Integrand size = 12, antiderivative size = 87 \[ \int \frac {e^{\text {sech}^{-1}\left (a x^p\right )}}{x} \, dx=-\frac {x^{-p}}{a p}-\frac {x^{-p} \sqrt {1-a x^p}}{a p \sqrt {\frac {1}{1+a x^p}}}-\frac {\sqrt {\frac {1}{1+a x^p}} \sqrt {1+a x^p} \arcsin \left (a x^p\right )}{p} \]
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Time = 0.05 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.22, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6469, 265, 352, 248, 283, 222} \[ \int \frac {e^{\text {sech}^{-1}\left (a x^p\right )}}{x} \, dx=-\frac {x^{-p} \sqrt {\frac {1}{a x^p+1}} \sqrt {a x^p+1} \sqrt {1-a^2 x^{2 p}}}{a p}-\frac {x^{-p}}{a p}-\frac {\sqrt {\frac {1}{a x^p+1}} \sqrt {a x^p+1} \csc ^{-1}\left (\frac {x^{-p}}{a}\right )}{p} \]
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Rule 222
Rule 248
Rule 265
Rule 283
Rule 352
Rule 6469
Rubi steps \begin{align*} \text {integral}& = -\frac {x^{-p}}{a p}+\frac {\left (\sqrt {\frac {1}{1+a x^p}} \sqrt {1+a x^p}\right ) \int x^{-1-p} \sqrt {1-a x^p} \sqrt {1+a x^p} \, dx}{a} \\ & = -\frac {x^{-p}}{a p}+\frac {\left (\sqrt {\frac {1}{1+a x^p}} \sqrt {1+a x^p}\right ) \int x^{-1-p} \sqrt {1-a^2 x^{2 p}} \, dx}{a} \\ & = -\frac {x^{-p}}{a p}-\frac {\left (\sqrt {\frac {1}{1+a x^p}} \sqrt {1+a x^p}\right ) \text {Subst}\left (\int \sqrt {1-\frac {a^2}{x^2}} \, dx,x,x^{-p}\right )}{a p} \\ & = -\frac {x^{-p}}{a p}+\frac {\left (\sqrt {\frac {1}{1+a x^p}} \sqrt {1+a x^p}\right ) \text {Subst}\left (\int \frac {\sqrt {1-a^2 x^2}}{x^2} \, dx,x,x^p\right )}{a p} \\ & = -\frac {x^{-p}}{a p}-\frac {x^{-p} \sqrt {\frac {1}{1+a x^p}} \sqrt {1+a x^p} \sqrt {1-a^2 x^{2 p}}}{a p}-\frac {\left (a \sqrt {\frac {1}{1+a x^p}} \sqrt {1+a x^p}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-a^2 x^2}} \, dx,x,x^p\right )}{p} \\ & = -\frac {x^{-p}}{a p}-\frac {x^{-p} \sqrt {\frac {1}{1+a x^p}} \sqrt {1+a x^p} \sqrt {1-a^2 x^{2 p}}}{a p}-\frac {\sqrt {\frac {1}{1+a x^p}} \sqrt {1+a x^p} \arcsin \left (a x^p\right )}{p} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.15 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.10 \[ \int \frac {e^{\text {sech}^{-1}\left (a x^p\right )}}{x} \, dx=-\frac {i \left (-i x^{-p}-i \left (a+x^{-p}\right ) \sqrt {\frac {1-a x^p}{1+a x^p}}+a \log \left (-2 i a x^p+2 \sqrt {\frac {1-a x^p}{1+a x^p}} \left (1+a x^p\right )\right )\right )}{a p} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.17 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.33
method | result | size |
derivativedivides | \(\frac {-\frac {\sqrt {-\frac {\left (a \,x^{p}-1\right ) x^{-p}}{a}}\, \sqrt {\frac {\left (1+a \,x^{p}\right ) x^{-p}}{a}}\, \left (\arctan \left (\frac {\operatorname {csgn}\left (a \right ) a \,x^{p}}{\sqrt {-a^{2} x^{2 p}+1}}\right ) a \,x^{p}+\sqrt {-a^{2} x^{2 p}+1}\, \operatorname {csgn}\left (a \right )\right ) \operatorname {csgn}\left (a \right )}{\sqrt {-a^{2} x^{2 p}+1}}-\frac {x^{-p}}{a}}{p}\) | \(116\) |
default | \(\frac {-\frac {\sqrt {-\frac {\left (a \,x^{p}-1\right ) x^{-p}}{a}}\, \sqrt {\frac {\left (1+a \,x^{p}\right ) x^{-p}}{a}}\, \left (\arctan \left (\frac {\operatorname {csgn}\left (a \right ) a \,x^{p}}{\sqrt {-a^{2} x^{2 p}+1}}\right ) a \,x^{p}+\sqrt {-a^{2} x^{2 p}+1}\, \operatorname {csgn}\left (a \right )\right ) \operatorname {csgn}\left (a \right )}{\sqrt {-a^{2} x^{2 p}+1}}-\frac {x^{-p}}{a}}{p}\) | \(116\) |
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Time = 0.28 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.17 \[ \int \frac {e^{\text {sech}^{-1}\left (a x^p\right )}}{x} \, dx=-\frac {a x^{p} \sqrt {\frac {a x^{p} + 1}{a x^{p}}} \sqrt {-\frac {a x^{p} - 1}{a x^{p}}} - a x^{p} \arctan \left (\sqrt {\frac {a x^{p} + 1}{a x^{p}}} \sqrt {-\frac {a x^{p} - 1}{a x^{p}}}\right ) + 1}{a p x^{p}} \]
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\[ \int \frac {e^{\text {sech}^{-1}\left (a x^p\right )}}{x} \, dx=\frac {\int \frac {x^{- p}}{x}\, dx + \int \frac {a \sqrt {-1 + \frac {x^{- p}}{a}} \sqrt {1 + \frac {x^{- p}}{a}}}{x}\, dx}{a} \]
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\[ \int \frac {e^{\text {sech}^{-1}\left (a x^p\right )}}{x} \, dx=\int { \frac {\sqrt {\frac {1}{a x^{p}} + 1} \sqrt {\frac {1}{a x^{p}} - 1} + \frac {1}{a x^{p}}}{x} \,d x } \]
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\[ \int \frac {e^{\text {sech}^{-1}\left (a x^p\right )}}{x} \, dx=\int { \frac {\sqrt {\frac {1}{a x^{p}} + 1} \sqrt {\frac {1}{a x^{p}} - 1} + \frac {1}{a x^{p}}}{x} \,d x } \]
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Timed out. \[ \int \frac {e^{\text {sech}^{-1}\left (a x^p\right )}}{x} \, dx=\int \frac {\sqrt {\frac {1}{a\,x^p}-1}\,\sqrt {\frac {1}{a\,x^p}+1}+\frac {1}{a\,x^p}}{x} \,d x \]
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