3.55 \(\int \frac {e^{\text {sech}^{-1}(a x^2)}}{x^3} \, dx\)
Optimal. Leaf size=118 \[ \frac {\sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \sqrt {1-a^2 x^4}}{4 a x^4}+\frac {1}{4} a \sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \tanh ^{-1}\left (\sqrt {1-a^2 x^4}\right )+\frac {1}{4 a x^4}-\frac {e^{\text {sech}^{-1}\left (a x^2\right )}}{2 x^2} \]
[Out]
1/4/a/x^4-1/2*(1/a/x^2+(1/a/x^2-1)^(1/2)*(1/a/x^2+1)^(1/2))/x^2+1/4*a*arctanh((-a^2*x^4+1)^(1/2))*(1/(a*x^2+1)
)^(1/2)*(a*x^2+1)^(1/2)+1/4*(1/(a*x^2+1))^(1/2)*(a*x^2+1)^(1/2)*(-a^2*x^4+1)^(1/2)/a/x^4
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 118, normalized size of antiderivative = 1.00,
number of steps used = 7, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used =
{6335, 30, 259, 266, 51, 63, 208} \[ \frac {\sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \sqrt {1-a^2 x^4}}{4 a x^4}+\frac {1}{4} a \sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \tanh ^{-1}\left (\sqrt {1-a^2 x^4}\right )+\frac {1}{4 a x^4}-\frac {e^{\text {sech}^{-1}\left (a x^2\right )}}{2 x^2} \]
Antiderivative was successfully verified.
[In]
Int[E^ArcSech[a*x^2]/x^3,x]
[Out]
1/(4*a*x^4) - E^ArcSech[a*x^2]/(2*x^2) + (Sqrt[(1 + a*x^2)^(-1)]*Sqrt[1 + a*x^2]*Sqrt[1 - a^2*x^4])/(4*a*x^4)
+ (a*Sqrt[(1 + a*x^2)^(-1)]*Sqrt[1 + a*x^2]*ArcTanh[Sqrt[1 - a^2*x^4]])/4
Rule 30
Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]
Rule 51
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]
Rule 63
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]
Rule 208
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]
Rule 259
Int[((c_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(n_))^(p_)*((a2_) + (b2_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[(c*x)
^m*(a1*a2 + b1*b2*x^(2*n))^p, x] /; FreeQ[{a1, b1, a2, b2, c, m, n, p}, x] && EqQ[a2*b1 + a1*b2, 0] && (Intege
rQ[p] || (GtQ[a1, 0] && GtQ[a2, 0]))
Rule 266
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Rule 6335
Int[E^ArcSech[(a_.)*(x_)^(p_.)]*(x_)^(m_.), x_Symbol] :> Simp[(x^(m + 1)*E^ArcSech[a*x^p])/(m + 1), x] + (Dist
[p/(a*(m + 1)), Int[x^(m - p), x], x] + Dist[(p*Sqrt[1 + a*x^p]*Sqrt[1/(1 + a*x^p)])/(a*(m + 1)), Int[x^(m - p
)/(Sqrt[1 + a*x^p]*Sqrt[1 - a*x^p]), x], x]) /; FreeQ[{a, m, p}, x] && NeQ[m, -1]
Rubi steps
