Integrand size = 24, antiderivative size = 195 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx=-\frac {(1-a x)^2}{a^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x}-\frac {2 (1-a x)^3}{5 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^2}+\frac {2 (1-a x)^3 (1+a x)}{15 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^3}-\frac {2 (1-a x)^3 (1+a x)^2 (28+13 a x)}{15 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}-\frac {2 (1-a x)^{5/2} (1+a x)^{5/2} \arcsin (a x)}{a^6 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5} \]
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Time = 0.32 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6302, 6294, 6264, 100, 155, 148, 41, 222} \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx=-\frac {(1-a x)^2}{a^2 x \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}-\frac {2 (a x+1)^{5/2} (1-a x)^{5/2} \arcsin (a x)}{a^6 x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}-\frac {2 (a x+1)^2 (13 a x+28) (1-a x)^3}{15 a^6 x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}+\frac {2 (a x+1) (1-a x)^3}{15 a^4 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}-\frac {2 (1-a x)^3}{5 a^3 x^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \]
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Rule 41
Rule 100
Rule 148
Rule 155
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 )^{5/2}} \, dx \\ & = -\frac {\left ((1-a x)^{5/2} (1+a x)^{5/2}\right ) \int \frac {e^{-2 \text {arctanh}(a x)} x^5}{(1-a x)^{5/2} (1+a x)^{5/2}} \, dx}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5} \\ & = -\frac {\left ((1-a x)^{5/2} (1+a x)^{5/2}\right ) \int \frac {x^5}{(1-a x)^{3/2} (1+a x)^{7/2}} \, dx}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5} \\ & = -\frac {(1-a x)^2}{a^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x}+\frac {\left ((1-a x)^{5/2} (1+a x)^{5/2}\right ) \int \frac {x^3 (4+2 a x)}{\sqrt {1-a x} (1+a x)^{7/2}} \, dx}{a^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5} \\ & = -\frac {(1-a x)^2}{a^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x}-\frac {2 (1-a x)^3}{5 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^2}+\frac {\left ((1-a x)^{5/2} (1+a x)^{5/2}\right ) \int \frac {x^2 \left (6 a+8 a^2 x\right )}{\sqrt {1-a x} (1+a x)^{5/2}} \, dx}{5 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5} \\ & = -\frac {(1-a x)^2}{a^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x}-\frac {2 (1-a x)^3}{5 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^2}+\frac {2 (1-a x)^3 (1+a x)}{15 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^3}+\frac {\left ((1-a x)^{5/2} (1+a x)^{5/2}\right ) \int \frac {x \left (-4 a^2+26 a^3 x\right )}{\sqrt {1-a x} (1+a x)^{3/2}} \, dx}{15 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5} \\ & = -\frac {(1-a x)^2}{a^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x}-\frac {2 (1-a x)^3}{5 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^2}+\frac {2 (1-a x)^3 (1+a x)}{15 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^3}-\frac {2 (1-a x)^3 (1+a x)^2 (28+13 a x)}{15 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}-\frac {\left (2 (1-a x)^{5/2} (1+a x)^{5/2}\right ) \int \frac {1}{\sqrt {1-a x} \sqrt {1+a x}} \, dx}{a^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5} \\ & = -\frac {(1-a x)^2}{a^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x}-\frac {2 (1-a x)^3}{5 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^2}+\frac {2 (1-a x)^3 (1+a x)}{15 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^3}-\frac {2 (1-a x)^3 (1+a x)^2 (28+13 a x)}{15 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}-\frac {\left (2 (1-a x)^{5/2} (1+a x)^{5/2}\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{a^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5} \\ & = -\frac {(1-a x)^2}{a^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x}-\frac {2 (1-a x)^3}{5 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^2}+\frac {2 (1-a x)^3 (1+a x)}{15 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^3}-\frac {2 (1-a x)^3 (1+a x)^2 (28+13 a x)}{15 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}-\frac {2 (1-a x)^{5/2} (1+a x)^{5/2} \arcsin (a x)}{a^6 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.54 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx=\frac {-56-82 a x+32 a^2 x^2+76 a^3 x^3+15 a^4 x^4-30 (1+a x)^2 \sqrt {-1+a^2 x^2} \log \left (a x+\sqrt {-1+a^2 x^2}\right )}{15 a^2 c^2 \sqrt {c-\frac {c}{a^2 x^2}} x (1+a x)^2} \]
