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

Optimal. Leaf size=40 \[ -\frac {\sqrt {\frac {1}{a^2 x^2}+1}}{2 x}-\frac {1}{2 a x^2}-\frac {1}{2} a \text {csch}^{-1}(a x) \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6336, 30, 335, 195, 215} \[ -\frac {\sqrt {\frac {1}{a^2 x^2}+1}}{2 x}-\frac {1}{2 a x^2}-\frac {1}{2} a \text {csch}^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 30

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

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

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 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]

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 43, normalized size = 1.08 \[ -\frac {a x \sqrt {\frac {1}{a^2 x^2}+1}+a^2 x^2 \sinh ^{-1}\left (\frac {1}{a x}\right )+1}{2 a x^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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fricas [B]  time = 0.46, size = 102, normalized size = 2.55 \[ -\frac {a^{2} x^{2} \log \left (a x \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} - a x + 1\right ) - a^{2} x^{2} \log \left (a x \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} - a x - 1\right ) + a x \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} + 1}{2 \, a x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.14, size = 82, normalized size = 2.05 \[ -\frac {a^{4} {\left | a \right |} \log \left (\sqrt {a^{2} x^{2} + 1} + 1\right ) \mathrm {sgn}\relax (x) - a^{4} {\left | a \right |} \log \left (\sqrt {a^{2} x^{2} + 1} - 1\right ) \mathrm {sgn}\relax (x) + \frac {2 \, {\left (\sqrt {a^{2} x^{2} + 1} a^{4} {\left | a \right |} \mathrm {sgn}\relax (x) + a^{5}\right )}}{a^{2} x^{2}}}{4 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(a^4*abs(a)*log(sqrt(a^2*x^2 + 1) + 1)*sgn(x) - a^4*abs(a)*log(sqrt(a^2*x^2 + 1) - 1)*sgn(x) + 2*(sqrt(a^
2*x^2 + 1)*a^4*abs(a)*sgn(x) + a^5)/(a^2*x^2))/a^4

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maple [B]  time = 0.05, size = 145, normalized size = 3.62 \[ -\frac {\sqrt {\frac {a^{2} x^{2}+1}{a^{2} x^{2}}}\, \left (a^{2} \left (\frac {a^{2} x^{2}+1}{a^{2}}\right )^{\frac {3}{2}} \sqrt {\frac {1}{a^{2}}}-\sqrt {\frac {1}{a^{2}}}\, \sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}\, x^{2} a^{2}+\ln \left (\frac {2 \sqrt {\frac {1}{a^{2}}}\, \sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}\, a^{2}+2}{a^{2} x}\right ) x^{2}\right )}{2 x \sqrt {\frac {1}{a^{2}}}\, \sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}}-\frac {1}{2 a \,x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.32, size = 86, normalized size = 2.15 \[ -\frac {a^{2} x \sqrt {\frac {1}{a^{2} x^{2}} + 1}}{2 \, {\left (a^{2} x^{2} {\left (\frac {1}{a^{2} x^{2}} + 1\right )} - 1\right )}} - \frac {1}{4} \, a \log \left (a x \sqrt {\frac {1}{a^{2} x^{2}} + 1} + 1\right ) + \frac {1}{4} \, a \log \left (a x \sqrt {\frac {1}{a^{2} x^{2}} + 1} - 1\right ) - \frac {1}{2 \, a x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 2.56, size = 42, normalized size = 1.05 \[ -\frac {\mathrm {asinh}\left (\frac {\sqrt {\frac {1}{a^2}}}{x}\right )}{2\,\sqrt {\frac {1}{a^2}}}-\frac {\sqrt {\frac {1}{a^2\,x^2}+1}}{2\,x}-\frac {1}{2\,a\,x^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 3.23, size = 36, normalized size = 0.90 \[ - \frac {a \operatorname {asinh}{\left (\frac {1}{a x} \right )}}{2} - \frac {\sqrt {1 + \frac {1}{a^{2} x^{2}}}}{2 x} - \frac {1}{2 a x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*asinh(1/(a*x))/2 - sqrt(1 + 1/(a**2*x**2))/(2*x) - 1/(2*a*x**2)

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