\begin {align*} \int \frac {e^{\text {sech}^{-1}\left (a x^2\right )}}{x^3} \, dx &=-\frac {e^{\text {sech}^{-1}\left (a x^2\right )}}{2 x^2}-\frac {\int \frac {1}{x^5} \, dx}{a}-\frac {\left (\sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \int \frac {1}{x^5 \sqrt {1-a x^2} \sqrt {1+a x^2}} \, dx}{a}\\ &=\frac {1}{4 a x^4}-\frac {e^{\text {sech}^{-1}\left (a x^2\right )}}{2 x^2}-\frac {\left (\sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \int \frac {1}{x^5 \sqrt {1-a^2 x^4}} \, dx}{a}\\ &=\frac {1}{4 a x^4}-\frac {e^{\text {sech}^{-1}\left (a x^2\right )}}{2 x^2}-\frac {\left (\sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {1-a^2 x}} \, dx,x,x^4\right )}{4 a}\\ &=\frac {1}{4 a x^4}-\frac {e^{\text {sech}^{-1}\left (a x^2\right )}}{2 x^2}+\frac {\sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} \sqrt {1-a^2 x^4}}{4 a x^4}-\frac {1}{8} \left (a \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^4\right )\\ &=\frac {1}{4 a x^4}-\frac {e^{\text {sech}^{-1}\left (a x^2\right )}}{2 x^2}+\frac {\sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} \sqrt {1-a^2 x^4}}{4 a x^4}+\frac {\left (\sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2}} \, dx,x,\sqrt {1-a^2 x^4}\right )}{4 a}\\ &=\frac {1}{4 a x^4}-\frac {e^{\text {sech}^{-1}\left (a x^2\right )}}{2 x^2}+\frac {\sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} \sqrt {1-a^2 x^4}}{4 a x^4}+\frac {1}{4} a \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} \tanh ^{-1}\left (\sqrt {1-a^2 x^4}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.27, size = 105, normalized size = 0.89 \[ -\frac {-\frac {a^2 \sqrt {\frac {1-a x^2}{a x^2+1}} \left (a x^2+1\right ) \tan ^{-1}\left (\sqrt {a^2 x^4-1}\right )}{\sqrt {a^2 x^4-1}}+\frac {\sqrt {\frac {1-a x^2}{a x^2+1}} \left (a x^2+1\right )}{x^4}+\frac {1}{x^4}}{4 a} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[E^ArcSech[a*x^2]/x^3,x]
[Out]
-1/4*(x^(-4) + (Sqrt[(1 - a*x^2)/(1 + a*x^2)]*(1 + a*x^2))/x^4 - (a^2*Sqrt[(1 - a*x^2)/(1 + a*x^2)]*(1 + a*x^2
)*ArcTan[Sqrt[-1 + a^2*x^4]])/Sqrt[-1 + a^2*x^4])/a
________________________________________________________________________________________
fricas [A] time = 0.86, size = 146, normalized size = 1.24 \[ \frac {a^{2} x^{4} \log \left (a x^{2} \sqrt {\frac {a x^{2} + 1}{a x^{2}}} \sqrt {-\frac {a x^{2} - 1}{a x^{2}}} + 1\right ) - a^{2} x^{4} \log \left (a x^{2} \sqrt {\frac {a x^{2} + 1}{a x^{2}}} \sqrt {-\frac {a x^{2} - 1}{a x^{2}}} - 1\right ) - 2 \, a x^{2} \sqrt {\frac {a x^{2} + 1}{a x^{2}}} \sqrt {-\frac {a x^{2} - 1}{a x^{2}}} - 2}{8 \, a x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((1/a/x^2+(1/a/x^2-1)^(1/2)*(1/a/x^2+1)^(1/2))/x^3,x, algorithm="fricas")
[Out]
1/8*(a^2*x^4*log(a*x^2*sqrt((a*x^2 + 1)/(a*x^2))*sqrt(-(a*x^2 - 1)/(a*x^2)) + 1) - a^2*x^4*log(a*x^2*sqrt((a*x
^2 + 1)/(a*x^2))*sqrt(-(a*x^2 - 1)/(a*x^2)) - 1) - 2*a*x^2*sqrt((a*x^2 + 1)/(a*x^2))*sqrt(-(a*x^2 - 1)/(a*x^2)
) - 2)/(a*x^4)
