3.51 \(\int e^{\text {sech}^{-1}(a x^2)} x \, dx\)
Optimal. Leaf size=68 \[ -\frac {\sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \tanh ^{-1}\left (\sqrt {1-a^2 x^4}\right )}{2 a}+\frac {1}{2} x^2 e^{\text {sech}^{-1}\left (a x^2\right )}+\frac {\log (x)}{a} \]
[Out]
1/2*(1/a/x^2+(1/a/x^2-1)^(1/2)*(1/a/x^2+1)^(1/2))*x^2+ln(x)/a-1/2*arctanh((-a^2*x^4+1)^(1/2))*(1/(a*x^2+1))^(1
/2)*(a*x^2+1)^(1/2)/a
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 68, normalized size of antiderivative = 1.00,
number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used =
{6335, 29, 259, 266, 63, 208} \[ -\frac {\sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \tanh ^{-1}\left (\sqrt {1-a^2 x^4}\right )}{2 a}+\frac {1}{2} x^2 e^{\text {sech}^{-1}\left (a x^2\right )}+\frac {\log (x)}{a} \]
Antiderivative was successfully verified.
[In]
Int[E^ArcSech[a*x^2]*x,x]
[Out]
(E^ArcSech[a*x^2]*x^2)/2 - (Sqrt[(1 + a*x^2)^(-1)]*Sqrt[1 + a*x^2]*ArcTanh[Sqrt[1 - a^2*x^4]])/(2*a) + Log[x]/
a
Rule 29
Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], 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 e^{\text {sech}^{-1}\left (a x^2\right )} x \, dx &=\frac {1}{2} e^{\text {sech}^{-1}\left (a x^2\right )} x^2+\frac {\int \frac {1}{x} \, dx}{a}+\frac {\left (\sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \int \frac {1}{x \sqrt {1-a x^2} \sqrt {1+a x^2}} \, dx}{a}\\ &=\frac {1}{2} e^{\text {sech}^{-1}\left (a x^2\right )} x^2+\frac {\log (x)}{a}+\frac {\left (\sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \int \frac {1}{x \sqrt {1-a^2 x^4}} \, dx}{a}\\ &=\frac {1}{2} e^{\text {sech}^{-1}\left (a x^2\right )} x^2+\frac {\log (x)}{a}+\frac {\left (\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 )}{4 a}\\ &=\frac {1}{2} e^{\text {sech}^{-1}\left (a x^2\right )} x^2+\frac {\log (x)}{a}-\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 )}{2 a^3}\\ &=\frac {1}{2} e^{\text {sech}^{-1}\left (a x^2\right )} x^2-\frac {\sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} \tanh ^{-1}\left (\sqrt {1-a^2 x^4}\right )}{2 a}+\frac {\log (x)}{a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.07, size = 100, normalized size = 1.47 \[ \frac {\sqrt {\frac {1-a x^2}{a x^2+1}} \left (a x^2+1\right )+2 \log \left (a x^2\right )-\log \left (a x^2 \sqrt {\frac {1-a x^2}{a x^2+1}}+\sqrt {\frac {1-a x^2}{a x^2+1}}+1\right )}{2 a} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[E^ArcSech[a*x^2]*x,x]
[Out]
(Sqrt[(1 - a*x^2)/(1 + a*x^2)]*(1 + a*x^2) + 2*Log[a*x^2] - Log[1 + Sqrt[(1 - a*x^2)/(1 + a*x^2)] + a*x^2*Sqrt
[(1 - a*x^2)/(1 + a*x^2)]])/(2*a)
________________________________________________________________________________________
fricas [B] time = 1.59, size = 133, normalized size = 1.96 \[ \frac {2 \, a x^{2} \sqrt {\frac {a x^{2} + 1}{a x^{2}}} \sqrt {-\frac {a x^{2} - 1}{a x^{2}}} - \log \left (a x^{2} \sqrt {\frac {a x^{2} + 1}{a x^{2}}} \sqrt {-\frac {a x^{2} - 1}{a x^{2}}} + 1\right ) + \log \left (a x^{2} \sqrt {\frac {a x^{2} + 1}{a x^{2}}} \sqrt {-\frac {a x^{2} - 1}{a x^{2}}} - 1\right ) + 4 \, \log \relax (x)}{4 \, a} \]
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,x, algorithm="fricas")
[Out]
1/4*(2*a*x^2*sqrt((a*x^2 + 1)/(a*x^2))*sqrt(-(a*x^2 - 1)/(a*x^2)) - log(a*x^2*sqrt((a*x^2 + 1)/(a*x^2))*sqrt(-
(a*x^2 - 1)/(a*x^2)) + 1) + log(a*x^2*sqrt((a*x^2 + 1)/(a*x^2))*sqrt(-(a*x^2 - 1)/(a*x^2)) - 1) + 4*log(x))/a
________________________________________________________________________________________
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,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,%%%{-4,[1,0]%%%},0,%%%{4,[4,4]%%%}] at parameters values [-23,65]schur row 1 1.68784e-10Franci
s algorithm not precise enough for[1.0,-96.6277521998,1945.1921865,-14619.0760005,37837.726424]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 [39.1803401988,-44]schur row 1 3.85284e
-10Francis algorithm not precise enough for[1.0,-1161.32542683,280974.322293,-25379093.0373,789465697.88]Bad c
onditionned root j= 2 value 96.7723338924 ratio 0.000375252022965 mindist 0.00603026989475Warning, choosing ro
