3.334 \(\int \frac {1}{(a+i b \sin ^{-1}(1-i d x^2))^{7/2}} \, dx\)
Optimal. Leaf size=389 \[ -\frac {\sqrt {d^2 x^4+2 i d x^2}}{15 b^3 d x \sqrt {a+i b \sin ^{-1}\left (1-i d x^2\right )}}-\frac {\sqrt {\pi } \left (-\frac {i}{b}\right )^{3/2} x \left (\cosh \left (\frac {a}{2 b}\right )-i \sinh \left (\frac {a}{2 b}\right )\right ) C\left (\frac {\sqrt {-\frac {i}{b}} \sqrt {a+i b \sin ^{-1}\left (1-i d x^2\right )}}{\sqrt {\pi }}\right )}{15 b^2 \left (\cos \left (\frac {1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )-\sin \left (\frac {1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )\right )}+\frac {\sqrt {\pi } \left (-\frac {i}{b}\right )^{3/2} x \left (\cosh \left (\frac {a}{2 b}\right )+i \sinh \left (\frac {a}{2 b}\right )\right ) S\left (\frac {\sqrt {-\frac {i}{b}} \sqrt {a+i b \sin ^{-1}\left (1-i d x^2\right )}}{\sqrt {\pi }}\right )}{15 b^2 \left (\cos \left (\frac {1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )-\sin \left (\frac {1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )\right )}-\frac {x}{15 b^2 \left (a+i b \sin ^{-1}\left (1-i d x^2\right )\right )^{3/2}}-\frac {\sqrt {d^2 x^4+2 i d x^2}}{5 b d x \left (a+i b \sin ^{-1}\left (1-i d x^2\right )\right )^{5/2}} \]
[Out]
-1/15*x/b^2/(a-I*b*arcsin(-1+I*d*x^2))^(3/2)-1/15*(-I/b)^(3/2)*x*FresnelC((-I/b)^(1/2)*(a-I*b*arcsin(-1+I*d*x^
2))^(1/2)/Pi^(1/2))*(cosh(1/2*a/b)-I*sinh(1/2*a/b))*Pi^(1/2)/b^2/(cos(1/2*arcsin(-1+I*d*x^2))+sin(1/2*arcsin(-
1+I*d*x^2)))+1/15*(-I/b)^(3/2)*x*FresnelS((-I/b)^(1/2)*(a-I*b*arcsin(-1+I*d*x^2))^(1/2)/Pi^(1/2))*(cosh(1/2*a/
b)+I*sinh(1/2*a/b))*Pi^(1/2)/b^2/(cos(1/2*arcsin(-1+I*d*x^2))+sin(1/2*arcsin(-1+I*d*x^2)))-1/5*(2*I*d*x^2+d^2*
x^4)^(1/2)/b/d/x/(a-I*b*arcsin(-1+I*d*x^2))^(5/2)-1/15*(2*I*d*x^2+d^2*x^4)^(1/2)/b^3/d/x/(a-I*b*arcsin(-1+I*d*
x^2))^(1/2)
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 389, normalized size of antiderivative = 1.00,
number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used =
{4828, 4822} \[ -\frac {\sqrt {d^2 x^4+2 i d x^2}}{15 b^3 d x \sqrt {a+i b \sin ^{-1}\left (1-i d x^2\right )}}-\frac {\sqrt {\pi } \left (-\frac {i}{b}\right )^{3/2} x \left (\cosh \left (\frac {a}{2 b}\right )-i \sinh \left (\frac {a}{2 b}\right )\right ) \text {FresnelC}\left (\frac {\sqrt {-\frac {i}{b}} \sqrt {a+i b \sin ^{-1}\left (1-i d x^2\right )}}{\sqrt {\pi }}\right )}{15 b^2 \left (\cos \left (\frac {1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )-\sin \left (\frac {1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )\right )}+\frac {\sqrt {\pi } \left (-\frac {i}{b}\right )^{3/2} x \left (\cosh \left (\frac {a}{2 b}\right )+i \sinh \left (\frac {a}{2 b}\right )\right ) S\left (\frac {\sqrt {-\frac {i}{b}} \sqrt {a+i b \sin ^{-1}\left (1-i d x^2\right )}}{\sqrt {\pi }}\right )}{15 b^2 \left (\cos \left (\frac {1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )-\sin \left (\frac {1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )\right )}-\frac {x}{15 b^2 \left (a+i b \sin ^{-1}\left (1-i d x^2\right )\right )^{3/2}}-\frac {\sqrt {d^2 x^4+2 i d x^2}}{5 b d x \left (a+i b \sin ^{-1}\left (1-i d x^2\right )\right )^{5/2}} \]
Antiderivative was successfully verified.
