3.74 \(\int \frac {e^{2 \text {sech}^{-1}(a x)}}{x^5} \, dx\)
Optimal. Leaf size=267 \[ \frac {3 a^4}{8 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )}+\frac {a^4}{8 \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )}-\frac {11 a^4}{8 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^2}-\frac {a^4}{8 \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^2}+\frac {8 a^4}{3 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^3}-\frac {3 a^4}{\left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^4}+\frac {2 a^4}{\left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^5}-\frac {2 a^4}{3 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^6}+\frac {1}{4} a^4 \tanh ^{-1}\left (\sqrt {\frac {1-a x}{a x+1}}\right ) \]
[Out]
1/4*a^4*arctanh(((-a*x+1)/(a*x+1))^(1/2))-2/3*a^4/(1-((-a*x+1)/(a*x+1))^(1/2))^6+2*a^4/(1-((-a*x+1)/(a*x+1))^(
1/2))^5-3*a^4/(1-((-a*x+1)/(a*x+1))^(1/2))^4+8/3*a^4/(1-((-a*x+1)/(a*x+1))^(1/2))^3-11/8*a^4/(1-((-a*x+1)/(a*x
+1))^(1/2))^2+3/8*a^4/(1-((-a*x+1)/(a*x+1))^(1/2))-1/8*a^4/(1+((-a*x+1)/(a*x+1))^(1/2))^2+1/8*a^4/(1+((-a*x+1)
/(a*x+1))^(1/2))
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Rubi [A] time = 0.54, antiderivative size = 267, 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} \[ \frac {3 a^4}{8 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )}+\frac {a^4}{8 \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )}-\frac {11 a^4}{8 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^2}-\frac {a^4}{8 \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^2}+\frac {8 a^4}{3 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^3}-\frac {3 a^4}{\left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^4}+\frac {2 a^4}{\left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^5}-\frac {2 a^4}{3 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^6}+\frac {1}{4} a^4 \tanh ^{-1}\left (\sqrt {\frac {1-a x}{a x+1}}\right ) \]
Antiderivative was successfully verified.
[In]
Int[E^(2*ArcSech[a*x])/x^5,x]
[Out]
(-2*a^4)/(3*(1 - Sqrt[(1 - a*x)/(1 + a*x)])^6) + (2*a^4)/(1 - Sqrt[(1 - a*x)/(1 + a*x)])^5 - (3*a^4)/(1 - Sqrt
[(1 - a*x)/(1 + a*x)])^4 + (8*a^4)/(3*(1 - Sqrt[(1 - a*x)/(1 + a*x)])^3) - (11*a^4)/(8*(1 - Sqrt[(1 - a*x)/(1
+ a*x)])^2) + (3*a^4)/(8*(1 - Sqrt[(1 - a*x)/(1 + a*x)])) - a^4/(8*(1 + Sqrt[(1 - a*x)/(1 + a*x)])^2) + a^4/(8
*(1 + Sqrt[(1 - a*x)/(1 + a*x)])) + (a^4*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^{2 \text {sech}^{-1}(a x)}}{x^5} \, 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^5} \, dx\\ &=(4 a) \operatorname {Subst}\left (\int \frac {x \left (a+a x^2\right )^3}{(-1+x)^7 (1+x)^3} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )\\ &=(4 a) \operatorname {Subst}\left (\int \left (\frac {a^3}{(-1+x)^7}+\frac {5 a^3}{2 (-1+x)^6}+\frac {3 a^3}{(-1+x)^5}+\frac {2 a^3}{(-1+x)^4}+\frac {11 a^3}{16 (-1+x)^3}+\frac {3 a^3}{32 (-1+x)^2}+\frac {a^3}{16 (1+x)^3}-\frac {a^3}{32 (1+x)^2}-\frac {a^3}{16 \left (-1+x^2\right )}\right ) \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )\\ &=-\frac {2 a^4}{3 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^6}+\frac {2 a^4}{\left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^5}-\frac {3 a^4}{\left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^4}+\frac {8 a^4}{3 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^3}-\frac {11 a^4}{8 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^2}+\frac {3 a^4}{8 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )}-\frac {a^4}{8 \left (1+\sqrt {\frac {1-a