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((1/a/x^2+(1/a/x^2-1)^(1/2)*(1/a/x^2+1)^(1/2))/x^3,x, algorithm="giac")
[Out]
Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(x)]schur row 1 2.33984e-10Francis algorithm not precise enough for[1.0,-1117.22141279,260038.267747,-2259602
4.9566,676199006.929]Warning, choosing root of [1,0,%%%{-12,[1,0]%%%},0,%%%{8,[4,4]%%%}+%%%{30,[2,0]%%%},0,%%%
{16,[5,4]%%%}+%%%{-28,[3,0]%%%},0,%%%{16,[8,8]%%%}+%%%{-24,[6,4]%%%}+%%%{9,[4,0]%%%}] at parameters values [54
.1277311612,-82]schur row 1 3.80414e-10Francis algorithm not precise enough for[1.0,-439.975588666,40328.85804
63,-1380066.57127,16264167.9132]Bad conditionned root j= 2 value 36.6628221508 ratio 0.000412274208284 mindist
0.00165644519952Bad conditionned root j= 0 value 36.66 ratio 0.00026134143357 mindist 0.0110443353806Bad cond
itionned root j= 2 value 36.67-0.004688*i ratio 0.00158404473284 mindist 0.009376Bad conditionned root j= 3 va
lue 36.67+0.004688*i ratio 0.00158404473284 mindist 0.009376Warning, choosing root of [1,0,%%%{-12,[1,0]%%%},0
,%%%{8,[4,4]%%%}+%%%{30,[2,0]%%%},0,%%%{16,[5,4]%%%}+%%%{-28,[3,0]%%%},0,%%%{16,[8,8]%%%}+%%%{-24,[6,4]%%%}+%%
%{9,[4,0]%%%}] at parameters values [82.1195442914,-89]schur row 1 2.26297e-10Francis algorithm not precise en
ough for[1.0,-310.806973653,20125.2030982,-486504.158708,4050237.99743]Unable to isolate roots number Vector [
0,1][0.259008132109614e2,0.259012999453233e2]Bad conditionned root j= 2 value 25.8996302569 ratio 0.0004236630
40072 mindist 0.00118295401739Warning, choosing root of [1,0,%%%{-12,[1,0]%%%},0,%%%{8,[4,4]%%%}+%%%{30,[2,0]%
%%},0,%%%{16,[5,4]%%%}+%%%{-28,[3,0]%%%},0,%%%{16,[8,8]%%%}+%%%{-24,[6,4]%%%}+%%%{9,[4,0]%%%}] at parameters v
alues [35.2935628123,-64]schur row 1 3.67828e-10Francis algorithm not precise enough for[1.0,-1024.27388138,21
8570.205017,-17412558.5081,477729345.21]Bad conditionned root j= 2 value 85.3520228111 ratio 0.000286084735534
mindist 0.0052898804365Warning, choosing root of [1,0,%%%{-12,[1,0]%%%},0,%%%{8,[4,4]%%%}+%%%{30,[2,0]%%%},0,
%%%{16,[5,4]%%%}+%%%{-28,[3,0]%%%},0,%%%{16,[8,8]%%%}+%%%{-24,[6,4]%%%}+%%%{9,[4,0]%%%}] at parameters values
[78.6493344628,42]schur row 1 1.40127e-11Francis algorithm not precise enough for[1.0,-550.918251291,63231.441
5845,-2709416.51745,39982152.0485]Bad conditionned root j= 2 value 45.9094216765 ratio 0.000696541081041 mindi
st 0.00106311994459Warning, choosing root of [1,0,%%%{-12,[1,0]%%%},0,%%%{8,[4,4]%%%}+%%%{30,[2,0]%%%},0,%%%{1
6,[5,4]%%%}+%%%{-28,[3,0]%%%},0,%%%{16,[8,8]%%%}+%%%{-24,[6,4]%%%}+%%%{9,[4,0]%%%}] at parameters values [62.4
