3.46 \(\int \frac{e^{\text{csch}^{-1}(a x^2)}}{x^4} \, dx\)

Optimal. Leaf size=181 \[ -\frac{\sqrt{a} \sqrt{\frac{a^2+\frac{1}{x^4}}{\left (a+\frac{1}{x^2}\right )^2}} \left (a+\frac{1}{x^2}\right ) \text{EllipticF}\left (2 \cot ^{-1}\left (\sqrt{a} x\right ),\frac{1}{2}\right )}{5 \sqrt{\frac{1}{a^2 x^4}+1}}-\frac{2 a^2 \sqrt{\frac{1}{a^2 x^4}+1}}{5 x \left (a+\frac{1}{x^2}\right )}-\frac{\sqrt{\frac{1}{a^2 x^4}+1}}{5 x^3}+\frac{2 \sqrt{a} \sqrt{\frac{a^2+\frac{1}{x^4}}{\left (a+\frac{1}{x^2}\right )^2}} \left (a+\frac{1}{x^2}\right ) E\left (2 \cot ^{-1}\left (\sqrt{a} x\right )|\frac{1}{2}\right )}{5 \sqrt{\frac{1}{a^2 x^4}+1}}-\frac{1}{5 a x^5} \]

[Out]

-1/(5*a*x^5) - Sqrt[1 + 1/(a^2*x^4)]/(5*x^3) - (2*a^2*Sqrt[1 + 1/(a^2*x^4)])/(5*(a + x^(-2))*x) + (2*Sqrt[a]*S
qrt[(a^2 + x^(-4))/(a + x^(-2))^2]*(a + x^(-2))*EllipticE[2*ArcCot[Sqrt[a]*x], 1/2])/(5*Sqrt[1 + 1/(a^2*x^4)])
- (Sqrt[a]*Sqrt[(a^2 + x^(-4))/(a + x^(-2))^2]*(a + x^(-2))*EllipticF[2*ArcCot[Sqrt[a]*x], 1/2])/(5*Sqrt[1 +
1/(a^2*x^4)])

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Rubi [A]  time = 0.0954163, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583, Rules used = {6336, 30, 335, 279, 305, 220, 1196} \[ -\frac{2 a^2 \sqrt{\frac{1}{a^2 x^4}+1}}{5 x \left (a+\frac{1}{x^2}\right )}-\frac{\sqrt{\frac{1}{a^2 x^4}+1}}{5 x^3}-\frac{\sqrt{a} \sqrt{\frac{a^2+\frac{1}{x^4}}{\left (a+\frac{1}{x^2}\right )^2}} \left (a+\frac{1}{x^2}\right ) F\left (2 \cot ^{-1}\left (\sqrt{a} x\right )|\frac{1}{2}\right )}{5 \sqrt{\frac{1}{a^2 x^4}+1}}+\frac{2 \sqrt{a} \sqrt{\frac{a^2+\frac{1}{x^4}}{\left (a+\frac{1}{x^2}\right )^2}} \left (a+\frac{1}{x^2}\right ) E\left (2 \cot ^{-1}\left (\sqrt{a} x\right )|\frac{1}{2}\right )}{5 \sqrt{\frac{1}{a^2 x^4}+1}}-\frac{1}{5 a x^5} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcCsch[a*x^2]/x^4,x]

[Out]

-1/(5*a*x^5) - Sqrt[1 + 1/(a^2*x^4)]/(5*x^3) - (2*a^2*Sqrt[1 + 1/(a^2*x^4)])/(5*(a + x^(-2))*x) + (2*Sqrt[a]*S
qrt[(a^2 + x^(-4))/(a + x^(-2))^2]*(a + x^(-2))*EllipticE[2*ArcCot[Sqrt[a]*x], 1/2])/(5*Sqrt[1 + 1/(a^2*x^4)])
- (Sqrt[a]*Sqrt[(a^2 + x^(-4))/(a + x^(-2))^2]*(a + x^(-2))*EllipticF[2*ArcCot[Sqrt[a]*x], 1/2])/(5*Sqrt[1 +
1/(a^2*x^4)])

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 335

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

Rule 279

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
n*p + 1)), x] + Dist[(a*n*p)/(m + n*p + 1), Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x]
&& IGtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 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 \frac{e^{\text{csch}^{-1}\left (a x^2\right )}}{x^4} \, dx &=\frac{\int \frac{1}{x^6} \, dx}{a}+\int \frac{\sqrt{1+\frac{1}{a^2 x^4}}}{x^4} \, dx\\ &=-\frac{1}{5 a x^5}-\operatorname{Subst}\left (\int x^2 \sqrt{1+\frac{x^4}{a^2}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{1}{5 a x^5}-\frac{\sqrt{1+\frac{1}{a^2 x^4}}}{5 x^3}-\frac{2}{5} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1+\frac{x^4}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{1}{5 a x^5}-\frac{\sqrt{1+\frac{1}{a^2 x^4}}}{5 x^3}-\frac{1}{5} (2 a) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^4}{a^2}}} \, dx,x,\frac{1}{x}\right )+\frac{1}{5} (2 a) \operatorname{Subst}\left (\int \frac{1-\frac{x^2}{a}}{\sqrt{1+\frac{x^4}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{1}{5 a x^5}-\frac{\sqrt{1+\frac{1}{a^2 x^4}}}{5 x^3}-\frac{2 a^2 \sqrt{1+\frac{1}{a^2 x^4}}}{5 \left (a+\frac{1}{x^2}\right ) x}+\frac{2 \sqrt{a} \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 )}{5 \sqrt{1+\frac{1}{a^2 x^4}}}-\frac{\sqrt{a} \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 )}{5 \sqrt{1+\frac{1}{a^2 x^4}}}\\ \end{align*}

