Integrand size = 24, antiderivative size = 108 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx=-\frac {5}{8} c x \sqrt {c-a^2 c x^2}-\frac {5 \left (c-a^2 c x^2\right )^{3/2}}{12 a}-\frac {(1-a x) \left (c-a^2 c x^2\right )^{3/2}}{4 a}-\frac {5 c^{3/2} \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{8 a} \]
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Time = 0.10 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {6302, 6277, 685, 655, 201, 223, 209} \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx=-\frac {5 c^{3/2} \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{8 a}-\frac {5}{8} c x \sqrt {c-a^2 c x^2}-\frac {(1-a x) \left (c-a^2 c x^2\right )^{3/2}}{4 a}-\frac {5 \left (c-a^2 c x^2\right )^{3/2}}{12 a} \]
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Rule 201
Rule 209
Rule 223
Rule 655
Rule 685
Rule 6277
Rule 6302
Rubi steps \begin{align*} \text {integral}& = -\int e^{-2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx \\ & = -\left (c \int (1-a x)^2 \sqrt {c-a^2 c x^2} \, dx\right ) \\ & = -\frac {(1-a x) \left (c-a^2 c x^2\right )^{3/2}}{4 a}-\frac {1}{4} (5 c) \int (1-a x) \sqrt {c-a^2 c x^2} \, dx \\ & = -\frac {5 \left (c-a^2 c x^2\right )^{3/2}}{12 a}-\frac {(1-a x) \left (c-a^2 c x^2\right )^{3/2}}{4 a}-\frac {1}{4} (5 c) \int \sqrt {c-a^2 c x^2} \, dx \\ & = -\frac {5}{8} c x \sqrt {c-a^2 c x^2}-\frac {5 \left (c-a^2 c x^2\right )^{3/2}}{12 a}-\frac {(1-a x) \left (c-a^2 c x^2\right )^{3/2}}{4 a}-\frac {1}{8} \left (5 c^2\right ) \int \frac {1}{\sqrt {c-a^2 c x^2}} \, dx \\ & = -\frac {5}{8} c x \sqrt {c-a^2 c x^2}-\frac {5 \left (c-a^2 c x^2\right )^{3/2}}{12 a}-\frac {(1-a x) \left (c-a^2 c x^2\right )^{3/2}}{4 a}-\frac {1}{8} \left (5 c^2\right ) \text {Subst}\left (\int \frac {1}{1+a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c-a^2 c x^2}}\right ) \\ & = -\frac {5}{8} c x \sqrt {c-a^2 c x^2}-\frac {5 \left (c-a^2 c x^2\right )^{3/2}}{12 a}-\frac {(1-a x) \left (c-a^2 c x^2\right )^{3/2}}{4 a}-\frac {5 c^{3/2} \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{8 a} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.08 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx=\frac {c \sqrt {c-a^2 c x^2} \left (\sqrt {1+a x} \left (-16+7 a x+25 a^2 x^2-22 a^3 x^3+6 a^4 x^4\right )+30 \sqrt {1-a x} \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{24 a \sqrt {1-a x} \sqrt {1-a^2 x^2}} \]
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Time = 0.63 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.83
method | result | size |
risch | \(\frac {\left (6 a^{3} x^{3}-16 a^{2} x^{2}+9 a x +16\right ) \left (a^{2} x^{2}-1\right ) c^{2}}{24 a \sqrt {-c \left (a^{2} x^{2}-1\right )}}-\frac {5 \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right ) c^{2}}{8 \sqrt {a^{2} c}}\) | \(90\) |
default | \(\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {a^{2} c}}\right )}{4}-\frac {2 \left (\frac {\left (-a^{2} c \left (x +\frac {1}{a}\right )^{2}+2 \left (x +\frac {1}{a}\right ) a c \right )^{\frac {3}{2}}}{3}+a c \left (-\frac {\left (-2 a^{2} c \left (x +\frac {1}{a}\right )+2 a c \right ) \sqrt {-a^{2} c \left (x +\frac {1}{a}\right )^{2}+2 \left (x +\frac {1}{a}\right ) a c}}{4 a^{2} c}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \left (x +\frac {1}{a}\right )^{2}+2 \left (x +\frac {1}{a}\right ) a c}}\right )}{2 \sqrt {a^{2} c}}\right )\right )}{a}\) | \(202\) |
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Time = 0.26 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.67 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx=\left [\frac {15 \, \sqrt {-c} c \log \left (2 \, a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right ) - 2 \, {\left (6 \, a^{3} c x^{3} - 16 \, a^{2} c x^{2} + 9 \, a c x + 16 \, c\right )} \sqrt {-a^{2} c x^{2} + c}}{48 \, a}, \frac {15 \, c^{\frac {3}{2}} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right ) - {\left (6 \, a^{3} c x^{3} - 16 \, a^{2} c x^{2} + 9 \, a c x + 16 \, c\right )} \sqrt {-a^{2} c x^{2} + c}}{24 \, a}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 248 vs. \(2 (97) = 194\).
