Integrand size = 24, antiderivative size = 121 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{9/2}} \, dx=\frac {2 (1-a x)}{9 a \left (c-a^2 c x^2\right )^{9/2}}-\frac {x}{9 c \left (c-a^2 c x^2\right )^{7/2}}-\frac {2 x}{15 c^2 \left (c-a^2 c x^2\right )^{5/2}}-\frac {8 x}{45 c^3 \left (c-a^2 c x^2\right )^{3/2}}-\frac {16 x}{45 c^4 \sqrt {c-a^2 c x^2}} \]
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Time = 0.10 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {6302, 6277, 667, 198, 197} \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{9/2}} \, dx=-\frac {16 x}{45 c^4 \sqrt {c-a^2 c x^2}}-\frac {8 x}{45 c^3 \left (c-a^2 c x^2\right )^{3/2}}-\frac {2 x}{15 c^2 \left (c-a^2 c x^2\right )^{5/2}}-\frac {x}{9 c \left (c-a^2 c x^2\right )^{7/2}}+\frac {2 (1-a x)}{9 a \left (c-a^2 c x^2\right )^{9/2}} \]
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Rule 197
Rule 198
Rule 667
Rule 6277
Rule 6302
Rubi steps \begin{align*} \text {integral}& = -\int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{9/2}} \, dx \\ & = -\left (c \int \frac {(1-a x)^2}{\left (c-a^2 c x^2\right )^{11/2}} \, dx\right ) \\ & = \frac {2 (1-a x)}{9 a \left (c-a^2 c x^2\right )^{9/2}}-\frac {7}{9} \int \frac {1}{\left (c-a^2 c x^2\right )^{9/2}} \, dx \\ & = \frac {2 (1-a x)}{9 a \left (c-a^2 c x^2\right )^{9/2}}-\frac {x}{9 c \left (c-a^2 c x^2\right )^{7/2}}-\frac {2 \int \frac {1}{\left (c-a^2 c x^2\right )^{7/2}} \, dx}{3 c} \\ & = \frac {2 (1-a x)}{9 a \left (c-a^2 c x^2\right )^{9/2}}-\frac {x}{9 c \left (c-a^2 c x^2\right )^{7/2}}-\frac {2 x}{15 c^2 \left (c-a^2 c x^2\right )^{5/2}}-\frac {8 \int \frac {1}{\left (c-a^2 c x^2\right )^{5/2}} \, dx}{15 c^2} \\ & = \frac {2 (1-a x)}{9 a \left (c-a^2 c x^2\right )^{9/2}}-\frac {x}{9 c \left (c-a^2 c x^2\right )^{7/2}}-\frac {2 x}{15 c^2 \left (c-a^2 c x^2\right )^{5/2}}-\frac {8 x}{45 c^3 \left (c-a^2 c x^2\right )^{3/2}}-\frac {16 \int \frac {1}{\left (c-a^2 c x^2\right )^{3/2}} \, dx}{45 c^3} \\ & = \frac {2 (1-a x)}{9 a \left (c-a^2 c x^2\right )^{9/2}}-\frac {x}{9 c \left (c-a^2 c x^2\right )^{7/2}}-\frac {2 x}{15 c^2 \left (c-a^2 c x^2\right )^{5/2}}-\frac {8 x}{45 c^3 \left (c-a^2 c x^2\right )^{3/2}}-\frac {16 x}{45 c^4 \sqrt {c-a^2 c x^2}} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.93 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{9/2}} \, dx=\frac {\sqrt {1-a^2 x^2} \left (10-25 a x-60 a^2 x^2+10 a^3 x^3+80 a^4 x^4+24 a^5 x^5-32 a^6 x^6-16 a^7 x^7\right )}{45 a c^4 (1-a x)^{5/2} (1+a x)^{9/2} \sqrt {c-a^2 c x^2}} \]
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Time = 0.62 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.66
method | result | size |
gosper | \(-\frac {\left (a x -1\right )^{2} \left (16 a^{7} x^{7}+32 a^{6} x^{6}-24 a^{5} x^{5}-80 a^{4} x^{4}-10 a^{3} x^{3}+60 a^{2} x^{2}+25 a x -10\right )}{45 a \left (-a^{2} c \,x^{2}+c \right )^{\frac {9}{2}}}\) | \(80\) |
trager | \(\frac {\left (16 a^{7} x^{7}+32 a^{6} x^{6}-24 a^{5} x^{5}-80 a^{4} x^{4}-10 a^{3} x^{3}+60 a^{2} x^{2}+25 a x -10\right ) \sqrt {-a^{2} c \,x^{2}+c}}{45 c^{5} \left (a x +1\right )^{5} \left (a x -1\right )^{3} a}\) | \(90\) |
