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

Optimal. Leaf size=43 \[ \frac {2 x \sqrt {\frac {1}{a^2 x^2}+1}}{a}+\frac {2 \log (x)}{a^2}-\frac {2 \text {csch}^{-1}(a x)}{a^2}+\frac {x^2}{2} \]

[Out]

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

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Rubi [A]  time = 0.16, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6338, 6742, 242, 277, 215} \[ \frac {2 x \sqrt {\frac {1}{a^2 x^2}+1}}{a}+\frac {2 \log (x)}{a^2}-\frac {2 \text {csch}^{-1}(a x)}{a^2}+\frac {x^2}{2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 242

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^2, x], x, 1/x] /; FreeQ[{a, b, p},
x] && ILtQ[n, 0]

Rule 277

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

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 6742

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

Rubi steps

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

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Mathematica [A]  time = 0.05, size = 44, normalized size = 1.02 \[ \frac {a x \left (4 \sqrt {\frac {1}{a^2 x^2}+1}+a x\right )-4 \sinh ^{-1}\left (\frac {1}{a x}\right )+4 \log (x)}{2 a^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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fricas [B]  time = 0.61, size = 99, normalized size = 2.30 \[ \frac {a^{2} x^{2} + 4 \, a x \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} - 4 \, \log \left (a x \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} - a x + 1\right ) + 4 \, \log \left (a x \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} - a x - 1\right ) + 4 \, \log \relax (x)}{2 \, a^{2}} \]

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]

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

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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,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 = 0.06, size = 120, normalized size = 2.79 \[ \frac {x^{2}}{2}+\frac {2 \ln \relax (x )}{a^{2}}+\frac {2 \sqrt {\frac {a^{2} x^{2}+1}{a^{2} x^{2}}}\, x \left (\sqrt {\frac {1}{a^{2}}}\, \sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}\, a^{2}-\ln \left (\frac {2 \sqrt {\frac {1}{a^{2}}}\, \sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}\, a^{2}+2}{a^{2} x}\right )\right )}{a^{3} \sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}\, \sqrt {\frac {1}{a^{2}}}} \]

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,x)

[Out]

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

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maxima [A]  time = 0.31, size = 75, normalized size = 1.74 \[ \frac {1}{2} \, x^{2} + \frac {2 \, x \sqrt {\frac {1}{a^{2} x^{2}} + 1} - \frac {\log \left (a x \sqrt {\frac {1}{a^{2} x^{2}} + 1} + 1\right )}{a} + \frac {\log \left (a x \sqrt {\frac {1}{a^{2} x^{2}} + 1} - 1\right )}{a}}{a} + \frac {2 \, \log \relax (x)}{a^{2}} \]

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

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mupad [B]  time = 2.20, size = 52, normalized size = 1.21 \[ \frac {x^2}{2}-\frac {2\,\ln \left (\frac {1}{x}\right )}{a^2}+\frac {2\,x\,\sqrt {\frac {1}{a^2\,x^2}+1}}{a}-\frac {2\,\mathrm {asinh}\left (\frac {\sqrt {\frac {1}{a^2}}}{x}\right )}{a^3\,\sqrt {\frac {1}{a^2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 3.79, size = 63, normalized size = 1.47 \[ \frac {x^{2}}{2} + \frac {2 x}{a \sqrt {1 + \frac {1}{a^{2} x^{2}}}} + \frac {2 \log {\relax (x )}}{a^{2}} - \frac {2 \operatorname {asinh}{\left (\frac {1}{a x} \right )}}{a^{2}} + \frac {2}{a^{3} x \sqrt {1 + \frac {1}{a^{2} x^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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