600259969,46]schur row 1 3.2626e-10Francis algorithm not precise enough for[1.0,-897.25122063,167720.781859,-1
1704597.0415,281302606.674]Unable to isolate roots number Vector [0,1][0.747726202076933e2,0.747726716167272e2
]Bad conditionned root j= 2 value 74.7675133332 ratio 0.00101141811991 mindist 0.00510687449423schur row 1 1.1
6125e-10Francis algorithm not precise enough for[1.0,-1092.17002563,248507.367685,-21109845.4104,617559117.938
]Bad conditionned root j= 2 value 91.0116865143 ratio 0.00109001405383 mindist 0.00219671084823Warning, choosi
ng root of [1,0,%%%{-12,[1,0]%%%},0,%%%{8,[4,4]%%%}+%%%{30,[2,0]%%%},0,%%%{16,[5,4]%%%}+%%%{-28,[3,0]%%%},0,%%
%{16,[8,8]%%%}+%%%{-24,[6,4]%%%}+%%%{9,[4,0]%%%}] at parameters values [33.9285577983,-49]schur row 1 7.18728e
-11Francis algorithm not precise enough for[1.0,-185.418596232,7162.51163099,-103293.777387,513015.728641]Warn
ing, choosing root of [1,0,%%%{-12,[1,0]%%%},0,%%%{8,[4,4]%%%}+%%%{30,[2,0]%%%},0,%%%{16,[5,4]%%%}+%%%{-28,[3,
0]%%%},0,%%%{16,[8,8]%%%}+%%%{-24,[6,4]%%%}+%%%{9,[4,0]%%%}] at parameters values [18.4052062202,63]schur row
1 2.75123e-11Francis algorithm not precise enough for[1.0,-619.731778616,80014.0577972,-3856786.44967,64022494
.4518]Bad conditionned root j= 2 value 51.6436427769 ratio 0.00161773509906 mindist 0.00178986679927Warning, c
hoosing root of [1,0,%%%{-12,[1,0]%%%},0,%%%{8,[4,4]%%%}+%%%{30,[2,0]%%%},0,%%%{16,[5,4]%%%}+%%%{-28,[3,0]%%%}
,0,%%%{16,[8,8]%%%}+%%%{-24,[6,4]%%%}+%%%{9,[4,0]%%%}] at parameters values [10.4309062702,-37]Warning, choosi
ng root of [1,0,%%%{-12,[1,0]%%%},0,%%%{8,[4,4]%%%}+%%%{30,[2,0]%%%},0,%%%{16,[5,4]%%%}+%%%{-28,[3,0]%%%},0,%%
%{16,[8,8]%%%}+%%%{-24,[6,4]%%%}+%%%{9,[4,0]%%%}] at parameters values [-23,65]schur row 1 1.68784e-10Francis
algorithm not precise enough for[1.0,-96.6277521998,1945.1921865,-14619.0760005,37837.726424]Warning, choosing
root of [1,0,%%%{-12,[1,0]%%%},0,%%%{8,[4,4]%%%}+%%%{30,[2,0]%%%},0,%%%{16,[5,4]%%%}+%%%{-28,[3,0]%%%},0,%%%{
16,[8,8]%%%}+%%%{-24,[6,4]%%%}+%%%{9,[4,0]%%%}] at parameters values [39.1803401988,-44]schur row 1 3.85284e-1
0Francis algorithm not precise enough for[1.0,-1161.32542683,280974.322293,-25379093.0373,789465697.88]Bad con
ditionned root j= 2 value 96.7723338924 ratio 0.000375252022965 mindist 0.00603026989475Warning, choosing root
of [1,0,%%%{-12,[1,0]%%%},0,%%%{8,[4,4]%%%}+%%%{30,[2,0]%%%},0,%%%{16,[5,4]%%%}+%%%{-28,[3,0]%%%},0,%%%{16,[8
,8]%%%}+%%%{-24,[6,4]%%%}+%%%{9,[4,0]%%%}] at parameters values [39.9828299829,31]schur row 1 3.46041e-10Franc
is algorithm not precise enough for[1.0,-1129.51443638,265792.262915,-23350148.7365,706455270.256]Bad conditio
