Integrand size = 12, antiderivative size = 73 \[ \int \frac {e^{\text {sech}^{-1}\left (a x^2\right )}}{x} \, dx=-\frac {1}{2} \sqrt {-1+\frac {1}{a x^2}} \sqrt {1+\frac {1}{a x^2}}-\frac {1}{2 a x^2}+\frac {1}{2} \arctan \left (\sqrt {-1+\frac {1}{a x^2}} \sqrt {1+\frac {1}{a x^2}}\right ) \] Output:
-1/2*(-1+1/a/x^2)^(1/2)*(1+1/a/x^2)^(1/2)-1/2/a/x^2+1/2*arctan((-1+1/a/x^2 )^(1/2)*(1+1/a/x^2)^(1/2))
Time = 0.11 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.30 \[ \int \frac {e^{\text {sech}^{-1}\left (a x^2\right )}}{x} \, dx=-\frac {1}{2} e^{\text {sech}^{-1}\left (a x^2\right )}+\arctan \left (e^{\text {sech}^{-1}\left (a x^2\right )}\right ) \] Input:
Integrate[E^ArcSech[a*x^2]/x,x]
Output:
-1/2*E^ArcSech[a*x^2] + ArcTan[E^ArcSech[a*x^2]]
Time = 0.44 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6888, 335, 807, 247, 223}
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.
\(\displaystyle \int \frac {e^{\text {sech}^{-1}\left (a x^2\right )}}{x} \, dx\) |
\(\Big \downarrow \) 6888 |
\(\displaystyle \frac {\sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \int \frac {\sqrt {1-a x^2} \sqrt {a x^2+1}}{x^3}dx}{a}-\frac {1}{2 a x^2}\) |
\(\Big \downarrow \) 335 |
\(\displaystyle \frac {\sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \int \frac {\sqrt {1-a^2 x^4}}{x^3}dx}{a}-\frac {1}{2 a x^2}\) |
\(\Big \downarrow \) 807 |
\(\displaystyle \frac {\sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \int \frac {\sqrt {1-a^2 x^4}}{x^4}dx^2}{2 a}-\frac {1}{2 a x^2}\) |
\(\Big \downarrow \) 247 |
\(\displaystyle \frac {\sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \left (a^2 \left (-\int \frac {1}{\sqrt {1-a^2 x^4}}dx^2\right )-\frac {\sqrt {1-a^2 x^4}}{x^2}\right )}{2 a}-\frac {1}{2 a x^2}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \frac {\sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \left (-\frac {\sqrt {1-a^2 x^4}}{x^2}-a \arcsin \left (a x^2\right )\right )}{2 a}-\frac {1}{2 a x^2}\) |
Input:
Int[E^ArcSech[a*x^2]/x,x]
Output:
-1/2*1/(a*x^2) + (Sqrt[(1 + a*x^2)^(-1)]*Sqrt[1 + a*x^2]*(-(Sqrt[1 - a^2*x ^4]/x^2) - a*ArcSin[a*x^2]))/(2*a)
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^p/(c*(m + 1))), x] - Simp[2*b*(p/(c^2*(m + 1))) Int[ (c*x)^(m + 2)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && LtQ[m, -1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(p _.), x_Symbol] :> Int[(e*x)^m*(a*c + b*d*x^4)^p, x] /; FreeQ[{a, b, c, d, e , m, p}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[c, 0] ))
