Integrand size = 24, antiderivative size = 124 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=-\frac {(1-a x)^2}{3 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x}+\frac {2 (1-a x)^2 (1+a x) (5+2 a x)}{3 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}+\frac {2 (1-a x)^{3/2} (1+a x)^{3/2} \arcsin (a x)}{a^4 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3} \]
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Time = 0.33 (sec) , antiderivative size = 124, 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, 6294, 6264, 100, 148, 41, 222} \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=-\frac {(1-a x)^2}{3 a^2 x \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}+\frac {2 (a x+1)^{3/2} (1-a x)^{3/2} \arcsin (a x)}{a^4 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}+\frac {2 (a x+1) (2 a x+5) (1-a x)^2}{3 a^4 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \]
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Rule 41
Rule 100
Rule 148
Rule 222
Rule 6264
Rule 6294
Rule 6302
Rubi steps \begin{align*} \text {integral}& = -\int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx \\ & = -\frac {\left ((1-a x)^{3/2} (1+a x)^{3/2}\right ) \int \frac {e^{-2 \text {arctanh}(a x)} x^3}{(1-a x)^{3/2} (1+a x)^{3/2}} \, dx}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3} \\ & = -\frac {\left ((1-a x)^{3/2} (1+a x)^{3/2}\right ) \int \frac {x^3}{\sqrt {1-a x} (1+a x)^{5/2}} \, dx}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3} \\ & = -\frac {(1-a x)^2}{3 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x}+\frac {\left ((1-a x)^{3/2} (1+a x)^{3/2}\right ) \int \frac {x (2-4 a x)}{\sqrt {1-a x} (1+a x)^{3/2}} \, dx}{3 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3} \\ & = -\frac {(1-a x)^2}{3 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x}+\frac {2 (1-a x)^2 (1+a x) (5+2 a x)}{3 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}+\frac {\left (2 (1-a x)^{3/2} (1+a x)^{3/2}\right ) \int \frac {1}{\sqrt {1-a x} \sqrt {1+a x}} \, dx}{a^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3} \\ & = -\frac {(1-a x)^2}{3 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x}+\frac {2 (1-a x)^2 (1+a x) (5+2 a x)}{3 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}+\frac {\left (2 (1-a x)^{3/2} (1+a x)^{3/2}\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{a^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3} \\ & = -\frac {(1-a x)^2}{3 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x}+\frac {2 (1-a x)^2 (1+a x) (5+2 a x)}{3 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}+\frac {2 (1-a x)^{3/2} (1+a x)^{3/2} \arcsin (a x)}{a^4 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.77 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\frac {-10-4 a x+11 a^2 x^2+3 a^3 x^3-6 (1+a x) \sqrt {-1+a^2 x^2} \log \left (a x+\sqrt {-1+a^2 x^2}\right )}{3 a^2 c \sqrt {c-\frac {c}{a^2 x^2}} x (1+a x)} \]
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Time = 0.54 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.63
method | result | size |
risch | \(\frac {a^{2} x^{2}-1}{a^{2} c x \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}+\frac {\left (-\frac {2 \ln \left (\frac {a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-c}\right )}{a^{3} \sqrt {a^{2} c}}-\frac {\sqrt {a^{2} c \left (x +\frac {1}{a}\right )^{2}-2 \left (x +\frac {1}{a}\right ) a c}}{3 a^{6} c \left (x +\frac {1}{a}\right )^{2}}+\frac {8 \sqrt {a^{2} c \left (x +\frac {1}{a}\right )^{2}-2 \left (x +\frac {1}{a}\right ) a c}}{3 a^{5} c \left (x +\frac {1}{a}\right )}\right ) a^{2} \sqrt {c \left (a^{2} x^{2}-1\right )}}{c x \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}\) | \(202\) |
default | \(\frac {\left (3 c^{\frac {3}{2}} \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}\, a^{3} x^{3}+15 x^{2} a^{2} c^{\frac {3}{2}} \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}-4 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, c^{\frac {3}{2}} a^{2} x^{2}-6 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, \ln \left (\sqrt {c}\, x +\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}\, a^{2} c x -4 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, c^{\frac {3}{2}} a x -6 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, \ln \left (\sqrt {c}\, x +\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}\, a c -12 c^{\frac {3}{2}} \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}+2 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, c^{\frac {3}{2}}\right ) \left (a x -1\right )}{3 \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}\, x^{3} {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )}^{\frac {3}{2}} a^{4} c^{\frac {3}{2}}}\) | \(326\) |
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Time = 0.27 (sec) , antiderivative size = 279, normalized size of antiderivative = 2.25 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\left [\frac {3 \, {\left (a^{2} x^{2} + 2 \, a x + 1\right )} \sqrt {c} \log \left (2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right ) + {\left (3 \, a^{3} x^{3} + 14 \, a^{2} x^{2} + 10 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{3 \, {\left (a^{3} c^{2} x^{2} + 2 \, a^{2} c^{2} x + a c^{2}\right )}}, \frac {6 \, {\left (a^{2} x^{2} + 2 \, a x + 1\right )} \sqrt {-c} \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) + {\left (3 \, a^{3} x^{3} + 14 \, a^{2} x^{2} + 10 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{3 \, {\left (a^{3} c^{2} x^{2} + 2 \, a^{2} c^{2} x + a c^{2}\right )}}\right ] \]
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\[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\int \frac {a x - 1}{\left (- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )\right )^{\frac {3}{2}} \left (a x + 1\right )}\, dx \]
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\[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\int { \frac {a x - 1}{{\left (a x + 1\right )} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {3}{2}}} \,d x } \]
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Exception generated. \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\int \frac {a\,x-1}{{\left (c-\frac {c}{a^2\,x^2}\right )}^{3/2}\,\left (a\,x+1\right )} \,d x \]
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