x}{1+a x}}\right )^2}+\frac {a^4}{8 \left (1+\sqrt {\frac {1-a x}{1+a x}}\right )}-\frac {1}{4} a^4 \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )\\ &=-\frac {2 a^4}{3 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^6}+\frac {2 a^4}{\left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^5}-\frac {3 a^4}{\left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^4}+\frac {8 a^4}{3 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^3}-\frac {11 a^4}{8 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^2}+\frac {3 a^4}{8 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )}-\frac {a^4}{8 \left (1+\sqrt {\frac {1-a x}{1+a x}}\right )^2}+\frac {a^4}{8 \left (1+\sqrt {\frac {1-a x}{1+a x}}\right )}+\frac {1}{4} a^4 \tanh ^{-1}\left (\sqrt {\frac {1-a x}{1+a x}}\right )\\ \end {align*}
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Mathematica [A] time = 0.15, size = 137, normalized size = 0.51 \[ \frac {-3 a^6 x^6 \log (x)+3 a^6 x^6 \log \left (a x \sqrt {\frac {1-a x}{a x+1}}+\sqrt {\frac {1-a x}{a x+1}}+1\right )+6 a^2 x^2+\sqrt {\frac {1-a x}{a x+1}} \left (3 a^5 x^5+3 a^4 x^4+2 a^3 x^3+2 a^2 x^2-8 a x-8\right )-8}{24 a^2 x^6} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[E^(2*ArcSech[a*x])/x^5,x]
[Out]
(-8 + 6*a^2*x^2 + 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)]])/(24*a^2*x^6)
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fricas [A] time = 0.55, size = 156, normalized size = 0.58 \[ \frac {3 \, a^{6} x^{6} \log \left (a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} + 1\right ) - 3 \, a^{6} x^{6} \log \left (a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 1\right ) + 12 \, a^{2} x^{2} + 2 \, {\left (3 \, a^{5} x^{5} + 2 \, a^{3} x^{3} - 8 \, a x\right )} \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 16}{48 \, a^{2} x^{6}} \]
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^5,x, algorithm="fricas")
[Out]
1/48*(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) + 12*a^2*x^2 + 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^2*x^6)
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}\right )}^{2}}{x^{5}}\,{d x} \]
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^5,x, algorithm="giac")
[Out]
integrate((sqrt(1/(a*x) + 1)*sqrt(1/(a*x) - 1) + 1/(a*x))^2/x^5, x)
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maple [A] time = 0.09, size = 153, normalized size = 0.57 \[ \frac {-\frac {1}{6 x^{6}}+\frac {a^{2}}{4 x^{4}}}{a^{2}}+\frac {\sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}\, \left (3 \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right ) x^{6} a^{6}+3 \sqrt {-a^{2} x^{2}+1}\, x^{4} a^{4}+2 a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}-8 \sqrt {-a^{2} x^{2}+1}\right )}{24 a \,x^{5} \sqrt {-a^{2} x^{2}+1}}-\frac {1}{6 a^{2} x^{6}} \]
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^5,x)
[Out]
1/a^2*(-1/6/x^6+1/4*a^2/x^4)+1/24/a*(-(a*x-1)/a/x)^(1/2)/x^5*((a*x+1)/a/x)^(1/2)*(3*arctanh(1/(-a^2*x^2+1)^(1/
2))*x^6*a^6+3*(-a^2*x^2+1)^(1/2)*x^4*a^4+2*a^2*x^2*(-a^2*x^2+1)^(1/2)-8*(-a^2*x^2+1)^(1/2))/(-a^2*x^2+1)^(1/2)
-1/6/a^2/x^6
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {2 \, {\left (\frac {1}{16} \, a^{6} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - \frac {1}{16} \, \sqrt {-a^{2} x^{2} + 1} a^{6} - \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{4}}{16 \, x^{2}} - \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{2}}{8 \, x^{4}} - \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{6 \, x^{6}}\right )}}{a^{2}} - \frac {1}{3 \, a^{2} x^{6}} - \int \frac {1}{x^{5}}\,{d x} \]
