3.72 \(\int \frac {e^{2 \text {sech}^{-1}(a x)}}{x^3} \, dx\)

Optimal. Leaf size=147 \[ \frac {a^2}{2 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )}-\frac {3 a^2}{2 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^2}+\frac {2 a^2}{\left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^3}-\frac {a^2}{\left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^4}+\frac {1}{2} a^2 \tanh ^{-1}\left (\sqrt {\frac {1-a x}{a x+1}}\right ) \]

[Out]

1/2*a^2*arctanh(((-a*x+1)/(a*x+1))^(1/2))-a^2/(1-((-a*x+1)/(a*x+1))^(1/2))^4+2*a^2/(1-((-a*x+1)/(a*x+1))^(1/2)
)^3-3/2*a^2/(1-((-a*x+1)/(a*x+1))^(1/2))^2+1/2*a^2/(1-((-a*x+1)/(a*x+1))^(1/2))

________________________________________________________________________________________

Rubi [A]  time = 0.45, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6337, 1612, 207} \[ \frac {a^2}{2 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )}-\frac {3 a^2}{2 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^2}+\frac {2 a^2}{\left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^3}-\frac {a^2}{\left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^4}+\frac {1}{2} a^2 \tanh ^{-1}\left (\sqrt {\frac {1-a x}{a x+1}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[E^(2*ArcSech[a*x])/x^3,x]

[Out]

-(a^2/(1 - Sqrt[(1 - a*x)/(1 + a*x)])^4) + (2*a^2)/(1 - Sqrt[(1 - a*x)/(1 + a*x)])^3 - (3*a^2)/(2*(1 - Sqrt[(1
- a*x)/(1 + a*x)])^2) + a^2/(2*(1 - Sqrt[(1 - a*x)/(1 + a*x)])) + (a^2*ArcTanh[Sqrt[(1 - a*x)/(1 + a*x)]])/2

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 1612

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[E
xpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && Poly
Q[Px, x] && IntegersQ[m, n]

Rule 6337

Int[E^(ArcSech[u_]*(n_.))*(x_)^(m_.), x_Symbol] :> Int[x^m*(1/u + Sqrt[(1 - u)/(1 + u)] + (1*Sqrt[(1 - u)/(1 +
u)])/u)^n, x] /; FreeQ[m, x] && IntegerQ[n]

Rubi steps

\begin {align*} \int \frac {e^{2 \text {sech}^{-1}(a x)}}{x^3} \, dx &=\int \frac {\left (\frac {1}{a x}+\sqrt {\frac {1-a x}{1+a x}}+\frac {\sqrt {\frac {1-a x}{1+a x}}}{a x}\right )^2}{x^3} \, dx\\ &=\left (4 a^2\right ) \operatorname {Subst}\left (\int \frac {x \left (1+x^2\right )}{(-1+x)^5 (1+x)} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )\\ &=\left (4 a^2\right ) \operatorname {Subst}\left (\int \left (\frac {1}{(-1+x)^5}+\frac {3}{2 (-1+x)^4}+\frac {3}{4 (-1+x)^3}+\frac {1}{8 (-1+x)^2}-\frac {1}{8 \left (-1+x^2\right )}\right ) \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )\\ &=-\frac {a^2}{\left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^4}+\frac {2 a^2}{\left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^3}-\frac {3 a^2}{2 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^2}+\frac {a^2}{2 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )}-\frac {1}{2} a^2 \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )\\ &=-\frac {a^2}{\left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^4}+\frac {2 a^2}{\left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^3}-\frac {3 a^2}{2 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^2}+\frac {a^2}{2 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )}+\frac {1}{2} a^2 \tanh ^{-1}\left (\sqrt {\frac {1-a x}{1+a x}}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.13, size = 121, normalized size = 0.82 \[ \frac {a^4 (-\log (x))+a^4 \log \left (a x \sqrt {\frac {1-a x}{a x+1}}+\sqrt {\frac {1-a x}{a x+1}}+1\right )+\frac {(a x+1) \left (a^2 x^2 \sqrt {\frac {1-a x}{a x+1}}+2 a x-2 \sqrt {\frac {1-a x}{a x+1}}-2\right )}{x^4}}{4 a^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(2*ArcSech[a*x])/x^3,x]