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
Int[E^ArcSech[(a_.)*(x_)^(p_.)]/(x_), x_Symbol] :> -Simp[(a*p*x^p)^(-1), x] + Simp[(Sqrt[1 + a*x^p]/a)*Sqrt[1/(1 + a*x^p)] Int[Sqrt[1 + a*x^p]*(Sqrt [1 - a*x^p]/x^(p + 1)), x], x] /; FreeQ[{a, p}, x]
Time = 0.12 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.41
method | result | size |
default | \(-\frac {\sqrt {-\frac {a \,x^{2}-1}{a \,x^{2}}}\, \sqrt {\frac {a \,x^{2}+1}{a \,x^{2}}}\, \left (\arctan \left (\frac {x^{2}}{\sqrt {-\frac {a^{2} x^{4}-1}{a^{2}}}}\right ) x^{2}+\sqrt {-\frac {a^{2} x^{4}-1}{a^{2}}}\right )}{2 \sqrt {-\frac {a^{2} x^{4}-1}{a^{2}}}}-\frac {1}{2 a \,x^{2}}\) | \(103\) |
Input:
int((1/a/x^2+(-1+1/a/x^2)^(1/2)*(1+1/a/x^2)^(1/2))/x,x,method=_RETURNVERBO SE)
Output:
-1/2*(-(a*x^2-1)/a/x^2)^(1/2)*((a*x^2+1)/a/x^2)^(1/2)*(arctan(x^2/(-(a^2*x ^4-1)/a^2)^(1/2))*x^2+(-(a^2*x^4-1)/a^2)^(1/2))/(-(a^2*x^4-1)/a^2)^(1/2)-1 /2/a/x^2
Time = 0.08 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.40 \[ \int \frac {e^{\text {sech}^{-1}\left (a x^2\right )}}{x} \, dx=-\frac {a x^{2} \sqrt {\frac {a x^{2} + 1}{a x^{2}}} \sqrt {-\frac {a x^{2} - 1}{a x^{2}}} - 2 \, a x^{2} \arctan \left (\frac {a x^{2} \sqrt {\frac {a x^{2} + 1}{a x^{2}}} \sqrt {-\frac {a x^{2} - 1}{a x^{2}}} - 1}{a x^{2}}\right ) + 1}{2 \, a x^{2}} \] Input:
integrate((1/a/x^2+(-1+1/a/x^2)^(1/2)*(1+1/a/x^2)^(1/2))/x,x, algorithm="f ricas")
Output:
-1/2*(a*x^2*sqrt((a*x^2 + 1)/(a*x^2))*sqrt(-(a*x^2 - 1)/(a*x^2)) - 2*a*x^2 *arctan((a*x^2*sqrt((a*x^2 + 1)/(a*x^2))*sqrt(-(a*x^2 - 1)/(a*x^2)) - 1)/( a*x^2)) + 1)/(a*x^2)
\[ \int \frac {e^{\text {sech}^{-1}\left (a x^2\right )}}{x} \, dx=\frac {\int \frac {1}{x^{3}}\, dx + \int \frac {a \sqrt {-1 + \frac {1}{a x^{2}}} \sqrt {1 + \frac {1}{a x^{2}}}}{x}\, dx}{a} \] Input:
integrate((1/a/x**2+(-1+1/a/x**2)**(1/2)*(1+1/a/x**2)**(1/2))/x,x)
Output:
(Integral(x**(-3), x) + Integral(a*sqrt(-1 + 1/(a*x**2))*sqrt(1 + 1/(a*x** 2))/x, x))/a
\[ \int \frac {e^{\text {sech}^{-1}\left (a x^2\right )}}{x} \, dx=\int { \frac {\sqrt {\frac {1}{a x^{2}} + 1} \sqrt {\frac {1}{a x^{2}} - 1} + \frac {1}{a x^{2}}}{x} \,d x } \] Input:
integrate((1/a/x^2+(-1+1/a/x^2)^(1/2)*(1+1/a/x^2)^(1/2))/x,x, algorithm="m axima")
Output:
integrate(sqrt(a*x^2 + 1)*sqrt(-a*x^2 + 1)/x^3, x)/a - 1/2/(a*x^2)
Leaf count of result is larger than twice the leaf count of optimal. 250 vs. \(2 (59) = 118\).
