3.87 \(\int \frac {e^{-\text {sech}^{-1}(a x)}}{x^7} \, dx\)

Optimal. Leaf size=353 \[ -\frac {5 a^6}{16 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )}-\frac {5 a^6}{16 \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )}+\frac {3 a^6}{8 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^2}+\frac {a^6}{\left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^2}-\frac {5 a^6}{12 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^3}-\frac {11 a^6}{6 \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^3}+\frac {a^6}{4 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^4}+\frac {9 a^6}{4 \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^4}-\frac {a^6}{10 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^5}-\frac {19 a^6}{10 \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^5}+\frac {a^6}{\left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^6}-\frac {2 a^6}{7 \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^7} \]

[Out]

-1/10*a^6/(1-((-a*x+1)/(a*x+1))^(1/2))^5+1/4*a^6/(1-((-a*x+1)/(a*x+1))^(1/2))^4-5/12*a^6/(1-((-a*x+1)/(a*x+1))
^(1/2))^3+3/8*a^6/(1-((-a*x+1)/(a*x+1))^(1/2))^2-5/16*a^6/(1-((-a*x+1)/(a*x+1))^(1/2))-2/7*a^6/(1+((-a*x+1)/(a
*x+1))^(1/2))^7+a^6/(1+((-a*x+1)/(a*x+1))^(1/2))^6-19/10*a^6/(1+((-a*x+1)/(a*x+1))^(1/2))^5+9/4*a^6/(1+((-a*x+
1)/(a*x+1))^(1/2))^4-11/6*a^6/(1+((-a*x+1)/(a*x+1))^(1/2))^3+a^6/(1+((-a*x+1)/(a*x+1))^(1/2))^2-5/16*a^6/(1+((
-a*x+1)/(a*x+1))^(1/2))

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Rubi [A]  time = 0.60, antiderivative size = 353, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6337, 1612} \[ -\frac {5 a^6}{16 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )}-\frac {5 a^6}{16 \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )}+\frac {3 a^6}{8 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^2}+\frac {a^6}{\left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^2}-\frac {5 a^6}{12 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^3}-\frac {11 a^6}{6 \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^3}+\frac {a^6}{4 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^4}+\frac {9 a^6}{4 \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^4}-\frac {a^6}{10 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^5}-\frac {19 a^6}{10 \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^5}+\frac {a^6}{\left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^6}-\frac {2 a^6}{7 \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^7} \]

Antiderivative was successfully verified.

[In]

Int[1/(E^ArcSech[a*x]*x^7),x]

[Out]

-a^6/(10*(1 - Sqrt[(1 - a*x)/(1 + a*x)])^5) + a^6/(4*(1 - Sqrt[(1 - a*x)/(1 + a*x)])^4) - (5*a^6)/(12*(1 - Sqr
t[(1 - a*x)/(1 + a*x)])^3) + (3*a^6)/(8*(1 - Sqrt[(1 - a*x)/(1 + a*x)])^2) - (5*a^6)/(16*(1 - Sqrt[(1 - a*x)/(
1 + a*x)])) - (2*a^6)/(7*(1 + Sqrt[(1 - a*x)/(1 + a*x)])^7) + a^6/(1 + Sqrt[(1 - a*x)/(1 + a*x)])^6 - (19*a^6)
/(10*(1 + Sqrt[(1 - a*x)/(1 + a*x)])^5) + (9*a^6)/(4*(1 + Sqrt[(1 - a*x)/(1 + a*x)])^4) - (11*a^6)/(6*(1 + Sqr
t[(1 - a*x)/(1 + a*x)])^3) + a^6/(1 + Sqrt[(1 - a*x)/(1 + a*x)])^2 - (5*a^6)/(16*(1 + Sqrt[(1 - a*x)/(1 + a*x)
]))

