### 3.42 $$\int e^{\text{csch}^{-1}(a x^2)} \, dx$$

Optimal. Leaf size=165 $-\frac{\sqrt{\frac{a^2+\frac{1}{x^4}}{\left (a+\frac{1}{x^2}\right )^2}} \left (a+\frac{1}{x^2}\right ) \text{EllipticF}\left (2 \cot ^{-1}\left (\sqrt{a} x\right ),\frac{1}{2}\right )}{a^{3/2} \sqrt{\frac{1}{a^2 x^4}+1}}+x \sqrt{\frac{1}{a^2 x^4}+1}-\frac{2 \sqrt{\frac{1}{a^2 x^4}+1}}{x \left (a+\frac{1}{x^2}\right )}+\frac{2 \sqrt{\frac{a^2+\frac{1}{x^4}}{\left (a+\frac{1}{x^2}\right )^2}} \left (a+\frac{1}{x^2}\right ) E\left (2 \cot ^{-1}\left (\sqrt{a} x\right )|\frac{1}{2}\right )}{a^{3/2} \sqrt{\frac{1}{a^2 x^4}+1}}-\frac{1}{a x}$

[Out]

-(1/(a*x)) - (2*Sqrt[1 + 1/(a^2*x^4)])/((a + x^(-2))*x) + Sqrt[1 + 1/(a^2*x^4)]*x + (2*Sqrt[(a^2 + x^(-4))/(a
+ x^(-2))^2]*(a + x^(-2))*EllipticE[2*ArcCot[Sqrt[a]*x], 1/2])/(a^(3/2)*Sqrt[1 + 1/(a^2*x^4)]) - (Sqrt[(a^2 +
x^(-4))/(a + x^(-2))^2]*(a + x^(-2))*EllipticF[2*ArcCot[Sqrt[a]*x], 1/2])/(a^(3/2)*Sqrt[1 + 1/(a^2*x^4)])

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Rubi [A]  time = 0.0803542, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 8, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.875, Rules used = {6331, 30, 242, 277, 305, 220, 1196} $x \sqrt{\frac{1}{a^2 x^4}+1}-\frac{2 \sqrt{\frac{1}{a^2 x^4}+1}}{x \left (a+\frac{1}{x^2}\right )}-\frac{\sqrt{\frac{a^2+\frac{1}{x^4}}{\left (a+\frac{1}{x^2}\right )^2}} \left (a+\frac{1}{x^2}\right ) F\left (2 \cot ^{-1}\left (\sqrt{a} x\right )|\frac{1}{2}\right )}{a^{3/2} \sqrt{\frac{1}{a^2 x^4}+1}}+\frac{2 \sqrt{\frac{a^2+\frac{1}{x^4}}{\left (a+\frac{1}{x^2}\right )^2}} \left (a+\frac{1}{x^2}\right ) E\left (2 \cot ^{-1}\left (\sqrt{a} x\right )|\frac{1}{2}\right )}{a^{3/2} \sqrt{\frac{1}{a^2 x^4}+1}}-\frac{1}{a x}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(1/(a*x)) - (2*Sqrt[1 + 1/(a^2*x^4)])/((a + x^(-2))*x) + Sqrt[1 + 1/(a^2*x^4)]*x + (2*Sqrt[(a^2 + x^(-4))/(a
+ x^(-2))^2]*(a + x^(-2))*EllipticE[2*ArcCot[Sqrt[a]*x], 1/2])/(a^(3/2)*Sqrt[1 + 1/(a^2*x^4)]) - (Sqrt[(a^2 +
x^(-4))/(a + x^(-2))^2]*(a + x^(-2))*EllipticF[2*ArcCot[Sqrt[a]*x], 1/2])/(a^(3/2)*Sqrt[1 + 1/(a^2*x^4)])

Rule 6331

Int[E^ArcCsch[(a_.)*(x_)^(p_.)], x_Symbol] :> Dist[1/a, Int[1/x^p, x], x] + Int[Sqrt[1 + 1/(a^2*x^(2*p))], 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 242

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

Rule 277

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

Rule 305

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 2]}, Dist[1/q, Int[1/Sqrt[a + b*x^4], x],
x] - Dist[1/q, Int[(1 - q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 220

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[((1 + q^2*x^2)*Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]*EllipticF[2*ArcTan[q*x], 1/2])/(2*q*Sqrt[a + b*x^4]), x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 1196

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, -Simp[(d*x*Sqrt[a + c
*x^4])/(a*(1 + q^2*x^2)), x] + Simp[(d*(1 + q^2*x^2)*Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]*EllipticE[2*ArcTan[
q*x], 1/2])/(q*Sqrt[a + c*x^4]), x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e}, x] && PosQ[c/a]