Time = 2.41 (sec) , antiderivative size = 248, normalized size of antiderivative = 2.30 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx=\begin {cases} \frac {2 c \left (\begin {cases} \left (\frac {a^{2} x^{2}}{3} - \frac {1}{3}\right ) \sqrt {- a^{2} c x^{2} + c} & \text {for}\: c \neq 0 \\\frac {a^{2} \sqrt {c} x^{2}}{2} & \text {otherwise} \end {cases}\right ) - c \left (\begin {cases} \frac {c \left (\begin {cases} \frac {\log {\left (- 2 a c x + 2 \sqrt {- c} \sqrt {- a^{2} c x^{2} + c} \right )}}{\sqrt {- c}} & \text {for}\: c \neq 0 \\\frac {a x \log {\left (a x \right )}}{\sqrt {- a^{2} c x^{2}}} & \text {otherwise} \end {cases}\right )}{8} + \left (\frac {a^{3} x^{3}}{4} - \frac {a x}{8}\right ) \sqrt {- a^{2} c x^{2} + c} & \text {for}\: c \neq 0 \\\frac {a^{3} \sqrt {c} x^{3}}{3} & \text {otherwise} \end {cases}\right ) - c \left (\begin {cases} \frac {a x \sqrt {- a^{2} c x^{2} + c}}{2} + \frac {c \left (\begin {cases} \frac {\log {\left (- 2 a c x + 2 \sqrt {- c} \sqrt {- a^{2} c x^{2} + c} \right )}}{\sqrt {- c}} & \text {for}\: c \neq 0 \\\frac {a x \log {\left (a x \right )}}{\sqrt {- a^{2} c x^{2}}} & \text {otherwise} \end {cases}\right )}{2} & \text {for}\: c \neq 0 \\a \sqrt {c} x & \text {otherwise} \end {cases}\right )}{a} & \text {for}\: a \neq 0 \\- c^{\frac {3}{2}} x & \text {otherwise} \end {cases} \]
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Time = 0.30 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.20 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx=\frac {1}{4} \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x - \sqrt {a^{2} c x^{2} + 4 \, a c x + 3 \, c} c x + \frac {3}{8} \, \sqrt {-a^{2} c x^{2} + c} c x + \frac {c^{3} \arcsin \left (a x + 2\right )}{a \left (-c\right )^{\frac {3}{2}}} + \frac {3 \, c^{\frac {3}{2}} \arcsin \left (a x\right )}{8 \, a} - \frac {2 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}{3 \, a} - \frac {2 \, \sqrt {a^{2} c x^{2} + 4 \, a c x + 3 \, c} c}{a} \]
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Time = 0.28 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.79 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx=-\frac {1}{24} \, \sqrt {-a^{2} c x^{2} + c} {\left ({\left (2 \, {\left (3 \, a^{2} c x - 8 \, a c\right )} x + 9 \, c\right )} x + \frac {16 \, c}{a}\right )} + \frac {5 \, c^{2} \log \left ({\left | -\sqrt {-a^{2} c} x + \sqrt {-a^{2} c x^{2} + c} \right |}\right )}{8 \, \sqrt {-c} {\left | a \right |}} \]
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Timed out. \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx=\int \frac {{\left (c-a^2\,c\,x^2\right )}^{3/2}\,\left (a\,x-1\right )}{a\,x+1} \,d x \]
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