\(\int \frac {e^{\text {sech}^{-1}(a x^p)}}{x^2} \, dx\) [64]

Optimal result

Integrand size = 12, antiderivative size = 107 \[ \int \frac {e^{\text {sech}^{-1}\left (a x^p\right )}}{x^2} \, dx=-\frac {e^{\text {sech}^{-1}\left (a x^p\right )}}{x}+\frac {p x^{-1-p}}{a (1+p)}+\frac {p x^{-1-p} \sqrt {\frac {1}{1+a x^p}} \sqrt {1+a x^p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {1+p}{2 p},-\frac {1-p}{2 p},a^2 x^{2 p}\right )}{a (1+p)} \]

[Out]

-(1/a/(x^p)+(1/a/(x^p)-1)^(1/2)*(1/a/(x^p)+1)^(1/2))/x+p*x^(-1-p)/a/(p+1)+p*x^(-1-p)*hypergeom([1/2, 1/2*(-1-p
)/p],[1/2*(-1+p)/p],a^2*x^(2*p))*(1/(1+a*x^p))^(1/2)*(1+a*x^p)^(1/2)/a/(p+1)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6470, 30, 265, 371} \[ \int \frac {e^{\text {sech}^{-1}\left (a x^p\right )}}{x^2} \, dx=\frac {p x^{-p-1} \sqrt {\frac {1}{a x^p+1}} \sqrt {a x^p+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {p+1}{2 p},-\frac {1-p}{2 p},a^2 x^{2 p}\right )}{a (p+1)}+\frac {p x^{-p-1}}{a (p+1)}-\frac {e^{\text {sech}^{-1}\left (a x^p\right )}}{x} \]

[In]

Int[E^ArcSech[a*x^p]/x^2,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]*Hype
rgeometric2F1[1/2, -1/2*(1 + p)/p, -1/2*(1 - p)/p, a^2*x^(2*p)])/(a*(1 + p))

Rule 30

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

Rule 265

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 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 6470

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]/(a*(m + 1)))*Sqrt[1/(1 + a*x^p)], Int[x^(m - p
)/(Sqrt[1 + a*x^p]*Sqrt[1 - a*x^p]), x], x]) /; FreeQ[{a, m, p}, x] && NeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = -\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} \operatorname {Hypergeometric2F1}\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] (verified)

Time = 0.24 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.55 \[ \int \frac {e^{\text {sech}^{-1}\left (a x^p\right )}}{x^2} \, dx=\frac {x^{-1-p} \left (-1-\sqrt {\frac {1-a x^p}{1+a x^p}}-a x^p \sqrt {\frac {1-a x^p}{1+a x^p}}+\frac {a^2 p x^{2 p} \sqrt {\frac {1-a x^p}{1+a x^p}} \sqrt {1-a^2 x^{2 p}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {-1+p}{2 p},\frac {3}{2}-\frac {1}{2 p},a^2 x^{2 p}\right )}{(-1+p) \left (-1+a x^p\right )}\right )}{a (1+p)} \]

[In]

Integrate[E^ArcSech[a*x^p]/x^2,x]

[Out]

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

Maple [F]

\[\int \frac {\frac {x^{-p}}{a}+\sqrt {\frac {x^{-p}}{a}-1}\, \sqrt {\frac {x^{-p}}{a}+1}}{x^{2}}d x\]

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {e^{\text {sech}^{-1}\left (a x^p\right )}}{x^2} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

Sympy [F]

\[ \int \frac {e^{\text {sech}^{-1}\left (a x^p\right )}}{x^2} \, dx=\frac {\int \frac {x^{- p}}{x^{2}}\, dx + \int \frac {a \sqrt {-1 + \frac {x^{- p}}{a}} \sqrt {1 + \frac {x^{- p}}{a}}}{x^{2}}\, dx}{a} \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {e^{\text {sech}^{-1}\left (a x^p\right )}}{x^2} \, dx=\int { \frac {\sqrt {\frac {1}{a x^{p}} + 1} \sqrt {\frac {1}{a x^{p}} - 1} + \frac {1}{a x^{p}}}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {e^{\text {sech}^{-1}\left (a x^p\right )}}{x^2} \, dx=\int { \frac {\sqrt {\frac {1}{a x^{p}} + 1} \sqrt {\frac {1}{a x^{p}} - 1} + \frac {1}{a x^{p}}}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{\text {sech}^{-1}\left (a x^p\right )}}{x^2} \, dx=\int \frac {\sqrt {\frac {1}{a\,x^p}-1}\,\sqrt {\frac {1}{a\,x^p}+1}+\frac {1}{a\,x^p}}{x^2} \,d x \]

[In]

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

[Out]

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