3.86 \(\int \frac {e^{-\text {sech}^{-1}(a x)}}{x^6} \, dx\)
Optimal. Leaf size=320 \[ \frac {a^5}{4 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )}+\frac {3 a^5}{8 \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )}-\frac {3 a^5}{8 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^2}-\frac {a^5}{\left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^2}+\frac {a^5}{4 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^3}+\frac {19 a^5}{12 \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^3}-\frac {a^5}{8 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^4}-\frac {13 a^5}{8 \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^4}+\frac {a^5}{\left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^5}-\frac {a^5}{3 \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^6}-\frac {1}{8} a^5 \tanh ^{-1}\left (\sqrt {\frac {1-a x}{a x+1}}\right ) \]
[Out]
-1/8*a^5*arctanh(((-a*x+1)/(a*x+1))^(1/2))-1/8*a^5/(1-((-a*x+1)/(a*x+1))^(1/2))^4+1/4*a^5/(1-((-a*x+1)/(a*x+1)
)^(1/2))^3-3/8*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))-1/3*a^5/(1+((-a*x+1)/(a
*x+1))^(1/2))^6+a^5/(1+((-a*x+1)/(a*x+1))^(1/2))^5-13/8*a^5/(1+((-a*x+1)/(a*x+1))^(1/2))^4+19/12*a^5/(1+((-a*x
+1)/(a*x+1))^(1/2))^3-a^5/(1+((-a*x+1)/(a*x+1))^(1/2))^2+3/8*a^5/(1+((-a*x+1)/(a*x+1))^(1/2))
________________________________________________________________________________________
Rubi [A] time = 0.57, antiderivative size = 320, 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} \[ \frac {a^5}{4 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )}+\frac {3 a^5}{8 \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )}-\frac {3 a^5}{8 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^2}-\frac {a^5}{\left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^2}+\frac {a^5}{4 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^3}+\frac {19 a^5}{12 \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^3}-\frac {a^5}{8 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^4}-\frac {13 a^5}{8 \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^4}+\frac {a^5}{\left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^5}-\frac {a^5}{3 \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^6}-\frac {1}{8} a^5 \tanh ^{-1}\left (\sqrt {\frac {1-a x}{a x+1}}\right ) \]
Antiderivative was successfully verified.
[In]
Int[1/(E^ArcSech[a*x]*x^6),x]
[Out]
-a^5/(8*(1 - Sqrt[(1 - a*x)/(1 + a*x)])^4) + a^5/(4*(1 - Sqrt[(1 - a*x)/(1 + a*x)])^3) - (3*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/(3*(1 + Sqrt[(1 - a*x)/(1 + a*x)])^6)
+ a^5/(1 + Sqrt[(1 - a*x)/(1 + a*x)])^5 - (13*a^5)/(8*(1 + Sqrt[(1 - a*x)/(1 + a*x)])^4) + (19*a^5)/(12*(1 +
Sqrt[(1 - a*x)/(1 + a*x)])^3) - a^5/(1 + Sqrt[(1 - a*x)/(1 + a*x)])^2 + (3*a^5)/(8*(1 + Sqrt[(1 - a*x)/(1 + a*
x)])) - (a^5*ArcTanh[Sqrt[(1 - a*x)/(1 + a*x)]])/8
