\(\int e^{\text {csch}^{-1}(a x)} \, dx\) [31]

   Optimal result
   Rubi [A] (verified)
   Mathematica [A] (verified)
   Maple [B] (verified)
   Fricas [B] (verification not implemented)
   Sympy [A] (verification not implemented)
   Maxima [A] (verification not implemented)
   Giac [A] (verification not implemented)
   Mupad [B] (verification not implemented)

Optimal result

Integrand size = 6, antiderivative size = 24 \[ \int e^{\text {csch}^{-1}(a x)} \, dx=e^{\text {csch}^{-1}(a x)} x-\frac {\text {csch}^{-1}(a x)}{a}+\frac {\log (x)}{a} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.29, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {6466, 29, 248, 283, 221} \[ \int e^{\text {csch}^{-1}(a x)} \, dx=x \sqrt {\frac {1}{a^2 x^2}+1}+\frac {\log (x)}{a}-\frac {\text {csch}^{-1}(a x)}{a} \]

[In]

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

[Out]

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

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

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 248

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 283

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 6466

Int[E^ArcCsch[(a_.)*(x_)^(p_.)], x_Symbol] :> Dist[1/a, Int[1/x^p, x], x] + Int[Sqrt[1 + 1/(a^2*x^(2*p))], x]
/; FreeQ[{a, p}, x]

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.46 \[ \int e^{\text {csch}^{-1}(a x)} \, dx=\frac {a \sqrt {1+\frac {1}{a^2 x^2}} x-\text {arcsinh}\left (\frac {1}{a x}\right )+\log (a x)}{a} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(112\) vs. \(2(37)=74\).

Time = 0.10 (sec) , antiderivative size = 113, normalized size of antiderivative = 4.71

method result size
default \(-\frac {\sqrt {\frac {a^{2} x^{2}+1}{a^{2} x^{2}}}\, x \left (-\sqrt {\frac {1}{a^{2}}}\, \sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}\, a^{2}+\ln \left (\frac {2 \sqrt {\frac {1}{a^{2}}}\, \sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}\, a^{2}+2}{a^{2} x}\right )\right )}{\sqrt {\frac {1}{a^{2}}}\, \sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}\, a^{2}}+\frac {\ln \left (x \right )}{a}\) \(113\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (37) = 74\).

Time = 0.24 (sec) , antiderivative size = 86, normalized size of antiderivative = 3.58 \[ \int e^{\text {csch}^{-1}(a x)} \, dx=\frac {a x \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} - \log \left (a x \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} - a x + 1\right ) + \log \left (a x \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} - a x - 1\right ) + \log \left (x\right )}{a} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.76 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.00 \[ \int e^{\text {csch}^{-1}(a x)} \, dx=\frac {x}{\sqrt {1 + \frac {1}{a^{2} x^{2}}}} + \frac {\log {\left (x \right )}}{a} - \frac {\operatorname {asinh}{\left (\frac {1}{a x} \right )}}{a} + \frac {1}{a^{2} x \sqrt {1 + \frac {1}{a^{2} x^{2}}}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.67 \[ \int e^{\text {csch}^{-1}(a x)} \, dx=x \sqrt {\frac {1}{a^{2} x^{2}} + 1} - \frac {\log \left (a x \sqrt {\frac {1}{a^{2} x^{2}} + 1} + 1\right )}{2 \, a} + \frac {\log \left (a x \sqrt {\frac {1}{a^{2} x^{2}} + 1} - 1\right )}{2 \, a} + \frac {\log \left (x\right )}{a} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.75 \[ \int e^{\text {csch}^{-1}(a x)} \, dx=-\frac {{\left (\log \left (\sqrt {a^{2} x^{2} + 1} + 1\right ) \mathrm {sgn}\left (x\right ) - \log \left (\sqrt {a^{2} x^{2} + 1} - 1\right ) \mathrm {sgn}\left (x\right ) - 2 \, \sqrt {a^{2} x^{2} + 1} \mathrm {sgn}\left (x\right )\right )} {\left | a \right |}}{2 \, a^{2}} + \frac {\log \left ({\left | x \right |}\right )}{a} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 4.89 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.50 \[ \int e^{\text {csch}^{-1}(a x)} \, dx=\frac {\ln \left (x\right )}{a}+x\,\sqrt {\frac {1}{a^2\,x^2}+1}+\frac {\mathrm {asin}\left (\frac {1{}\mathrm {i}}{a\,x}\right )\,1{}\mathrm {i}}{a} \]

[In]

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

[Out]

log(x)/a + (asin(1i/(a*x))*1i)/a + x*(1/(a^2*x^2) + 1)^(1/2)