ot 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-10Fra
ncis algorithm not precise enough for[1.0,-1129.51443638,265792.262915,-23350148.7365,706455270.256]Bad condit
ionned 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
algorithm not precise enough for[1.0,-637.349737572,84628.0599964,-4195152.25343,71619085.3875]Warning, choos
ing 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 ro
ot 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 [55.0
343274642,0]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]%%%}+%%%{-3
2,[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,-1
7]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 p
arameters 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 ro
ot 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 [61.7
431004322,-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,%%%{-3
2,[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 p
arameters 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 r
oot 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
]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 pa
rameters 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 n
ot 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 a
lgorithm not precise enough for[1.0,-1159.70905962,280192.729784,-25273270.3354,785079658.236]Warning, choosin
g 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 [25,45.0210851603]Sign error (%%%{-2*a,2
%%%}+%%%{undef,3%%%})Evaluation time: 35.6Limit: Max order reached or unable to make series expansion Error: B
ad Argument Value
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maple [C] time = 0.19, size = 127, normalized size = 1.87 \[ \frac {\sqrt {-\frac {a \,x^{2}-1}{a \,x^{2}}}\, x^{2} \sqrt {\frac {a \,x^{2}+1}{a \,x^{2}}}\, \left (\mathrm {csgn}\left (\frac {1}{a}\right ) a \sqrt {-\frac {a^{2} x^{4}-1}{a^{2}}}-\ln \left (\frac {2 \,\mathrm {csgn}\left (\frac {1}{a}\right ) a \sqrt {-\frac {a^{2} x^{4}-1}{a^{2}}}+2}{a^{2} x^{2}}\right )\right ) \mathrm {csgn}\left (\frac {1}{a}\right )}{2 a \sqrt {-\frac {a^{2} x^{4}-1}{a^{2}}}}+\frac {\ln \relax (x )}{a} \]
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,x)
[Out]
1/2*(-(a*x^2-1)/a/x^2)^(1/2)*x^2*((a*x^2+1)/a/x^2)^(1/2)*(csgn(1/a)*a*(-(a^2*x^4-1)/a^2)^(1/2)-ln(2*(csgn(1/a)
*a*(-(a^2*x^4-1)/a^2)^(1/2)+1)/a^2/x^2))*csgn(1/a)/a/(-(a^2*x^4-1)/a^2)^(1/2)+ln(x)/a
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\frac {1}{2} \, \sqrt {-a^{2} x^{4} + 1} - \frac {1}{2} \, \log \left (\frac {2 \, \sqrt {-a^{2} x^{4} + 1}}{x^{2}} + \frac {2}{x^{2}}\right )}{a} + \frac {\log \relax (x)}{a} \]
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,x, algorithm="maxima")
[Out]
integrate(sqrt(a*x^2 + 1)*sqrt(-a*x^2 + 1)/x, x)/a + log(x)/a
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mupad [B] time = 3.20, size = 182, normalized size = 2.68 \[ \frac {\ln \relax (x)}{a}-\frac {2\,\mathrm {atanh}\left (\frac {\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}}{\sqrt {\frac {1}{a\,x^2}+1}-1}\right )}{a}+\frac {\frac {5\,{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^2}+1}{\frac {8\,a\,\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}{\sqrt {\frac {1}{a\,x^2}+1}-1}+\frac {8\,a\,{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^3}{{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^3}}+\frac {\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}}{8\,a\,\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x*((1/(a*x^2) - 1)^(1/2)*(1/(a*x^2) + 1)^(1/2) + 1/(a*x^2)),x)
[Out]
log(x)/a - (2*atanh(((1/(a*x^2) - 1)^(1/2) - 1i)/((1/(a*x^2) + 1)^(1/2) - 1)))/a + ((5*((1/(a*x^2) - 1)^(1/2)
- 1i)^2)/((1/(a*x^2) + 1)^(1/2) - 1)^2 + 1)/((8*a*((1/(a*x^2) - 1)^(1/2) - 1i))/((1/(a*x^2) + 1)^(1/2) - 1) +
(8*a*((1/(a*x^2) - 1)^(1/2) - 1i)^3)/((1/(a*x^2) + 1)^(1/2) - 1)^3) + ((1/(a*x^2) - 1)^(1/2) - 1i)/(8*a*((1/(a
*x^2) + 1)^(1/2) - 1))
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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,x)
[Out]
Timed out
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