Integrand size = 10, antiderivative size = 61 \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x \, dx=\frac {\left (1+a x^2\right ) \sqrt {-1+\frac {2}{1+a x^2}}}{2 a}-\frac {\text {arctanh}\left (\sqrt {-1+\frac {2}{1+a x^2}}\right )}{a}+\frac {\log (x)}{a} \] Output:
1/2*(a*x^2+1)*(-1+2/(a*x^2+1))^(1/2)/a-arctanh((-1+2/(a*x^2+1))^(1/2))/a+l n(x)/a
Time = 0.08 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.64 \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x \, dx=\frac {\sqrt {\frac {1-a x^2}{1+a x^2}} \left (1+a x^2\right )+2 \log \left (a x^2\right )-\log \left (1+\sqrt {\frac {1-a x^2}{1+a x^2}}+a x^2 \sqrt {\frac {1-a x^2}{1+a x^2}}\right )}{2 a} \] Input:
Integrate[E^ArcSech[a*x^2]*x,x]
Output:
(Sqrt[(1 - a*x^2)/(1 + a*x^2)]*(1 + a*x^2) + 2*Log[a*x^2] - Log[1 + Sqrt[( 1 - a*x^2)/(1 + a*x^2)] + a*x^2*Sqrt[(1 - a*x^2)/(1 + a*x^2)]])/(2*a)
Time = 0.45 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.11, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6889, 14, 335, 798, 73, 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.
\(\displaystyle \int x e^{\text {sech}^{-1}\left (a x^2\right )} \, dx\) |
\(\Big \downarrow \) 6889 |
\(\displaystyle \frac {\sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \int \frac {1}{x \sqrt {1-a x^2} \sqrt {a x^2+1}}dx}{a}+\frac {\int \frac {1}{x}dx}{a}+\frac {1}{2} x^2 e^{\text {sech}^{-1}\left (a x^2\right )}\) |
\(\Big \downarrow \) 14 |
\(\displaystyle \frac {\sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \int \frac {1}{x \sqrt {1-a x^2} \sqrt {a x^2+1}}dx}{a}+\frac {1}{2} x^2 e^{\text {sech}^{-1}\left (a x^2\right )}+\frac {\log (x)}{a}\) |
\(\Big \downarrow \) 335 |
\(\displaystyle \frac {\sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \int \frac {1}{x \sqrt {1-a^2 x^4}}dx}{a}+\frac {1}{2} x^2 e^{\text {sech}^{-1}\left (a x^2\right )}+\frac {\log (x)}{a}\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {\sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \int \frac {1}{x^4 \sqrt {1-a^2 x^4}}dx^4}{4 a}+\frac {1}{2} x^2 e^{\text {sech}^{-1}\left (a x^2\right )}+\frac {\log (x)}{a}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\frac {\sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \int \frac {1}{\frac {1}{a^2}-\frac {x^8}{a^2}}d\sqrt {1-a^2 x^4}}{2 a^3}+\frac {1}{2} x^2 e^{\text {sech}^{-1}\left (a x^2\right )}+\frac {\log (x)}{a}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {\sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \text {arctanh}\left (\sqrt {1-a^2 x^4}\right )}{2 a}+\frac {1}{2} x^2 e^{\text {sech}^{-1}\left (a x^2\right )}+\frac {\log (x)}{a}\) |
Input:
Int[E^ArcSech[a*x^2]*x,x]
Output:
(E^ArcSech[a*x^2]*x^2)/2 - (Sqrt[(1 + a*x^2)^(-1)]*Sqrt[1 + a*x^2]*ArcTanh [Sqrt[1 - a^2*x^4]])/(2*a) + Log[x]/a
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
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_.)*(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] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.11 (sec) , antiderivative size = 127, normalized size of antiderivative = 2.08
method | result | size |
default | \(\frac {\sqrt {-\frac {a \,x^{2}-1}{a \,x^{2}}}\, x^{2} \sqrt {\frac {a \,x^{2}+1}{a \,x^{2}}}\, \left (\operatorname {csgn}\left (\frac {1}{a}\right ) a \sqrt {-\frac {a^{2} x^{4}-1}{a^{2}}}-\ln \left (\frac {2 \,\operatorname {csgn}\left (\frac {1}{a}\right ) a \sqrt {-\frac {a^{2} x^{4}-1}{a^{2}}}+2}{a^{2} x^{2}}\right )\right ) \operatorname {csgn}\left (\frac {1}{a}\right )}{2 a \sqrt {-\frac {a^{2} x^{4}-1}{a^{2}}}}+\frac {\ln \left (x \right )}{a}\) | \(127\) |
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)*x^2*((a*x^2+1)/a/x^2)^(1/2)*(csgn(1/a)*a*(-(a ^2*x^4-1)/a^2)^(1/2)-ln(2*(csgn(1/a)*a*(-(a^2*x^4-1)/a^2)^(1/2)+1)/a^2/x^2 ))*csgn(1/a)/a/(-(a^2*x^4-1)/a^2)^(1/2)+ln(x)/a
Leaf count of result is larger than twice the leaf count of optimal. 133 vs. \(2 (55) = 110\).
