Integrand size = 24, antiderivative size = 97 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=-\frac {2 (1+a x)}{7 a \left (c-a^2 c x^2\right )^{7/2}}-\frac {x}{7 c \left (c-a^2 c x^2\right )^{5/2}}-\frac {4 x}{21 c^2 \left (c-a^2 c x^2\right )^{3/2}}-\frac {8 x}{21 c^3 \sqrt {c-a^2 c x^2}} \]
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Time = 0.09 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {6302, 6276, 667, 198, 197} \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=-\frac {8 x}{21 c^3 \sqrt {c-a^2 c x^2}}-\frac {4 x}{21 c^2 \left (c-a^2 c x^2\right )^{3/2}}-\frac {x}{7 c \left (c-a^2 c x^2\right )^{5/2}}-\frac {2 (a x+1)}{7 a \left (c-a^2 c x^2\right )^{7/2}} \]
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Rule 197
Rule 198
Rule 667
Rule 6276
Rule 6302
Rubi steps \begin{align*} \text {integral}& = -\int \frac {e^{2 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx \\ & = -\left (c \int \frac {(1+a x)^2}{\left (c-a^2 c x^2\right )^{9/2}} \, dx\right ) \\ & = -\frac {2 (1+a x)}{7 a \left (c-a^2 c x^2\right )^{7/2}}-\frac {5}{7} \int \frac {1}{\left (c-a^2 c x^2\right )^{7/2}} \, dx \\ & = -\frac {2 (1+a x)}{7 a \left (c-a^2 c x^2\right )^{7/2}}-\frac {x}{7 c \left (c-a^2 c x^2\right )^{5/2}}-\frac {4 \int \frac {1}{\left (c-a^2 c x^2\right )^{5/2}} \, dx}{7 c} \\ & = -\frac {2 (1+a x)}{7 a \left (c-a^2 c x^2\right )^{7/2}}-\frac {x}{7 c \left (c-a^2 c x^2\right )^{5/2}}-\frac {4 x}{21 c^2 \left (c-a^2 c x^2\right )^{3/2}}-\frac {8 \int \frac {1}{\left (c-a^2 c x^2\right )^{3/2}} \, dx}{21 c^2} \\ & = -\frac {2 (1+a x)}{7 a \left (c-a^2 c x^2\right )^{7/2}}-\frac {x}{7 c \left (c-a^2 c x^2\right )^{5/2}}-\frac {4 x}{21 c^2 \left (c-a^2 c x^2\right )^{3/2}}-\frac {8 x}{21 c^3 \sqrt {c-a^2 c x^2}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.99 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=-\frac {\sqrt {1-a^2 x^2} \left (6+9 a x-24 a^2 x^2+4 a^3 x^3+16 a^4 x^4-8 a^5 x^5\right )}{21 a c^3 (1-a x)^{7/2} (1+a x)^{3/2} \sqrt {c-a^2 c x^2}} \]
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Time = 0.63 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.66
method | result | size |
gosper | \(\frac {\left (a x +1\right )^{2} \left (8 a^{5} x^{5}-16 a^{4} x^{4}-4 a^{3} x^{3}+24 a^{2} x^{2}-9 a x -6\right )}{21 a \left (-a^{2} c \,x^{2}+c \right )^{\frac {7}{2}}}\) | \(64\) |
trager | \(\frac {\left (8 a^{5} x^{5}-16 a^{4} x^{4}-4 a^{3} x^{3}+24 a^{2} x^{2}-9 a x -6\right ) \sqrt {-a^{2} c \,x^{2}+c}}{21 c^{4} \left (a x -1\right )^{4} \left (a x +1\right )^{2} a}\) | \(74\) |
default | \(\frac {x}{5 c \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 c \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}+\frac {8 x}{15 c^{2} \sqrt {-a^{2} c \,x^{2}+c}}}{c}+\frac {\frac {2}{7 a c \left (x -\frac {1}{a}\right ) \left (-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c \right )^{\frac {5}{2}}}-\frac {12 a \left (-\frac {-2 a^{2} c \left (x -\frac {1}{a}\right )-2 a c}{10 a^{2} c^{2} \left (-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c \right )^{\frac {5}{2}}}+\frac {-\frac {2 \left (-2 a^{2} c \left (x -\frac {1}{a}\right )-2 a c \right )}{15 a^{2} c^{2} \left (-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c \right )^{\frac {3}{2}}}-\frac {4 \left (-2 a^{2} c \left (x -\frac {1}{a}\right )-2 a c \right )}{15 a^{2} c^{3} \sqrt {-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c}}}{c}\right )}{7}}{a}\) | \(292\) |
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Time = 0.36 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.28 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\frac {{\left (8 \, a^{5} x^{5} - 16 \, a^{4} x^{4} - 4 \, a^{3} x^{3} + 24 \, a^{2} x^{2} - 9 \, a x - 6\right )} \sqrt {-a^{2} c x^{2} + c}}{21 \, {\left (a^{7} c^{4} x^{6} - 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} + 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right )}} \]
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\[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\int \frac {a x + 1}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {7}{2}} \left (a x - 1\right )}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.02 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\frac {2}{7 \, {\left ({\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} a^{2} c x - {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} a c\right )}} - \frac {8 \, x}{21 \, \sqrt {-a^{2} c x^{2} + c} c^{3}} - \frac {4 \, x}{21 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} c^{2}} - \frac {x}{7 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} c} \]
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\[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\int { \frac {a x + 1}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}} {\left (a x - 1\right )}} \,d x } \]
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Time = 4.07 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.38 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\frac {\sqrt {c-a^2\,c\,x^2}}{14\,a\,c^4\,{\left (a\,x-1\right )}^3}-\frac {\sqrt {c-a^2\,c\,x^2}}{28\,a\,c^4\,{\left (a\,x-1\right )}^4}-\frac {\sqrt {c-a^2\,c\,x^2}\,\left (\frac {11\,x}{42\,c^4}+\frac {5}{28\,a\,c^4}\right )}{{\left (a\,x-1\right )}^2\,{\left (a\,x+1\right )}^2}+\frac {8\,x\,\sqrt {c-a^2\,c\,x^2}}{21\,c^4\,\left (a\,x-1\right )\,\left (a\,x+1\right )} \]
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