Optimal. Leaf size=87 \[ -\frac{x^{-p} \sqrt{1-a x^p}}{a p \sqrt{\frac{1}{a x^p+1}}}-\frac{x^{-p}}{a p}-\frac{\sqrt{\frac{1}{a x^p+1}} \sqrt{a x^p+1} \sin ^{-1}\left (a x^p\right )}{p} \]
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Rubi [A] time = 0.0689316, antiderivative size = 106, normalized size of antiderivative = 1.22, number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, 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 6334
Rule 259
Rule 345
Rule 242
Rule 277
Rule 216
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*}
Mathematica [C] time = 0.137093, size = 96, normalized size = 1.1 \[ -\frac{i \left (-i \left (a+x^{-p}\right ) \sqrt{\frac{1-a x^p}{a x^p+1}}+a \log \left (2 \sqrt{\frac{1-a x^p}{a x^p+1}} \left (a x^p+1\right )-2 i a x^p\right )-i x^{-p}\right )}{a p} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.326, size = 145, normalized size = 1.7 \begin{align*} -{\frac{{\it csgn} \left ( a \right ){x}^{p}a}{p}\sqrt{-{\frac{a{x}^{p}-1}{a{x}^{p}}}}\sqrt{{\frac{1+a{x}^{p}}{a{x}^{p}}}}\arctan \left ({{\it csgn} \left ( a \right ){x}^{p}a{\frac{1}{\sqrt{- \left ({x}^{p} \right ) ^{2}{a}^{2}+1}}}} \right ){\frac{1}{\sqrt{- \left ({x}^{p} \right ) ^{2}{a}^{2}+1}}}}-{\frac{ \left ({\it csgn} \left ( a \right ) \right ) ^{2}}{p}\sqrt{-{\frac{a{x}^{p}-1}{a{x}^{p}}}}\sqrt{{\frac{1+a{x}^{p}}{a{x}^{p}}}}}-{\frac{1}{ap{x}^{p}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.45426, size = 197, normalized size = 2.26 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\frac{1}{a x^{p}} + 1} \sqrt{\frac{1}{a x^{p}} - 1} + \frac{1}{a x^{p}}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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