3.1.56 \(\int \frac {e^{2 \text {csch}^{-1}(a x)}}{x^3} \, dx\) [56]

Optimal. Leaf size=73 \[ -\frac {1}{2 a^2 x^4}-\frac {\sqrt {1+\frac {1}{a^2 x^2}}}{2 a x^3}-\frac {1}{2 x^2}-\frac {a \sqrt {1+\frac {1}{a^2 x^2}}}{4 x}+\frac {1}{4} a^2 \text {csch}^{-1}(a x) \]

[Out]

-1/2/a^2/x^4-1/2/x^2+1/4*a^2*arccsch(a*x)-1/2*(1+1/a^2/x^2)^(1/2)/a/x^3-1/4*a*(1+1/a^2/x^2)^(1/2)/x

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Rubi [A]
time = 0.15, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6473, 6874, 342, 285, 327, 221} \begin {gather*} -\frac {1}{2 a^2 x^4}-\frac {a \sqrt {\frac {1}{a^2 x^2}+1}}{4 x}-\frac {\sqrt {\frac {1}{a^2 x^2}+1}}{2 a x^3}+\frac {1}{4} a^2 \text {csch}^{-1}(a x)-\frac {1}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(2*ArcCsch[a*x])/x^3,x]

[Out]

-1/2*1/(a^2*x^4) - Sqrt[1 + 1/(a^2*x^2)]/(2*a*x^3) - 1/(2*x^2) - (a*Sqrt[1 + 1/(a^2*x^2)])/(4*x) + (a^2*ArcCsc
h[a*x])/4

Rule 221

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 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 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

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 6473

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 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int \frac {e^{2 \text {csch}^{-1}(a x)}}{x^3} \, dx &=\int \frac {\left (\sqrt {1+\frac {1}{a^2 x^2}}+\frac {1}{a x}\right )^2}{x^3} \, dx\\ &=\int \left (\frac {2}{a^2 x^5}+\frac {2 \sqrt {1+\frac {1}{a^2 x^2}}}{a x^4}+\frac {1}{x^3}\right ) \, dx\\ &=-\frac {1}{2 a^2 x^4}-\frac {1}{2 x^2}+\frac {2 \int \frac {\sqrt {1+\frac {1}{a^2 x^2}}}{x^4} \, dx}{a}\\ &=-\frac {1}{2 a^2 x^4}-\frac {1}{2 x^2}-\frac {2 \text {Subst}\left (\int x^2 \sqrt {1+\frac {x^2}{a^2}} \, dx,x,\frac {1}{x}\right )}{a}\\ &=-\frac {1}{2 a^2 x^4}-\frac {\sqrt {1+\frac {1}{a^2 x^2}}}{2 a x^3}-\frac {1}{2 x^2}-\frac {\text {Subst}\left (\int \frac {x^2}{\sqrt {1+\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{2 a}\\ &=-\frac {1}{2 a^2 x^4}-\frac {\sqrt {1+\frac {1}{a^2 x^2}}}{2 a x^3}-\frac {1}{2 x^2}-\frac {a \sqrt {1+\frac {1}{a^2 x^2}}}{4 x}+\frac {1}{4} a \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {1}{2 a^2 x^4}-\frac {\sqrt {1+\frac {1}{a^2 x^2}}}{2 a x^3}-\frac {1}{2 x^2}-\frac {a \sqrt {1+\frac {1}{a^2 x^2}}}{4 x}+\frac {1}{4} a^2 \text {csch}^{-1}(a x)\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 73, normalized size = 1.00 \begin {gather*} -\frac {1}{2 a^2 x^4}-\frac {1}{2 x^2}+\left (-\frac {1}{2 a x^3}-\frac {a}{4 x}\right ) \sqrt {\frac {1+a^2 x^2}{a^2 x^2}}+\frac {1}{4} a^2 \sinh ^{-1}\left (\frac {1}{a x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(2*ArcCsch[a*x])/x^3,x]

[Out]

-1/2*1/(a^2*x^4) - 1/(2*x^2) + (-1/2*1/(a*x^3) - a/(4*x))*Sqrt[(1 + a^2*x^2)/(a^2*x^2)] + (a^2*ArcSinh[1/(a*x)
])/4

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(188\) vs. \(2(59)=118\).
time = 0.04, size = 189, normalized size = 2.59

method	result	size
default	\(\frac {-\frac {1}{4 x^{4}}-\frac {a^{2}}{2 x^{2}}}{a^{2}}+\frac {a \sqrt {\frac {a^{2} x^{2}+1}{a^{2} x^{2}}}\, \left (\left (\frac {a^{2} x^{2}+1}{a^{2}}\right )^{\frac {3}{2}} \sqrt {\frac {1}{a^{2}}}\, a^{2} x^{2}-\sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}\, \sqrt {\frac {1}{a^{2}}}\, a^{2} x^{4}+\ln \left (\frac {2 \sqrt {\frac {1}{a^{2}}}\, \sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}\, a^{2}+2}{a^{2} x}\right ) x^{4}-2 \left (\frac {a^{2} x^{2}+1}{a^{2}}\right )^{\frac {3}{2}} \sqrt {\frac {1}{a^{2}}}\right )}{4 x^{3} \sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}\, \sqrt {\frac {1}{a^{2}}}}-\frac {1}{4 a^{2} x^{4}}\)	\(189\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a^2*(-1/4/x^4-1/2*a^2/x^2)+1/4*a*((a^2*x^2+1)/a^2/x^2)^(1/2)/x^3*(((a^2*x^2+1)/a^2)^(3/2)*(1/a^2)^(1/2)*a^2*
x^2-((a^2*x^2+1)/a^2)^(1/2)*(1/a^2)^(1/2)*a^2*x^4+ln(2*((1/a^2)^(1/2)*((a^2*x^2+1)/a^2)^(1/2)*a^2+1)/x/a^2)*x^
4-2*((a^2*x^2+1)/a^2)^(3/2)*(1/a^2)^(1/2))/((a^2*x^2+1)/a^2)^(1/2)/(1/a^2)^(1/2)-1/4/a^2/x^4