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 1.02, size = 146, normalized size = 0.99 \[ \frac {a^{4} x^{4} \log \left (a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} + 1\right ) - a^{4} x^{4} \log \left (a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 1\right ) + 4 \, a^{2} x^{2} + 2 \, {\left (a^{3} x^{3} - 2 \, a x\right )} \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 4}{8 \, a^{2} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2))^2/x^3,x, algorithm="fricas")

[Out]

1/8*(a^4*x^4*log(a*x*sqrt((a*x + 1)/(a*x))*sqrt(-(a*x - 1)/(a*x)) + 1) - a^4*x^4*log(a*x*sqrt((a*x + 1)/(a*x))
*sqrt(-(a*x - 1)/(a*x)) - 1) + 4*a^2*x^2 + 2*(a^3*x^3 - 2*a*x)*sqrt((a*x + 1)/(a*x))*sqrt(-(a*x - 1)/(a*x)) -
4)/(a^2*x^4)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}\right )}^{2}}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2))^2/x^3,x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.06, size = 131, normalized size = 0.89 \[ \frac {\frac {a^{2}}{2 x^{2}}-\frac {1}{4 x^{4}}}{a^{2}}+\frac {\sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}\, \left (\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right ) x^{4} a^{4}+a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}-2 \sqrt {-a^{2} x^{2}+1}\right )}{4 a \,x^{3} \sqrt {-a^{2} x^{2}+1}}-\frac {1}{4 a^{2} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a^2*(1/2*a^2/x^2-1/4/x^4)+1/4/a*(-(a*x-1)/a/x)^(1/2)/x^3*((a*x+1)/a/x)^(1/2)*(arctanh(1/(-a^2*x^2+1)^(1/2))*
x^4*a^4+a^2*x^2*(-a^2*x^2+1)^(1/2)-2*(-a^2*x^2+1)^(1/2))/(-a^2*x^2+1)^(1/2)-1/4/a^2/x^4

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {2 \, {\left (\frac {1}{8} \, a^{4} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - \frac {1}{8} \, \sqrt {-a^{2} x^{2} + 1} a^{4} - \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{2}}{8 \, x^{2}} - \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{4 \, x^{4}}\right )}}{a^{2}} - \frac {1}{2 \, a^{2} x^{4}} - \int \frac {1}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2))^2/x^3,x, algorithm="maxima")

[Out]

2*integrate(sqrt(a*x + 1)*sqrt(-a*x + 1)/x^5, x)/a^2 - 1/2/(a^2*x^4) - integrate(x^(-3), x)

