Integrand size = 27, antiderivative size = 181 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^5} \, dx=\frac {6}{5} a^4 \sqrt {c-\frac {c}{a^2 x^2}}+\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{5 x^4}-\frac {a \sqrt {c-\frac {c}{a^2 x^2}}}{2 x^3}+\frac {3 a^2 \sqrt {c-\frac {c}{a^2 x^2}}}{5 x^2}-\frac {3 a^3 \sqrt {c-\frac {c}{a^2 x^2}}}{4 x}-\frac {3 a^5 \sqrt {c-\frac {c}{a^2 x^2}} x \text {arctanh}\left (\sqrt {1-a x} \sqrt {1+a x}\right )}{4 \sqrt {1-a x} \sqrt {1+a x}} \]
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Time = 0.46 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {6302, 6294, 6264, 100, 156, 12, 94, 214} \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^5} \, dx=\frac {3 a^2 \sqrt {c-\frac {c}{a^2 x^2}}}{5 x^2}+\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{5 x^4}-\frac {a \sqrt {c-\frac {c}{a^2 x^2}}}{2 x^3}-\frac {3 a^5 x \text {arctanh}\left (\sqrt {1-a x} \sqrt {a x+1}\right ) \sqrt {c-\frac {c}{a^2 x^2}}}{4 \sqrt {1-a x} \sqrt {a x+1}}+\frac {6}{5} a^4 \sqrt {c-\frac {c}{a^2 x^2}}-\frac {3 a^3 \sqrt {c-\frac {c}{a^2 x^2}}}{4 x} \]
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Rule 12
Rule 94
Rule 100
Rule 156
Rule 214
Rule 6264
Rule 6294
Rule 6302
Rubi steps \begin{align*} \text {integral}& = -\int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^5} \, dx \\ & = -\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {1-a x} \sqrt {1+a x}}{x^6} \, dx}{\sqrt {1-a x} \sqrt {1+a x}} \\ & = -\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {(1-a x)^{3/2}}{x^6 \sqrt {1+a x}} \, dx}{\sqrt {1-a x} \sqrt {1+a x}} \\ & = \frac {\sqrt {c-\frac {c}{a^2 x^2}}}{5 x^4}+\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {10 a-9 a^2 x}{x^5 \sqrt {1-a x} \sqrt {1+a x}} \, dx}{5 \sqrt {1-a x} \sqrt {1+a x}} \\ & = \frac {\sqrt {c-\frac {c}{a^2 x^2}}}{5 x^4}-\frac {a \sqrt {c-\frac {c}{a^2 x^2}}}{2 x^3}-\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {36 a^2-30 a^3 x}{x^4 \sqrt {1-a x} \sqrt {1+a x}} \, dx}{20 \sqrt {1-a x} \sqrt {1+a x}} \\ & = \frac {\sqrt {c-\frac {c}{a^2 x^2}}}{5 x^4}-\frac {a \sqrt {c-\frac {c}{a^2 x^2}}}{2 x^3}+\frac {3 a^2 \sqrt {c-\frac {c}{a^2 x^2}}}{5 x^2}+\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {90 a^3-72 a^4 x}{x^3 \sqrt {1-a x} \sqrt {1+a x}} \, dx}{60 \sqrt {1-a x} \sqrt {1+a x}} \\ & = \frac {\sqrt {c-\frac {c}{a^2 x^2}}}{5 x^4}-\frac {a \sqrt {c-\frac {c}{a^2 x^2}}}{2 x^3}+\frac {3 a^2 \sqrt {c-\frac {c}{a^2 x^2}}}{5 x^2}-\frac {3 a^3 \sqrt {c-\frac {c}{a^2 x^2}}}{4 x}-\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {144 a^4-90 a^5 x}{x^2 \sqrt {1-a x} \sqrt {1+a x}} \, dx}{120 \sqrt {1-a x} \sqrt {1+a x}} \\ & = \frac {6}{5} a^4 \sqrt {c-\frac {c}{a^2 x^2}}+\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{5 x^4}-\frac {a \sqrt {c-\frac {c}{a^2 x^2}}}{2 x^3}+\frac {3 a^2 \sqrt {c-\frac {c}{a^2 x^2}}}{5 x^2}-\frac {3 a^3 \sqrt {c-\frac {c}{a^2 x^2}}}{4 x}+\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {90 a^5}{x \sqrt {1-a x} \sqrt {1+a x}} \, dx}{120 \sqrt {1-a x} \sqrt {1+a x}} \\ & = \frac {6}{5} a^4 \sqrt {c-\frac {c}{a^2 x^2}}+\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{5 x^4}-\frac {a \sqrt {c-\frac {c}{a^2 x^2}}}{2 x^3}+\frac {3 a^2 \sqrt {c-\frac {c}{a^2 x^2}}}{5 x^2}-\frac {3 a^3 \sqrt {c-\frac {c}{a^2 x^2}}}{4 x}+\frac {\left (3 a^5 \sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {1}{x \sqrt {1-a x} \sqrt {1+a x}} \, dx}{4 \sqrt {1-a x} \sqrt {1+a x}} \\ & = \frac {6}{5} a^4 \sqrt {c-\frac {c}{a^2 x^2}}+\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{5 x^4}-\frac {a \sqrt {c-\frac {c}{a^2 x^2}}}{2 x^3}+\frac {3 a^2 \sqrt {c-\frac {c}{a^2 x^2}}}{5 x^2}-\frac {3 a^3 \sqrt {c-\frac {c}{a^2 x^2}}}{4 x}-\frac {\left (3 a^6 \sqrt {c-\frac {c}{a^2 x^2}} x\right ) \text {Subst}\left (\int \frac {1}{a-a x^2} \, dx,x,\sqrt {1-a x} \sqrt {1+a x}\right )}{4 \sqrt {1-a x} \sqrt {1+a x}} \\ & = \frac {6}{5} a^4 \sqrt {c-\frac {c}{a^2 x^2}}+\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{5 x^4}-\frac {a \sqrt {c-\frac {c}{a^2 x^2}}}{2 x^3}+\frac {3 a^2 \sqrt {c-\frac {c}{a^2 x^2}}}{5 x^2}-\frac {3 a^3 \sqrt {c-\frac {c}{a^2 x^2}}}{4 x}-\frac {3 a^5 \sqrt {c-\frac {c}{a^2 x^2}} x \text {arctanh}\left (\sqrt {1-a x} \sqrt {1+a x}\right )}{4 \sqrt {1-a x} \sqrt {1+a x}} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.56 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^5} \, dx=\frac {\sqrt {c-\frac {c}{a^2 x^2}} \left (\sqrt {-1+a^2 x^2} \left (4-10 a x+12 a^2 x^2-15 a^3 x^3+24 a^4 x^4\right )+15 a^5 x^5 \arctan \left (\frac {1}{\sqrt {-1+a^2 x^2}}\right )\right )}{20 x^4 \sqrt {-1+a^2 x^2}} \]
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Time = 0.61 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.92
method | result | size |
risch | \(\frac {\left (24 a^{6} x^{6}-15 a^{5} x^{5}-12 a^{4} x^{4}+5 a^{3} x^{3}-8 a^{2} x^{2}+10 a x -4\right ) \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}{20 x^{4} \left (a^{2} x^{2}-1\right )}+\frac {3 a^{5} \ln \left (\frac {-2 c +2 \sqrt {-c}\, \sqrt {a^{2} c \,x^{2}-c}}{x}\right ) \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, \sqrt {c \left (a^{2} x^{2}-1\right )}\, x}{4 \sqrt {-c}\, \left (a^{2} x^{2}-1\right )}\) | \(167\) |
default | \(-\frac {\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, a^{2} \left (-40 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{4} c \,x^{6}+40 \sqrt {-\frac {c}{a^{2}}}\, {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} a^{4} x^{4}+40 \sqrt {-\frac {c}{a^{2}}}\, c^{\frac {3}{2}} \ln \left (\sqrt {c}\, x +\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) a^{2} x^{5}-40 \sqrt {-\frac {c}{a^{2}}}\, c^{\frac {3}{2}} \ln \left (\frac {\sqrt {c}\, \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}+c x}{\sqrt {c}}\right ) a^{2} x^{5}+40 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}\, a^{3} c \,x^{5}-15 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{3} c \,x^{5}-25 \sqrt {-\frac {c}{a^{2}}}\, {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} a^{3} x^{3}-15 \ln \left (\frac {2 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2}-2 c}{a^{2} x}\right ) a \,c^{2} x^{5}+16 \sqrt {-\frac {c}{a^{2}}}\, {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} a^{2} x^{2}-10 a {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} x \sqrt {-\frac {c}{a^{2}}}+4 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} \sqrt {-\frac {c}{a^{2}}}\right )}{20 x^{4} \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, c \sqrt {-\frac {c}{a^{2}}}}\) | \(447\) |