Time = 165.25 (sec) , antiderivative size = 250, normalized size of antiderivative = 3.42 \[ \int \frac {e^{\text {sech}^{-1}\left (a x^2\right )}}{x} \, dx=-\frac {1}{2} \, a {\left (\frac {\pi + 2 \, \arctan \left (\frac {\sqrt {a^{2} x^{2} + a} {\left (\frac {{\left (\sqrt {2} \sqrt {a} - \sqrt {-a^{2} x^{2} + a}\right )}^{2}}{a^{2} x^{2} + a} - 1\right )}}{2 \, {\left (\sqrt {2} \sqrt {a} - \sqrt {-a^{2} x^{2} + a}\right )}}\right )}{a} + \frac {4 \, {\left (\frac {\sqrt {2} \sqrt {a} - \sqrt {-a^{2} x^{2} + a}}{\sqrt {a^{2} x^{2} + a}} - \frac {\sqrt {a^{2} x^{2} + a}}{\sqrt {2} \sqrt {a} - \sqrt {-a^{2} x^{2} + a}}\right )}}{{\left ({\left (\frac {\sqrt {2} \sqrt {a} - \sqrt {-a^{2} x^{2} + a}}{\sqrt {a^{2} x^{2} + a}} - \frac {\sqrt {a^{2} x^{2} + a}}{\sqrt {2} \sqrt {a} - \sqrt {-a^{2} x^{2} + a}}\right )}^{2} - 4\right )} a} + \frac {1}{a^{2} x^{2}}\right )} \] Input:
integrate((1/a/x^2+(-1+1/a/x^2)^(1/2)*(1+1/a/x^2)^(1/2))/x,x, algorithm="g iac")
Output:
-1/2*a*((pi + 2*arctan(1/2*sqrt(a^2*x^2 + a)*((sqrt(2)*sqrt(a) - sqrt(-a^2 *x^2 + a))^2/(a^2*x^2 + a) - 1)/(sqrt(2)*sqrt(a) - sqrt(-a^2*x^2 + a))))/a + 4*((sqrt(2)*sqrt(a) - sqrt(-a^2*x^2 + a))/sqrt(a^2*x^2 + a) - sqrt(a^2* x^2 + a)/(sqrt(2)*sqrt(a) - sqrt(-a^2*x^2 + a)))/((((sqrt(2)*sqrt(a) - sqr t(-a^2*x^2 + a))/sqrt(a^2*x^2 + a) - sqrt(a^2*x^2 + a)/(sqrt(2)*sqrt(a) - sqrt(-a^2*x^2 + a)))^2 - 4)*a) + 1/(a^2*x^2))
Time = 26.91 (sec) , antiderivative size = 185, normalized size of antiderivative = 2.53 \[ \int \frac {e^{\text {sech}^{-1}\left (a x^2\right )}}{x} \, dx=-\frac {\ln \left (\frac {{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^2}+1\right )\,1{}\mathrm {i}}{2}+\frac {\ln \left (\frac {\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}}{\sqrt {\frac {1}{a\,x^2}+1}-1}\right )\,1{}\mathrm {i}}{2}-\frac {1}{2\,a\,x^2}+\frac {{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^2\,8{}\mathrm {i}}{{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^2\,\left (2+\frac {2\,{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^4}-\frac {4\,{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^2}\right )} \] Input:
int(((1/(a*x^2) - 1)^(1/2)*(1/(a*x^2) + 1)^(1/2) + 1/(a*x^2))/x,x)
Output:
(log(((1/(a*x^2) - 1)^(1/2) - 1i)/((1/(a*x^2) + 1)^(1/2) - 1))*1i)/2 - (lo g(((1/(a*x^2) - 1)^(1/2) - 1i)^2/((1/(a*x^2) + 1)^(1/2) - 1)^2 + 1)*1i)/2 - 1/(2*a*x^2) + (((1/(a*x^2) - 1)^(1/2) - 1i)^2*8i)/(((1/(a*x^2) + 1)^(1/2 ) - 1)^2*((2*((1/(a*x^2) - 1)^(1/2) - 1i)^4)/((1/(a*x^2) + 1)^(1/2) - 1)^4 - (4*((1/(a*x^2) - 1)^(1/2) - 1i)^2)/((1/(a*x^2) + 1)^(1/2) - 1)^2 + 2))
Time = 0.15 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.86 \[ \int \frac {e^{\text {sech}^{-1}\left (a x^2\right )}}{x} \, dx=\frac {2 \mathit {atan} \left (\frac {\sqrt {a \,x^{2}+1}\, \sqrt {-a \,x^{2}+1}}{a \,x^{2}-1}\right ) a \,x^{2}-\sqrt {a \,x^{2}+1}\, \sqrt {-a \,x^{2}+1}-1}{2 a \,x^{2}} \] Input:
int((1/a/x^2+(-1+1/a/x^2)^(1/2)*(1+1/a/x^2)^(1/2))/x,x)
Output:
(2*atan((sqrt(a*x**2 + 1)*sqrt( - a*x**2 + 1))/(a*x**2 - 1))*a*x**2 - sqrt (a*x**2 + 1)*sqrt( - a*x**2 + 1) - 1)/(2*a*x**2)