Integrand size = 8, antiderivative size = 65 \[ \int e^{-\text {sech}^{-1}(a x)} \, dx=-\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x)}{a}+\frac {\log (1+a x)}{a}+\frac {2 \log \left (1+\sqrt {\frac {1-a x}{1+a x}}\right )}{a} \]
Time = 0.05 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.11 \[ \int e^{-\text {sech}^{-1}(a x)} \, dx=\frac {-\sqrt {\frac {1-a x}{1+a x}} (1+a x)+\log \left (1+\sqrt {\frac {1-a x}{1+a x}}+a x \sqrt {\frac {1-a x}{1+a x}}\right )}{a} \]
(-(Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)) + Log[1 + Sqrt[(1 - a*x)/(1 + a*x) ] + a*x*Sqrt[(1 - a*x)/(1 + a*x)]])/a
Time = 0.43 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.43, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6886, 7268, 25, 2178, 657, 2009}
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.
\(\displaystyle \int e^{-\text {sech}^{-1}(a x)} \, dx\) |
\(\Big \downarrow \) 6886 |
\(\displaystyle \int \frac {1}{\frac {\sqrt {\frac {1-a x}{a x+1}}}{a x}+\sqrt {\frac {1-a x}{a x+1}}+\frac {1}{a x}}dx\) |
\(\Big \downarrow \) 7268 |
\(\displaystyle \frac {4 \int -\frac {\sqrt {\frac {1-a x}{a x+1}} \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )}{\left (\sqrt {\frac {1-a x}{a x+1}}+1\right ) \left (\frac {1-a x}{a x+1}+1\right )^2}d\sqrt {\frac {1-a x}{a x+1}}}{a}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {4 \int \frac {\sqrt {\frac {1-a x}{a x+1}} \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )}{\left (\sqrt {\frac {1-a x}{a x+1}}+1\right ) \left (\frac {1-a x}{a x+1}+1\right )^2}d\sqrt {\frac {1-a x}{a x+1}}}{a}\) |
\(\Big \downarrow \) 2178 |
\(\displaystyle \frac {4 \left (\frac {1}{2} \int \frac {1-\sqrt {\frac {1-a x}{a x+1}}}{\left (\sqrt {\frac {1-a x}{a x+1}}+1\right ) \left (\frac {1-a x}{a x+1}+1\right )}d\sqrt {\frac {1-a x}{a x+1}}-\frac {\sqrt {\frac {1-a x}{a x+1}}}{2 \left (\frac {1-a x}{a x+1}+1\right )}\right )}{a}\) |
\(\Big \downarrow \) 657 |
\(\displaystyle \frac {4 \left (\frac {1}{2} \int \left (\frac {1}{\sqrt {\frac {1-a x}{a x+1}}+1}-\frac {\sqrt {\frac {1-a x}{a x+1}}}{\frac {1-a x}{a x+1}+1}\right )d\sqrt {\frac {1-a x}{a x+1}}-\frac {\sqrt {\frac {1-a x}{a x+1}}}{2 \left (\frac {1-a x}{a x+1}+1\right )}\right )}{a}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {4 \left (\frac {1}{2} \left (\log \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )-\frac {1}{2} \log \left (\frac {1-a x}{a x+1}+1\right )\right )-\frac {\sqrt {\frac {1-a x}{a x+1}}}{2 \left (\frac {1-a x}{a x+1}+1\right )}\right )}{a}\) |
(4*(-1/2*Sqrt[(1 - a*x)/(1 + a*x)]/(1 + (1 - a*x)/(1 + a*x)) + (Log[1 + Sq rt[(1 - a*x)/(1 + a*x)]] - Log[1 + (1 - a*x)/(1 + a*x)]/2)/2))/a
3.1.80.3.1 Defintions of rubi rules used
Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_) + (c_.)*( x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*((f + g*x)^n/(a + c*x^ 2)), x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IntegersQ[n]
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]]
Leaf count of result is larger than twice the leaf count of optimal. \(2611\) vs. \(2(61)=122\).
