3.62 \(\int e^{\text{sech}^{-1}(a x^p)} \, dx\)

Optimal. Leaf size=105 \[ \frac{p x^{1-p} \sqrt{\frac{1}{a x^p+1}} \sqrt{a x^p+1} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1}{2} \left (\frac{1}{p}-1\right ),\frac{p+1}{2 p},a^2 x^{2 p}\right )}{a (1-p)}+\frac{p x^{1-p}}{a (1-p)}+x e^{\text{sech}^{-1}\left (a x^p\right )} \]

[Out]

E^ArcSech[a*x^p]*x + (p*x^(1 - p))/(a*(1 - p)) + (p*x^(1 - p)*Sqrt[(1 + a*x^p)^(-1)]*Sqrt[1 + a*x^p]*Hypergeom
etric2F1[1/2, (-1 + p^(-1))/2, (1 + p)/(2*p), a^2*x^(2*p)])/(a*(1 - p))

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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.

[In]

Int[E^ArcSech[a*x^p],x]

[Out]

E^ArcSech[a*x^p]*x + (p*x^(1 - p))/(a*(1 - p)) + (p*x^(1 - p)*Sqrt[(1 + a*x^p)^(-1)]*Sqrt[1 + a*x^p]*Hypergeom
etric2F1[1/2, (-1 + p^(-1))/2, (1 + p)/(2*p), a^2*x^(2*p)])/(a*(1 - p))

Rule 6330

Int[E^ArcSech[(a_.)*(x_)^(p_)], x_Symbol] :> Simp[x*E^ArcSech[a*x^p], x] + (Dist[p/a, Int[1/x^p, x], x] + Dist
[(p*Sqrt[1 + a*x^p]*Sqrt[1/(1 + a*x^p)])/a, Int[1/(x^p*Sqrt[1 + a*x^p]*Sqrt[1 - a*x^p]), x], x]) /; FreeQ[{a,
p}, x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 259

Int[((c_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(n_))^(p_)*((a2_) + (b2_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[(c*x)
^m*(a1*a2 + b1*b2*x^(2*n))^p, x] /; FreeQ[{a1, b1, a2, b2, c, m, n, p}, x] && EqQ[a2*b1 + a1*b2, 0] && (Intege
rQ[p] || (GtQ[a1, 0] && GtQ[a2, 0]))

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

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.

[In]

Integrate[E^ArcSech[a*x^p],x]

[Out]

(x*(x^(-p) + (a + x^(-p))*Sqrt[(1 - a*x^p)/(1 + a*x^p)] - (a^2*p*x^p*Sqrt[(1 - a*x^p)/(1 + a*x^p)]*Sqrt[1 - a^
2*x^(2*p)]*Hypergeometric2F1[1/2, (1 + p)/(2*p), (3 + p^(-1))/2, a^2*x^(2*p)])/((1 + p)*(-1 + a*x^p))))/(a - a
*p)

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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.

[In]

int(1/a/(x^p)+(1/a/(x^p)-1)^(1/2)*(1/a/(x^p)+1)^(1/2),x)

[Out]

int(1/a/(x^p)+(1/a/(x^p)-1)^(1/2)*(1/a/(x^p)+1)^(1/2),x)

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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.

[In]

integrate(1/a/(x^p)+(1/a/(x^p)-1)^(1/2)*(1/a/(x^p)+1)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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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.

[In]

integrate(1/a/(x^p)+(1/a/(x^p)-1)^(1/2)*(1/a/(x^p)+1)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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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.

[In]

integrate(1/a/(x**p)+(1/a/(x**p)-1)**(1/2)*(1/a/(x**p)+1)**(1/2),x)

[Out]

(Integral(x**(-p), x) + Integral(a*sqrt(-1 + x**(-p)/a)*sqrt(1 + x**(-p)/a), x))/a

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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.

[In]

integrate(1/a/(x^p)+(1/a/(x^p)-1)^(1/2)*(1/a/(x^p)+1)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(1/(a*x^p) + 1)*sqrt(1/(a*x^p) - 1) + 1/(a*x^p), x)