Integrand size = 10, antiderivative size = 53 \[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^2} \, dx=-\frac {1}{2 a x^2}-\frac {\sqrt {-1+\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{2 x}+\frac {1}{2} a \text {sech}^{-1}(a x) \] Output:
-1/2/a/x^2-1/2*(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2)/x+1/2*a*arcsech(a*x)
Time = 0.13 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.75 \[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^2} \, dx=\frac {1}{2} \left (-\frac {1}{a x^2}-\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x)}{a x^2}-a \log (x)+a \log \left (1+\sqrt {\frac {1-a x}{1+a x}}+a x \sqrt {\frac {1-a x}{1+a x}}\right )\right ) \] Input:
Integrate[E^ArcSech[a*x]/x^2,x]
Output:
(-(1/(a*x^2)) - (Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x))/(a*x^2) - a*Log[x] + a*Log[1 + Sqrt[(1 - a*x)/(1 + a*x)] + a*x*Sqrt[(1 - a*x)/(1 + a*x)]])/2
Time = 0.45 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.92, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {6889, 15, 114, 25, 27, 103, 221}
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.
\(\displaystyle \int \frac {e^{\text {sech}^{-1}(a x)}}{x^2} \, dx\) |
\(\Big \downarrow \) 6889 |
\(\displaystyle -\frac {\int \frac {1}{x^3}dx}{a}-\frac {\sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \int \frac {1}{x^3 \sqrt {1-a x} \sqrt {a x+1}}dx}{a}-\frac {e^{\text {sech}^{-1}(a x)}}{x}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle -\frac {\sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \int \frac {1}{x^3 \sqrt {1-a x} \sqrt {a x+1}}dx}{a}+\frac {1}{2 a x^2}-\frac {e^{\text {sech}^{-1}(a x)}}{x}\) |
\(\Big \downarrow \) 114 |
\(\displaystyle -\frac {\sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \left (-\frac {1}{2} \int -\frac {a^2}{x \sqrt {1-a x} \sqrt {a x+1}}dx-\frac {\sqrt {1-a x} \sqrt {a x+1}}{2 x^2}\right )}{a}+\frac {1}{2 a x^2}-\frac {e^{\text {sech}^{-1}(a x)}}{x}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \left (\frac {1}{2} \int \frac {a^2}{x \sqrt {1-a x} \sqrt {a x+1}}dx-\frac {\sqrt {1-a x} \sqrt {a x+1}}{2 x^2}\right )}{a}+\frac {1}{2 a x^2}-\frac {e^{\text {sech}^{-1}(a x)}}{x}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \left (\frac {1}{2} a^2 \int \frac {1}{x \sqrt {1-a x} \sqrt {a x+1}}dx-\frac {\sqrt {1-a x} \sqrt {a x+1}}{2 x^2}\right )}{a}+\frac {1}{2 a x^2}-\frac {e^{\text {sech}^{-1}(a x)}}{x}\) |
\(\Big \downarrow \) 103 |
\(\displaystyle -\frac {\sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \left (-\frac {1}{2} a^3 \int \frac {1}{a-a (1-a x) (a x+1)}d\left (\sqrt {1-a x} \sqrt {a x+1}\right )-\frac {\sqrt {1-a x} \sqrt {a x+1}}{2 x^2}\right )}{a}+\frac {1}{2 a x^2}-\frac {e^{\text {sech}^{-1}(a x)}}{x}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {\sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \left (-\frac {1}{2} a^2 \text {arctanh}\left (\sqrt {1-a x} \sqrt {a x+1}\right )-\frac {\sqrt {1-a x} \sqrt {a x+1}}{2 x^2}\right )}{a}+\frac {1}{2 a x^2}-\frac {e^{\text {sech}^{-1}(a x)}}{x}\) |
Input:
Int[E^ArcSech[a*x]/x^2,x]
Output:
1/(2*a*x^2) - E^ArcSech[a*x]/x - (Sqrt[(1 + a*x)^(-1)]*Sqrt[1 + a*x]*(-1/2 *(Sqrt[1 - a*x]*Sqrt[1 + a*x])/x^2 - (a^2*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a *x]])/2))/a
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_ ))), x_] :> Simp[b*f Subst[Int[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sq rt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[2*b*d *e - f*(b*c + a*d), 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[E^ArcSech[(a_.)*(x_)^(p_.)]*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*(E^ ArcSech[a*x^p]/(m + 1)), x] + (Simp[p/(a*(m + 1)) Int[x^(m - p), x], x] + Simp[p*(Sqrt[1 + a*x^p]/(a*(m + 1)))*Sqrt[1/(1 + a*x^p)] Int[x^(m - p)/( Sqrt[1 + a*x^p]*Sqrt[1 - a*x^p]), x], x]) /; FreeQ[{a, m, p}, x] && NeQ[m, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(90\) vs. \(2(43)=86\).