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 139 vs. \(2 (59) = 118\).
time = 0.27, size = 139, normalized size = 1.90 \begin {gather*} \frac {a^{3} \log \left (a x \sqrt {\frac {1}{a^{2} x^{2}} + 1} + 1\right ) - a^{3} \log \left (a x \sqrt {\frac {1}{a^{2} x^{2}} + 1} - 1\right ) - \frac {2 \, {\left (a^{6} x^{3} {\left (\frac {1}{a^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + a^{4} x \sqrt {\frac {1}{a^{2} x^{2}} + 1}\right )}}{a^{4} x^{4} {\left (\frac {1}{a^{2} x^{2}} + 1\right )}^{2} - 2 \, a^{2} x^{2} {\left (\frac {1}{a^{2} x^{2}} + 1\right )} + 1}}{8 \, a} - \frac {1}{2 \, x^{2}} - \frac {1}{2 \, a^{2} x^{4}} \end {gather*}

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^3,x, algorithm="maxima")

[Out]

1/8*(a^3*log(a*x*sqrt(1/(a^2*x^2) + 1) + 1) - a^3*log(a*x*sqrt(1/(a^2*x^2) + 1) - 1) - 2*(a^6*x^3*(1/(a^2*x^2)
+ 1)^(3/2) + a^4*x*sqrt(1/(a^2*x^2) + 1))/(a^4*x^4*(1/(a^2*x^2) + 1)^2 - 2*a^2*x^2*(1/(a^2*x^2) + 1) + 1))/a
- 1/2/x^2 - 1/2/(a^2*x^4)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 121 vs. \(2 (59) = 118\).
time = 0.41, size = 121, normalized size = 1.66 \begin {gather*} \frac {a^{4} x^{4} \log \left (a x \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} - a x + 1\right ) - a^{4} x^{4} \log \left (a x \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} - a x - 1\right ) - 2 \, a^{2} x^{2} - {\left (a^{3} x^{3} + 2 \, a x\right )} \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} - 2}{4 \, a^{2} x^{4}} \end {gather*}

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^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 3.15, size = 92, normalized size = 1.26 \begin {gather*} \frac {a^{2} \operatorname {asinh}{\left (\frac {1}{a x} \right )}}{4} - \frac {a}{4 x \sqrt {1 + \frac {1}{a^{2} x^{2}}}} - \frac {1}{2 x^{2}} - \frac {3}{4 a x^{3} \sqrt {1 + \frac {1}{a^{2} x^{2}}}} - \frac {1}{2 a^{2} x^{4}} - \frac {1}{2 a^{3} x^{5} \sqrt {1 + \frac {1}{a^{2} x^{2}}}} \end {gather*}

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**3,x)

[Out]

a**2*asinh(1/(a*x))/4 - a/(4*x*sqrt(1 + 1/(a**2*x**2))) - 1/(2*x**2) - 3/(4*a*x**3*sqrt(1 + 1/(a**2*x**2))) -
1/(2*a**2*x**4) - 1/(2*a**3*x**5*sqrt(1 + 1/(a**2*x**2)))

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Giac [A]
time = 0.41, size = 112, normalized size = 1.53 \begin {gather*} \frac {a^{6} {\left | a \right |} \log \left (\sqrt {a^{2} x^{2} + 1} + 1\right ) \mathrm {sgn}\left (x\right ) - a^{6} {\left | a \right |} \log \left (\sqrt {a^{2} x^{2} + 1} - 1\right ) \mathrm {sgn}\left (x\right ) - \frac {2 \, {\left ({\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{6} {\left | a \right |} \mathrm {sgn}\left (x\right ) + \sqrt {a^{2} x^{2} + 1} a^{6} {\left | a \right |} \mathrm {sgn}\left (x\right ) + 2 \, {\left (a^{2} x^{2} + 1\right )} a^{7}\right )}}{a^{4} x^{4}}}{8 \, a^{5}} \end {gather*}

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^3,x, algorithm="giac")

[Out]

1/8*(a^6*abs(a)*log(sqrt(a^2*x^2 + 1) + 1)*sgn(x) - a^6*abs(a)*log(sqrt(a^2*x^2 + 1) - 1)*sgn(x) - 2*((a^2*x^2
+ 1)^(3/2)*a^6*abs(a)*sgn(x) + sqrt(a^2*x^2 + 1)*a^6*abs(a)*sgn(x) + 2*(a^2*x^2 + 1)*a^7)/(a^4*x^4))/a^5

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Mupad [B]
time = 2.31, size = 68, normalized size = 0.93 \begin {gather*} \frac {a\,\mathrm {asinh}\left (\frac {\sqrt {\frac {1}{a^2}}}{x}\right )}{4\,\sqrt {\frac {1}{a^2}}}-\frac {1}{2\,a^2\,x^4}-\frac {a\,\sqrt {\frac {1}{a^2\,x^2}+1}}{4\,x}-\frac {1}{2\,x^2}-\frac {\sqrt {\frac {1}{a^2\,x^2}+1}}{2\,a\,x^3} \end {gather*}

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^3,x)

[Out]

(a*asinh((1/a^2)^(1/2)/x))/(4*(1/a^2)^(1/2)) - 1/(2*a^2*x^4) - (a*(1/(a^2*x^2) + 1)^(1/2))/(4*x) - 1/(2*x^2) -
(1/(a^2*x^2) + 1)^(1/2)/(2*a*x^3)

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