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^5,x, algorithm="maxima")
[Out]
2*integrate(sqrt(a*x + 1)*sqrt(-a*x + 1)/x^7, x)/a^2 - 1/3/(a^2*x^6) - integrate(x^(-5), x)
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mupad [B] time = 65.19, size = 2480, normalized size = 9.29 \[ \text {result too large to display} \]
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^5,x)
[Out]
((311*a^4*((1/(a*x) - 1)^(1/2) - 1i)^5)/(2*((1/(a*x) + 1)^(1/2) - 1)^5) - (175*a^4*((1/(a*x) - 1)^(1/2) - 1i)^
3)/(6*((1/(a*x) + 1)^(1/2) - 1)^3) + (8361*a^4*((1/(a*x) - 1)^(1/2) - 1i)^7)/(2*((1/(a*x) + 1)^(1/2) - 1)^7) +
(42259*a^4*((1/(a*x) - 1)^(1/2) - 1i)^9)/(3*((1/(a*x) + 1)^(1/2) - 1)^9) + (25295*a^4*((1/(a*x) - 1)^(1/2) -
1i)^11)/((1/(a*x) + 1)^(1/2) - 1)^11 + (25295*a^4*((1/(a*x) - 1)^(1/2) - 1i)^13)/((1/(a*x) + 1)^(1/2) - 1)^13
+ (42259*a^4*((1/(a*x) - 1)^(1/2) - 1i)^15)/(3*((1/(a*x) + 1)^(1/2) - 1)^15) + (8361*a^4*((1/(a*x) - 1)^(1/2)
- 1i)^17)/(2*((1/(a*x) + 1)^(1/2) - 1)^17) + (311*a^4*((1/(a*x) - 1)^(1/2) - 1i)^19)/(2*((1/(a*x) + 1)^(1/2) -
1)^19) - (175*a^4*((1/(a*x) - 1)^(1/2) - 1i)^21)/(6*((1/(a*x) + 1)^(1/2) - 1)^21) + (5*a^4*((1/(a*x) - 1)^(1/
2) - 1i)^23)/(2*((1/(a*x) + 1)^(1/2) - 1)^23) + (5*a^4*((1/(a*x) - 1)^(1/2) - 1i))/(2*((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)^2
0 - (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) + (a^4*atanh(((1/(a*x) - 1)^(1/2) - 1i)/((1/(a*x) + 1)^(1/2) - 1)))/2 - ((a^4*((1/(a*x
) - 1)^(1/2) - 1i)^6*4096i)/(3*((1/(a*x) + 1)^(1/2) - 1)^6) + (a^4*((1/(a*x) - 1)^(1/2) - 1i)^8*8192i)/(3*((1/
(a*x) + 1)^(1/2) - 1)^8) + (a^4*((1/(a*x) - 1)^(1/2) - 1i)^10*24576i)/(5*((1/(a*x) + 1)^(1/2) - 1)^10) + (a^4*
((1/(a*x) - 1)^(1/2) - 1i)^12*8192i)/(3*((1/(a*x) + 1)^(1/2) - 1)^12) + (a^4*((1/(a*x) - 1)^(1/2) - 1i)^14*409
6i)/(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)^1
0)/((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)^20/
((1/(a*x) + 1)^(1/2) - 1)^20 + 1) + ((a^4*((1/(a*x) - 1)^(1/2) - 1i)^6*20480i)/((1/(a*x) + 1)^(1/2) - 1)^6 + (
a^4*((1/(a*x) - 1)^(1/2) - 1i)^8*40960i)/((1/(a*x) + 1)^(1/2) - 1)^8 + (a^4*((1/(a*x) - 1)^(1/2) - 1i)^10*7372
8i)/((1/(a*x) + 1)^(1/2) - 1)^10 + (a^4*((1/(a*x) - 1)^(1/2) - 1i)^12*40960i)/((1/(a*x) + 1)^(1/2) - 1)^12 + (
a^4*((1/(a*x) - 1)^(1/2) - 1i)^14*20480i)/((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)^20/((1/(a*x) + 1)^(1/2) - 1)^20 + 1)) + ((23*a^4*((1/(a*x) - 1)^(1/2) - 1i)
^3)/((1/(a*x) + 1)^(1/2) - 1)^3 + (333*a^4*((1/(a*x) - 1)^(1/2) - 1i)^5)/((1/(a*x) + 1)^(1/2) - 1)^5 + (671*a^
4*((1/(a*x) - 1)^(1/2) - 1i)^7)/((1/(a*x) + 1)^(1/2) - 1)^7 + (671*a^4*((1/(a*x) - 1)^(1/2) - 1i)^9)/((1/(a*x)
+ 1)^(1/2) - 1)^9 + (333*a^4*((1/(a*x) - 1)^(1/2) - 1i)^11)/((1/(a*x) + 1)^(1/2) - 1)^11 + (23*a^4*((1/(a*x)
- 1)^(1/2) - 1i)^13)/((1/(a*x) + 1)^(1/2) - 1)^13 - (3*a^4*((1/(a*x) - 1)^(1/2) - 1i)^15)/((1/(a*x) + 1)^(1/2)
- 1)^15 - (3*a^4*((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) - 1
i)^16/((1/(a*x) + 1)^(1/2) - 1)^16 + 1) + 1/(4*x^4) - 1/(3*a^2*x^6)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {2}{x^{7}}\, dx + \int \left (- \frac {a^{2}}{x^{5}}\right )\, dx + \int \frac {2 a \sqrt {-1 + \frac {1}{a x}} \sqrt {1 + \frac {1}{a x}}}{x^{6}}\, dx}{a^{2}} \]
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**5,x)
[Out]
(Integral(2/x**7, x) + Integral(-a**2/x**5, x) + Integral(2*a*sqrt(-1 + 1/(a*x))*sqrt(1 + 1/(a*x))/x**6, x))/a
**2
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