### 3.26 $$\int e^{\text{csch}^{-1}(a x)} x^m \, dx$$

Optimal. Leaf size=52 $\frac{x^{m+1} \text{Hypergeometric2F1}\left (-\frac{1}{2},\frac{1}{2} (-m-1),\frac{1-m}{2},-\frac{1}{a^2 x^2}\right )}{m+1}+\frac{x^m}{a m}$

[Out]

x^m/(a*m) + (x^(1 + m)*Hypergeometric2F1[-1/2, (-1 - m)/2, (1 - m)/2, -(1/(a^2*x^2))])/(1 + m)

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Rubi [A]  time = 0.0398986, antiderivative size = 52, 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 = {6336, 30, 339, 364} $\frac{x^{m+1} \, _2F_1\left (-\frac{1}{2},\frac{1}{2} (-m-1);\frac{1-m}{2};-\frac{1}{a^2 x^2}\right )}{m+1}+\frac{x^m}{a m}$

Antiderivative was successfully veriﬁed.

[In]

Int[E^ArcCsch[a*x]*x^m,x]

[Out]

x^m/(a*m) + (x^(1 + m)*Hypergeometric2F1[-1/2, (-1 - m)/2, (1 - m)/2, -(1/(a^2*x^2))])/(1 + m)

Rule 6336

Int[E^ArcCsch[(a_.)*(x_)^(p_.)]*(x_)^(m_.), x_Symbol] :> Dist[1/a, Int[x^(m - p), x], x] + Int[x^m*Sqrt[1 + 1/
(a^2*x^(2*p))], x] /; FreeQ[{a, m, p}, x]

Rule 30

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

Rule 339

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Dist[((c*x)^(m + 1)*(1/x)^(m + 1))/c, Subst
[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x], x] /; FreeQ[{a, b, c, m, p}, x] && ILtQ[n, 0] &&  !RationalQ[m]

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{csch}^{-1}(a x)} x^m \, dx &=\frac{\int x^{-1+m} \, dx}{a}+\int \sqrt{1+\frac{1}{a^2 x^2}} x^m \, dx\\ &=\frac{x^m}{a m}-\left (\left (\frac{1}{x}\right )^m x^m\right ) \operatorname{Subst}\left (\int x^{-2-m} \sqrt{1+\frac{x^2}{a^2}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{x^m}{a m}+\frac{x^{1+m} \, _2F_1\left (-\frac{1}{2},\frac{1}{2} (-1-m);\frac{1-m}{2};-\frac{1}{a^2 x^2}\right )}{1+m}\\ \end{align*}

Mathematica [A]  time = 0.0454976, size = 54, normalized size = 1.04 $\frac{x^{m+1} \text{Hypergeometric2F1}\left (-\frac{1}{2},\frac{1}{2} (-m-1),\frac{1}{2} (-m-1)+1,-\frac{1}{a^2 x^2}\right )}{m+1}+\frac{x^m}{a m}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[E^ArcCsch[a*x]*x^m,x]

[Out]

x^m/(a*m) + (x^(1 + m)*Hypergeometric2F1[-1/2, (-1 - m)/2, 1 + (-1 - m)/2, -(1/(a^2*x^2))])/(1 + m)

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Maple [F]  time = 0.255, size = 0, normalized size = 0. \begin{align*} \int \left ({\frac{1}{ax}}+\sqrt{1+{\frac{1}{{a}^{2}{x}^{2}}}} \right ){x}^{m}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((1/a/x+(1+1/a^2/x^2)^(1/2))*x^m,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+(1+1/a^2/x^2)^(1/2))*x^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{a x x^{m} \sqrt{\frac{a^{2} x^{2} + 1}{a^{2} x^{2}}} + x^{m}}{a x}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((a*x*x^m*sqrt((a^2*x^2 + 1)/(a^2*x^2)) + x^m)/(a*x), x)

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Sympy [A]  time = 8.6337, size = 51, normalized size = 0.98 \begin{align*} - \frac{x^{m} \Gamma \left (- \frac{m}{2}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{m}{2} \\ \frac{m}{2} + 1 \end{matrix}\middle |{a^{2} x^{2} e^{i \pi }} \right )}}{2 a \Gamma \left (1 - \frac{m}{2}\right )} + \frac{\begin{cases} \frac{x^{m}}{m} & \text{for}\: m \neq 0 \\\log{\left (x \right )} & \text{otherwise} \end{cases}}{a} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/a/x+(1+1/a**2/x**2)**(1/2))*x**m,x)

[Out]

-x**m*gamma(-m/2)*hyper((-1/2, m/2), (m/2 + 1,), a**2*x**2*exp_polar(I*pi))/(2*a*gamma(1 - m/2)) + Piecewise((
x**m/m, Ne(m, 0)), (log(x), True))/a

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m}{\left (\sqrt{\frac{1}{a^{2} x^{2}} + 1} + \frac{1}{a x}\right )}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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