Integrand size = 24, antiderivative size = 294 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \, dx=-\frac {5 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^3}{8 (1-a x)^2}+\frac {25 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}{8 (1-a x)^2 (1+a x)^2}-\frac {17 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^4}{12 (1-a x)^2 (1+a x)}+\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^2 (1+a x)}{6 (1-a x)^2}+\frac {\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x (1+a x)}{4 (1-a x)}-\frac {2 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5 \arcsin (a x)}{(1-a x)^{5/2} (1+a x)^{5/2}}-\frac {9 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5 \text {arctanh}\left (\sqrt {1-a x} \sqrt {1+a x}\right )}{8 (1-a x)^{5/2} (1+a x)^{5/2}} \]
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Time = 0.37 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {6302, 6294, 6264, 99, 154, 159, 163, 41, 222, 94, 214} \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \, dx=\frac {a x^2 (a x+1) \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{6 (1-a x)^2}+\frac {x (a x+1) \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{4 (1-a x)}-\frac {5 a^2 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{8 (1-a x)^2}-\frac {2 a^4 x^5 \arcsin (a x) \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{(1-a x)^{5/2} (a x+1)^{5/2}}-\frac {9 a^4 x^5 \text {arctanh}\left (\sqrt {1-a x} \sqrt {a x+1}\right ) \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{8 (1-a x)^{5/2} (a x+1)^{5/2}}+\frac {25 a^4 x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{8 (1-a x)^2 (a x+1)^2}-\frac {17 a^3 x^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{12 (1-a x)^2 (a x+1)} \]
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Rule 41
Rule 94
Rule 99
Rule 154
Rule 159
Rule 163
Rule 214
Rule 222
Rule 6264
Rule 6294
Rule 6302
Rubi steps \begin{align*} \text {integral}& = -\int e^{2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \, dx \\ & = -\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac {e^{2 \text {arctanh}(a x)} (1-a x)^{5/2} (1+a x)^{5/2}}{x^5} \, dx}{(1-a x)^{5/2} (1+a x)^{5/2}} \\ & = -\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac {(1-a x)^{3/2} (1+a x)^{7/2}}{x^5} \, dx}{(1-a x)^{5/2} (1+a x)^{5/2}} \\ & = \frac {\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x (1+a x)}{4 (1-a x)}-\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac {\sqrt {1-a x} (1+a x)^{5/2} \left (2 a-5 a^2 x\right )}{x^4} \, dx}{4 (1-a x)^{5/2} (1+a x)^{5/2}} \\ & = \frac {a \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^2 (1+a x)}{6 (1-a x)^2}+\frac {\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x (1+a x)}{4 (1-a x)}-\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac {(1+a x)^{5/2} \left (-15 a^2+13 a^3 x\right )}{x^3 \sqrt {1-a x}} \, dx}{12 (1-a x)^{5/2} (1+a x)^{5/2}} \\ & = -\frac {5 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^3}{8 (1-a x)^2}+\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^2 (1+a x)}{6 (1-a x)^2}+\frac {\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x (1+a x)}{4 (1-a x)}-\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac {(1+a x)^{3/2} \left (-34 a^3+41 a^4 x\right )}{x^2 \sqrt {1-a x}} \, dx}{24 (1-a x)^{5/2} (1+a x)^{5/2}} \\ & = -\frac {5 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^3}{8 (1-a x)^2}-\frac {17 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^4}{12 (1-a x)^2 (1+a x)}+\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^2 (1+a x)}{6 (1-a x)^2}+\frac {\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x (1+a x)}{4 (1-a x)}-\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac {\sqrt {1+a x} \left (-27 a^4+75 a^5 x\right )}{x \sqrt {1-a x}} \, dx}{24 (1-a x)^{5/2} (1+a x)^{5/2}} \\ & = -\frac {5 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^3}{8 (1-a x)^2}+\frac {25 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}{8 (1-a x)^2 (1+a x)^2}-\frac {17 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^4}{12 (1-a x)^2 (1+a x)}+\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^2 (1+a x)}{6 (1-a x)^2}+\frac {\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x (1+a x)}{4 (1-a x)}+\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac {27 a^5-48 a^6 x}{x \sqrt {1-a x} \sqrt {1+a x}} \, dx}{24 a (1-a x)^{5/2} (1+a x)^{5/2}} \\ & = -\frac {5 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^3}{8 (1-a x)^2}+\frac {25 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}{8 (1-a x)^2 (1+a x)^2}-\frac {17 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^4}{12 (1-a x)^2 (1+a x)}+\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^2 (1+a x)}{6 (1-a x)^2}+\frac {\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x (1+a x)}{4 (1-a x)}+\frac {\left (9 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac {1}{x \sqrt {1-a x} \sqrt {1+a x}} \, dx}{8 (1-a x)^{5/2} (1+a x)^{5/2}}-\frac {\left (2 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac {1}{\sqrt {1-a x} \sqrt {1+a x}} \, dx}{(1-a x)^{5/2} (1+a x)^{5/2}} \\ & = -\frac {5 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^3}{8 (1-a x)^2}+\frac {25 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}{8 (1-a x)^2 (1+a x)^2}-\frac {17 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^4}{12 (1-a x)^2 (1+a x)}+\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^2 (1+a x)}{6 (1-a x)^2}+\frac {\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x (1+a x)}{4 (1-a x)}-\frac {\left (9 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5\right ) \text {Subst}\left (\int \frac {1}{a-a x^2} \, dx,x,\sqrt {1-a x} \sqrt {1+a x}\right )}{8 (1-a x)^{5/2} (1+a x)^{5/2}}-\frac {\left (2 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{(1-a x)^{5/2} (1+a x)^{5/2}} \\ & = -\frac {5 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^3}{8 (1-a x)^2}+\frac {25 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}{8 (1-a x)^2 (1+a x)^2}-\frac {17 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^4}{12 (1-a x)^2 (1+a x)}+\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^2 (1+a x)}{6 (1-a x)^2}+\frac {\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x (1+a x)}{4 (1-a x)}-\frac {2 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5 \arcsin (a x)}{(1-a x)^{5/2} (1+a x)^{5/2}}-\frac {9 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5 \text {arctanh}\left (\sqrt {1-a x} \sqrt {1+a x}\right )}{8 (1-a x)^{5/2} (1+a x)^{5/2}} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.46 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \, dx=\frac {c^2 \sqrt {c-\frac {c}{a^2 x^2}} \left (\sqrt {-1+a^2 x^2} \left (6+16 a x-3 a^2 x^2-64 a^3 x^3+24 a^4 x^4\right )+27 a^4 x^4 \arctan \left (\frac {1}{\sqrt {-1+a^2 x^2}}\right )+48 a^4 x^4 \log \left (a x+\sqrt {-1+a^2 x^2}\right )\right )}{24 a^4 x^3 \sqrt {-1+a^2 x^2}} \]
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Time = 0.62 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.80
method | result | size |
risch | \(-\frac {\left (64 a^{5} x^{5}+3 a^{4} x^{4}-80 a^{3} x^{3}-9 a^{2} x^{2}+16 a x +6\right ) c^{2} \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}{24 x^{3} a^{4} \left (a^{2} x^{2}-1\right )}+\frac {\left (\frac {2 a^{5} \ln \left (\frac {a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-c}\right )}{\sqrt {a^{2} c}}+\frac {9 a^{4} \ln \left (\frac {-2 c +2 \sqrt {-c}\, \sqrt {a^{2} c \,x^{2}-c}}{x}\right )}{8 \sqrt {-c}}+\frac {a^{4} \sqrt {c \left (a^{2} x^{2}-1\right )}}{c}\right ) c^{2} \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x \sqrt {c \left (a^{2} x^{2}-1\right )}}{a^{4} \left (a^{2} x^{2}-1\right )}\) | \(234\) |
default | \(\frac {{\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )}^{\frac {5}{2}} x \left (-80 \sqrt {-\frac {c}{a^{2}}}\, {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {5}{2}} a^{7} c \,x^{5}+80 \sqrt {-\frac {c}{a^{2}}}\, {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {7}{2}} a^{7} x^{3}+48 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {5}{2}} a^{6} c \,x^{4}+27 \sqrt {-\frac {c}{a^{2}}}\, {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {5}{2}} a^{6} c \,x^{4}+60 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {3}{2}} a^{5} c^{2} x^{5}-75 \sqrt {-\frac {c}{a^{2}}}\, {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {7}{2}} a^{6} x^{2}+100 \sqrt {-\frac {c}{a^{2}}}\, {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} a^{5} c^{2} x^{5}-80 \sqrt {-\frac {c}{a^{2}}}\, {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {7}{2}} a^{5} x -45 \sqrt {-\frac {c}{a^{2}}}\, {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} a^{4} c^{2} x^{4}-90 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}\, a^{3} c^{3} x^{5}-150 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{3} c^{3} x^{5}-30 a^{4} {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {7}{2}} \sqrt {-\frac {c}{a^{2}}}+150 \sqrt {-\frac {c}{a^{2}}}\, c^{\frac {7}{2}} \ln \left (\sqrt {c}\, x +\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) a \,x^{4}+90 \sqrt {-\frac {c}{a^{2}}}\, c^{\frac {7}{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^{4}+135 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2} c^{3} x^{4}+135 \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^{4} x^{4}\right )}{120 a^{2} \sqrt {-\frac {c}{a^{2}}}\, {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {5}{2}} c}\) | \(625\) |
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Time = 0.27 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.34 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \, dx=\left [-\frac {96 \, a^{3} \sqrt {-c} c^{2} x^{3} \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) - 27 \, a^{3} \sqrt {-c} c^{2} x^{3} \log \left (-\frac {a^{2} c x^{2} - 2 \, a \sqrt {-c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right ) - 2 \, {\left (24 \, a^{4} c^{2} x^{4} - 64 \, a^{3} c^{2} x^{3} - 3 \, a^{2} c^{2} x^{2} + 16 \, a c^{2} x + 6 \, c^{2}\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{48 \, a^{4} x^{3}}, \frac {27 \, a^{3} c^{\frac {5}{2}} x^{3} \arctan \left (\frac {a \sqrt {c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) + 24 \, a^{3} c^{\frac {5}{2}} x^{3} \log \left (2 \, a^{2} c x^{2} + 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right ) + {\left (24 \, a^{4} c^{2} x^{4} - 64 \, a^{3} c^{2} x^{3} - 3 \, a^{2} c^{2} x^{2} + 16 \, a c^{2} x + 6 \, c^{2}\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{24 \, a^{4} x^{3}}\right ] \]
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Result contains complex when optimal does not.
