3.75 \(\int \frac {e^{2 \text {sech}^{-1}(a x)}}{x^6} \, dx\)

Optimal. Leaf size=301 \[ -\frac {a^5}{4 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )}-\frac {a^5}{4 \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )}+\frac {11 a^5}{8 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^2}+\frac {a^5}{8 \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^2}-\frac {35 a^5}{12 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^3}-\frac {a^5}{12 \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^3}+\frac {4 a^5}{\left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^4}-\frac {18 a^5}{5 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^5}+\frac {2 a^5}{\left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^6}-\frac {4 a^5}{7 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^7} \]

[Out]

-4/7*a^5/(1-((-a*x+1)/(a*x+1))^(1/2))^7+2*a^5/(1-((-a*x+1)/(a*x+1))^(1/2))^6-18/5*a^5/(1-((-a*x+1)/(a*x+1))^(1
/2))^5+4*a^5/(1-((-a*x+1)/(a*x+1))^(1/2))^4-35/12*a^5/(1-((-a*x+1)/(a*x+1))^(1/2))^3+11/8*a^5/(1-((-a*x+1)/(a*
x+1))^(1/2))^2-1/4*a^5/(1-((-a*x+1)/(a*x+1))^(1/2))-1/12*a^5/(1+((-a*x+1)/(a*x+1))^(1/2))^3+1/8*a^5/(1+((-a*x+
1)/(a*x+1))^(1/2))^2-1/4*a^5/(1+((-a*x+1)/(a*x+1))^(1/2))

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Rubi [A]  time = 0.58, antiderivative size = 301, 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} \[ -\frac {a^5}{4 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )}-\frac {a^5}{4 \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )}+\frac {11 a^5}{8 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^2}+\frac {a^5}{8 \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^2}-\frac {35 a^5}{12 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^3}-\frac {a^5}{12 \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^3}+\frac {4 a^5}{\left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^4}-\frac {18 a^5}{5 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^5}+\frac {2 a^5}{\left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^6}-\frac {4 a^5}{7 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^7} \]

Antiderivative was successfully verified.

[In]

Int[E^(2*ArcSech[a*x])/x^6,x]

[Out]

(-4*a^5)/(7*(1 - Sqrt[(1 - a*x)/(1 + a*x)])^7) + (2*a^5)/(1 - Sqrt[(1 - a*x)/(1 + a*x)])^6 - (18*a^5)/(5*(1 -
Sqrt[(1 - a*x)/(1 + a*x)])^5) + (4*a^5)/(1 - Sqrt[(1 - a*x)/(1 + a*x)])^4 - (35*a^5)/(12*(1 - Sqrt[(1 - a*x)/(
1 + a*x)])^3) + (11*a^5)/(8*(1 - Sqrt[(1 - a*x)/(1 + a*x)])^2) - a^5/(4*(1 - Sqrt[(1 - a*x)/(1 + a*x)])) - a^5
/(12*(1 + Sqrt[(1 - a*x)/(1 + a*x)])^3) + a^5/(8*(1 + Sqrt[(1 - a*x)/(1 + a*x)])^2) - a^5/(4*(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^{2 \text {sech}^{-1}(a x)}}{x^6} \, 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^6} \, dx\\ &=-\left ((4 a) \operatorname {Subst}\left (\int \frac {x \left (a+a x^2\right )^4}{(-1+x)^8 (1+x)^4} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )\right )\\ &=-\left ((4 a) \operatorname {Subst}\left (\int \left (\frac {a^4}{(-1+x)^8}+\frac {3 a^4}{(-1+x)^7}+\frac {9 a^4}{2 (-1+x)^6}+\frac {4 a^4}{(-1+x)^5}+\frac {35 a^4}{16 (-1+x)^4}+\frac {11 a^4}{16 (-1+x)^3}+\frac {a^4}{16 (-1+x)^2}-\frac {a^4}{16 (1+x)^4}+\frac {a^4}{16 (1+x)^3}-\frac {a^4}{16 (1+x)^2}\right ) \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )\right )\\ &=-\frac {4 a^5}{7 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^7}+\frac {2 a^5}{\left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^6}-\frac {18 a^5}{5 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^5}+\frac {4 a^5}{\left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^4}-\frac {35 a^5}{12 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^3}+\frac {11 a^5}{8 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^2}-\frac {a^5}{4 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )}-\frac {a^5}{12 \left (1+\sqrt {\frac {1-a x}{1+a x}}\right )^3}+\frac {a^5}{8 \left (1+\sqrt {\frac {1-a x}{1+a x}}\right )^2}-\frac {a^5}{4 \left (1+\sqrt {\frac {1-a x}{1+a x}}\right )}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 85, normalized size = 0.28 \[ \frac {21 a^2 x^2+2 \sqrt {\frac {1-a x}{a x+1}} (a x+1)^2 \left (8 a^5 x^5-8 a^4 x^4+12 a^3 x^3-12 a^2 x^2+15 a x-15\right )-30}{105 a^2 x^7} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(2*ArcSech[a*x])/x^6,x]

