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

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

[Out]

-1/a/x-2*(1+1/a^2/x^4)^(1/2)/(a+1/x^2)/x+x*(1+1/a^2/x^4)^(1/2)+2*(a+1/x^2)*(cos(2*arccot(x*a^(1/2)))^2)^(1/2)/
cos(2*arccot(x*a^(1/2)))*EllipticE(sin(2*arccot(x*a^(1/2))),1/2*2^(1/2))*((a^2+1/x^4)/(a+1/x^2)^2)^(1/2)/a^(3/
2)/(1+1/a^2/x^4)^(1/2)-(a+1/x^2)*(cos(2*arccot(x*a^(1/2)))^2)^(1/2)/cos(2*arccot(x*a^(1/2)))*EllipticF(sin(2*a
rccot(x*a^(1/2))),1/2*2^(1/2))*((a^2+1/x^4)/(a+1/x^2)^2)^(1/2)/a^(3/2)/(1+1/a^2/x^4)^(1/2)

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Rubi [A]  time = 0.08, antiderivative size = 165, normalized size of antiderivative = 1.00, 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 30

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

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 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 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]

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]

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*}

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Mathematica [C]  time = 0.15, size = 96, normalized size = 0.58 \[ \frac {\sqrt {2} x e^{\text {csch}^{-1}\left (a x^2\right )} \sqrt {\frac {e^{\text {csch}^{-1}\left (a x^2\right )}}{e^{2 \text {csch}^{-1}\left (a x^2\right )}-1}} \left (4 \sqrt {1-e^{2 \text {csch}^{-1}\left (a x^2\right )}} \, _2F_1\left (\frac {3}{4},\frac {3}{2};\frac {7}{4};e^{2 \text {csch}^{-1}\left (a x^2\right )}\right )-3\right )}{3 \sqrt {a x^2}} \]

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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fricas [F]  time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a x^{2} \sqrt {\frac {a^{2} x^{4} + 1}{a^{2} x^{4}}} + 1}{a x^{2}}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\frac {1}{a^{2} x^{4}} + 1} + \frac {1}{a x^{2}}\,{d x} \]

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)

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maple [C]  time = 0.06, size = 146, normalized size = 0.88 \[ \frac {\sqrt {\frac {a^{2} x^{4}+1}{a^{2} x^{4}}}\, x \left (-\sqrt {i a}\, x^{4} a^{2}+2 i \sqrt {-i a \,x^{2}+1}\, \sqrt {i a \,x^{2}+1}\, x \EllipticF \left (x \sqrt {i a}, i\right ) a -2 i \sqrt {-i a \,x^{2}+1}\, \sqrt {i a \,x^{2}+1}\, x \EllipticE \left (x \sqrt {i a}, i\right ) a -\sqrt {i a}\right )}{\left (a^{2} x^{4}+1\right ) \sqrt {i a}}-\frac {1}{a x} \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \frac {\frac {\Gamma \left (-\frac {1}{4}\right ) \,_2F_1\left (\begin {matrix} -\frac {1}{2},-\frac {1}{4} \\ \frac {3}{4} \end {matrix} ; -a^{2} x^{4} \right )}{4 \, x \Gamma \left (\frac {3}{4}\right )}}{a} - \frac {1}{a x} \]

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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mupad [B]  time = 2.33, size = 24, normalized size = 0.15 \[ x\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{2},-\frac {1}{4};\ \frac {3}{4};\ -\frac {1}{a^2\,x^4}\right )-\frac {1}{a\,x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*hypergeom([-1/2, -1/4], 3/4, -1/(a^2*x^4)) - 1/(a*x)

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sympy [C]  time = 0.78, size = 42, normalized size = 0.25 \[ - \frac {x \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {e^{i \pi }}{a^{2} x^{4}}} \right )}}{4 \Gamma \left (\frac {3}{4}\right )} - \frac {1}{a x} \]

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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