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^6} \, dx &=\int \frac {1}{x^6 \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 )^4}{(-1+x)^5 (1+x)^7} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )\\ &=(4 a) \operatorname {Subst}\left (\int \left (\frac {a^4}{8 (-1+x)^5}+\frac {3 a^4}{16 (-1+x)^4}+\frac {3 a^4}{16 (-1+x)^3}+\frac {a^4}{16 (-1+x)^2}+\frac {a^4}{2 (1+x)^7}-\frac {5 a^4}{4 (1+x)^6}+\frac {13 a^4}{8 (1+x)^5}-\frac {19 a^4}{16 (1+x)^4}+\frac {a^4}{2 (1+x)^3}-\frac {3 a^4}{32 (1+x)^2}+\frac {a^4}{32 \left (-1+x^2\right )}\right ) \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )\\ &=-\frac {a^5}{8 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^4}+\frac {a^5}{4 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^3}-\frac {3 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}{3 \left (1+\sqrt {\frac {1-a x}{1+a x}}\right )^6}+\frac {a^5}{\left (1+\sqrt {\frac {1-a x}{1+a x}}\right )^5}-\frac {13 a^5}{8 \left (1+\sqrt {\frac {1-a x}{1+a x}}\right )^4}+\frac {19 a^5}{12 \left (1+\sqrt {\frac {1-a x}{1+a x}}\right )^3}-\frac {a^5}{\left (1+\sqrt {\frac {1-a x}{1+a x}}\right )^2}+\frac {3 a^5}{8 \left (1+\sqrt {\frac {1-a x}{1+a x}}\right )}+\frac {1}{8} a^5 \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )\\ &=-\frac {a^5}{8 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^4}+\frac {a^5}{4 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^3}-\frac {3 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}{3 \left (1+\sqrt {\frac {1-a x}{1+a x}}\right )^6}+\frac {a^5}{\left (1+\sqrt {\frac {1-a x}{1+a x}}\right )^5}-\frac {13 a^5}{8 \left (1+\sqrt {\frac {1-a x}{1+a x}}\right )^4}+\frac {19 a^5}{12 \left (1+\sqrt {\frac {1-a x}{1+a x}}\right )^3}-\frac {a^5}{\left (1+\sqrt {\frac {1-a x}{1+a x}}\right )^2}+\frac {3 a^5}{8 \left (1+\sqrt {\frac {1-a x}{1+a x}}\right )}-\frac {1}{8} a^5 \tanh ^{-1}\left (\sqrt {\frac {1-a x}{1+a x}}\right )\\ \end {align*}
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Mathematica [A] time = 0.14, size = 129, normalized size = 0.40 \[ -\frac {-3 a^6 x^6 \log (x)+3 a^6 x^6 \log \left (a x \sqrt {\frac {1-a x}{a x+1}}+\sqrt {\frac {1-a x}{a x+1}}+1\right )+\sqrt {\frac {1-a x}{a x+1}} \left (3 a^5 x^5+3 a^4 x^4+2 a^3 x^3+2 a^2 x^2-8 a x-8\right )+8}{48 a x^6} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[1/(E^ArcSech[a*x]*x^6),x]
[Out]
-1/48*(8 + Sqrt[(1 - a*x)/(1 + a*x)]*(-8 - 8*a*x + 2*a^2*x^2 + 2*a^3*x^3 + 3*a^4*x^4 + 3*a^5*x^5) - 3*a^6*x^6*
Log[x] + 3*a^6*x^6*Log[1 + Sqrt[(1 - a*x)/(1 + a*x)] + a*x*Sqrt[(1 - a*x)/(1 + a*x)]])/(a*x^6)
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fricas [A] time = 0.83, size = 148, normalized size = 0.46 \[ -\frac {3 \, a^{6} x^{6} \log \left (a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} + 1\right ) - 3 \, a^{6} x^{6} \log \left (a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 1\right ) + 2 \, {\left (3 \, a^{5} x^{5} + 2 \, a^{3} x^{3} - 8 \, a x\right )} \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} + 16}{96 \, a x^{6}} \]
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^6,x, algorithm="fricas")
[Out]
