Integrand size = 20, antiderivative size = 40 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx=-\frac {4}{7 a (c-a c x)^{7/2}}+\frac {2}{5 a c (c-a c x)^{5/2}} \]
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Time = 0.07 (sec) , antiderivative size = 40, 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 = {6302, 6265, 21, 45} \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx=\frac {2}{5 a c (c-a c x)^{5/2}}-\frac {4}{7 a (c-a c x)^{7/2}} \]
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Rule 21
Rule 45
Rule 6265
Rule 6302
Rubi steps \begin{align*} \text {integral}& = -\int \frac {e^{2 \text {arctanh}(a x)}}{(c-a c x)^{7/2}} \, dx \\ & = -\int \frac {1+a x}{(1-a x) (c-a c x)^{7/2}} \, dx \\ & = -\left (c \int \frac {1+a x}{(c-a c x)^{9/2}} \, dx\right ) \\ & = -\left (c \int \left (\frac {2}{(c-a c x)^{9/2}}-\frac {1}{c (c-a c x)^{7/2}}\right ) \, dx\right ) \\ & = -\frac {4}{7 a (c-a c x)^{7/2}}+\frac {2}{5 a c (c-a c x)^{5/2}} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.85 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx=\frac {6+14 a x}{35 a c^3 (-1+a x)^3 \sqrt {c-a c x}} \]
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Time = 0.51 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.52
method | result | size |
gosper | \(-\frac {2 \left (7 a x +3\right )}{35 a \left (-a c x +c \right )^{\frac {7}{2}}}\) | \(21\) |
trager | \(-\frac {2 \left (7 a x +3\right ) \sqrt {-a c x +c}}{35 c^{4} \left (a x -1\right )^{4} a}\) | \(31\) |
pseudoelliptic | \(\frac {\frac {2 a x}{5}+\frac {6}{35}}{a \,c^{3} \left (a x -1\right )^{3} \sqrt {-c \left (a x -1\right )}}\) | \(32\) |
derivativedivides | \(-\frac {2 \left (-\frac {1}{5 \left (-a c x +c \right )^{\frac {5}{2}}}+\frac {2 c}{7 \left (-a c x +c \right )^{\frac {7}{2}}}\right )}{c a}\) | \(33\) |
default | \(\frac {\frac {2}{5 \left (-a c x +c \right )^{\frac {5}{2}}}-\frac {4 c}{7 \left (-a c x +c \right )^{\frac {7}{2}}}}{a c}\) | \(33\) |
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Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (32) = 64\).
Time = 0.25 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.65 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx=-\frac {2 \, \sqrt {-a c x + c} {\left (7 \, a x + 3\right )}}{35 \, {\left (a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, a^{3} c^{4} x^{2} - 4 \, a^{2} c^{4} x + a c^{4}\right )}} \]
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Time = 1.92 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.45 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx=\begin {cases} - \frac {2 \cdot \left (\frac {2 c}{7 \left (- a c x + c\right )^{\frac {7}{2}}} - \frac {1}{5 \left (- a c x + c\right )^{\frac {5}{2}}}\right )}{a c} & \text {for}\: a c \neq 0 \\\frac {\begin {cases} - x & \text {for}\: a = 0 \\\frac {a x + 2 \log {\left (a x - 1 \right )} - 1}{a} & \text {otherwise} \end {cases}}{c^{\frac {7}{2}}} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.65 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx=-\frac {2 \, {\left (7 \, a c x + 3 \, c\right )}}{35 \, {\left (-a c x + c\right )}^{\frac {7}{2}} a c} \]
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Time = 0.27 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.90 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx=\frac {2 \, {\left (7 \, a c x + 3 \, c\right )}}{35 \, {\left (a c x - c\right )}^{3} \sqrt {-a c x + c} a c} \]
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Time = 3.93 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.50 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx=-\frac {14\,a\,x+6}{35\,a\,{\left (c-a\,c\,x\right )}^{7/2}} \]
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