Time = 0.28 (sec) , antiderivative size = 2612, normalized size of antiderivative = 40.18
1/2*(a*x+1)/x^3*(x^2*ln(a^2*x^2)*(-(a*x-1)/a/x)^(3/2)*((a*x+1)/a/x)^(1/2)* (-a^2*x^2+1)^(1/2)*a^2+ln(-2*(-((-a*x^2*(-(a*x-1)/a/x)^(1/2)*((a*x+1)/a/x) ^(1/2)+(-(a*x-1)*x)^(1/2)*((a*x+1)*x)^(1/2))*(a*x^2*(-(a*x-1)/a/x)^(1/2)*( (a*x+1)/a/x)^(1/2)+(-(a*x-1)*x)^(1/2)*((a*x+1)*x)^(1/2))*a^2/(a*x-1)/(a*x+ 1)/x^2)^(1/2)*x+(-a^2*x^2+1)^(1/2)+1)*a^2/(-a^2*x+((-a*x^2*(-(a*x-1)/a/x)^ (1/2)*((a*x+1)/a/x)^(1/2)+(-(a*x-1)*x)^(1/2)*((a*x+1)*x)^(1/2))*(a*x^2*(-( a*x-1)/a/x)^(1/2)*((a*x+1)/a/x)^(1/2)+(-(a*x-1)*x)^(1/2)*((a*x+1)*x)^(1/2) )*a^2/(a*x-1)/(a*x+1)/x^2)^(1/2)))*a^3*x^3+ln(2*(((-a*x^2*(-(a*x-1)/a/x)^( 1/2)*((a*x+1)/a/x)^(1/2)+(-(a*x-1)*x)^(1/2)*((a*x+1)*x)^(1/2))*(a*x^2*(-(a *x-1)/a/x)^(1/2)*((a*x+1)/a/x)^(1/2)+(-(a*x-1)*x)^(1/2)*((a*x+1)*x)^(1/2)) *a^2/(a*x-1)/(a*x+1)/x^2)^(1/2)*x+(-a^2*x^2+1)^(1/2)+1)*a^2/(a^2*x+((-a*x^ 2*(-(a*x-1)/a/x)^(1/2)*((a*x+1)/a/x)^(1/2)+(-(a*x-1)*x)^(1/2)*((a*x+1)*x)^ (1/2))*(a*x^2*(-(a*x-1)/a/x)^(1/2)*((a*x+1)/a/x)^(1/2)+(-(a*x-1)*x)^(1/2)* ((a*x+1)*x)^(1/2))*a^2/(a*x-1)/(a*x+1)/x^2)^(1/2)))*a^3*x^3-2*(-a^2*x^2+1) ^(1/2)*a^3*x^3-x^2*ln(-2*(-((-a*x^2*(-(a*x-1)/a/x)^(1/2)*((a*x+1)/a/x)^(1/ 2)+(-(a*x-1)*x)^(1/2)*((a*x+1)*x)^(1/2))*(a*x^2*(-(a*x-1)/a/x)^(1/2)*((a*x +1)/a/x)^(1/2)+(-(a*x-1)*x)^(1/2)*((a*x+1)*x)^(1/2))*a^2/(a*x-1)/(a*x+1)/x ^2)^(1/2)*x+(-a^2*x^2+1)^(1/2)+1)*a^2/(-a^2*x+((-a*x^2*(-(a*x-1)/a/x)^(1/2 )*((a*x+1)/a/x)^(1/2)+(-(a*x-1)*x)^(1/2)*((a*x+1)*x)^(1/2))*(a*x^2*(-(a*x- 1)/a/x)^(1/2)*((a*x+1)/a/x)^(1/2)+(-(a*x-1)*x)^(1/2)*((a*x+1)*x)^(1/2))...
Time = 0.26 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.77 \[ \int e^{-\text {sech}^{-1}(a x)} \, dx=-\frac {2 \, a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - \log \left (a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} + 1\right ) + \log \left (a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 1\right ) - 2 \, \log \left (x\right )}{2 \, a} \]
-1/2*(2*a*x*sqrt((a*x + 1)/(a*x))*sqrt(-(a*x - 1)/(a*x)) - log(a*x*sqrt((a *x + 1)/(a*x))*sqrt(-(a*x - 1)/(a*x)) + 1) + log(a*x*sqrt((a*x + 1)/(a*x)) *sqrt(-(a*x - 1)/(a*x)) - 1) - 2*log(x))/a
\[ \int e^{-\text {sech}^{-1}(a x)} \, dx=a \int \frac {x}{a x \sqrt {-1 + \frac {1}{a x}} \sqrt {1 + \frac {1}{a x}} + 1}\, dx \]
\[ \int e^{-\text {sech}^{-1}(a x)} \, dx=\int { \frac {1}{\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}} \,d x } \]
\[ \int e^{-\text {sech}^{-1}(a x)} \, dx=\int { \frac {1}{\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}} \,d x } \]
Time = 7.36 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.72 \[ \int e^{-\text {sech}^{-1}(a x)} \, dx=\frac {\mathrm {acosh}\left (\frac {1}{a\,x}\right )}{a}-\frac {\ln \left (\frac {1}{x}\right )}{a}-x\,\sqrt {\frac {1}{a\,x}-1}\,\sqrt {\frac {1}{a\,x}+1} \]