nned root j= 2 value 94.1217752457 ratio 0.000267509068199 mindist 0.00575040663189Warning, choosing root of [
1,0,%%%{-12,[1,0]%%%},0,%%%{8,[4,4]%%%}+%%%{30,[2,0]%%%},0,%%%{16,[5,4]%%%}+%%%{-28,[3,0]%%%},0,%%%{16,[8,8]%%
%}+%%%{-24,[6,4]%%%}+%%%{9,[4,0]%%%}] at parameters values [83.4865739918,-66]schur row 1 3.76847e-10Francis a
lgorithm not precise enough for[1.0,-637.349737572,84628.0599964,-4195152.25343,71619085.3875]Warning, choosin
g root of [1,0,%%%{-12,[1,0]%%%},0,%%%{8,[4,4]%%%}+%%%{30,[2,0]%%%},0,%%%{16,[5,4]%%%}+%%%{-28,[3,0]%%%},0,%%%
{16,[8,8]%%%}+%%%{-24,[6,4]%%%}+%%%{9,[4,0]%%%}] at parameters values [6.82230772497,79]Warning, choosing root
of [1,0,%%%{-8,[1,2]%%%}+%%%{-4,[1,0]%%%},0,%%%{8,[4,0]%%%}+%%%{16,[2,4]%%%}+%%%{8,[2,2]%%%}+%%%{6,[2,0]%%%},
0,%%%{-32,[5,2]%%%}+%%%{48,[5,0]%%%}+%%%{-32,[3,4]%%%}+%%%{8,[3,2]%%%}+%%%{-4,[3,0]%%%},0,%%%{16,[8,0]%%%}+%%%
{-32,[6,2]%%%}+%%%{8,[6,0]%%%}+%%%{16,[4,4]%%%}+%%%{-8,[4,2]%%%}+%%%{1,[4,0]%%%}] at parameters values [55.034
3274642,0]Warning, choosing root of [1,0,%%%{-8,[1,2]%%%}+%%%{-4,[1,0]%%%},0,%%%{8,[4,0]%%%}+%%%{16,[2,4]%%%}+
%%%{8,[2,2]%%%}+%%%{6,[2,0]%%%},0,%%%{-32,[5,2]%%%}+%%%{48,[5,0]%%%}+%%%{-32,[3,4]%%%}+%%%{8,[3,2]%%%}+%%%{-4,
[3,0]%%%},0,%%%{16,[8,0]%%%}+%%%{-32,[6,2]%%%}+%%%{8,[6,0]%%%}+%%%{16,[4,4]%%%}+%%%{-8,[4,2]%%%}+%%%{1,[4,0]%%
%}] at parameters values [66.0382199469,-8]Warning, choosing root of [1,0,%%%{-8,[1,2]%%%}+%%%{-4,[1,0]%%%},0,
%%%{8,[4,0]%%%}+%%%{16,[2,4]%%%}+%%%{8,[2,2]%%%}+%%%{6,[2,0]%%%},0,%%%{-32,[5,2]%%%}+%%%{48,[5,0]%%%}+%%%{-32,
[3,4]%%%}+%%%{8,[3,2]%%%}+%%%{-4,[3,0]%%%},0,%%%{16,[8,0]%%%}+%%%{-32,[6,2]%%%}+%%%{8,[6,0]%%%}+%%%{16,[4,4]%%
%}+%%%{-8,[4,2]%%%}+%%%{1,[4,0]%%%}] at parameters values [4.66774101928,97]Warning, choosing root of [1,0,%%%
{-8,[1,2]%%%}+%%%{-4,[1,0]%%%},0,%%%{8,[4,0]%%%}+%%%{16,[2,4]%%%}+%%%{8,[2,2]%%%}+%%%{6,[2,0]%%%},0,%%%{-32,[5
,2]%%%}+%%%{48,[5,0]%%%}+%%%{-32,[3,4]%%%}+%%%{8,[3,2]%%%}+%%%{-4,[3,0]%%%},0,%%%{16,[8,0]%%%}+%%%{-32,[6,2]%%
%}+%%%{8,[6,0]%%%}+%%%{16,[4,4]%%%}+%%%{-8,[4,2]%%%}+%%%{1,[4,0]%%%}] at parameters values [70.9232513234,-17]
Warning, choosing root of [1,0,%%%{-8,[1,2]%%%}+%%%{-4,[1,0]%%%},0,%%%{8,[4,0]%%%}+%%%{16,[2,4]%%%}+%%%{8,[2,2
]%%%}+%%%{6,[2,0]%%%},0,%%%{-32,[5,2]%%%}+%%%{48,[5,0]%%%}+%%%{-32,[3,4]%%%}+%%%{8,[3,2]%%%}+%%%{-4,[3,0]%%%},
0,%%%{16,[8,0]%%%}+%%%{-32,[6,2]%%%}+%%%{8,[6,0]%%%}+%%%{16,[4,4]%%%}+%%%{-8,[4,2]%%%}+%%%{1,[4,0]%%%}] at par
ameters values [82.4264548342,0]Warning, choosing root of [1,0,%%%{-8,[1,2]%%%}+%%%{-4,[1,0]%%%},0,%%%{8,[4,0]
%%%}+%%%{16,[2,4]%%%}+%%%{8,[2,2]%%%}+%%%{6,[2,0]%%%},0,%%%{-32,[5,2]%%%}+%%%{48,[5,0]%%%}+%%%{-32,[3,4]%%%}+%
