3.34 \(\int \frac {e^{\text {csch}^{-1}(a x)}}{x^3} \, dx\)

Optimal. Leaf size=31 \[ -\frac {1}{3} a^2 \left (\frac {1}{a^2 x^2}+1\right )^{3/2}-\frac {1}{3 a x^3} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6336, 30, 261} \[ -\frac {1}{3} a^2 \left (\frac {1}{a^2 x^2}+1\right )^{3/2}-\frac {1}{3 a x^3} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcCsch[a*x]/x^3,x]

[Out]

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

Rule 30

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

Rule 261

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

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^3} \, dx &=\frac {\int \frac {1}{x^4} \, dx}{a}+\int \frac {\sqrt {1+\frac {1}{a^2 x^2}}}{x^3} \, dx\\ &=-\frac {1}{3} a^2 \left (1+\frac {1}{a^2 x^2}\right )^{3/2}-\frac {1}{3 a x^3}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 37, normalized size = 1.19 \[ -\frac {a x \sqrt {\frac {1}{a^2 x^2}+1} \left (a^2 x^2+1\right )+1}{3 a x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[E^ArcCsch[a*x]/x^3,x]

[Out]

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

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fricas [A]  time = 0.70, size = 47, normalized size = 1.52 \[ -\frac {a^{3} x^{3} + {\left (a^{3} x^{3} + a x\right )} \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} + 1}{3 \, a x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/a/x+(1+1/a^2/x^2)^(1/2))/x^3,x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.17, size = 69, normalized size = 2.23 \[ \frac {2 \, {\left (3 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} + 1}\right )}^{4} a^{2} \mathrm {sgn}\relax (x) + a^{2} \mathrm {sgn}\relax (x)\right )}}{3 \, {\left ({\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} + 1}\right )}^{2} - 1\right )}^{3}} - \frac {1}{3 \, a x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/a/x+(1+1/a^2/x^2)^(1/2))/x^3,x, algorithm="giac")

[Out]

2/3*(3*(x*abs(a) - sqrt(a^2*x^2 + 1))^4*a^2*sgn(x) + a^2*sgn(x))/((x*abs(a) - sqrt(a^2*x^2 + 1))^2 - 1)^3 - 1/
3/(a*x^3)

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maple [A]  time = 0.06, size = 42, normalized size = 1.35 \[ -\frac {\sqrt {\frac {a^{2} x^{2}+1}{a^{2} x^{2}}}\, \left (a^{2} x^{2}+1\right )}{3 x^{2}}-\frac {1}{3 x^{3} a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1/a/x+(1+1/x^2/a^2)^(1/2))/x^3,x)

[Out]

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

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maxima [A]  time = 0.31, size = 25, normalized size = 0.81 \[ -\frac {1}{3} \, a^{2} {\left (\frac {1}{a^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - \frac {1}{3 \, a x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/a/x+(1+1/a^2/x^2)^(1/2))/x^3,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 2.16, size = 42, normalized size = 1.35 \[ -\frac {\frac {x\,\sqrt {\frac {1}{a^2\,x^2}+1}}{3}+\frac {1}{3\,a}}{x^3}-\frac {a^2\,\sqrt {\frac {1}{a^2\,x^2}+1}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((1/(a^2*x^2) + 1)^(1/2) + 1/(a*x))/x^3,x)

[Out]

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

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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))/x**3,x)

[Out]

Exception raised: TypeError

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