Optimal. Leaf size=105 \[ \frac{p x^{1-p} \sqrt{\frac{1}{a x^p+1}} \sqrt{a x^p+1} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1}{2} \left (\frac{1}{p}-1\right ),\frac{p+1}{2 p},a^2 x^{2 p}\right )}{a (1-p)}+\frac{p x^{1-p}}{a (1-p)}+x e^{\text{sech}^{-1}\left (a x^p\right )} \]
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Rubi [A] time = 0.0468155, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6330, 30, 259, 364} \[ \frac{p x^{1-p} \sqrt{\frac{1}{a x^p+1}} \sqrt{a x^p+1} \, _2F_1\left (\frac{1}{2},\frac{1}{2} \left (\frac{1}{p}-1\right );\frac{p+1}{2 p};a^2 x^{2 p}\right )}{a (1-p)}+\frac{p x^{1-p}}{a (1-p)}+x e^{\text{sech}^{-1}\left (a x^p\right )} \]
Antiderivative was successfully verified.
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Rule 6330
Rule 30
Rule 259
Rule 364
Rubi steps
\begin{align*} \int e^{\text{sech}^{-1}\left (a x^p\right )} \, dx &=e^{\text{sech}^{-1}\left (a x^p\right )} x+\frac{p \int x^{-p} \, dx}{a}+\frac{\left (p \sqrt{\frac{1}{1+a x^p}} \sqrt{1+a x^p}\right ) \int \frac{x^{-p}}{\sqrt{1-a x^p} \sqrt{1+a x^p}} \, dx}{a}\\ &=e^{\text{sech}^{-1}\left (a x^p\right )} x+\frac{p x^{1-p}}{a (1-p)}+\frac{\left (p \sqrt{\frac{1}{1+a x^p}} \sqrt{1+a x^p}\right ) \int \frac{x^{-p}}{\sqrt{1-a^2 x^{2 p}}} \, dx}{a}\\ &=e^{\text{sech}^{-1}\left (a x^p\right )} x+\frac{p x^{1-p}}{a (1-p)}+\frac{p x^{1-p} \sqrt{\frac{1}{1+a x^p}} \sqrt{1+a x^p} \, _2F_1\left (\frac{1}{2},\frac{1}{2} \left (-1+\frac{1}{p}\right );\frac{1+p}{2 p};a^2 x^{2 p}\right )}{a (1-p)}\\ \end{align*}
Mathematica [A] time = 0.385518, size = 139, normalized size = 1.32 \[ \frac{x \left (-\frac{a^2 p x^p \sqrt{\frac{1-a x^p}{a x^p+1}} \sqrt{1-a^2 x^{2 p}} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{p+1}{2 p},\frac{1}{2} \left (\frac{1}{p}+3\right ),a^2 x^{2 p}\right )}{(p+1) \left (a x^p-1\right )}+\left (a+x^{-p}\right ) \sqrt{\frac{1-a x^p}{a x^p+1}}+x^{-p}\right )}{a-a p} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 1.154, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{a{x}^{p}}}+\sqrt{{\frac{1}{a{x}^{p}}}-1}\sqrt{{\frac{1}{a{x}^{p}}}+1}\, dx \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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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 x^{- p}\, dx + \int a \sqrt{-1 + \frac{x^{- p}}{a}} \sqrt{1 + \frac{x^{- p}}{a}}\, 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 \sqrt{\frac{1}{a x^{p}} + 1} \sqrt{\frac{1}{a x^{p}} - 1} + \frac{1}{a x^{p}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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