Integrand size = 24, antiderivative size = 46 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx=\frac {(1+a x)^4 \left (c-a^2 c x^2\right )^{3/2}}{4 a^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3} \]
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Time = 0.14 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6327, 6328, 32} \[ \int e^{3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx=\frac {(a x+1)^4 \left (c-a^2 c x^2\right )^{3/2}}{4 a^4 x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}} \]
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Rule 32
Rule 6327
Rule 6328
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c-a^2 c x^2\right )^{3/2} \int e^{3 \coth ^{-1}(a x)} \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3 \, dx}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3} \\ & = \frac {\left (c-a^2 c x^2\right )^{3/2} \int (1+a x)^3 \, dx}{a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3} \\ & = \frac {(1+a x)^4 \left (c-a^2 c x^2\right )^{3/2}}{4 a^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.26 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx=-\frac {c \sqrt {c-a^2 c x^2} \left (4+6 a x+4 a^2 x^2+a^3 x^3\right )}{4 a \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Time = 0.51 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.04
method | result | size |
default | \(-\frac {\left (a x -1\right ) \left (a x +1\right )^{2} \sqrt {-c \left (a^{2} x^{2}-1\right )}\, c}{4 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} a}\) | \(48\) |
gosper | \(\frac {x \left (a^{3} x^{3}+4 a^{2} x^{2}+6 a x +4\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{4 \left (a x +1\right )^{3} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}\) | \(60\) |
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none
Time = 0.26 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.91 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx=-\frac {{\left (a^{3} c x^{4} + 4 \, a^{2} c x^{3} + 6 \, a c x^{2} + 4 \, c x\right )} \sqrt {-a^{2} c}}{4 \, a} \]
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\[ \int e^{3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx=\int \frac {\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (40) = 80\).
Time = 0.21 (sec) , antiderivative size = 97, normalized size of antiderivative = 2.11 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx=-\frac {{\left (a^{5} \sqrt {-c} c x^{5} + 3 \, a^{4} \sqrt {-c} c x^{4} + 2 \, a^{3} \sqrt {-c} c x^{3} - 2 \, a^{2} \sqrt {-c} c x^{2} - 4 \, \sqrt {-c} c\right )} {\left (a x + 1\right )}^{2}}{4 \, {\left (a^{3} x^{2} + 2 \, a^{2} x + a\right )} {\left (a x - 1\right )}} \]
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\[ \int e^{3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx=\int { \frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int e^{3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx=\int \frac {{\left (c-a^2\,c\,x^2\right )}^{3/2}}{{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \,d x \]
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