3.84 \(\int \frac {e^{-\text {sech}^{-1}(a x)}}{x^4} \, dx\)

Optimal. Leaf size=200 \[ \frac {a^3}{4 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )}+\frac {a^3}{2 \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )}-\frac {a^3}{4 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^2}-\frac {a^3}{\left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^2}+\frac {a^3}{\left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^3}-\frac {a^3}{2 \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^4}-\frac {1}{4} a^3 \tanh ^{-1}\left (\sqrt {\frac {1-a x}{a x+1}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.50, antiderivative size = 200, 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} \[ \frac {a^3}{4 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )}+\frac {a^3}{2 \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )}-\frac {a^3}{4 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^2}-\frac {a^3}{\left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^2}+\frac {a^3}{\left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^3}-\frac {a^3}{2 \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^4}-\frac {1}{4} a^3 \tanh ^{-1}\left (\sqrt {\frac {1-a x}{a x+1}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(E^ArcSech[a*x]*x^4),x]

[Out]

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

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^{-\text {sech}^{-1}(a x)}}{x^4} \, dx &=\int \frac {1}{x^4 \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 )} \, dx\\ &=(4 a) \operatorname {Subst}\left (\int \frac {x \left (a+a x^2\right )^2}{(-1+x)^3 (1+x)^5} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )\\ &=(4 a) \operatorname {Subst}\left (\int \left (\frac {a^2}{8 (-1+x)^3}+\frac {a^2}{16 (-1+x)^2}+\frac {a^2}{2 (1+x)^5}-\frac {3 a^2}{4 (1+x)^4}+\frac {a^2}{2 (1+x)^3}-\frac {a^2}{8 (1+x)^2}+\frac {a^2}{16 \left (-1+x^2\right )}\right ) \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )\\ &=-\frac {a^3}{4 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^2}+\frac {a^3}{4 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )}-\frac {a^3}{2 \left (1+\sqrt {\frac {1-a x}{1+a x}}\right )^4}+\frac {a^3}{\left (1+\sqrt {\frac {1-a x}{1+a x}}\right )^3}-\frac {a^3}{\left (1+\sqrt {\frac {1-a x}{1+a x}}\right )^2}+\frac {a^3}{2 \left (1+\sqrt {\frac {1-a x}{1+a x}}\right )}+\frac {1}{4} a^3 \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )\\ &=-\frac {a^3}{4 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^2}+\frac {a^3}{4 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )}-\frac {a^3}{2 \left (1+\sqrt {\frac {1-a x}{1+a x}}\right )^4}+\frac {a^3}{\left (1+\sqrt {\frac {1-a x}{1+a x}}\right )^3}-\frac {a^3}{\left (1+\sqrt {\frac {1-a x}{1+a x}}\right )^2}+\frac {a^3}{2 \left (1+\sqrt {\frac {1-a x}{1+a x}}\right )}-\frac {1}{4} a^3 \tanh ^{-1}\left (\sqrt {\frac {1-a x}{1+a x}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 110, normalized size = 0.55 \[ -\frac {-a^4 x^4 \log (x)+a^4 x^4 \log \left (a x \sqrt {\frac {1-a x}{a x+1}}+\sqrt {\frac {1-a x}{a x+1}}+1\right )+\sqrt {\frac {1-a x}{a x+1}} \left (a^3 x^3+a^2 x^2-2 a x-2\right )+2}{8 a x^4} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(E^ArcSech[a*x]*x^4),x]

[Out]

-1/8*(2 + Sqrt[(1 - a*x)/(1 + a*x)]*(-2 - 2*a*x + a^2*x^2 + a^3*x^3) - a^4*x^4*Log[x] + a^4*x^4*Log[1 + Sqrt[(
1 - a*x)/(1 + a*x)] + a*x*Sqrt[(1 - a*x)/(1 + a*x)]])/(a*x^4)

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fricas [A]  time = 0.63, size = 138, normalized size = 0.69 \[ -\frac {a^{4} x^{4} \log \left (a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} + 1\right ) - a^{4} x^{4} \log \left (a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 1\right ) + 2 \, {\left (a^{3} x^{3} - 2 \, a x\right )} \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} + 4}{16 \, a x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2))/x^4,x, algorithm="fricas")

[Out]

-1/16*(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) + 2*(a^3*x^3 - 2*a*x)*sqrt((a*x + 1)/(a*x))*sqrt(-(a*x - 1)/(a*x)) + 4)/(a*x^4)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2))/x^4,x, algorithm="giac")

[Out]

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

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maple [F(-2)]  time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right ) x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2))/x^4,x)

[Out]

int(1/(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2))/x^4,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2))/x^4,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 43.71, size = 1511, normalized size = 7.56 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4*((1/(a*x) - 1)^(1/2)*(1/(a*x) + 1)^(1/2) + 1/(a*x))),x)