[Out]

(-30 + 21*a^2*x^2 + 2*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))/(105*a^2*x^7)

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fricas [A]  time = 0.76, size = 78, normalized size = 0.26 \[ \frac {21 \, a^{2} x^{2} + 2 \, {\left (8 \, a^{7} x^{7} + 4 \, a^{5} x^{5} + 3 \, a^{3} x^{3} - 15 \, a x\right )} \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 30}{105 \, a^{2} x^{7}} \]

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^6,x, algorithm="fricas")

[Out]

1/105*(21*a^2*x^2 + 2*(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)
) - 30)/(a^2*x^7)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}\right )}^{2}}{x^{6}}\,{d x} \]

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^6,x, algorithm="giac")

[Out]

integrate((sqrt(1/(a*x) + 1)*sqrt(1/(a*x) - 1) + 1/(a*x))^2/x^6, x)

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maple [A]  time = 0.07, size = 92, normalized size = 0.31 \[ \frac {-\frac {1}{7 x^{7}}+\frac {a^{2}}{5 x^{5}}}{a^{2}}+\frac {2 \sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}\, \left (a^{2} x^{2}-1\right ) \left (8 x^{4} a^{4}+12 a^{2} x^{2}+15\right )}{105 a \,x^{6}}-\frac {1}{7 a^{2} x^{7}} \]

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^6,x)

[Out]

1/a^2*(-1/7/x^7+1/5*a^2/x^5)+2/105/a*(-(a*x-1)/a/x)^(1/2)/x^6*((a*x+1)/a/x)^(1/2)*(a^2*x^2-1)*(8*a^4*x^4+12*a^
2*x^2+15)-1/7/a^2/x^7

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maxima [A]  time = 0.40, size = 65, normalized size = 0.22 \[ \frac {1}{5 \, x^{5}} + \frac {2 \, {\left (8 \, a^{6} x^{7} + 4 \, a^{4} x^{5} + 3 \, a^{2} x^{3} - 15 \, x\right )} \sqrt {a x + 1} \sqrt {-a x + 1}}{105 \, a^{2} x^{8}} - \frac {2}{7 \, a^{2} x^{7}} \]

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^6,x, algorithm="maxima")

[Out]

1/5/x^5 + 2/105*(8*a^6*x^7 + 4*a^4*x^5 + 3*a^2*x^3 - 15*x)*sqrt(a*x + 1)*sqrt(-a*x + 1)/(a^2*x^8) - 2/7/(a^2*x
^7)

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mupad [B]  time = 2.12, size = 105, normalized size = 0.35 \[ \frac {\frac {a^2\,x^2}{5}-\frac {2}{7}}{a^2\,x^7}+\frac {\sqrt {\frac {1}{a\,x}-1}\,\left (\frac {2\,a\,x^2\,\sqrt {\frac {1}{a\,x}+1}}{35}-\frac {2\,\sqrt {\frac {1}{a\,x}+1}}{7\,a}+\frac {8\,a^3\,x^4\,\sqrt {\frac {1}{a\,x}+1}}{105}+\frac {16\,a^5\,x^6\,\sqrt {\frac {1}{a\,x}+1}}{105}\right )}{x^6} \]

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^6,x)

[Out]

((a^2*x^2)/5 - 2/7)/(a^2*x^7) + ((1/(a*x) - 1)^(1/2)*((2*a*x^2*(1/(a*x) + 1)^(1/2))/35 - (2*(1/(a*x) + 1)^(1/2
))/(7*a) + (8*a^3*x^4*(1/(a*x) + 1)^(1/2))/105 + (16*a^5*x^6*(1/(a*x) + 1)^(1/2))/105))/x^6

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {2}{x^{8}}\, dx + \int \left (- \frac {a^{2}}{x^{6}}\right )\, dx + \int \frac {2 a \sqrt {-1 + \frac {1}{a x}} \sqrt {1 + \frac {1}{a x}}}{x^{7}}\, dx}{a^{2}} \]

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**6,x)

[Out]

(Integral(2/x**8, x) + Integral(-a**2/x**6, x) + Integral(2*a*sqrt(-1 + 1/(a*x))*sqrt(1 + 1/(a*x))/x**7, x))/a
**2

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