Time = 0.08 (sec) , antiderivative size = 133, normalized size of antiderivative = 2.18 \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x \, dx=\frac {2 \, a x^{2} \sqrt {\frac {a x^{2} + 1}{a x^{2}}} \sqrt {-\frac {a x^{2} - 1}{a x^{2}}} - \log \left (a x^{2} \sqrt {\frac {a x^{2} + 1}{a x^{2}}} \sqrt {-\frac {a x^{2} - 1}{a x^{2}}} + 1\right ) + \log \left (a x^{2} \sqrt {\frac {a x^{2} + 1}{a x^{2}}} \sqrt {-\frac {a x^{2} - 1}{a x^{2}}} - 1\right ) + 4 \, \log \left (x\right )}{4 \, a} \] 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:
1/4*(2*a*x^2*sqrt((a*x^2 + 1)/(a*x^2))*sqrt(-(a*x^2 - 1)/(a*x^2)) - log(a* x^2*sqrt((a*x^2 + 1)/(a*x^2))*sqrt(-(a*x^2 - 1)/(a*x^2)) + 1) + log(a*x^2* sqrt((a*x^2 + 1)/(a*x^2))*sqrt(-(a*x^2 - 1)/(a*x^2)) - 1) + 4*log(x))/a
\[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x \, dx=\frac {\int \frac {1}{x}\, dx + \int a x \sqrt {-1 + \frac {1}{a x^{2}}} \sqrt {1 + \frac {1}{a x^{2}}}\, dx}{a} \] Input:
integrate((1/a/x**2+(-1+1/a/x**2)**(1/2)*(1+1/a/x**2)**(1/2))*x,x)
Output:
(Integral(1/x, x) + Integral(a*x*sqrt(-1 + 1/(a*x**2))*sqrt(1 + 1/(a*x**2) ), x))/a
\[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x \, dx=\int { x {\left (\sqrt {\frac {1}{a x^{2}} + 1} \sqrt {\frac {1}{a x^{2}} - 1} + \frac {1}{a x^{2}}\right )} \,d x } \] 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, x)/a + log(x)/a
Exception generated. \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x \, dx=\text {Exception raised: TypeError} \] 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:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Limit: Max order reached or unable to make series expansion Error: Bad Argument Value
Time = 26.35 (sec) , antiderivative size = 182, normalized size of antiderivative = 2.98 \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x \, dx=\frac {\ln \left (x\right )}{a}-\frac {2\,\mathrm {atanh}\left (\frac {\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}}{\sqrt {\frac {1}{a\,x^2}+1}-1}\right )}{a}+\frac {\frac {5\,{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^2}+1}{\frac {8\,a\,\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}{\sqrt {\frac {1}{a\,x^2}+1}-1}+\frac {8\,a\,{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^3}{{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^3}}+\frac {\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}}{8\,a\,\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )} \] Input:
int(x*((1/(a*x^2) - 1)^(1/2)*(1/(a*x^2) + 1)^(1/2) + 1/(a*x^2)),x)
Output:
log(x)/a - (2*atanh(((1/(a*x^2) - 1)^(1/2) - 1i)/((1/(a*x^2) + 1)^(1/2) - 1)))/a + ((5*((1/(a*x^2) - 1)^(1/2) - 1i)^2)/((1/(a*x^2) + 1)^(1/2) - 1)^2 + 1)/((8*a*((1/(a*x^2) - 1)^(1/2) - 1i))/((1/(a*x^2) + 1)^(1/2) - 1) + (8 *a*((1/(a*x^2) - 1)^(1/2) - 1i)^3)/((1/(a*x^2) + 1)^(1/2) - 1)^3) + ((1/(a *x^2) - 1)^(1/2) - 1i)/(8*a*((1/(a*x^2) + 1)^(1/2) - 1))
Time = 0.15 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.77 \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x \, dx=\frac {\sqrt {a \,x^{2}+1}\, \sqrt {-a \,x^{2}+1}+2 \,\mathrm {log}\left (\sqrt {-a \,x^{2}+1}-\sqrt {a \,x^{2}+1}\right )}{2 a} \] Input:
int((1/a/x^2+(-1+1/a/x^2)^(1/2)*(1+1/a/x^2)^(1/2))*x,x)
Output:
(sqrt(a*x**2 + 1)*sqrt( - a*x**2 + 1) + 2*log(sqrt( - a*x**2 + 1) - sqrt(a *x**2 + 1)))/(2*a)