default | \(\frac {x}{7 c \left (-a^{2} c \,x^{2}+c \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 c \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 c \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}+\frac {8 x}{15 c^{2} \sqrt {-a^{2} c \,x^{2}+c}}\right )}{7 c}}{c}-\frac {2 \left (-\frac {1}{9 a c \left (x +\frac {1}{a}\right ) \left (-a^{2} c \left (x +\frac {1}{a}\right )^{2}+2 \left (x +\frac {1}{a}\right ) a c \right )^{\frac {7}{2}}}+\frac {8 a \left (-\frac {-2 a^{2} c \left (x +\frac {1}{a}\right )+2 a c}{14 a^{2} c^{2} \left (-a^{2} c \left (x +\frac {1}{a}\right )^{2}+2 \left (x +\frac {1}{a}\right ) a c \right )^{\frac {7}{2}}}+\frac {-\frac {3 \left (-2 a^{2} c \left (x +\frac {1}{a}\right )+2 a c \right )}{35 a^{2} c^{2} \left (-a^{2} c \left (x +\frac {1}{a}\right )^{2}+2 \left (x +\frac {1}{a}\right ) a c \right )^{\frac {5}{2}}}+\frac {6 \left (-\frac {2 \left (-2 a^{2} c \left (x +\frac {1}{a}\right )+2 a c \right )}{15 a^{2} c^{2} \left (-a^{2} c \left (x +\frac {1}{a}\right )^{2}+2 \left (x +\frac {1}{a}\right ) a c \right )^{\frac {3}{2}}}-\frac {4 \left (-2 a^{2} c \left (x +\frac {1}{a}\right )+2 a c \right )}{15 a^{2} c^{3} \sqrt {-a^{2} c \left (x +\frac {1}{a}\right )^{2}+2 \left (x +\frac {1}{a}\right ) a c}}\right )}{7 c}}{c}\right )}{9}\right )}{a}\) | \(348\) |
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Time = 0.52 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.26 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{9/2}} \, dx=\frac {{\left (16 \, a^{7} x^{7} + 32 \, a^{6} x^{6} - 24 \, a^{5} x^{5} - 80 \, a^{4} x^{4} - 10 \, a^{3} x^{3} + 60 \, a^{2} x^{2} + 25 \, a x - 10\right )} \sqrt {-a^{2} c x^{2} + c}}{45 \, {\left (a^{9} c^{5} x^{8} + 2 \, a^{8} c^{5} x^{7} - 2 \, a^{7} c^{5} x^{6} - 6 \, a^{6} c^{5} x^{5} + 6 \, a^{4} c^{5} x^{3} + 2 \, a^{3} c^{5} x^{2} - 2 \, a^{2} c^{5} x - a c^{5}\right )}} \]
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\[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{9/2}} \, dx=\int \frac {a x - 1}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {9}{2}} \left (a x + 1\right )}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.97 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{9/2}} \, dx=\frac {2}{9 \, {\left ({\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}} a^{2} c x + {\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}} a c\right )}} - \frac {16 \, x}{45 \, \sqrt {-a^{2} c x^{2} + c} c^{4}} - \frac {8 \, x}{45 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} c^{3}} - \frac {2 \, x}{15 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} c^{2}} - \frac {x}{9 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}} c} \]
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\[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{9/2}} \, dx=\int { \frac {a x - 1}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {9}{2}} {\left (a x + 1\right )}} \,d x } \]
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Time = 4.26 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.46 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{9/2}} \, dx=\frac {5\,\sqrt {c-a^2\,c\,x^2}}{144\,a\,c^5\,{\left (a\,x+1\right )}^4}+\frac {\sqrt {c-a^2\,c\,x^2}}{72\,a\,c^5\,{\left (a\,x+1\right )}^5}+\frac {\sqrt {c-a^2\,c\,x^2}\,\left (\frac {31\,x}{120\,c^5}-\frac {5}{24\,a\,c^5}\right )}{{\left (a\,x-1\right )}^3\,{\left (a\,x+1\right )}^3}-\frac {\sqrt {c-a^2\,c\,x^2}\,\left (\frac {8\,x}{45\,c^5}+\frac {5}{144\,a\,c^5}\right )}{{\left (a\,x-1\right )}^2\,{\left (a\,x+1\right )}^2}+\frac {16\,x\,\sqrt {c-a^2\,c\,x^2}}{45\,c^5\,\left (a\,x-1\right )\,\left (a\,x+1\right )} \]
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