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Time = 0.26 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.28 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^5} \, dx=\left [\frac {15 \, a^{4} \sqrt {-c} x^{4} \log \left (-\frac {a^{2} c x^{2} - 2 \, a \sqrt {-c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right ) + 2 \, {\left (24 \, a^{4} x^{4} - 15 \, a^{3} x^{3} + 12 \, a^{2} x^{2} - 10 \, a x + 4\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{40 \, x^{4}}, \frac {15 \, a^{4} \sqrt {c} x^{4} \arctan \left (\frac {a \sqrt {c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) + {\left (24 \, a^{4} x^{4} - 15 \, a^{3} x^{3} + 12 \, a^{2} x^{2} - 10 \, a x + 4\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{20 \, x^{4}}\right ] \]
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\[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^5} \, dx=\int \frac {\sqrt {- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )} \left (a x - 1\right )}{x^{5} \left (a x + 1\right )}\, dx \]
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\[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^5} \, dx=\int { \frac {{\left (a x - 1\right )} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{{\left (a x + 1\right )} x^{5}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 362 vs. \(2 (149) = 298\).
Time = 2.57 (sec) , antiderivative size = 362, normalized size of antiderivative = 2.00 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^5} \, dx=-\frac {1}{10} \, {\left (15 \, a^{3} \sqrt {c} \arctan \left (-\frac {\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}}{\sqrt {c}}\right ) \mathrm {sgn}\left (x\right ) - \frac {15 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{9} a^{3} c \mathrm {sgn}\left (x\right ) + 70 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{7} a^{3} c^{2} \mathrm {sgn}\left (x\right ) + 40 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{6} a^{2} c^{\frac {5}{2}} {\left | a \right |} \mathrm {sgn}\left (x\right ) + 200 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{4} a^{2} c^{\frac {7}{2}} {\left | a \right |} \mathrm {sgn}\left (x\right ) - 70 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{3} a^{3} c^{4} \mathrm {sgn}\left (x\right ) + 120 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} a^{2} c^{\frac {9}{2}} {\left | a \right |} \mathrm {sgn}\left (x\right ) - 15 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )} a^{3} c^{5} \mathrm {sgn}\left (x\right ) + 24 \, a^{2} c^{\frac {11}{2}} {\left | a \right |} \mathrm {sgn}\left (x\right )}{{\left ({\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} + c\right )}^{5}}\right )} {\left | a \right |} \]
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Timed out. \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^5} \, dx=\int \frac {\sqrt {c-\frac {c}{a^2\,x^2}}\,\left (a\,x-1\right )}{x^5\,\left (a\,x+1\right )} \,d x \]
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