Mathematica [C]  time = 0.192641, size = 114, normalized size = 0.63 \[ \frac{\left (a x^2\right )^{3/2} \left (4 e^{2 \text{csch}^{-1}\left (a x^2\right )} \text{Hypergeometric2F1}\left (-\frac{1}{2},\frac{3}{4},\frac{7}{4},e^{2 \text{csch}^{-1}\left (a x^2\right )}\right )+3 \left (1-e^{2 \text{csch}^{-1}\left (a x^2\right )}\right )^{3/2}\right )}{6 x^3 \sqrt{2-2 e^{2 \text{csch}^{-1}\left (a x^2\right )}} \sqrt{\frac{e^{\text{csch}^{-1}\left (a x^2\right )}}{e^{2 \text{csch}^{-1}\left (a x^2\right )}-1}}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^ArcCsch[a*x^2]/x^4,x]

[Out]

((a*x^2)^(3/2)*(3*(1 - E^(2*ArcCsch[a*x^2]))^(3/2) + 4*E^(2*ArcCsch[a*x^2])*Hypergeometric2F1[-1/2, 3/4, 7/4,
E^(2*ArcCsch[a*x^2])]))/(6*Sqrt[2 - 2*E^(2*ArcCsch[a*x^2])]*Sqrt[E^ArcCsch[a*x^2]/(-1 + E^(2*ArcCsch[a*x^2]))]
*x^3)

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Maple [C]  time = 0.215, size = 171, normalized size = 0.9 \begin{align*}{\frac{1}{5\,{x}^{3} \left ({a}^{2}{x}^{4}+1 \right ) }\sqrt{{\frac{{a}^{2}{x}^{4}+1}{{a}^{2}{x}^{4}}}} \left ( -2\,\sqrt{ia}{x}^{8}{a}^{4}+2\,i{a}^{3}\sqrt{1-ia{x}^{2}}\sqrt{1+ia{x}^{2}}{x}^{5}{\it EllipticF} \left ( x\sqrt{ia},i \right ) -2\,i{a}^{3}\sqrt{1-ia{x}^{2}}\sqrt{1+ia{x}^{2}}{x}^{5}{\it EllipticE} \left ( x\sqrt{ia},i \right ) -3\,\sqrt{ia}{x}^{4}{a}^{2}-\sqrt{ia} \right ){\frac{1}{\sqrt{ia}}}}-{\frac{1}{5\,a{x}^{5}}} \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^4,x)

[Out]

1/5*((a^2*x^4+1)/a^2/x^4)^(1/2)*(-2*(I*a)^(1/2)*x^8*a^4+2*I*a^3*(1-I*a*x^2)^(1/2)*(1+I*a*x^2)^(1/2)*x^5*Ellipt
icF(x*(I*a)^(1/2),I)-2*I*a^3*(1-I*a*x^2)^(1/2)*(1+I*a*x^2)^(1/2)*x^5*EllipticE(x*(I*a)^(1/2),I)-3*(I*a)^(1/2)*
x^4*a^2-(I*a)^(1/2))/x^3/(a^2*x^4+1)/(I*a)^(1/2)-1/5/a/x^5

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\frac{\Gamma \left (-\frac{5}{4}\right ) \,_2F_1\left (\begin{matrix} -\frac{5}{4},-\frac{1}{2} \\ -\frac{1}{4} \end{matrix} ; -a^{2} x^{4} \right )}{4 \, x^{5} \Gamma \left (-\frac{1}{4}\right )}}{a} - \frac{1}{5 \, a x^{5}} \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^4,x, algorithm="maxima")

[Out]

integrate(sqrt(a^2*x^4 + 1)/x^6, x)/a - 1/5/(a*x^5)

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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^{6}}, 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^4,x, algorithm="fricas")

[Out]

integral((a*x^2*sqrt((a^2*x^4 + 1)/(a^2*x^4)) + 1)/(a*x^6), x)

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Sympy [C]  time = 3.46518, size = 44, normalized size = 0.24 \begin{align*} - \frac{\Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{e^{i \pi }}{a^{2} x^{4}}} \right )}}{4 x^{3} \Gamma \left (\frac{7}{4}\right )} - \frac{1}{5 a x^{5}} \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**4,x)

[Out]

-gamma(3/4)*hyper((-1/2, 3/4), (7/4,), exp_polar(I*pi)/(a**2*x**4))/(4*x**3*gamma(7/4)) - 1/(5*a*x**5)

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

[Out]

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