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

Optimal. Leaf size=119 \[ \frac{p x^{2-p} \sqrt{\frac{1}{a x^p+1}} \sqrt{a x^p+1} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1}{2} \left (\frac{2}{p}-1\right ),\frac{1}{2} \left (\frac{2}{p}+1\right ),a^2 x^{2 p}\right )}{2 a (2-p)}+\frac{p x^{2-p}}{2 a (2-p)}+\frac{1}{2} x^2 e^{\text{sech}^{-1}\left (a x^p\right )} \]

[Out]

(E^ArcSech[a*x^p]*x^2)/2 + (p*x^(2 - p))/(2*a*(2 - p)) + (p*x^(2 - p)*Sqrt[(1 + a*x^p)^(-1)]*Sqrt[1 + a*x^p]*H
ypergeometric2F1[1/2, (-1 + 2/p)/2, (1 + 2/p)/2, a^2*x^(2*p)])/(2*a*(2 - p))

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Rubi [A]  time = 0.0528343, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {6335, 30, 259, 364} \[ \frac{p x^{2-p} \sqrt{\frac{1}{a x^p+1}} \sqrt{a x^p+1} \, _2F_1\left (\frac{1}{2},\frac{1}{2} \left (\frac{2}{p}-1\right );\frac{1}{2} \left (1+\frac{2}{p}\right );a^2 x^{2 p}\right )}{2 a (2-p)}+\frac{p x^{2-p}}{2 a (2-p)}+\frac{1}{2} x^2 e^{\text{sech}^{-1}\left (a x^p\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(E^ArcSech[a*x^p]*x^2)/2 + (p*x^(2 - p))/(2*a*(2 - p)) + (p*x^(2 - p)*Sqrt[(1 + a*x^p)^(-1)]*Sqrt[1 + a*x^p]*H
ypergeometric2F1[1/2, (-1 + 2/p)/2, (1 + 2/p)/2, a^2*x^(2*p)])/(2*a*(2 - p))

Rule 6335

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

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 )} x \, dx &=\frac{1}{2} e^{\text{sech}^{-1}\left (a x^p\right )} x^2+\frac{p \int x^{1-p} \, dx}{2 a}+\frac{\left (p \sqrt{\frac{1}{1+a x^p}} \sqrt{1+a x^p}\right ) \int \frac{x^{1-p}}{\sqrt{1-a x^p} \sqrt{1+a x^p}} \, dx}{2 a}\\ &=\frac{1}{2} e^{\text{sech}^{-1}\left (a x^p\right )} x^2+\frac{p x^{2-p}}{2 a (2-p)}+\frac{\left (p \sqrt{\frac{1}{1+a x^p}} \sqrt{1+a x^p}\right ) \int \frac{x^{1-p}}{\sqrt{1-a^2 x^{2 p}}} \, dx}{2 a}\\ &=\frac{1}{2} e^{\text{sech}^{-1}\left (a x^p\right )} x^2+\frac{p x^{2-p}}{2 a (2-p)}+\frac{p x^{2-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{2}{p}\right );\frac{1}{2} \left (1+\frac{2}{p}\right );a^2 x^{2 p}\right )}{2 a (2-p)}\\ \end{align*}

Mathematica [A]  time = 0.29688, size = 159, normalized size = 1.34 \[ \frac{x^{2-p} \left (\frac{a^2 p x^{2 p} \sqrt{\frac{1-a x^p}{a x^p+1}} \sqrt{1-a^2 x^{2 p}} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1}{p}+\frac{1}{2},\frac{1}{p}+\frac{3}{2},a^2 x^{2 p}\right )}{(p+2) \left (a x^p-1\right )}-a x^p \sqrt{\frac{1-a x^p}{a x^p+1}}-\sqrt{\frac{1-a x^p}{a x^p+1}}-1\right )}{a (p-2)} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]  time = 1.102, size = 0, normalized size = 0. \begin{align*} \int \left ({\frac{1}{a{x}^{p}}}+\sqrt{{\frac{1}{a{x}^{p}}}-1}\sqrt{{\frac{1}{a{x}^{p}}}+1} \right ) x\, 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,x)

[Out]

int((1/a/(x^p)+(1/a/(x^p)-1)^(1/2)*(1/a/(x^p)+1)^(1/2))*x,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,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,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 x^{- p}\, dx + \int a x \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,x)

[Out]

(Integral(x*x**(-p), x) + Integral(a*x*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 x{\left (\sqrt{\frac{1}{a x^{p}} + 1} \sqrt{\frac{1}{a x^{p}} - 1} + \frac{1}{a x^{p}}\right )}\,{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,x, algorithm="giac")

[Out]

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