[Out]

((a^3*((1/(a*x) - 1)^(1/2) - 1i)^4*192i)/((1/(a*x) + 1)^(1/2) - 1)^4 + (a^3*((1/(a*x) - 1)^(1/2) - 1i)^6*128i)
/((1/(a*x) + 1)^(1/2) - 1)^6 + (a^3*((1/(a*x) - 1)^(1/2) - 1i)^8*192i)/((1/(a*x) + 1)^(1/2) - 1)^8)/(3*((15*((
1/(a*x) - 1)^(1/2) - 1i)^4)/((1/(a*x) + 1)^(1/2) - 1)^4 - (6*((1/(a*x) - 1)^(1/2) - 1i)^2)/((1/(a*x) + 1)^(1/2
) - 1)^2 - (20*((1/(a*x) - 1)^(1/2) - 1i)^6)/((1/(a*x) + 1)^(1/2) - 1)^6 + (15*((1/(a*x) - 1)^(1/2) - 1i)^8)/(
(1/(a*x) + 1)^(1/2) - 1)^8 - (6*((1/(a*x) - 1)^(1/2) - 1i)^10)/((1/(a*x) + 1)^(1/2) - 1)^10 + ((1/(a*x) - 1)^(
1/2) - 1i)^12/((1/(a*x) + 1)^(1/2) - 1)^12 + 1)) - ((a^3*((1/(a*x) - 1)^(1/2) - 1i)^4*64i)/((1/(a*x) + 1)^(1/2
) - 1)^4 + (a^3*((1/(a*x) - 1)^(1/2) - 1i)^6*128i)/(3*((1/(a*x) + 1)^(1/2) - 1)^6) + (a^3*((1/(a*x) - 1)^(1/2)
- 1i)^8*64i)/((1/(a*x) + 1)^(1/2) - 1)^8)/((15*((1/(a*x) - 1)^(1/2) - 1i)^4)/((1/(a*x) + 1)^(1/2) - 1)^4 - (6
*((1/(a*x) - 1)^(1/2) - 1i)^2)/((1/(a*x) + 1)^(1/2) - 1)^2 - (20*((1/(a*x) - 1)^(1/2) - 1i)^6)/((1/(a*x) + 1)^
(1/2) - 1)^6 + (15*((1/(a*x) - 1)^(1/2) - 1i)^8)/((1/(a*x) + 1)^(1/2) - 1)^8 - (6*((1/(a*x) - 1)^(1/2) - 1i)^1
0)/((1/(a*x) + 1)^(1/2) - 1)^10 + ((1/(a*x) - 1)^(1/2) - 1i)^12/((1/(a*x) + 1)^(1/2) - 1)^12 + 1) - (a^3*atanh
(((1/(a*x) - 1)^(1/2) - 1i)/((1/(a*x) + 1)^(1/2) - 1)))/2 + ((14*a^3*((1/(a*x) - 1)^(1/2) - 1i)^3)/((1/(a*x) +
1)^(1/2) - 1)^3 + (14*a^3*((1/(a*x) - 1)^(1/2) - 1i)^5)/((1/(a*x) + 1)^(1/2) - 1)^5 + (2*a^3*((1/(a*x) - 1)^(
1/2) - 1i)^7)/((1/(a*x) + 1)^(1/2) - 1)^7 + (2*a^3*((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^3*((1/(a*x) - 1)^(1/2) - 1i)^3)/(2*((1/(a*x) + 1)^(1/2) - 1)^3) + (333*a^
3*((1/(a*x) - 1)^(1/2) - 1i)^5)/(2*((1/(a*x) + 1)^(1/2) - 1)^5) + (671*a^3*((1/(a*x) - 1)^(1/2) - 1i)^7)/(2*((
1/(a*x) + 1)^(1/2) - 1)^7) + (671*a^3*((1/(a*x) - 1)^(1/2) - 1i)^9)/(2*((1/(a*x) + 1)^(1/2) - 1)^9) + (333*a^3
*((1/(a*x) - 1)^(1/2) - 1i)^11)/(2*((1/(a*x) + 1)^(1/2) - 1)^11) + (23*a^3*((1/(a*x) - 1)^(1/2) - 1i)^13)/(2*(
(1/(a*x) + 1)^(1/2) - 1)^13) - (3*a^3*((1/(a*x) - 1)^(1/2) - 1i)^15)/(2*((1/(a*x) + 1)^(1/2) - 1)^15) - (3*a^3
*((1/(a*x) - 1)^(1/2) - 1i))/(2*((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/(4*a*x^4)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1/a/x+(1/a/x-1)**(1/2)*(1+1/a/x)**(1/2))/x**4,x)

[Out]

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

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