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

Optimal. Leaf size=47 \[ -\frac {2 \sqrt {\frac {1}{a^2 x^2}+1}}{a}+\frac {2 \tanh ^{-1}\left (\sqrt {\frac {1}{a^2 x^2}+1}\right )}{a}-\frac {2}{a^2 x}+x \]

[Out]

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

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Rubi [A]  time = 0.07, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6333, 6742, 266, 50, 63, 208} \[ -\frac {2 \sqrt {\frac {1}{a^2 x^2}+1}}{a}+\frac {2 \tanh ^{-1}\left (\sqrt {\frac {1}{a^2 x^2}+1}\right )}{a}-\frac {2}{a^2 x}+x \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 266

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

Rule 6333

Int[E^(ArcCsch[u_]*(n_.)), x_Symbol] :> Int[(1/u + Sqrt[1 + 1/u^2])^n, x] /; IntegerQ[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)} \, dx &=\int \left (\sqrt {1+\frac {1}{a^2 x^2}}+\frac {1}{a x}\right )^2 \, dx\\ &=\int \left (1+\frac {2}{a^2 x^2}+\frac {2 \sqrt {1+\frac {1}{a^2 x^2}}}{a x}\right ) \, dx\\ &=-\frac {2}{a^2 x}+x+\frac {2 \int \frac {\sqrt {1+\frac {1}{a^2 x^2}}}{x} \, dx}{a}\\ &=-\frac {2}{a^2 x}+x-\frac {\operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {x}{a^2}}}{x} \, dx,x,\frac {1}{x^2}\right )}{a}\\ &=-\frac {2 \sqrt {1+\frac {1}{a^2 x^2}}}{a}-\frac {2}{a^2 x}+x-\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{a}\\ &=-\frac {2 \sqrt {1+\frac {1}{a^2 x^2}}}{a}-\frac {2}{a^2 x}+x-(2 a) \operatorname {Subst}\left (\int \frac {1}{-a^2+a^2 x^2} \, dx,x,\sqrt {1+\frac {1}{a^2 x^2}}\right )\\ &=-\frac {2 \sqrt {1+\frac {1}{a^2 x^2}}}{a}-\frac {2}{a^2 x}+x+\frac {2 \tanh ^{-1}\left (\sqrt {1+\frac {1}{a^2 x^2}}\right )}{a}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 52, normalized size = 1.11 \[ -\frac {2 \sqrt {\frac {1}{a^2 x^2}+1}}{a}+\frac {2 \log \left (a x \left (\sqrt {\frac {1}{a^2 x^2}+1}+1\right )\right )}{a}-\frac {2}{a^2 x}+x \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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fricas [A]  time = 0.94, size = 73, normalized size = 1.55 \[ \frac {a^{2} x^{2} - 2 \, a x \log \left (a x \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} - a x\right ) - 2 \, a x \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} - 2 \, a x - 2}{a^{2} x} \]

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, algorithm="fricas")

[Out]

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

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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, 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(x)]Error: Bad Argument Type

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maple [B]  time = 0.05, size = 112, normalized size = 2.38 \[ x -\frac {2}{a^{2} x}+\frac {2 \sqrt {\frac {a^{2} x^{2}+1}{a^{2} x^{2}}}\, \left (-a^{2} \left (\frac {a^{2} x^{2}+1}{a^{2}}\right )^{\frac {3}{2}}+\sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}\, x^{2} a^{2}+\ln \left (x +\sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}\right ) x \right )}{a \sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}} \]

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)

[Out]

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

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maxima [A]  time = 0.31, size = 59, normalized size = 1.26 \[ x - \frac {2 \, \sqrt {\frac {1}{a^{2} x^{2}} + 1} - \log \left (\sqrt {\frac {1}{a^{2} x^{2}} + 1} + 1\right ) + \log \left (\sqrt {\frac {1}{a^{2} x^{2}} + 1} - 1\right )}{a} - \frac {2}{a^{2} x} \]

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, algorithm="maxima")

[Out]

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

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mupad [B]  time = 2.26, size = 47, normalized size = 1.00 \[ x-\frac {2\,\sqrt {\frac {1}{a^2\,x^2}+1}}{a}-\frac {2}{a^2\,x}-\frac {\mathrm {atan}\left (\sqrt {\frac {1}{a^2\,x^2}+1}\,1{}\mathrm {i}\right )\,2{}\mathrm {i}}{a} \]

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)

[Out]

x - (atan((1/(a^2*x^2) + 1)^(1/2)*1i)*2i)/a - (2*(1/(a^2*x^2) + 1)^(1/2))/a - 2/(a^2*x)

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sympy [A]  time = 3.63, size = 49, normalized size = 1.04 \[ x - \frac {2 x}{\sqrt {a^{2} x^{2} + 1}} + \frac {2 \operatorname {asinh}{\left (a x \right )}}{a} - \frac {2}{a^{2} x} - \frac {2}{a^{2} x \sqrt {a^{2} x^{2} + 1}} \]

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)

[Out]

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

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