Time = 0.10 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.72
method | result | size |
default | \(\frac {\sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}\, \left (a^{2} x^{2} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )-\sqrt {-a^{2} x^{2}+1}\right )}{2 x \sqrt {-a^{2} x^{2}+1}}-\frac {1}{2 a \,x^{2}}\) | \(91\) |
Input:
int((1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2))/x^2,x,method=_RETURNVERBOSE)
Output:
1/2*(-(a*x-1)/a/x)^(1/2)/x*((a*x+1)/a/x)^(1/2)*(a^2*x^2*arctanh(1/(-a^2*x^ 2+1)^(1/2))-(-a^2*x^2+1)^(1/2))/(-a^2*x^2+1)^(1/2)-1/2/a/x^2
Leaf count of result is larger than twice the leaf count of optimal. 128 vs. \(2 (43) = 86\).
Time = 0.09 (sec) , antiderivative size = 128, normalized size of antiderivative = 2.42 \[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^2} \, dx=\frac {a^{2} x^{2} \log \left (a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} + 1\right ) - a^{2} x^{2} \log \left (a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 1\right ) - 2 \, a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 2}{4 \, a x^{2}} \] Input:
integrate((1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2))/x^2,x, algorithm="frica s")
Output:
1/4*(a^2*x^2*log(a*x*sqrt((a*x + 1)/(a*x))*sqrt(-(a*x - 1)/(a*x)) + 1) - a ^2*x^2*log(a*x*sqrt((a*x + 1)/(a*x))*sqrt(-(a*x - 1)/(a*x)) - 1) - 2*a*x*s qrt((a*x + 1)/(a*x))*sqrt(-(a*x - 1)/(a*x)) - 2)/(a*x^2)
Time = 2.67 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.34 \[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^2} \, dx=- a \left (2 \sqrt {-1 + \frac {1}{a x}} \left (\frac {\left (1 + \frac {1}{a x}\right )^{\frac {3}{2}}}{4} - \frac {\sqrt {1 + \frac {1}{a x}}}{4}\right ) - \log {\left (2 \sqrt {-1 + \frac {1}{a x}} + 2 \sqrt {1 + \frac {1}{a x}} \right )}\right ) - \frac {1}{2 a x^{2}} \] Input:
integrate((1/a/x+(-1+1/a/x)**(1/2)*(1+1/a/x)**(1/2))/x**2,x)
Output:
-a*(2*sqrt(-1 + 1/(a*x))*((1 + 1/(a*x))**(3/2)/4 - sqrt(1 + 1/(a*x))/4) - log(2*sqrt(-1 + 1/(a*x)) + 2*sqrt(1 + 1/(a*x)))) - 1/(2*a*x**2)
\[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^2} \, dx=\int { \frac {\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}}{x^{2}} \,d x } \] Input:
integrate((1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2))/x^2,x, algorithm="maxim a")
Output:
integrate(sqrt(a*x + 1)*sqrt(-a*x + 1)/x^3, x)/a - 1/2/(a*x^2)
\[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^2} \, dx=\int { \frac {\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}}{x^{2}} \,d x } \] Input:
integrate((1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2))/x^2,x, algorithm="giac" )
Output:
integrate((sqrt(1/(a*x) + 1)*sqrt(1/(a*x) - 1) + 1/(a*x))/x^2, x)
Time = 24.97 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.34 \[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^2} \, dx=\frac {a\,\ln \left (\sqrt {\frac {1}{a\,x}-1}\,\sqrt {\frac {1}{a\,x}+1}+\frac {1}{a\,x}\right )}{2}-\frac {1}{2\,a\,x^2}-\frac {\sqrt {\frac {1}{a\,x}-1}\,\sqrt {\frac {1}{a\,x}+1}}{2\,x} \] Input:
int(((1/(a*x) - 1)^(1/2)*(1/(a*x) + 1)^(1/2) + 1/(a*x))/x^2,x)
Output:
(a*log((1/(a*x) - 1)^(1/2)*(1/(a*x) + 1)^(1/2) + 1/(a*x)))/2 - 1/(2*a*x^2) - ((1/(a*x) - 1)^(1/2)*(1/(a*x) + 1)^(1/2))/(2*x)
Time = 0.15 (sec) , antiderivative size = 143, normalized size of antiderivative = 2.70 \[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^2} \, dx=\frac {-\sqrt {a x +1}\, \sqrt {-a x +1}+\mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )-1\right ) a^{2} x^{2}-\mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )+1\right ) a^{2} x^{2}+\mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )-1\right ) a^{2} x^{2}-\mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )+1\right ) a^{2} x^{2}-1}{2 a \,x^{2}} \] Input:
int((1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2))/x^2,x)
Output:
( - sqrt(a*x + 1)*sqrt( - a*x + 1) + log( - sqrt(2) + tan(asin(sqrt( - a*x + 1)/sqrt(2))/2) - 1)*a**2*x**2 - log( - sqrt(2) + tan(asin(sqrt( - a*x + 1)/sqrt(2))/2) + 1)*a**2*x**2 + log(sqrt(2) + tan(asin(sqrt( - a*x + 1)/s qrt(2))/2) - 1)*a**2*x**2 - log(sqrt(2) + tan(asin(sqrt( - a*x + 1)/sqrt(2 ))/2) + 1)*a**2*x**2 - 1)/(2*a*x**2)