-1/96*(3*a^6*x^6*log(a*x*sqrt((a*x + 1)/(a*x))*sqrt(-(a*x - 1)/(a*x)) + 1) - 3*a^6*x^6*log(a*x*sqrt((a*x + 1)/
(a*x))*sqrt(-(a*x - 1)/(a*x)) - 1) + 2*(3*a^5*x^5 + 2*a^3*x^3 - 8*a*x)*sqrt((a*x + 1)/(a*x))*sqrt(-(a*x - 1)/(
a*x)) + 16)/(a*x^6)
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{6} {\left (\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}\right )}}\,{d x} \]
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^6,x, algorithm="giac")
[Out]
integrate(1/(x^6*(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 \[ \int \frac {1}{\left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right ) x^{6}}\, dx \]
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^6,x)
[Out]
int(1/(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2))/x^6,x)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{6} {\left (\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}\right )}}\,{d x} \]
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^6,x, algorithm="maxima")
[Out]
integrate(1/(x^6*(sqrt(1/(a*x) + 1)*sqrt(1/(a*x) - 1) + 1/(a*x))), x)
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mupad [B] time = 65.40, size = 2479, normalized size = 7.75 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(x^6*((1/(a*x) - 1)^(1/2)*(1/(a*x) + 1)^(1/2) + 1/(a*x))),x)
[Out]
((a^5*((1/(a*x) - 1)^(1/2) - 1i)^6*10240i)/((1/(a*x) + 1)^(1/2) - 1)^6 + (a^5*((1/(a*x) - 1)^(1/2) - 1i)^8*204
80i)/((1/(a*x) + 1)^(1/2) - 1)^8 + (a^5*((1/(a*x) - 1)^(1/2) - 1i)^10*36864i)/((1/(a*x) + 1)^(1/2) - 1)^10 + (
a^5*((1/(a*x) - 1)^(1/2) - 1i)^12*20480i)/((1/(a*x) + 1)^(1/2) - 1)^12 + (a^5*((1/(a*x) - 1)^(1/2) - 1i)^14*10
240i)/((1/(a*x) + 1)^(1/2) - 1)^14)/(15*((45*((1/(a*x) - 1)^(1/2) - 1i)^4)/((1/(a*x) + 1)^(1/2) - 1)^4 - (10*(
(1/(a*x) - 1)^(1/2) - 1i)^2)/((1/(a*x) + 1)^(1/2) - 1)^2 - (120*((1/(a*x) - 1)^(1/2) - 1i)^6)/((1/(a*x) + 1)^(
1/2) - 1)^6 + (210*((1/(a*x) - 1)^(1/2) - 1i)^8)/((1/(a*x) + 1)^(1/2) - 1)^8 - (252*((1/(a*x) - 1)^(1/2) - 1i)
^10)/((1/(a*x) + 1)^(1/2) - 1)^10 + (210*((1/(a*x) - 1)^(1/2) - 1i)^12)/((1/(a*x) + 1)^(1/2) - 1)^12 - (120*((
1/(a*x) - 1)^(1/2) - 1i)^14)/((1/(a*x) + 1)^(1/2) - 1)^14 + (45*((1/(a*x) - 1)^(1/2) - 1i)^16)/((1/(a*x) + 1)^
(1/2) - 1)^16 - (10*((1/(a*x) - 1)^(1/2) - 1i)^18)/((1/(a*x) + 1)^(1/2) - 1)^18 + ((1/(a*x) - 1)^(1/2) - 1i)^2
0/((1/(a*x) + 1)^(1/2) - 1)^20 + 1)) - (a^5*atanh(((1/(a*x) - 1)^(1/2) - 1i)/((1/(a*x) + 1)^(1/2) - 1)))/4 - (
(a^5*((1/(a*x) - 1)^(1/2) - 1i)^6*2048i)/(3*((1/(a*x) + 1)^(1/2) - 1)^6) + (a^5*((1/(a*x) - 1)^(1/2) - 1i)^8*4
096i)/(3*((1/(a*x) + 1)^(1/2) - 1)^8) + (a^5*((1/(a*x) - 1)^(1/2) - 1i)^10*12288i)/(5*((1/(a*x) + 1)^(1/2) - 1
)^10) + (a^5*((1/(a*x) - 1)^(1/2) - 1i)^12*4096i)/(3*((1/(a*x) + 1)^(1/2) - 1)^12) + (a^5*((1/(a*x) - 1)^(1/2)
- 1i)^14*2048i)/(3*((1/(a*x) + 1)^(1/2) - 1)^14))/((45*((1/(a*x) - 1)^(1/2) - 1i)^4)/((1/(a*x) + 1)^(1/2) - 1
)^4 - (10*((1/(a*x) - 1)^(1/2) - 1i)^2)/((1/(a*x) + 1)^(1/2) - 1)^2 - (120*((1/(a*x) - 1)^(1/2) - 1i)^6)/((1/(
a*x) + 1)^(1/2) - 1)^6 + (210*((1/(a*x) - 1)^(1/2) - 1i)^8)/((1/(a*x) + 1)^(1/2) - 1)^8 - (252*((1/(a*x) - 1)^
