### 3.37 $$\int e^{\text{csch}^{-1}(a x^2)} x^m \, dx$$

Optimal. Leaf size=59 $\frac{x^{m+1} \text{Hypergeometric2F1}\left (-\frac{1}{2},\frac{1}{4} (-m-1),\frac{3-m}{4},-\frac{1}{a^2 x^4}\right )}{m+1}-\frac{x^{m-1}}{a (1-m)}$

[Out]

-(x^(-1 + m)/(a*(1 - m))) + (x^(1 + m)*Hypergeometric2F1[-1/2, (-1 - m)/4, (3 - m)/4, -(1/(a^2*x^4))])/(1 + m)

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Rubi [A]  time = 0.0415475, antiderivative size = 59, 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 = {6336, 30, 339, 364} $\frac{x^{m+1} \, _2F_1\left (-\frac{1}{2},\frac{1}{4} (-m-1);\frac{3-m}{4};-\frac{1}{a^2 x^4}\right )}{m+1}-\frac{x^{m-1}}{a (1-m)}$

Antiderivative was successfully veriﬁed.

[In]

Int[E^ArcCsch[a*x^2]*x^m,x]

[Out]

-(x^(-1 + m)/(a*(1 - m))) + (x^(1 + m)*Hypergeometric2F1[-1/2, (-1 - m)/4, (3 - m)/4, -(1/(a^2*x^4))])/(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}\left (a x^2\right )} x^m \, dx &=\frac{\int x^{-2+m} \, dx}{a}+\int \sqrt{1+\frac{1}{a^2 x^4}} x^m \, dx\\ &=-\frac{x^{-1+m}}{a (1-m)}-\left (\left (\frac{1}{x}\right )^m x^m\right ) \operatorname{Subst}\left (\int x^{-2-m} \sqrt{1+\frac{x^4}{a^2}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{x^{-1+m}}{a (1-m)}+\frac{x^{1+m} \, _2F_1\left (-\frac{1}{2},\frac{1}{4} (-1-m);\frac{3-m}{4};-\frac{1}{a^2 x^4}\right )}{1+m}\\ \end{align*}

Mathematica [A]  time = 0.0612119, size = 55, normalized size = 0.93 $x^{m-1} \left (\frac{x^2 \text{Hypergeometric2F1}\left (-\frac{1}{2},-\frac{m}{4}-\frac{1}{4},\frac{3}{4}-\frac{m}{4},-\frac{1}{a^2 x^4}\right )}{m+1}+\frac{1}{a (m-1)}\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[E^ArcCsch[a*x^2]*x^m,x]

[Out]

x^(-1 + m)*(1/(a*(-1 + m)) + (x^2*Hypergeometric2F1[-1/2, -1/4 - m/4, 3/4 - m/4, -(1/(a^2*x^4))])/(1 + m))

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Maple [F]  time = 0.199, size = 0, normalized size = 0. \begin{align*} \int \left ({\frac{1}{a{x}^{2}}}+\sqrt{1+{\frac{1}{{a}^{2}{x}^{4}}}} \right ){x}^{m}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((1/a/x^2+(1+1/a^2/x^4)^(1/2))*x^m,x)

[Out]

int((1/a/x^2+(1+1/a^2/x^4)^(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^2+(1+1/a^2/x^4)^(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^{2} x^{m} \sqrt{\frac{a^{2} x^{4} + 1}{a^{2} x^{4}}} + x^{m}}{a x^{2}}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/a/x^2+(1+1/a^2/x^4)^(1/2))*x^m,x, algorithm="fricas")

[Out]

integral((a*x^2*x^m*sqrt((a^2*x^4 + 1)/(a^2*x^4)) + x^m)/(a*x^2), x)

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Sympy [A]  time = 19.5374, size = 66, normalized size = 1.12 \begin{align*} - \frac{x x^{m} \Gamma \left (- \frac{m}{4} - \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, - \frac{m}{4} - \frac{1}{4} \\ \frac{3}{4} - \frac{m}{4} \end{matrix}\middle |{\frac{e^{i \pi }}{a^{2} x^{4}}} \right )}}{4 \Gamma \left (\frac{3}{4} - \frac{m}{4}\right )} + \frac{\begin{cases} \frac{x^{m}}{m x - x} & \text{for}\: m \neq 1 \\\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**2+(1+1/a**2/x**4)**(1/2))*x**m,x)

[Out]

-x*x**m*gamma(-m/4 - 1/4)*hyper((-1/2, -m/4 - 1/4), (3/4 - m/4,), exp_polar(I*pi)/(a**2*x**4))/(4*gamma(3/4 -
m/4)) + Piecewise((x**m/(m*x - x), Ne(m, 1)), (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^{4}} + 1} + \frac{1}{a x^{2}}\right )}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/a/x^2+(1+1/a^2/x^4)^(1/2))*x^m,x, algorithm="giac")

[Out]

integrate(x^m*(sqrt(1/(a^2*x^4) + 1) + 1/(a*x^2)), x)