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^7} \, dx &=\int \frac {1}{x^7 \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\\ &=-\left ((4 a) \operatorname {Subst}\left (\int \frac {x \left (a+a x^2\right )^5}{(-1+x)^6 (1+x)^8} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )\right )\\ &=-\left ((4 a) \operatorname {Subst}\left (\int \left (\frac {a^5}{8 (-1+x)^6}+\frac {a^5}{4 (-1+x)^5}+\frac {5 a^5}{16 (-1+x)^4}+\frac {3 a^5}{16 (-1+x)^3}+\frac {5 a^5}{64 (-1+x)^2}-\frac {a^5}{2 (1+x)^8}+\frac {3 a^5}{2 (1+x)^7}-\frac {19 a^5}{8 (1+x)^6}+\frac {9 a^5}{4 (1+x)^5}-\frac {11 a^5}{8 (1+x)^4}+\frac {a^5}{2 (1+x)^3}-\frac {5 a^5}{64 (1+x)^2}\right ) \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )\right )\\ &=-\frac {a^6}{10 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^5}+\frac {a^6}{4 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^4}-\frac {5 a^6}{12 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^3}+\frac {3 a^6}{8 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^2}-\frac {5 a^6}{16 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )}-\frac {2 a^6}{7 \left (1+\sqrt {\frac {1-a x}{1+a x}}\right )^7}+\frac {a^6}{\left (1+\sqrt {\frac {1-a x}{1+a x}}\right )^6}-\frac {19 a^6}{10 \left (1+\sqrt {\frac {1-a x}{1+a x}}\right )^5}+\frac {9 a^6}{4 \left (1+\sqrt {\frac {1-a x}{1+a x}}\right )^4}-\frac {11 a^6}{6 \left (1+\sqrt {\frac {1-a x}{1+a x}}\right )^3}+\frac {a^6}{\left (1+\sqrt {\frac {1-a x}{1+a x}}\right )^2}-\frac {5 a^6}{16 \left (1+\sqrt {\frac {1-a x}{1+a x}}\right )}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 76, normalized size = 0.22 \[ -\frac {\sqrt {\frac {1-a x}{a x+1}} \left (8 a^5 x^5-8 a^4 x^4+12 a^3 x^3-12 a^2 x^2+15 a x-15\right ) (a x+1)^2+15}{105 a x^7} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(E^ArcSech[a*x]*x^7),x]

[Out]

-1/105*(15 + Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)^2*(-15 + 15*a*x - 12*a^2*x^2 + 12*a^3*x^3 - 8*a^4*x^4 + 8*a^5
*x^5))/(a*x^7)

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fricas [A]  time = 0.81, size = 69, normalized size = 0.20 \[ -\frac {{\left (8 \, a^{7} x^{7} + 4 \, a^{5} x^{5} + 3 \, a^{3} x^{3} - 15 \, a x\right )} \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} + 15}{105 \, a x^{7}} \]

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^7,x, algorithm="fricas")

[Out]

-1/105*((8*a^7*x^7 + 4*a^5*x^5 + 3*a^3*x^3 - 15*a*x)*sqrt((a*x + 1)/(a*x))*sqrt(-(a*x - 1)/(a*x)) + 15)/(a*x^7
)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{7} {\left (\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}\right )}}\,{d x} \]

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^7,x, algorithm="giac")

[Out]

integrate(1/(x^7*(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 \[ \int \frac {1}{\left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right ) x^{7}}\, dx \]

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^7,x)

[Out]

int(1/(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2))/x^7,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{7} {\left (\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}\right )}}\,{d x} \]

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^7,x, algorithm="maxima")

[Out]

integrate(1/(x^7*(sqrt(1/(a*x) + 1)*sqrt(1/(a*x) - 1) + 1/(a*x))), x)

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mupad [B]  time = 2.59, size = 91, normalized size = 0.26 \[ -\frac {1}{7\,a\,x^7}-\frac {\sqrt {\frac {1}{a\,x}-1}\,\left (\frac {a\,x^2}{35}-\frac {x}{7}-\frac {1}{7\,a}+\frac {a^2\,x^3}{35}+\frac {4\,a^3\,x^4}{105}+\frac {4\,a^4\,x^5}{105}+\frac {8\,a^5\,x^6}{105}+\frac {8\,a^6\,x^7}{105}\right )}{x^7\,\sqrt {\frac {1}{a\,x}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^7*((1/(a*x) - 1)^(1/2)*(1/(a*x) + 1)^(1/2) + 1/(a*x))),x)

[Out]

- 1/(7*a*x^7) - ((1/(a*x) - 1)^(1/2)*((a*x^2)/35 - x/7 - 1/(7*a) + (a^2*x^3)/35 + (4*a^3*x^4)/105 + (4*a^4*x^5
)/105 + (8*a^5*x^6)/105 + (8*a^6*x^7)/105))/(x^7*(1/(a*x) + 1)^(1/2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ a \int \frac {1}{a x^{7} \sqrt {-1 + \frac {1}{a x}} \sqrt {1 + \frac {1}{a x}} + x^{6}}\, dx \]

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**7,x)

[Out]

a*Integral(1/(a*x**7*sqrt(-1 + 1/(a*x))*sqrt(1 + 1/(a*x)) + x**6), x)

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