%%{8,[3,2]%%%}+%%%{-4,[3,0]%%%},0,%%%{16,[8,0]%%%}+%%%{-32,[6,2]%%%}+%%%{8,[6,0]%%%}+%%%{16,[4,4]%%%}+%%%{-8,[
4,2]%%%}+%%%{1,[4,0]%%%}] at parameters values [59.4272477375,89]schur row 3 1.36691e-10Warning, choosing root
of [1,0,%%%{-8,[1,2]%%%}+%%%{-4,[1,0]%%%},0,%%%{8,[4,0]%%%}+%%%{16,[2,4]%%%}+%%%{8,[2,2]%%%}+%%%{6,[2,0]%%%},
0,%%%{-32,[5,2]%%%}+%%%{48,[5,0]%%%}+%%%{-32,[3,4]%%%}+%%%{8,[3,2]%%%}+%%%{-4,[3,0]%%%},0,%%%{16,[8,0]%%%}+%%%
{-32,[6,2]%%%}+%%%{8,[6,0]%%%}+%%%{16,[4,4]%%%}+%%%{-8,[4,2]%%%}+%%%{1,[4,0]%%%}] at parameters values [61.743
1004322,-65]Warning, choosing root of [1,0,%%%{-8,[1,2]%%%}+%%%{-4,[1,0]%%%},0,%%%{8,[4,0]%%%}+%%%{16,[2,4]%%%
}+%%%{8,[2,2]%%%}+%%%{6,[2,0]%%%},0,%%%{-32,[5,2]%%%}+%%%{48,[5,0]%%%}+%%%{-32,[3,4]%%%}+%%%{8,[3,2]%%%}+%%%{-
4,[3,0]%%%},0,%%%{16,[8,0]%%%}+%%%{-32,[6,2]%%%}+%%%{8,[6,0]%%%}+%%%{16,[4,4]%%%}+%%%{-8,[4,2]%%%}+%%%{1,[4,0]
%%%}] at parameters values [58.4409598615,-10]Warning, choosing root of [1,0,%%%{-8,[1,2]%%%}+%%%{-4,[1,0]%%%}
,0,%%%{8,[4,0]%%%}+%%%{16,[2,4]%%%}+%%%{8,[2,2]%%%}+%%%{6,[2,0]%%%},0,%%%{-32,[5,2]%%%}+%%%{48,[5,0]%%%}+%%%{-
32,[3,4]%%%}+%%%{8,[3,2]%%%}+%%%{-4,[3,0]%%%},0,%%%{16,[8,0]%%%}+%%%{-32,[6,2]%%%}+%%%{8,[6,0]%%%}+%%%{16,[4,4
]%%%}+%%%{-8,[4,2]%%%}+%%%{1,[4,0]%%%}] at parameters values [18.9804396471,0]Warning, choosing root of [1,0,%
%%{-8,[1,2]%%%}+%%%{-4,[1,0]%%%},0,%%%{8,[4,0]%%%}+%%%{16,[2,4]%%%}+%%%{8,[2,2]%%%}+%%%{6,[2,0]%%%},0,%%%{-32,
[5,2]%%%}+%%%{48,[5,0]%%%}+%%%{-32,[3,4]%%%}+%%%{8,[3,2]%%%}+%%%{-4,[3,0]%%%},0,%%%{16,[8,0]%%%}+%%%{-32,[6,2]
%%%}+%%%{8,[6,0]%%%}+%%%{16,[4,4]%%%}+%%%{-8,[4,2]%%%}+%%%{1,[4,0]%%%}] at parameters values [70.2045348478,0]
Warning, choosing root of [1,0,%%%{-8,[2,1]%%%}+%%%{-4,[0,1]%%%},0,%%%{16,[4,2]%%%}+%%%{8,[2,2]%%%}+%%%{8,[0,4
]%%%}+%%%{6,[0,2]%%%},0,%%%{-32,[4,3]%%%}+%%%{-32,[2,5]%%%}+%%%{8,[2,3]%%%}+%%%{48,[0,5]%%%}+%%%{-4,[0,3]%%%},
0,%%%{16,[4,4]%%%}+%%%{-32,[2,6]%%%}+%%%{-8,[2,4]%%%}+%%%{16,[0,8]%%%}+%%%{8,[0,6]%%%}+%%%{1,[0,4]%%%}] at par
ameters values [0,57.2153722499]schur row 3 2.56736e-11Warning, choosing root of [1,0,%%%{-8,[2,1]%%%}+%%%{-4,
[0,1]%%%},0,%%%{16,[4,2]%%%}+%%%{8,[2,2]%%%}+%%%{8,[0,4]%%%}+%%%{6,[0,2]%%%},0,%%%{-32,[4,3]%%%}+%%%{-32,[2,5]
%%%}+%%%{8,[2,3]%%%}+%%%{48,[0,5]%%%}+%%%{-4,[0,3]%%%},0,%%%{16,[4,4]%%%}+%%%{-32,[2,6]%%%}+%%%{-8,[2,4]%%%}+%
%%{16,[0,8]%%%}+%%%{8,[0,6]%%%}+%%%{1,[0,4]%%%}] at parameters values [-58,54.6372379069]Warning, choosing roo
t of [1,0,%%%{-8,[2,1]%%%}+%%%{-4,[0,1]%%%},0,%%%{16,[4,2]%%%}+%%%{8,[2,2]%%%}+%%%{8,[0,4]%%%}+%%%{6,[0,2]%%%}