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Time = 0.55 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.47
method | result | size |
risch | \(\frac {a^{2} x^{2}-1}{a^{2} c^{2} x \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}+\frac {\left (-\frac {2 \ln \left (\frac {a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-c}\right )}{a^{5} \sqrt {a^{2} c}}-\frac {41 \sqrt {a^{2} c \left (x +\frac {1}{a}\right )^{2}-2 \left (x +\frac {1}{a}\right ) a c}}{60 a^{8} c \left (x +\frac {1}{a}\right )^{2}}+\frac {383 \sqrt {a^{2} c \left (x +\frac {1}{a}\right )^{2}-2 \left (x +\frac {1}{a}\right ) a c}}{120 a^{7} c \left (x +\frac {1}{a}\right )}-\frac {\sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+2 \left (x -\frac {1}{a}\right ) a c}}{8 a^{7} c \left (x -\frac {1}{a}\right )}+\frac {\sqrt {a^{2} c \left (x +\frac {1}{a}\right )^{2}-2 \left (x +\frac {1}{a}\right ) a c}}{10 a^{9} c \left (x +\frac {1}{a}\right )^{3}}\right ) a^{4} \sqrt {c \left (a^{2} x^{2}-1\right )}}{c^{2} x \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}\) | \(286\) |
default | \(\frac {\left (15 c^{\frac {5}{2}} \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {3}{2}} a^{5} x^{5}+45 x^{4} c^{\frac {5}{2}} a^{4} \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {3}{2}}+16 c^{\frac {5}{2}} {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} a^{4} x^{4}-60 c^{\frac {5}{2}} \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {3}{2}} a^{3} x^{3}+16 c^{\frac {5}{2}} {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} a^{3} x^{3}-30 \ln \left (\sqrt {c}\, x +\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {3}{2}} {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} a^{4} c x -90 c^{\frac {5}{2}} \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {3}{2}} a^{2} x^{2}-24 c^{\frac {5}{2}} {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} a^{2} x^{2}-30 \ln \left (\sqrt {c}\, x +\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {3}{2}} {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} a^{3} c +50 c^{\frac {5}{2}} \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {3}{2}} a x -24 c^{\frac {5}{2}} {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} a x +50 c^{\frac {5}{2}} \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {3}{2}}+6 c^{\frac {5}{2}} {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}}\right ) \left (a x -1\right )}{15 \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {3}{2}} x^{5} {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )}^{\frac {5}{2}} a^{6} c^{\frac {5}{2}}}\) | \(462\) |
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Time = 0.28 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.80 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx=\left [\frac {15 \, {\left (a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1\right )} \sqrt {c} \log \left (2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right ) + {\left (15 \, a^{5} x^{5} + 76 \, a^{4} x^{4} + 32 \, a^{3} x^{3} - 82 \, a^{2} x^{2} - 56 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{15 \, {\left (a^{5} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{3} - 2 \, a^{2} c^{3} x - a c^{3}\right )}}, \frac {30 \, {\left (a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1\right )} \sqrt {-c} \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) + {\left (15 \, a^{5} x^{5} + 76 \, a^{4} x^{4} + 32 \, a^{3} x^{3} - 82 \, a^{2} x^{2} - 56 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{15 \, {\left (a^{5} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{3} - 2 \, a^{2} c^{3} x - a c^{3}\right )}}\right ] \]
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\[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx=\int \frac {a x - 1}{\left (- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )\right )^{\frac {5}{2}} \left (a x + 1\right )}\, dx \]
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\[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx=\int { \frac {a x - 1}{{\left (a x + 1\right )} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {5}{2}}} \,d x } \]
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Exception generated. \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx=\int \frac {a\,x-1}{{\left (c-\frac {c}{a^2\,x^2}\right )}^{5/2}\,\left (a\,x+1\right )} \,d x \]
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