Optimal. Leaf size=87 \[ -\frac {x^{-p}}{a p}-\frac {\sqrt {\frac {1}{a x^p+1}} \sqrt {a x^p+1} \sin ^{-1}\left (a x^p\right )}{p}-\frac {x^{-p} \sqrt {1-a x^p}}{a p \sqrt {\frac {1}{a x^p+1}}} \]
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Rubi [A] time = 0.07, antiderivative size = 106, normalized size of antiderivative = 1.22, number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6334, 259, 345, 242, 277, 216} \[ -\frac {x^{-p} \sqrt {\frac {1}{a x^p+1}} \sqrt {a x^p+1} \sqrt {1-a^2 x^{2 p}}}{a p}-\frac {x^{-p}}{a p}-\frac {\sqrt {\frac {1}{a x^p+1}} \sqrt {a x^p+1} \csc ^{-1}\left (\frac {x^{-p}}{a}\right )}{p} \]
Warning: Unable to verify antiderivative.
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Rule 216
Rule 242
Rule 259
Rule 277
Rule 345
Rule 6334
Rubi steps
\begin {align*} \int \frac {e^{\text {sech}^{-1}\left (a x^p\right )}}{x} \, dx &=-\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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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} \sin ^{-1}\left (a x^p\right )}{p}\\ \end {align*}
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Mathematica [C] time = 0.16, size = 96, normalized size = 1.10 \[ -\frac {i \left (a \log \left (2 \sqrt {\frac {1-a x^p}{a x^p+1}} \left (a x^p+1\right )-2 i a x^p\right )-i \left (a+x^{-p}\right ) \sqrt {\frac {1-a x^p}{a x^p+1}}-i x^{-p}\right )}{a p} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.70, size = 102, normalized size = 1.17 \[ -\frac {a x^{p} \sqrt {\frac {a x^{p} + 1}{a x^{p}}} \sqrt {-\frac {a x^{p} - 1}{a x^{p}}} - a x^{p} \arctan \left (\sqrt {\frac {a x^{p} + 1}{a x^{p}}} \sqrt {-\frac {a x^{p} - 1}{a x^{p}}}\right ) + 1}{a p x^{p}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\frac {1}{a x^{p}} + 1} \sqrt {\frac {1}{a x^{p}} - 1} + \frac {1}{a x^{p}}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.15, size = 145, normalized size = 1.67 \[ -\frac {\sqrt {-\frac {\left (a \,x^{p}-1\right ) x^{-p}}{a}}\, \sqrt {\frac {\left (1+a \,x^{p}\right ) x^{-p}}{a}}\, \mathrm {csgn}\relax (a ) \arctan \left (\frac {\mathrm {csgn}\relax (a ) a \,x^{p}}{\sqrt {1-a^{2} x^{2 p}}}\right ) x^{p} a}{p \sqrt {1-a^{2} x^{2 p}}}-\frac {\sqrt {-\frac {\left (a \,x^{p}-1\right ) x^{-p}}{a}}\, \sqrt {\frac {\left (1+a \,x^{p}\right ) x^{-p}}{a}}\, \mathrm {csgn}\relax (a )^{2}}{p}-\frac {x^{-p}}{a p} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\sqrt {a x^{p} + 1} \sqrt {-a x^{p} + 1}}{x x^{p}}\,{d x}}{a} - \frac {1}{a p x^{p}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {\frac {1}{a\,x^p}-1}\,\sqrt {\frac {1}{a\,x^p}+1}+\frac {1}{a\,x^p}}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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