,0,%%%{-32,[4,3]%%%}+%%%{-32,[2,5]%%%}+%%%{8,[2,3]%%%}+%%%{48,[0,5]%%%}+%%%{-4,[0,3]%%%},0,%%%{16,[4,4]%%%}+%%
%{-32,[2,6]%%%}+%%%{-8,[2,4]%%%}+%%%{16,[0,8]%%%}+%%%{8,[0,6]%%%}+%%%{1,[0,4]%%%}] at parameters values [71,86
.2839511861]Warning, choosing root of [1,0,%%%{-8,[2,1]%%%}+%%%{-4,[0,1]%%%},0,%%%{16,[4,2]%%%}+%%%{8,[2,2]%%%
}+%%%{8,[0,4]%%%}+%%%{6,[0,2]%%%},0,%%%{-32,[4,3]%%%}+%%%{-32,[2,5]%%%}+%%%{8,[2,3]%%%}+%%%{48,[0,5]%%%}+%%%{-
4,[0,3]%%%},0,%%%{16,[4,4]%%%}+%%%{-32,[2,6]%%%}+%%%{-8,[2,4]%%%}+%%%{16,[0,8]%%%}+%%%{8,[0,6]%%%}+%%%{1,[0,4]
%%%}] at parameters values [11,80.4553440167]Warning, choosing root of [1,0,%%%{-8,[2,1]%%%}+%%%{-4,[0,1]%%%},
0,%%%{16,[4,2]%%%}+%%%{8,[2,2]%%%}+%%%{8,[0,4]%%%}+%%%{6,[0,2]%%%},0,%%%{-32,[4,3]%%%}+%%%{-32,[2,5]%%%}+%%%{8
,[2,3]%%%}+%%%{48,[0,5]%%%}+%%%{-4,[0,3]%%%},0,%%%{16,[4,4]%%%}+%%%{-32,[2,6]%%%}+%%%{-8,[2,4]%%%}+%%%{16,[0,8
]%%%}+%%%{8,[0,6]%%%}+%%%{1,[0,4]%%%}] at parameters values [0,45.716705855]Warning, choosing root of [1,0,%%%
{-8,[2,1]%%%}+%%%{-4,[0,1]%%%},0,%%%{16,[4,2]%%%}+%%%{8,[2,2]%%%}+%%%{8,[0,4]%%%}+%%%{6,[0,2]%%%},0,%%%{-32,[4
,3]%%%}+%%%{-32,[2,5]%%%}+%%%{8,[2,3]%%%}+%%%{48,[0,5]%%%}+%%%{-4,[0,3]%%%},0,%%%{16,[4,4]%%%}+%%%{-32,[2,6]%%
%}+%%%{-8,[2,4]%%%}+%%%{16,[0,8]%%%}+%%%{8,[0,6]%%%}+%%%{1,[0,4]%%%}] at parameters values [81,87.5126850624]W
arning, choosing root of [1,0,%%%{-8,[2,1]%%%}+%%%{-4,[0,1]%%%},0,%%%{16,[4,2]%%%}+%%%{8,[2,2]%%%}+%%%{8,[0,4]
%%%}+%%%{6,[0,2]%%%},0,%%%{-32,[4,3]%%%}+%%%{-32,[2,5]%%%}+%%%{8,[2,3]%%%}+%%%{48,[0,5]%%%}+%%%{-4,[0,3]%%%},0
,%%%{16,[4,4]%%%}+%%%{-32,[2,6]%%%}+%%%{-8,[2,4]%%%}+%%%{16,[0,8]%%%}+%%%{8,[0,6]%%%}+%%%{1,[0,4]%%%}] at para
meters values [-11,23.9552401127]Warning, choosing root of [1,0,%%%{-8,[2,1]%%%}+%%%{-4,[0,1]%%%},0,%%%{16,[4,
2]%%%}+%%%{8,[2,2]%%%}+%%%{8,[0,4]%%%}+%%%{6,[0,2]%%%},0,%%%{-32,[4,3]%%%}+%%%{-32,[2,5]%%%}+%%%{8,[2,3]%%%}+%
%%{48,[0,5]%%%}+%%%{-4,[0,3]%%%},0,%%%{16,[4,4]%%%}+%%%{-32,[2,6]%%%}+%%%{-8,[2,4]%%%}+%%%{16,[0,8]%%%}+%%%{8,
[0,6]%%%}+%%%{1,[0,4]%%%}] at parameters values [93,41.1512670754]schur row 1 1.99488e-10Francis algorithm not
precise enough for[1.0,-729.896147886,110989.247229,-6300826.31183,123186130.005]Warning, choosing root of [1
,0,%%%{-12,[0,1]%%%},0,%%%{8,[4,4]%%%}+%%%{30,[0,2]%%%},0,%%%{16,[4,5]%%%}+%%%{-28,[0,3]%%%},0,%%%{16,[8,8]%%%
}+%%%{-24,[4,6]%%%}+%%%{9,[0,4]%%%}] at parameters values [-26,75.876540896]schur row 1 3.66933e-10Francis alg
orithm not precise enough for[1.0,-1159.70905962,280192.729784,-25273270.3354,785079658.236]Warning, choosing
root of [1,0,%%%{-12,[0,1]%%%},0,%%%{8,[4,4]%%%}+%%%{30,[0,2]%%%},0,%%%{16,[4,5]%%%}+%%%{-28,[0,3]%%%},0,%%%{1
