3.1.46 \(\int \frac {e^{\text {csch}^{-1}(a x^2)}}{x^4} \, dx\) [46]

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

[Out]

-1/5/a/x^5-1/5*(1+1/a^2/x^4)^(1/2)/x^3-2/5*a^2*(1+1/a^2/x^4)^(1/2)/(a+1/x^2)/x+2/5*(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^(1/2)*((a^2+1/x^4
)/(a+1/x^2)^2)^(1/2)/(1+1/a^2/x^4)^(1/2)-1/5*(a+1/x^2)*(cos(2*arccot(x*a^(1/2)))^2)^(1/2)/cos(2*arccot(x*a^(1/
2)))*EllipticF(sin(2*arccot(x*a^(1/2))),1/2*2^(1/2))*a^(1/2)*((a^2+1/x^4)/(a+1/x^2)^2)^(1/2)/(1+1/a^2/x^4)^(1/
2)

________________________________________________________________________________________

Rubi [A]
time = 0.07, antiderivative size = 181, normalized size of antiderivative = 1.00, 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 = {6471, 30, 342, 285, 311, 226, 1210} \begin {gather*} -\frac {\sqrt {\frac {1}{a^2 x^4}+1}}{5 x^3}-\frac {2 a^2 \sqrt {\frac {1}{a^2 x^4}+1}}{5 x \left (a+\frac {1}{x^2}\right )}-\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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/5*1/(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]
*Sqrt[(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 30

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

Rule 226

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)]/(2*q*Sqrt[a + b*x^4]))*EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 285

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 311

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 342

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 1210

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)]/(q*Sqrt[a + c*x^4
]))*EllipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e}, x] && PosQ[c/a]

Rule 6471

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]

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}-\text {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} \text {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) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^4}{a^2}}} \, dx,x,\frac {1}{x}\right )+\frac {1}{5} (2 a) \text {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] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 0.14, size = 114, normalized size = 0.63 \begin {gather*} \frac {\left (a x^2\right )^{3/2} \left (3 \left (1-e^{2 \text {csch}^{-1}\left (a x^2\right )}\right )^{3/2}+4 e^{2 \text {csch}^{-1}\left (a x^2\right )} \, _2F_1\left (-\frac {1}{2},\frac {3}{4};\frac {7}{4};e^{2 \text {csch}^{-1}\left (a x^2\right )}\right )\right )}{6 \sqrt {2-2 e^{2 \text {csch}^{-1}\left (a x^2\right )}} \sqrt {\frac {e^{\text {csch}^{-1}\left (a x^2\right )}}{-1+e^{2 \text {csch}^{-1}\left (a x^2\right )}}} x^3} \end {gather*}

Antiderivative was successfully verified.

[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)

________________________________________________________________________________________

Maple [C] Result contains complex when optimal does not.
time = 0.05, size = 171, normalized size = 0.94

method	result	size
default	\(\frac {\sqrt {\frac {a^{2} x^{4}+1}{a^{2} x^{4}}}\, \left (-2 \sqrt {i a}\, a^{4} x^{8}+2 i a^{3} \sqrt {-i a \,x^{2}+1}\, \sqrt {i a \,x^{2}+1}\, x^{5} \EllipticF \left (x \sqrt {i a}, i\right )-2 i a^{3} \sqrt {-i a \,x^{2}+1}\, \sqrt {i a \,x^{2}+1}\, x^{5} \EllipticE \left (x \sqrt {i a}, i\right )-3 \sqrt {i a}\, x^{4} a^{2}-\sqrt {i a}\right )}{5 x^{3} \left (a^{2} x^{4}+1\right ) \sqrt {i a}}-\frac {1}{5 a \,x^{5}}\)	\(171\)

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,method=_RETURNVERBOSE)

[Out]

1/5*((a^2*x^4+1)/a^2/x^4)^(1/2)*(-2*(I*a)^(1/2)*a^4*x^8+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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)

________________________________________________________________________________________

Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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]

Exception raised: TypeError >> Symbolic function elliptic_ec takes exactly 1 arguments (2 given)

________________________________________________________________________________________

Sympy [C] Result contains complex when optimal does not.
time = 1.48, size = 44, normalized size = 0.24 \begin {gather*} - \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 {gather*}

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)

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {\frac {1}{a^2\,x^4}+1}+\frac {1}{a\,x^2}}{x^4} \,d x \end {gather*}

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^4,x)

[Out]

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

________________________________________________________________________________________