Optimal. Leaf size=107 \[ \frac{p x^{-p-1} \sqrt{\frac{1}{a x^p+1}} \sqrt{a x^p+1} \text{Hypergeometric2F1}\left (\frac{1}{2},-\frac{p+1}{2 p},-\frac{1-p}{2 p},a^2 x^{2 p}\right )}{a (p+1)}+\frac{p x^{-p-1}}{a (p+1)}-\frac{e^{\text{sech}^{-1}\left (a x^p\right )}}{x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0604132, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6335, 30, 259, 364} \[ \frac{p x^{-p-1} \sqrt{\frac{1}{a x^p+1}} \sqrt{a x^p+1} \, _2F_1\left (\frac{1}{2},-\frac{p+1}{2 p};-\frac{1-p}{2 p};a^2 x^{2 p}\right )}{a (p+1)}+\frac{p x^{-p-1}}{a (p+1)}-\frac{e^{\text{sech}^{-1}\left (a x^p\right )}}{x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6335
Rule 30
Rule 259
Rule 364
Rubi steps
\begin{align*} \int \frac{e^{\text{sech}^{-1}\left (a x^p\right )}}{x^2} \, dx &=-\frac{e^{\text{sech}^{-1}\left (a x^p\right )}}{x}-\frac{p \int x^{-2-p} \, dx}{a}-\frac{\left (p \sqrt{\frac{1}{1+a x^p}} \sqrt{1+a x^p}\right ) \int \frac{x^{-2-p}}{\sqrt{1-a x^p} \sqrt{1+a x^p}} \, dx}{a}\\ &=-\frac{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^{-2-p}}{\sqrt{1-a^2 x^{2 p}}} \, dx}{a}\\ &=-\frac{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+p}{2 p};-\frac{1-p}{2 p};a^2 x^{2 p}\right )}{a (1+p)}\\ \end{align*}
Mathematica [A] time = 0.258708, size = 156, normalized size = 1.46 \[ x^{-p-1} \left (\frac{a p x^{2 p} \sqrt{\frac{1-a x^p}{a x^p+1}} \sqrt{1-a^2 x^{2 p}} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{p-1}{2 p},\frac{3}{2}-\frac{1}{2 p},a^2 x^{2 p}\right )}{(p-1) (p+1) \left (a x^p-1\right )}-\frac{\sqrt{\frac{1-a x^p}{a x^p+1}} \left (a x^p+1\right )}{a (p+1)}-\frac{1}{a p+a}\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 1.182, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \left ({\frac{1}{a{x}^{p}}}+\sqrt{{\frac{1}{a{x}^{p}}}-1}\sqrt{{\frac{1}{a{x}^{p}}}+1} \right ) }\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{x^{- p}}{x^{2}}\, dx + \int \frac{a \sqrt{-1 + \frac{x^{- p}}{a}} \sqrt{1 + \frac{x^{- p}}{a}}}{x^{2}}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]