sqrt {-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c}}\right )}{5}}{a}\) | \(206\) |
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Time = 0.29 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.01 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\frac {{\left (2 \, a^{3} x^{3} - 4 \, a^{2} x^{2} + a x + 2\right )} \sqrt {-a^{2} c x^{2} + c}}{5 \, {\left (a^{5} c^{3} x^{4} - 2 \, a^{4} c^{3} x^{3} + 2 \, a^{2} c^{3} x - a c^{3}\right )}} \]
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\[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int \frac {a x + 1}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}} \left (a x - 1\right )}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.08 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\frac {2}{5 \, {\left ({\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} a^{2} c x - {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} a c\right )}} - \frac {2 \, x}{5 \, \sqrt {-a^{2} c x^{2} + c} c^{2}} - \frac {x}{5 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} c} \]
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\[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {a x + 1}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} {\left (a x - 1\right )}} \,d x } \]
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Time = 4.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.76 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\frac {\sqrt {c-a^2\,c\,x^2}\,\left (2\,a^3\,x^3-4\,a^2\,x^2+a\,x+2\right )}{5\,a\,c^3\,{\left (a\,x-1\right )}^3\,\left (a\,x+1\right )} \]
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