3.1.54 \(\int \frac {e^{2 \text {csch}^{-1}(a x)}}{x} \, dx\) [54]

Optimal. Leaf size=38 \[ -\frac {1}{a^2 x^2}-\frac {\sqrt {1+\frac {1}{a^2 x^2}}}{a x}-\text {csch}^{-1}(a x)+\log (x) \]

[Out]

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

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Rubi [A]
time = 0.14, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6473, 6874, 342, 201, 221} \begin {gather*} -\frac {\sqrt {\frac {1}{a^2 x^2}+1}}{a x}-\frac {1}{a^2 x^2}-\text {csch}^{-1}(a x)+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 201

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

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Mathematica [A]
time = 0.04, size = 39, normalized size = 1.03 \begin {gather*} -\frac {1+a \sqrt {1+\frac {1}{a^2 x^2}} x}{a^2 x^2}-\sinh ^{-1}\left (\frac {1}{a x}\right )+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(163\) vs. \(2(36)=72\).
time = 0.04, size = 164, normalized size = 4.32

method	result	size
default	\(\frac {-\frac {1}{2 x^{2}}+a^{2} \ln \left (x \right )}{a^{2}}-\frac {\sqrt {\frac {a^{2} x^{2}+1}{a^{2} x^{2}}}\, \left (a^{2} \left (\frac {a^{2} x^{2}+1}{a^{2}}\right )^{\frac {3}{2}} \sqrt {\frac {1}{a^{2}}}-\sqrt {\frac {1}{a^{2}}}\, \sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}\, a^{2} x^{2}+\ln \left (\frac {2 \sqrt {\frac {1}{a^{2}}}\, \sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}\, a^{2}+2}{a^{2} x}\right ) x^{2}\right )}{a x \sqrt {\frac {1}{a^{2}}}\, \sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}}-\frac {1}{2 a^{2} x^{2}}\)	\(164\)

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,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 93 vs. \(2 (36) = 72\).
time = 0.26, size = 93, normalized size = 2.45 \begin {gather*} -\frac {\frac {2 \, a^{2} x \sqrt {\frac {1}{a^{2} x^{2}} + 1}}{a^{2} x^{2} {\left (\frac {1}{a^{2} x^{2}} + 1\right )} - 1} + a \log \left (a x \sqrt {\frac {1}{a^{2} x^{2}} + 1} + 1\right ) - a \log \left (a x \sqrt {\frac {1}{a^{2} x^{2}} + 1} - 1\right )}{2 \, a} - \frac {1}{a^{2} x^{2}} + \log \left (x\right ) \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,x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (36) = 72\).
time = 0.35, size = 112, normalized size = 2.95 \begin {gather*} -\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^{2} x^{2} \log \left (x\right ) + a x \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} + 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,x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 2.44, size = 34, normalized size = 0.89 \begin {gather*} \log {\left (x \right )} - \operatorname {asinh}{\left (\frac {1}{a x} \right )} - \frac {\sqrt {1 + \frac {1}{a^{2} x^{2}}}}{a x} - \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,x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 118 vs. \(2 (36) = 72\).
time = 0.42, size = 118, normalized size = 3.11 \begin {gather*} -\frac {{\left (a^{4} {\left | a \right |} \mathrm {sgn}\left (x\right ) - a^{5}\right )} \log \left (\sqrt {a^{2} x^{2} + 1} + 1\right ) - {\left (a^{4} {\left | a \right |} \mathrm {sgn}\left (x\right ) + a^{5}\right )} \log \left (\sqrt {a^{2} x^{2} + 1} - 1\right ) + \frac {2 \, {\left (\sqrt {a^{2} x^{2} + 1} a^{4} {\left | a \right |} \mathrm {sgn}\left (x\right ) + a^{5}\right )}}{{\left (\sqrt {a^{2} x^{2} + 1} + 1\right )} {\left (\sqrt {a^{2} x^{2} + 1} - 1\right )}}}{2 \, 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,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 2.25, size = 44, normalized size = 1.16 \begin {gather*} -\ln \left (\frac {1}{x}\right )-\mathrm {asinh}\left (\frac {1}{a\,x}\right )-\frac {1}{a^2\,x^2}-\frac {\sqrt {\frac {1}{a^2\,x^2}+1}}{a\,x} \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,x)

[Out]

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

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