Rubi steps

\begin{align*} \int e^{\text{csch}^{-1}\left (a x^2\right )} \, dx &=\frac{\int \frac{1}{x^2} \, dx}{a}+\int \sqrt{1+\frac{1}{a^2 x^4}} \, dx\\ &=-\frac{1}{a x}-\operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x^4}{a^2}}}{x^2} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{1}{a x}+\sqrt{1+\frac{1}{a^2 x^4}} x-\frac{2 \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1+\frac{x^4}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a^2}\\ &=-\frac{1}{a x}+\sqrt{1+\frac{1}{a^2 x^4}} x-\frac{2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^4}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a}+\frac{2 \operatorname{Subst}\left (\int \frac{1-\frac{x^2}{a}}{\sqrt{1+\frac{x^4}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a}\\ &=-\frac{1}{a x}-\frac{2 \sqrt{1+\frac{1}{a^2 x^4}}}{\left (a+\frac{1}{x^2}\right ) x}+\sqrt{1+\frac{1}{a^2 x^4}} x+\frac{2 \sqrt{\frac{a^2+\frac{1}{x^4}}{\left (a+\frac{1}{x^2}\right )^2}} \left (a+\frac{1}{x^2}\right ) E\left (2 \cot ^{-1}\left (\sqrt{a} x\right )|\frac{1}{2}\right )}{a^{3/2} \sqrt{1+\frac{1}{a^2 x^4}}}-\frac{\sqrt{\frac{a^2+\frac{1}{x^4}}{\left (a+\frac{1}{x^2}\right )^2}} \left (a+\frac{1}{x^2}\right ) F\left (2 \cot ^{-1}\left (\sqrt{a} x\right )|\frac{1}{2}\right )}{a^{3/2} \sqrt{1+\frac{1}{a^2 x^4}}}\\ \end{align*}

Mathematica [C]  time = 0.141479, size = 96, normalized size = 0.58 $\frac{\sqrt{2} x e^{\text{csch}^{-1}\left (a x^2\right )} \sqrt{\frac{e^{\text{csch}^{-1}\left (a x^2\right )}}{e^{2 \text{csch}^{-1}\left (a x^2\right )}-1}} \left (4 \sqrt{1-e^{2 \text{csch}^{-1}\left (a x^2\right )}} \text{Hypergeometric2F1}\left (\frac{3}{4},\frac{3}{2},\frac{7}{4},e^{2 \text{csch}^{-1}\left (a x^2\right )}\right )-3\right )}{3 \sqrt{a x^2}}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[2]*E^ArcCsch[a*x^2]*Sqrt[E^ArcCsch[a*x^2]/(-1 + E^(2*ArcCsch[a*x^2]))]*x*(-3 + 4*Sqrt[1 - E^(2*ArcCsch[a
*x^2])]*Hypergeometric2F1[3/4, 3/2, 7/4, E^(2*ArcCsch[a*x^2])]))/(3*Sqrt[a*x^2])

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Maple [C]  time = 0.18, size = 146, normalized size = 0.9 \begin{align*}{\frac{x}{{a}^{2}{x}^{4}+1}\sqrt{{\frac{{a}^{2}{x}^{4}+1}{{a}^{2}{x}^{4}}}} \left ( -\sqrt{ia}{x}^{4}{a}^{2}+2\,i\sqrt{1-ia{x}^{2}}\sqrt{1+ia{x}^{2}}x{\it EllipticF} \left ( x\sqrt{ia},i \right ) a-2\,i\sqrt{1-ia{x}^{2}}\sqrt{1+ia{x}^{2}}x{\it EllipticE} \left ( x\sqrt{ia},i \right ) a-\sqrt{ia} \right ){\frac{1}{\sqrt{ia}}}}-{\frac{1}{ax}} \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)

[Out]

((a^2*x^4+1)/a^2/x^4)^(1/2)*x*(-(I*a)^(1/2)*x^4*a^2+2*I*(1-I*a*x^2)^(1/2)*(1+I*a*x^2)^(1/2)*x*EllipticF(x*(I*a
)^(1/2),I)*a-2*I*(1-I*a*x^2)^(1/2)*(1+I*a*x^2)^(1/2)*x*EllipticE(x*(I*a)^(1/2),I)*a-(I*a)^(1/2))/(a^2*x^4+1)/(
I*a)^(1/2)-1/a/x

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\frac{\Gamma \left (-\frac{1}{4}\right ) \,_2F_1\left (\begin{matrix} -\frac{1}{2},-\frac{1}{4} \\ \frac{3}{4} \end{matrix} ; -a^{2} x^{4} \right )}{4 \, x \Gamma \left (\frac{3}{4}\right )}}{a} - \frac{1}{a 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, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{a x^{2} \sqrt{\frac{a^{2} x^{4} + 1}{a^{2} x^{4}}} + 1}{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, algorithm="fricas")

[Out]

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

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Sympy [C]  time = 0.908202, size = 42, normalized size = 0.25 \begin{align*} - \frac{x \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{4} \\ \frac{3}{4} \end{matrix}\middle |{\frac{e^{i \pi }}{a^{2} x^{4}}} \right )}}{4 \Gamma \left (\frac{3}{4}\right )} - \frac{1}{a 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)

[Out]

-x*gamma(-1/4)*hyper((-1/2, -1/4), (3/4,), exp_polar(I*pi)/(a**2*x**4))/(4*gamma(3/4)) - 1/(a*x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\frac{1}{a^{2} x^{4}} + 1} + \frac{1}{a x^{2}}\,{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, algorithm="giac")

[Out]

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