________________________________________________________________________________________

mupad [B]  time = 46.99, size = 885, normalized size = 6.02 \[ a^2\,\mathrm {atanh}\left (\frac {\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}}{\sqrt {\frac {1}{a\,x}+1}-1}\right )-\frac {\frac {28\,a^2\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^3}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^3}+\frac {28\,a^2\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^5}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^5}+\frac {4\,a^2\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^7}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^7}+\frac {4\,a^2\,\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}{\sqrt {\frac {1}{a\,x}+1}-1}}{1+\frac {6\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^4}-\frac {4\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^6}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^6}+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^8}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^8}-\frac {4\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2}}-\frac {\frac {23\,a^2\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^3}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^3}+\frac {333\,a^2\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^5}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^5}+\frac {671\,a^2\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^7}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^7}+\frac {671\,a^2\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^9}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^9}+\frac {333\,a^2\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{11}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{11}}+\frac {23\,a^2\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{13}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{13}}-\frac {3\,a^2\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{15}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{15}}-\frac {3\,a^2\,\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}{\sqrt {\frac {1}{a\,x}+1}-1}}{1+\frac {28\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^4}-\frac {56\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^6}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^6}+\frac {70\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^8}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^8}-\frac {56\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{10}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{10}}+\frac {28\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{12}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{12}}-\frac {8\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{14}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{14}}+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{16}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{16}}-\frac {8\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2}}+\frac {1}{2\,x^2}-\frac {1}{2\,a^2\,x^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^2*atanh(((1/(a*x) - 1)^(1/2) - 1i)/((1/(a*x) + 1)^(1/2) - 1)) - ((28*a^2*((1/(a*x) - 1)^(1/2) - 1i)^3)/((1/(
a*x) + 1)^(1/2) - 1)^3 + (28*a^2*((1/(a*x) - 1)^(1/2) - 1i)^5)/((1/(a*x) + 1)^(1/2) - 1)^5 + (4*a^2*((1/(a*x)
- 1)^(1/2) - 1i)^7)/((1/(a*x) + 1)^(1/2) - 1)^7 + (4*a^2*((1/(a*x) - 1)^(1/2) - 1i))/((1/(a*x) + 1)^(1/2) - 1)
)/((6*((1/(a*x) - 1)^(1/2) - 1i)^4)/((1/(a*x) + 1)^(1/2) - 1)^4 - (4*((1/(a*x) - 1)^(1/2) - 1i)^2)/((1/(a*x) +
1)^(1/2) - 1)^2 - (4*((1/(a*x) - 1)^(1/2) - 1i)^6)/((1/(a*x) + 1)^(1/2) - 1)^6 + ((1/(a*x) - 1)^(1/2) - 1i)^8
/((1/(a*x) + 1)^(1/2) - 1)^8 + 1) - ((23*a^2*((1/(a*x) - 1)^(1/2) - 1i)^3)/((1/(a*x) + 1)^(1/2) - 1)^3 + (333*
a^2*((1/(a*x) - 1)^(1/2) - 1i)^5)/((1/(a*x) + 1)^(1/2) - 1)^5 + (671*a^2*((1/(a*x) - 1)^(1/2) - 1i)^7)/((1/(a*
x) + 1)^(1/2) - 1)^7 + (671*a^2*((1/(a*x) - 1)^(1/2) - 1i)^9)/((1/(a*x) + 1)^(1/2) - 1)^9 + (333*a^2*((1/(a*x)
- 1)^(1/2) - 1i)^11)/((1/(a*x) + 1)^(1/2) - 1)^11 + (23*a^2*((1/(a*x) - 1)^(1/2) - 1i)^13)/((1/(a*x) + 1)^(1/
2) - 1)^13 - (3*a^2*((1/(a*x) - 1)^(1/2) - 1i)^15)/((1/(a*x) + 1)^(1/2) - 1)^15 - (3*a^2*((1/(a*x) - 1)^(1/2)
- 1i))/((1/(a*x) + 1)^(1/2) - 1))/((28*((1/(a*x) - 1)^(1/2) - 1i)^4)/((1/(a*x) + 1)^(1/2) - 1)^4 - (8*((1/(a*x
) - 1)^(1/2) - 1i)^2)/((1/(a*x) + 1)^(1/2) - 1)^2 - (56*((1/(a*x) - 1)^(1/2) - 1i)^6)/((1/(a*x) + 1)^(1/2) - 1
)^6 + (70*((1/(a*x) - 1)^(1/2) - 1i)^8)/((1/(a*x) + 1)^(1/2) - 1)^8 - (56*((1/(a*x) - 1)^(1/2) - 1i)^10)/((1/(
a*x) + 1)^(1/2) - 1)^10 + (28*((1/(a*x) - 1)^(1/2) - 1i)^12)/((1/(a*x) + 1)^(1/2) - 1)^12 - (8*((1/(a*x) - 1)^
(1/2) - 1i)^14)/((1/(a*x) + 1)^(1/2) - 1)^14 + ((1/(a*x) - 1)^(1/2) - 1i)^16/((1/(a*x) + 1)^(1/2) - 1)^16 + 1)
+ 1/(2*x^2) - 1/(2*a^2*x^4)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {2}{x^{5}}\, dx + \int \left (- \frac {a^{2}}{x^{3}}\right )\, dx + \int \frac {2 a \sqrt {-1 + \frac {1}{a x}} \sqrt {1 + \frac {1}{a x}}}{x^{4}}\, dx}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(Integral(2/x**5, x) + Integral(-a**2/x**3, x) + Integral(2*a*sqrt(-1 + 1/(a*x))*sqrt(1 + 1/(a*x))/x**4, x))/a
**2

________________________________________________________________________________________