(1/2) - 1i)^10)/((1/(a*x) + 1)^(1/2) - 1)^10 + (210*((1/(a*x) - 1)^(1/2) - 1i)^12)/((1/(a*x) + 1)^(1/2) - 1)^1
2 - (120*((1/(a*x) - 1)^(1/2) - 1i)^14)/((1/(a*x) + 1)^(1/2) - 1)^14 + (45*((1/(a*x) - 1)^(1/2) - 1i)^16)/((1/
(a*x) + 1)^(1/2) - 1)^16 - (10*((1/(a*x) - 1)^(1/2) - 1i)^18)/((1/(a*x) + 1)^(1/2) - 1)^18 + ((1/(a*x) - 1)^(1
/2) - 1i)^20/((1/(a*x) + 1)^(1/2) - 1)^20 + 1) - ((311*a^5*((1/(a*x) - 1)^(1/2) - 1i)^5)/(4*((1/(a*x) + 1)^(1/
2) - 1)^5) - (175*a^5*((1/(a*x) - 1)^(1/2) - 1i)^3)/(12*((1/(a*x) + 1)^(1/2) - 1)^3) + (8361*a^5*((1/(a*x) - 1
)^(1/2) - 1i)^7)/(4*((1/(a*x) + 1)^(1/2) - 1)^7) + (42259*a^5*((1/(a*x) - 1)^(1/2) - 1i)^9)/(6*((1/(a*x) + 1)^
(1/2) - 1)^9) + (25295*a^5*((1/(a*x) - 1)^(1/2) - 1i)^11)/(2*((1/(a*x) + 1)^(1/2) - 1)^11) + (25295*a^5*((1/(a
*x) - 1)^(1/2) - 1i)^13)/(2*((1/(a*x) + 1)^(1/2) - 1)^13) + (42259*a^5*((1/(a*x) - 1)^(1/2) - 1i)^15)/(6*((1/(
a*x) + 1)^(1/2) - 1)^15) + (8361*a^5*((1/(a*x) - 1)^(1/2) - 1i)^17)/(4*((1/(a*x) + 1)^(1/2) - 1)^17) + (311*a^
5*((1/(a*x) - 1)^(1/2) - 1i)^19)/(4*((1/(a*x) + 1)^(1/2) - 1)^19) - (175*a^5*((1/(a*x) - 1)^(1/2) - 1i)^21)/(1
2*((1/(a*x) + 1)^(1/2) - 1)^21) + (5*a^5*((1/(a*x) - 1)^(1/2) - 1i)^23)/(4*((1/(a*x) + 1)^(1/2) - 1)^23) + (5*
a^5*((1/(a*x) - 1)^(1/2) - 1i))/(4*((1/(a*x) + 1)^(1/2) - 1)))/((66*((1/(a*x) - 1)^(1/2) - 1i)^4)/((1/(a*x) +
1)^(1/2) - 1)^4 - (12*((1/(a*x) - 1)^(1/2) - 1i)^2)/((1/(a*x) + 1)^(1/2) - 1)^2 - (220*((1/(a*x) - 1)^(1/2) -
1i)^6)/((1/(a*x) + 1)^(1/2) - 1)^6 + (495*((1/(a*x) - 1)^(1/2) - 1i)^8)/((1/(a*x) + 1)^(1/2) - 1)^8 - (792*((1
/(a*x) - 1)^(1/2) - 1i)^10)/((1/(a*x) + 1)^(1/2) - 1)^10 + (924*((1/(a*x) - 1)^(1/2) - 1i)^12)/((1/(a*x) + 1)^
(1/2) - 1)^12 - (792*((1/(a*x) - 1)^(1/2) - 1i)^14)/((1/(a*x) + 1)^(1/2) - 1)^14 + (495*((1/(a*x) - 1)^(1/2) -
1i)^16)/((1/(a*x) + 1)^(1/2) - 1)^16 - (220*((1/(a*x) - 1)^(1/2) - 1i)^18)/((1/(a*x) + 1)^(1/2) - 1)^18 + (66
*((1/(a*x) - 1)^(1/2) - 1i)^20)/((1/(a*x) + 1)^(1/2) - 1)^20 - (12*((1/(a*x) - 1)^(1/2) - 1i)^22)/((1/(a*x) +
1)^(1/2) - 1)^22 + ((1/(a*x) - 1)^(1/2) - 1i)^24/((1/(a*x) + 1)^(1/2) - 1)^24 + 1) - ((23*a^5*((1/(a*x) - 1)^(
1/2) - 1i)^3)/(2*((1/(a*x) + 1)^(1/2) - 1)^3) + (333*a^5*((1/(a*x) - 1)^(1/2) - 1i)^5)/(2*((1/(a*x) + 1)^(1/2)
- 1)^5) + (671*a^5*((1/(a*x) - 1)^(1/2) - 1i)^7)/(2*((1/(a*x) + 1)^(1/2) - 1)^7) + (671*a^5*((1/(a*x) - 1)^(1
/2) - 1i)^9)/(2*((1/(a*x) + 1)^(1/2) - 1)^9) + (333*a^5*((1/(a*x) - 1)^(1/2) - 1i)^11)/(2*((1/(a*x) + 1)^(1/2)
- 1)^11) + (23*a^5*((1/(a*x) - 1)^(1/2) - 1i)^13)/(2*((1/(a*x) + 1)^(1/2) - 1)^13) - (3*a^5*((1/(a*x) - 1)^(1
/2) - 1i)^15)/(2*((1/(a*x) + 1)^(1/2) - 1)^15) - (3*a^5*((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/(6*a*x^6)
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ a \int \frac {1}{a x^{6} \sqrt {-1 + \frac {1}{a x}} \sqrt {1 + \frac {1}{a x}} + x^{5}}\, dx \]
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**6,x)
[Out]
a*Integral(1/(a*x**6*sqrt(-1 + 1/(a*x))*sqrt(1 + 1/(a*x)) + x**5), x)
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