3.53 \(\int \frac {e^{\text {sech}^{-1}(a x^2)}}{x} \, dx\)

Optimal. Leaf size=80 \[ -\frac {\sqrt {1-a x^2}}{2 a x^2 \sqrt {\frac {1}{a x^2+1}}}-\frac {1}{2 a x^2}-\frac {1}{2} \sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \sin ^{-1}\left (a x^2\right ) \]

[Out]

-1/2/a/x^2-1/2*(-a*x^2+1)^(1/2)/a/x^2/(1/(a*x^2+1))^(1/2)-1/2*arcsin(a*x^2)*(1/(a*x^2+1))^(1/2)*(a*x^2+1)^(1/2
)

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Rubi [A]  time = 0.05, antiderivative size = 93, normalized size of antiderivative = 1.16, number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6334, 259, 275, 277, 216} \[ -\frac {\sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \sqrt {1-a^2 x^4}}{2 a x^2}-\frac {1}{2 a x^2}-\frac {1}{2} \sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \sin ^{-1}\left (a x^2\right ) \]

Warning: Unable to verify antiderivative.

[In]

Int[E^ArcSech[a*x^2]/x,x]

[Out]

-1/(2*a*x^2) - (Sqrt[(1 + a*x^2)^(-1)]*Sqrt[1 + a*x^2]*Sqrt[1 - a^2*x^4])/(2*a*x^2) - (Sqrt[(1 + a*x^2)^(-1)]*
Sqrt[1 + a*x^2]*ArcSin[a*x^2])/2

Rule 216

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]

Rule 259

Int[((c_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(n_))^(p_)*((a2_) + (b2_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[(c*x)
^m*(a1*a2 + b1*b2*x^(2*n))^p, x] /; FreeQ[{a1, b1, a2, b2, c, m, n, p}, x] && EqQ[a2*b1 + a1*b2, 0] && (Intege
rQ[p] || (GtQ[a1, 0] && GtQ[a2, 0]))

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[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]

Rule 277

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
1)), x] - Dist[(b*n*p)/(c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 6334

Int[E^ArcSech[(a_.)*(x_)^(p_.)]/(x_), x_Symbol] :> -Simp[(a*p*x^p)^(-1), x] + Dist[(Sqrt[1 + a*x^p]*Sqrt[1/(1
+ a*x^p)])/a, Int[(Sqrt[1 + a*x^p]*Sqrt[1 - a*x^p])/x^(p + 1), x], x] /; FreeQ[{a, p}, x]

Rubi steps

\begin {align*} \int \frac {e^{\text {sech}^{-1}\left (a x^2\right )}}{x} \, dx &=-\frac {1}{2 a x^2}+\frac {\left (\sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \int \frac {\sqrt {1-a x^2} \sqrt {1+a x^2}}{x^3} \, dx}{a}\\ &=-\frac {1}{2 a x^2}+\frac {\left (\sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \int \frac {\sqrt {1-a^2 x^4}}{x^3} \, dx}{a}\\ &=-\frac {1}{2 a x^2}+\frac {\left (\sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1-a^2 x^2}}{x^2} \, dx,x,x^2\right )}{2 a}\\ &=-\frac {1}{2 a x^2}-\frac {\sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} \sqrt {1-a^2 x^4}}{2 a x^2}-\frac {1}{2} \left (a \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-a^2 x^2}} \, dx,x,x^2\right )\\ &=-\frac {1}{2 a x^2}-\frac {\sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} \sqrt {1-a^2 x^4}}{2 a x^2}-\frac {1}{2} \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} \sin ^{-1}\left (a x^2\right )\\ \end {align*}

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Mathematica [A]  time = 0.15, size = 113, normalized size = 1.41 \[ \frac {\sqrt {\frac {1-a x^2}{a x^2+1}} \left (a x^2+1\right ) \tanh ^{-1}\left (\frac {a x^2}{\sqrt {a^2 x^4-1}}\right )}{2 \sqrt {a^2 x^4-1}}+\sqrt {\frac {1-a x^2}{a x^2+1}} \left (-\frac {1}{2 a x^2}-\frac {1}{2}\right )-\frac {1}{2 a x^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^ArcSech[a*x^2]/x,x]

[Out]

-1/2*1/(a*x^2) + (-1/2 - 1/(2*a*x^2))*Sqrt[(1 - a*x^2)/(1 + a*x^2)] + (Sqrt[(1 - a*x^2)/(1 + a*x^2)]*(1 + a*x^
2)*ArcTanh[(a*x^2)/Sqrt[-1 + a^2*x^4]])/(2*Sqrt[-1 + a^2*x^4])

