Integrand size = 27, antiderivative size = 111 \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^2} \, dx=\frac {3}{2} a \sqrt {c-\frac {c}{a^2 x^2}}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} (1+a x)}{2 x}+\frac {3 a^2 \sqrt {c-\frac {c}{a^2 x^2}} x \text {arctanh}\left (\sqrt {1-a x} \sqrt {1+a x}\right )}{2 \sqrt {1-a x} \sqrt {1+a x}} \]
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Time = 0.39 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6302, 6294, 6264, 96, 94, 214} \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^2} \, dx=\frac {3 a^2 x \text {arctanh}\left (\sqrt {1-a x} \sqrt {a x+1}\right ) \sqrt {c-\frac {c}{a^2 x^2}}}{2 \sqrt {1-a x} \sqrt {a x+1}}+\frac {3}{2} a \sqrt {c-\frac {c}{a^2 x^2}}+\frac {(a x+1) \sqrt {c-\frac {c}{a^2 x^2}}}{2 x} \]
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Rule 94
Rule 96
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^2} \, 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^3} \, 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^3 \sqrt {1-a x}} \, dx}{\sqrt {1-a x} \sqrt {1+a x}} \\ & = \frac {\sqrt {c-\frac {c}{a^2 x^2}} (1+a x)}{2 x}-\frac {\left (3 a \sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {\sqrt {1+a x}}{x^2 \sqrt {1-a x}} \, dx}{2 \sqrt {1-a x} \sqrt {1+a x}} \\ & = \frac {3}{2} a \sqrt {c-\frac {c}{a^2 x^2}}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} (1+a x)}{2 x}-\frac {\left (3 a^2 \sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {1}{x \sqrt {1-a x} \sqrt {1+a x}} \, dx}{2 \sqrt {1-a x} \sqrt {1+a x}} \\ & = \frac {3}{2} a \sqrt {c-\frac {c}{a^2 x^2}}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} (1+a x)}{2 x}+\frac {\left (3 a^3 \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 )}{2 \sqrt {1-a x} \sqrt {1+a x}} \\ & = \frac {3}{2} a \sqrt {c-\frac {c}{a^2 x^2}}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} (1+a x)}{2 x}+\frac {3 a^2 \sqrt {c-\frac {c}{a^2 x^2}} x \text {arctanh}\left (\sqrt {1-a x} \sqrt {1+a x}\right )}{2 \sqrt {1-a x} \sqrt {1+a x}} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.70 \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^2} \, dx=\frac {\sqrt {c-\frac {c}{a^2 x^2}} \left ((1+4 a x) \sqrt {-1+a^2 x^2}-3 a^2 x^2 \arctan \left (\frac {1}{\sqrt {-1+a^2 x^2}}\right )\right )}{2 x \sqrt {-1+a^2 x^2}} \]
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Time = 0.61 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.28
method | result | size |
risch | \(\frac {\left (4 a^{3} x^{3}+a^{2} x^{2}-4 a x -1\right ) \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}{2 x \left (a^{2} x^{2}-1\right )}-\frac {3 a^{2} \ln \left (\frac {-2 c +2 \sqrt {-c}\, \sqrt {a^{2} c \,x^{2}-c}}{x}\right ) \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, \sqrt {c \left (a^{2} x^{2}-1\right )}\, x}{2 \sqrt {-c}\, \left (a^{2} x^{2}-1\right )}\) | \(142\) |
default | \(-\frac {\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, \left (-4 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{3} c \,x^{3}+4 \sqrt {-\frac {c}{a^{2}}}\, {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} a^{3} x +4 \sqrt {-\frac {c}{a^{2}}}\, c^{\frac {3}{2}} \ln \left (\sqrt {c}\, x +\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) a \,x^{2}-4 \sqrt {-\frac {c}{a^{2}}}\, c^{\frac {3}{2}} \ln \left (\frac {\sqrt {c}\, \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}+c x}{\sqrt {c}}\right ) a \,x^{2}-4 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}\, a^{2} c \,x^{2}+3 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2} c \,x^{2}+a^{2} {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} \sqrt {-\frac {c}{a^{2}}}+3 \ln \left (\frac {2 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2}-2 c}{a^{2} x}\right ) c^{2} x^{2}\right )}{2 x \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, c}\) | \(347\) |
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none
Time = 0.25 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.59 \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^2} \, dx=\left [\frac {3 \, a \sqrt {-c} x \log \left (-\frac {a^{2} c x^{2} + 2 \, a \sqrt {-c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right ) + 2 \, {\left (4 \, a x + 1\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{4 \, x}, -\frac {3 \, a \sqrt {c} x \arctan \left (\frac {a \sqrt {c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) - {\left (4 \, a x + 1\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, x}\right ] \]
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\[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^2} \, dx=\int \frac {\sqrt {- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )} \left (a x + 1\right )}{x^{2} \left (a x - 1\right )}\, dx \]
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\[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^2} \, dx=\int { \frac {{\left (a x + 1\right )} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{{\left (a x - 1\right )} x^{2}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 194 vs. \(2 (91) = 182\).
Time = 0.38 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.75 \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^2} \, dx={\left (3 \, \sqrt {c} \arctan \left (-\frac {\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}}{\sqrt {c}}\right ) \mathrm {sgn}\left (x\right ) - \frac {{\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{3} a c \mathrm {sgn}\left (x\right ) - 4 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} c^{\frac {3}{2}} {\left | a \right |} \mathrm {sgn}\left (x\right ) - {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )} a c^{2} \mathrm {sgn}\left (x\right ) - 4 \, c^{\frac {5}{2}} {\left | a \right |} \mathrm {sgn}\left (x\right )}{{\left ({\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} + c\right )}^{2} a}\right )} {\left | a \right |} \]
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Timed out. \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^2} \, dx=\int \frac {\sqrt {c-\frac {c}{a^2\,x^2}}\,\left (a\,x+1\right )}{x^2\,\left (a\,x-1\right )} \,d x \]
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