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

Optimal. Leaf size=34 \[ -\frac {2}{3 a^2 x^3}-\frac {2}{3} a \left (\frac {1}{a^2 x^2}+1\right )^{3/2}-\frac {1}{x} \]

[Out]

-2/3*a*(1+1/a^2/x^2)^(3/2)-2/3/a^2/x^3-1/x

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Rubi [A]  time = 0.20, antiderivative size = 54, normalized size of antiderivative = 1.59, number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6338, 6715, 2117} \[ -\frac {1}{6} a \left (\sqrt {\frac {1}{a^2 x^2}+1}+\frac {1}{a x}\right )^3-\frac {1}{2} a \sqrt {\frac {1}{a^2 x^2}+1}-\frac {1}{2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*Sqrt[1 + 1/(a^2*x^2)])/2 - (a*(Sqrt[1 + 1/(a^2*x^2)] + 1/(a*x))^3)/6 - 1/(2*x)

Rule 2117

Int[((g_.) + (h_.)*((d_.) + (e_.)*(x_) + (f_.)*Sqrt[(a_) + (c_.)*(x_)^2])^(n_))^(p_.), x_Symbol] :> Dist[1/(2*
e), Subst[Int[((g + h*x^n)^p*(d^2 + a*f^2 - 2*d*x + x^2))/(d - x)^2, x], x, d + e*x + f*Sqrt[a + c*x^2]], x] /
; FreeQ[{a, c, d, e, f, g, h, n}, x] && EqQ[e^2 - c*f^2, 0] && IntegerQ[p]

Rule 6338

Int[E^(ArcCsch[u_]*(n_.))*(x_)^(m_.), x_Symbol] :> Int[x^m*(1/u + Sqrt[1 + 1/u^2])^n, x] /; FreeQ[m, x] && Int
egerQ[n]

Rule 6715

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

Rubi steps

\begin {align*} \int \frac {e^{2 \text {csch}^{-1}(a x)}}{x^2} \, dx &=\int \frac {\left (\sqrt {1+\frac {1}{a^2 x^2}}+\frac {1}{a x}\right )^2}{x^2} \, dx\\ &=-\operatorname {Subst}\left (\int \left (\frac {x}{a}+\sqrt {1+\frac {x^2}{a^2}}\right )^2 \, dx,x,\frac {1}{x}\right )\\ &=-\left (\frac {1}{2} a \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\sqrt {1+\frac {1}{a^2 x^2}}+\frac {1}{a x}\right )\right )\\ &=-\frac {1}{2} a \sqrt {1+\frac {1}{a^2 x^2}}-\frac {1}{6} a \left (\sqrt {1+\frac {1}{a^2 x^2}}+\frac {1}{a x}\right )^3-\frac {1}{2 x}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 46, normalized size = 1.35 \[ -\frac {3 a^2 x^2+2 a x \sqrt {\frac {1}{a^2 x^2}+1} \left (a^2 x^2+1\right )+2}{3 a^2 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(2 + 3*a^2*x^2 + 2*a*Sqrt[1 + 1/(a^2*x^2)]*x*(1 + a^2*x^2))/(a^2*x^3)

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fricas [B]  time = 0.59, size = 57, normalized size = 1.68 \[ -\frac {2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + 2 \, {\left (a^{3} x^{3} + a x\right )} \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} + 2}{3 \, a^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(2*a^3*x^3 + 3*a^2*x^2 + 2*(a^3*x^3 + a*x)*sqrt((a^2*x^2 + 1)/(a^2*x^2)) + 2)/(a^2*x^3)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(t_nostep)]Error: Bad Argument Type

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maple [B]  time = 0.05, size = 63, normalized size = 1.85 \[ \frac {-\frac {a^{2}}{x}-\frac {1}{3 x^{3}}}{a^{2}}-\frac {2 \sqrt {\frac {a^{2} x^{2}+1}{a^{2} x^{2}}}\, \left (a^{2} x^{2}+1\right )}{3 a \,x^{2}}-\frac {1}{3 x^{3} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a^2*(-a^2/x-1/3/x^3)-2/3/a*((a^2*x^2+1)/a^2/x^2)^(1/2)/x^2*(a^2*x^2+1)-1/3/x^3/a^2

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maxima [A]  time = 0.31, size = 28, normalized size = 0.82 \[ -\frac {2}{3} \, a {\left (\frac {1}{a^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - \frac {1}{x} - \frac {2}{3 \, a^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/a/x+(1+1/a^2/x^2)^(1/2))^2/x^2,x, algorithm="maxima")

[Out]

-2/3*a*(1/(a^2*x^2) + 1)^(3/2) - 1/x - 2/3/(a^2*x^3)

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mupad [B]  time = 2.21, size = 51, normalized size = 1.50 \[ -\frac {\frac {2}{3\,a^2}+\frac {2\,x\,\sqrt {\frac {1}{a^2\,x^2}+1}}{3\,a}}{x^3}-\frac {\frac {2\,a\,x\,\sqrt {\frac {1}{a^2\,x^2}+1}}{3}+1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- (2/(3*a^2) + (2*x*(1/(a^2*x^2) + 1)^(1/2))/(3*a))/x^3 - ((2*a*x*(1/(a^2*x^2) + 1)^(1/2))/3 + 1)/x

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/a/x+(1+1/a**2/x**2)**(1/2))**2/x**2,x)

[Out]

Exception raised: TypeError

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