6,[8,8]%%%}+%%%{-24,[4,6]%%%}+%%%{9,[0,4]%%%}] at parameters values [25,45.0210851603]Sign error (%%%{-2*a,2%%
%}+%%%{undef,3%%%})Evaluation time: 44.85Limit: Max order reached or unable to make series expansion Error: Ba
d Argument Value
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maple [C] time = 0.18, size = 129, normalized size = 1.09 \[ -\frac {\sqrt {-\frac {a \,x^{2}-1}{a \,x^{2}}}\, \sqrt {\frac {a \,x^{2}+1}{a \,x^{2}}}\, \left (-\ln \left (\frac {2 \,\mathrm {csgn}\left (\frac {1}{a}\right ) a \sqrt {-\frac {a^{2} x^{4}-1}{a^{2}}}+2}{a^{2} x^{2}}\right ) x^{4} a +\sqrt {-\frac {a^{2} x^{4}-1}{a^{2}}}\, \mathrm {csgn}\left (\frac {1}{a}\right )\right ) \mathrm {csgn}\left (\frac {1}{a}\right )}{4 x^{2} \sqrt {-\frac {a^{2} x^{4}-1}{a^{2}}}}-\frac {1}{4 x^{4} a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((1/a/x^2+(1/a/x^2-1)^(1/2)*(1/a/x^2+1)^(1/2))/x^3,x)
[Out]
-1/4*(-(a*x^2-1)/a/x^2)^(1/2)/x^2*((a*x^2+1)/a/x^2)^(1/2)*(-ln(2*(csgn(1/a)*a*(-(a^2*x^4-1)/a^2)^(1/2)+1)/a^2/
x^2)*x^4*a+(-(a^2*x^4-1)/a^2)^(1/2)*csgn(1/a))*csgn(1/a)/(-(a^2*x^4-1)/a^2)^(1/2)-1/4/x^4/a
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\frac {1}{4} \, a^{2} \log \left (\frac {2 \, \sqrt {-a^{2} x^{4} + 1}}{x^{2}} + \frac {2}{x^{2}}\right ) - \frac {1}{4} \, \sqrt {-a^{2} x^{4} + 1} a^{2} - \frac {{\left (-a^{2} x^{4} + 1\right )}^{\frac {3}{2}}}{4 \, x^{4}}}{a} - \frac {1}{4 \, a x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((1/a/x^2+(1/a/x^2-1)^(1/2)*(1/a/x^2+1)^(1/2))/x^3,x, algorithm="maxima")
[Out]
integrate(sqrt(a*x^2 + 1)*sqrt(-a*x^2 + 1)/x^5, x)/a - 1/4/(a*x^4)
________________________________________________________________________________________
mupad [B] time = 2.02, size = 71, normalized size = 0.60 \[ \frac {a\,\ln \left (\sqrt {\frac {1}{a\,x^2}-1}\,\sqrt {\frac {1}{a\,x^2}+1}+\frac {1}{a\,x^2}\right )}{4}-\frac {1}{4\,a\,x^4}-\frac {\sqrt {\frac {1}{a\,x^2}-1}\,\sqrt {\frac {1}{a\,x^2}+1}}{4\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((1/(a*x^2) - 1)^(1/2)*(1/(a*x^2) + 1)^(1/2) + 1/(a*x^2))/x^3,x)
[Out]
(a*log((1/(a*x^2) - 1)^(1/2)*(1/(a*x^2) + 1)^(1/2) + 1/(a*x^2)))/4 - 1/(4*a*x^4) - ((1/(a*x^2) - 1)^(1/2)*(1/(
a*x^2) + 1)^(1/2))/(4*x^2)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {1}{x^{5}}\, dx + \int \frac {a \sqrt {-1 + \frac {1}{a x^{2}}} \sqrt {1 + \frac {1}{a x^{2}}}}{x^{3}}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((1/a/x**2+(1/a/x**2-1)**(1/2)*(1/a/x**2+1)**(1/2))/x**3,x)
[Out]
(Integral(x**(-5), x) + Integral(a*sqrt(-1 + 1/(a*x**2))*sqrt(1 + 1/(a*x**2))/x**3, x))/a
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