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fricas [B]  time = 0.97, size = 102, normalized size = 1.28 \[ -\frac {a x^{2} \sqrt {\frac {a x^{2} + 1}{a x^{2}}} \sqrt {-\frac {a x^{2} - 1}{a x^{2}}} - 2 \, a x^{2} \arctan \left (\frac {a x^{2} \sqrt {\frac {a x^{2} + 1}{a x^{2}}} \sqrt {-\frac {a x^{2} - 1}{a x^{2}}} - 1}{a x^{2}}\right ) + 1}{2 \, a x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/a/x^2+(1/a/x^2-1)^(1/2)*(1/a/x^2+1)^(1/2))/x,x, algorithm="fricas")

[Out]

-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)

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giac [B]  time = 3.32, size = 252, normalized size = 3.15 \[ -\frac {{\left (\pi + 2 \, \arctan \left (\frac {\sqrt {a^{2} x^{2} + a} {\left (\frac {{\left (\sqrt {2} \sqrt {a} - \sqrt {-a^{2} x^{2} + a}\right )}^{2}}{a^{2} x^{2} + a} - 1\right )}}{2 \, {\left (\sqrt {2} \sqrt {a} - \sqrt {-a^{2} x^{2} + a}\right )}}\right )\right )} a^{3} + \frac {4 \, a^{3} {\left (\frac {\sqrt {2} \sqrt {a} - \sqrt {-a^{2} x^{2} + a}}{\sqrt {a^{2} x^{2} + a}} - \frac {\sqrt {a^{2} x^{2} + a}}{\sqrt {2} \sqrt {a} - \sqrt {-a^{2} x^{2} + a}}\right )}}{{\left (\frac {\sqrt {2} \sqrt {a} - \sqrt {-a^{2} x^{2} + a}}{\sqrt {a^{2} x^{2} + a}} - \frac {\sqrt {a^{2} x^{2} + a}}{\sqrt {2} \sqrt {a} - \sqrt {-a^{2} x^{2} + a}}\right )}^{2} - 4} + \frac {a^{2}}{x^{2}}}{2 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/a/x^2+(1/a/x^2-1)^(1/2)*(1/a/x^2+1)^(1/2))/x,x, algorithm="giac")

[Out]

-1/2*((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^3 + 4*a^3*((sqrt(2)*sqrt(a) - sqrt(-a^2*x^2 + a))/sqrt(a^2*x^2 + a) - sqr
t(a^2*x^2 + a)/(sqrt(2)*sqrt(a) - sqrt(-a^2*x^2 + a)))/(((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)))^2 - 4) + a^2/x^2)/a^3

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maple [A]  time = 0.16, size = 103, normalized size = 1.29 \[ -\frac {\sqrt {-\frac {a \,x^{2}-1}{a \,x^{2}}}\, \sqrt {\frac {a \,x^{2}+1}{a \,x^{2}}}\, \left (\arctan \left (\frac {x^{2}}{\sqrt {-\frac {a^{2} x^{4}-1}{a^{2}}}}\right ) x^{2}+\sqrt {-\frac {a^{2} x^{4}-1}{a^{2}}}\right )}{2 \sqrt {-\frac {a^{2} x^{4}-1}{a^{2}}}}-\frac {1}{2 a \,x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1/a/x^2+(1/a/x^2-1)^(1/2)*(1/a/x^2+1)^(1/2))/x,x)

[Out]

-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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {-\frac {1}{2} \, a \arcsin \left (a x^{2}\right ) - \frac {\sqrt {-a^{2} x^{4} + 1}}{2 \, x^{2}}}{a} - \frac {1}{2 \, a x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/a/x^2+(1/a/x^2-1)^(1/2)*(1/a/x^2+1)^(1/2))/x,x, algorithm="maxima")

[Out]

integrate(sqrt(a*x^2 + 1)*sqrt(-a*x^2 + 1)/x^3, x)/a - 1/2/(a*x^2)

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mupad [B]  time = 4.00, size = 185, normalized size = 2.31 \[ -\frac {\ln \left (\frac {{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^2}+1\right )\,1{}\mathrm {i}}{2}+\frac {\ln \left (\frac {\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}}{\sqrt {\frac {1}{a\,x^2}+1}-1}\right )\,1{}\mathrm {i}}{2}-\frac {1}{2\,a\,x^2}+\frac {{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^2\,8{}\mathrm {i}}{{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^2\,\left (2+\frac {2\,{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^4}-\frac {4\,{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((1/(a*x^2) - 1)^(1/2)*(1/(a*x^2) + 1)^(1/2) + 1/(a*x^2))/x,x)

[Out]

(log(((1/(a*x^2) - 1)^(1/2) - 1i)/((1/(a*x^2) + 1)^(1/2) - 1))*1i)/2 - (log(((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))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {1}{x^{3}}\, dx + \int \frac {a \sqrt {-1 + \frac {1}{a x^{2}}} \sqrt {1 + \frac {1}{a x^{2}}}}{x}\, dx}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/a/x**2+(1/a/x**2-1)**(1/2)*(1/a/x**2+1)**(1/2))/x,x)

[Out]

(Integral(x**(-3), x) + Integral(a*sqrt(-1 + 1/(a*x**2))*sqrt(1 + 1/(a*x**2))/x, x))/a

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