Integrand size = 8, antiderivative size = 57 \[ \int e^{2 \text {sech}^{-1}(a x)} \, dx=-x-\frac {4}{a \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )}+\frac {4 \arctan \left (\sqrt {\frac {1-a x}{1+a x}}\right )}{a} \] Output:
-x-4/a/(1-((-a*x+1)/(a*x+1))^(1/2))+4*arctan(((-a*x+1)/(a*x+1))^(1/2))/a
Time = 0.13 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.32 \[ \int e^{2 \text {sech}^{-1}(a x)} \, dx=-\frac {2+a^2 x^2+2 \sqrt {\frac {1-a x}{1+a x}} (1+a x)+2 a x \arctan \left (\frac {a x}{\sqrt {\frac {1-a x}{1+a x}} (1+a x)}\right )}{a^2 x} \] Input:
Integrate[E^(2*ArcSech[a*x]),x]
Output:
-((2 + a^2*x^2 + 2*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x) + 2*a*x*ArcTan[(a*x )/(Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x))])/(a^2*x))
Time = 0.74 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.44, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6886, 7268, 2178, 27, 594, 216}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int e^{2 \text {sech}^{-1}(a x)} \, dx\) |
\(\Big \downarrow \) 6886 |
\(\displaystyle \int \left (\frac {\sqrt {\frac {1-a x}{a x+1}}}{a x}+\sqrt {\frac {1-a x}{a x+1}}+\frac {1}{a x}\right )^2dx\) |
\(\Big \downarrow \) 7268 |
\(\displaystyle -\frac {4 \int \frac {\sqrt {\frac {1-a x}{a x+1}} \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^2}{\left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^2 \left (\frac {1-a x}{a x+1}+1\right )^2}d\sqrt {\frac {1-a x}{a x+1}}}{a}\) |
\(\Big \downarrow \) 2178 |
\(\displaystyle -\frac {4 \left (\frac {1}{2 \left (\frac {1-a x}{a x+1}+1\right )}-\frac {1}{2} \int -\frac {4 \sqrt {\frac {1-a x}{a x+1}}}{\left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^2 \left (\frac {1-a x}{a x+1}+1\right )}d\sqrt {\frac {1-a x}{a x+1}}\right )}{a}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {4 \left (2 \int \frac {\sqrt {\frac {1-a x}{a x+1}}}{\left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^2 \left (\frac {1-a x}{a x+1}+1\right )}d\sqrt {\frac {1-a x}{a x+1}}+\frac {1}{2 \left (\frac {1-a x}{a x+1}+1\right )}\right )}{a}\) |
\(\Big \downarrow \) 594 |
\(\displaystyle -\frac {4 \left (2 \left (\frac {1}{2 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )}-\frac {1}{2} \int \frac {1}{\frac {1-a x}{a x+1}+1}d\sqrt {\frac {1-a x}{a x+1}}\right )+\frac {1}{2 \left (\frac {1-a x}{a x+1}+1\right )}\right )}{a}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle -\frac {4 \left (2 \left (\frac {1}{2 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )}-\frac {1}{2} \arctan \left (\sqrt {\frac {1-a x}{a x+1}}\right )\right )+\frac {1}{2 \left (\frac {1-a x}{a x+1}+1\right )}\right )}{a}\) |
Input:
Int[E^(2*ArcSech[a*x]),x]
Output:
(-4*(1/(2*(1 + (1 - a*x)/(1 + a*x))) + 2*(1/(2*(1 - Sqrt[(1 - a*x)/(1 + a* x)])) - ArcTan[Sqrt[(1 - a*x)/(1 + a*x)]]/2)))/a
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[(x_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-c)*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/((n + 1)*(b*c^2 + a*d^2))) , x] + Simp[1/((n + 1)*(b*c^2 + a*d^2)) Int[(c + d*x)^(n + 1)*(a + b*x^2) ^p*(a*d*(n + 1) + b*c*(n + 2*p + 3)*x), x], x] /; FreeQ[{a, b, c, d, p}, x] && LtQ[n, -1] && NeQ[b*c^2 + a*d^2, 0]
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : > With[{Qx = PolynomialQuotient[(d + e*x)^m*Pq, a + b*x^2, x], R = Coeff[Po lynomialRemainder[(d + e*x)^m*Pq, a + b*x^2, x], x, 0], S = Coeff[Polynomia lRemainder[(d + e*x)^m*Pq, a + b*x^2, x], x, 1]}, Simp[(a*S - b*R*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*b*(p + 1)) Int[(d + e*x )^m*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*b*(p + 1)*Qx)/(d + e*x)^m + (b*R*( 2*p + 3))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, d, e}, x] && PolyQ[Pq, x ] && NeQ[b*d^2 + a*e^2, 0] && LtQ[p, -1] && ILtQ[m, 0]
Int[E^(ArcSech[u_]*(n_.)), x_Symbol] :> Int[(1/u + Sqrt[(1 - u)/(1 + u)] + (1/u)*Sqrt[(1 - u)/(1 + u)])^n, x] /; IntegerQ[n]
Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfQuotientOfLinears [u, x]}, Simp[lst[[2]]*lst[[4]] Subst[Int[lst[[1]], x], x, lst[[3]]^(1/ls t[[2]])], x] /; !FalseQ[lst]]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.14 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.95
method | result | size |
default | \(\frac {-a^{2} x -\frac {1}{x}}{a^{2}}-\frac {2 \sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}\, \left (\arctan \left (\frac {\operatorname {csgn}\left (a \right ) a x}{\sqrt {-a^{2} x^{2}+1}}\right ) a x +\sqrt {-a^{2} x^{2}+1}\, \operatorname {csgn}\left (a \right )\right ) \operatorname {csgn}\left (a \right )}{a \sqrt {-a^{2} x^{2}+1}}-\frac {1}{a^{2} x}\) | \(111\) |
Input:
int((1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2))^2,x,method=_RETURNVERBOSE)
Output:
1/a^2*(-a^2*x-1/x)-2/a*(-(a*x-1)/a/x)^(1/2)*((a*x+1)/a/x)^(1/2)*(arctan(cs gn(a)*a*x/(-a^2*x^2+1)^(1/2))*a*x+(-a^2*x^2+1)^(1/2)*csgn(a))*csgn(a)/(-a^ 2*x^2+1)^(1/2)-1/a^2/x
Time = 0.13 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.49 \[ \int e^{2 \text {sech}^{-1}(a x)} \, dx=-\frac {a^{2} x^{2} + 2 \, a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 2 \, a x \arctan \left (\sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}}\right ) + 2}{a^{2} x} \] Input:
integrate((1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2))^2,x, algorithm="fricas" )
Output:
-(a^2*x^2 + 2*a*x*sqrt((a*x + 1)/(a*x))*sqrt(-(a*x - 1)/(a*x)) - 2*a*x*arc tan(sqrt((a*x + 1)/(a*x))*sqrt(-(a*x - 1)/(a*x))) + 2)/(a^2*x)
\[ \int e^{2 \text {sech}^{-1}(a x)} \, dx=\frac {\int \left (- a^{2}\right )\, dx + \int \frac {2}{x^{2}}\, dx + \int \frac {2 a \sqrt {-1 + \frac {1}{a x}} \sqrt {1 + \frac {1}{a x}}}{x}\, dx}{a^{2}} \] Input:
integrate((1/a/x+(-1+1/a/x)**(1/2)*(1+1/a/x)**(1/2))**2,x)
Output:
(Integral(-a**2, x) + Integral(2/x**2, x) + Integral(2*a*sqrt(-1 + 1/(a*x) )*sqrt(1 + 1/(a*x))/x, x))/a**2
\[ \int e^{2 \text {sech}^{-1}(a x)} \, dx=\int { {\left (\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}\right )}^{2} \,d x } \] Input:
integrate((1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2))^2,x, algorithm="maxima" )
Output:
-x + 2*integrate(sqrt(a*x + 1)*sqrt(-a*x + 1)/x^2, x)/a^2 + integrate(x^(- 2), x)/a^2 - 1/(a^2*x)
\[ \int e^{2 \text {sech}^{-1}(a x)} \, dx=\int { {\left (\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}\right )}^{2} \,d x } \] Input:
integrate((1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2))^2,x, algorithm="giac")
Output:
integrate((sqrt(1/(a*x) + 1)*sqrt(1/(a*x) - 1) + 1/(a*x))^2, x)
Time = 26.68 (sec) , antiderivative size = 162, normalized size of antiderivative = 2.84 \[ \int e^{2 \text {sech}^{-1}(a x)} \, dx=-x-\frac {\left (\ln \left (\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2}+1\right )-\ln \left (\frac {\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}}{\sqrt {\frac {1}{a\,x}+1}-1}\right )\right )\,2{}\mathrm {i}}{a}-\frac {2}{a^2\,x}+\frac {{\left (1+\sqrt {-\frac {a-\frac {1}{x}}{a}}\,1{}\mathrm {i}\right )}^2\,{\left (\sqrt {\frac {a+\frac {1}{x}}{a}}-1\right )}^2\,4{}\mathrm {i}}{a\,{\left (\sqrt {\frac {a+\frac {1}{x}}{a}}\,1{}\mathrm {i}+\sqrt {-\frac {a-\frac {1}{x}}{a}}-2{}\mathrm {i}\right )}^2} \] Input:
int(((1/(a*x) - 1)^(1/2)*(1/(a*x) + 1)^(1/2) + 1/(a*x))^2,x)
Output:
(((-(a - 1/x)/a)^(1/2)*1i + 1)^2*(((a + 1/x)/a)^(1/2) - 1)^2*4i)/(a*(((a + 1/x)/a)^(1/2)*1i + (-(a - 1/x)/a)^(1/2) - 2i)^2) - ((log(((1/(a*x) - 1)^( 1/2) - 1i)^2/((1/(a*x) + 1)^(1/2) - 1)^2 + 1) - log(((1/(a*x) - 1)^(1/2) - 1i)/((1/(a*x) + 1)^(1/2) - 1)))*2i)/a - 2/(a^2*x) - x
Time = 0.15 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.86 \[ \int e^{2 \text {sech}^{-1}(a x)} \, dx=\frac {4 \mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right ) a x -2 \sqrt {a x +1}\, \sqrt {-a x +1}-a^{2} x^{2}-2}{a^{2} x} \] Input:
int((1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2))^2,x)
Output:
(4*asin(sqrt( - a*x + 1)/sqrt(2))*a*x - 2*sqrt(a*x + 1)*sqrt( - a*x + 1) - a**2*x**2 - 2)/(a**2*x)