Time = 10.05 (sec) , antiderivative size = 500, normalized size of antiderivative = 1.70 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \, dx=c^{2} \left (\begin {cases} \frac {\sqrt {c} \sqrt {a^{2} x^{2} - 1}}{a} - \frac {i \sqrt {c} \log {\left (a x \right )}}{a} + \frac {i \sqrt {c} \log {\left (a^{2} x^{2} \right )}}{2 a} + \frac {\sqrt {c} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{a} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {i \sqrt {c} \sqrt {- a^{2} x^{2} + 1}}{a} + \frac {i \sqrt {c} \log {\left (a^{2} x^{2} \right )}}{2 a} - \frac {i \sqrt {c} \log {\left (\sqrt {- a^{2} x^{2} + 1} + 1 \right )}}{a} & \text {otherwise} \end {cases}\right ) + \frac {2 c^{2} \left (\begin {cases} - \frac {a \sqrt {c} x}{\sqrt {a^{2} x^{2} - 1}} + \sqrt {c} \operatorname {acosh}{\left (a x \right )} + \frac {\sqrt {c}}{a x \sqrt {a^{2} x^{2} - 1}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {i a \sqrt {c} x}{\sqrt {- a^{2} x^{2} + 1}} - i \sqrt {c} \operatorname {asin}{\left (a x \right )} - \frac {i \sqrt {c}}{a x \sqrt {- a^{2} x^{2} + 1}} & \text {otherwise} \end {cases}\right )}{a} - \frac {2 c^{2} \left (\begin {cases} 0 & \text {for}\: c = 0 \\\frac {a^{2} \left (c - \frac {c}{a^{2} x^{2}}\right )^{\frac {3}{2}}}{3 c} & \text {otherwise} \end {cases}\right )}{a^{3}} - \frac {c^{2} \left (\begin {cases} \frac {i a^{3} \sqrt {c} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{8} - \frac {i a^{2} \sqrt {c}}{8 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} + \frac {3 i \sqrt {c}}{8 x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {i \sqrt {c}}{4 a^{2} x^{5} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\- \frac {a^{3} \sqrt {c} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{8} + \frac {a^{2} \sqrt {c}}{8 x \sqrt {1 - \frac {1}{a^{2} x^{2}}}} - \frac {3 \sqrt {c}}{8 x^{3} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {\sqrt {c}}{4 a^{2} x^{5} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} & \text {otherwise} \end {cases}\right )}{a^{4}} \]
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\[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \, dx=\int { \frac {{\left (a x + 1\right )} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {5}{2}}}{a x - 1} \,d x } \]
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Time = 1.81 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.41 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \, dx=-\frac {1}{12} \, {\left (\frac {27 \, c^{\frac {5}{2}} \arctan \left (-\frac {\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}}{\sqrt {c}}\right ) \mathrm {sgn}\left (x\right )}{a^{2}} + \frac {24 \, c^{\frac {5}{2}} \log \left ({\left | -\sqrt {a^{2} c} x + \sqrt {a^{2} c x^{2} - c} \right |}\right ) \mathrm {sgn}\left (x\right )}{a {\left | a \right |}} - \frac {12 \, \sqrt {a^{2} c x^{2} - c} c^{2} \mathrm {sgn}\left (x\right )}{a^{2}} - \frac {3 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{7} c^{3} {\left | a \right |} \mathrm {sgn}\left (x\right ) - 96 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{6} a c^{\frac {7}{2}} \mathrm {sgn}\left (x\right ) - 21 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{5} c^{4} {\left | a \right |} \mathrm {sgn}\left (x\right ) - 192 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{4} a c^{\frac {9}{2}} \mathrm {sgn}\left (x\right ) + 21 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{3} c^{5} {\left | a \right |} \mathrm {sgn}\left (x\right ) - 160 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} a c^{\frac {11}{2}} \mathrm {sgn}\left (x\right ) - 3 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )} c^{6} {\left | a \right |} \mathrm {sgn}\left (x\right ) - 64 \, a c^{\frac {13}{2}} \mathrm {sgn}\left (x\right )}{{\left ({\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} + c\right )}^{4} a^{2} {\left | a \right |}}\right )} {\left | a \right |} \]
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Timed out. \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \, dx=\int \frac {{\left (c-\frac {c}{a^2\,x^2}\right )}^{5/2}\,\left (a\,x+1\right )}{a\,x-1} \,d x \]
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