[In]
Int[(a + I*b*ArcSin[1 - I*d*x^2])^(-7/2),x]
[Out]
-Sqrt[(2*I)*d*x^2 + d^2*x^4]/(5*b*d*x*(a + I*b*ArcSin[1 - I*d*x^2])^(5/2)) - x/(15*b^2*(a + I*b*ArcSin[1 - I*d
*x^2])^(3/2)) - Sqrt[(2*I)*d*x^2 + d^2*x^4]/(15*b^3*d*x*Sqrt[a + I*b*ArcSin[1 - I*d*x^2]]) - (((-I)/b)^(3/2)*S
qrt[Pi]*x*FresnelC[(Sqrt[(-I)/b]*Sqrt[a + I*b*ArcSin[1 - I*d*x^2]])/Sqrt[Pi]]*(Cosh[a/(2*b)] - I*Sinh[a/(2*b)]
))/(15*b^2*(Cos[ArcSin[1 - I*d*x^2]/2] - Sin[ArcSin[1 - I*d*x^2]/2])) + (((-I)/b)^(3/2)*Sqrt[Pi]*x*FresnelS[(S
qrt[(-I)/b]*Sqrt[a + I*b*ArcSin[1 - I*d*x^2]])/Sqrt[Pi]]*(Cosh[a/(2*b)] + I*Sinh[a/(2*b)]))/(15*b^2*(Cos[ArcSi
n[1 - I*d*x^2]/2] - Sin[ArcSin[1 - I*d*x^2]/2]))
Rule 4822
Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)^2]*(b_.))^(-3/2), x_Symbol] :> -Simp[Sqrt[-2*c*d*x^2 - d^2*x^4]/(b*d*x*S
qrt[a + b*ArcSin[c + d*x^2]]), x] + (-Simp[((c/b)^(3/2)*Sqrt[Pi]*x*(Cos[a/(2*b)] + c*Sin[a/(2*b)])*FresnelC[Sq
rt[c/(Pi*b)]*Sqrt[a + b*ArcSin[c + d*x^2]]])/(Cos[(1/2)*ArcSin[c + d*x^2]] - c*Sin[ArcSin[c + d*x^2]/2]), x] +
Simp[((c/b)^(3/2)*Sqrt[Pi]*x*(Cos[a/(2*b)] - c*Sin[a/(2*b)])*FresnelS[Sqrt[c/(Pi*b)]*Sqrt[a + b*ArcSin[c + d*
x^2]]])/(Cos[(1/2)*ArcSin[c + d*x^2]] - c*Sin[ArcSin[c + d*x^2]/2]), x]) /; FreeQ[{a, b, c, d}, x] && EqQ[c^2,
1]
Rule 4828
Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)^2]*(b_.))^(n_), x_Symbol] :> Simp[(x*(a + b*ArcSin[c + d*x^2])^(n + 2))/
(4*b^2*(n + 1)*(n + 2)), x] + (-Dist[1/(4*b^2*(n + 1)*(n + 2)), Int[(a + b*ArcSin[c + d*x^2])^(n + 2), x], x]
+ Simp[(Sqrt[-2*c*d*x^2 - d^2*x^4]*(a + b*ArcSin[c + d*x^2])^(n + 1))/(2*b*d*(n + 1)*x), x]) /; FreeQ[{a, b, c
, d}, x] && EqQ[c^2, 1] && LtQ[n, -1] && NeQ[n, -2]
Rubi steps
\begin {align*} \int \frac {1}{\left (a+i b \sin ^{-1}\left (1-i d x^2\right )\right )^{7/2}} \, dx &=-\frac {\sqrt {2 i d x^2+d^2 x^4}}{5 b d x \left (a+i b \sin ^{-1}\left (1-i d x^2\right )\right )^{5/2}}-\frac {x}{15 b^2 \left (a+i b \sin ^{-1}\left (1-i d x^2\right )\right )^{3/2}}+\frac {\int \frac {1}{\left (a+i b \sin ^{-1}\left (1-i d x^2\right )\right )^{3/2}} \, dx}{15 b^2}\\ &=-\frac {\sqrt {2 i d x^2+d^2 x^4}}{5 b d x \left (a+i b \sin ^{-1}\left (1-i d x^2\right )\right )^{5/2}}-\frac {x}{15 b^2 \left (a+i b \sin ^{-1}\left (1-i d x^2\right )\right )^{3/2}}-\frac {\sqrt {2 i d x^2+d^2 x^4}}{15 b^3 d x \sqrt {a+i b \sin ^{-1}\left (1-i d x^2\right )}}-\frac {\left (-\frac {i}{b}\right )^{3/2} \sqrt {\pi } x C\left (\frac {\sqrt {-\frac {i}{b}} \sqrt {a+i b \sin ^{-1}\left (1-i d x^2\right )}}{\sqrt {\pi }}\right ) \left (\cosh \left (\frac {a}{2 b}\right )-i \sinh \left (\frac {a}{2 b}\right )\right )}{15 b^2 \left (\cos \left (\frac {1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )-\sin \left (\frac {1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )\right )}+\frac {\left (-\frac {i}{b}\right )^{3/2} \sqrt {\pi } x S\left (\frac {\sqrt {-\frac {i}{b}} \sqrt {a+i b \sin ^{-1}\left (1-i d x^2\right )}}{\sqrt {\pi }}\right ) \left (\cosh \left (\frac {a}{2 b}\right )+i \sinh \left (\frac {a}{2 b}\right )\right )}{15 b^2 \left (\cos \left (\frac {1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )-\sin \left (\frac {1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.99, size = 365, normalized size = 0.94 \[ \frac {-\frac {\sqrt {\pi } \left (-\frac {i}{b}\right )^{3/2} x \left (\cosh \left (\frac {a}{2 b}\right )-i \sinh \left (\frac {a}{2 b}\right )\right ) C\left (\frac {\sqrt {-\frac {i}{b}} \sqrt {a+i b \sin ^{-1}\left (1-i d x^2\right )}}{\sqrt {\pi }}\right )}{\cos \left (\frac {1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )-\sin \left (\frac {1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )}+\frac {\sqrt {\pi } \left (-\frac {i}{b}\right )^{3/2} x \left (\cosh \left (\frac {a}{2 b}\right )+i \sinh \left (\frac {a}{2 b}\right )\right ) S\left (\frac {\sqrt {-\frac {i}{b}} \sqrt {a+i b \sin ^{-1}\left (1-i d x^2\right )}}{\sqrt {\pi }}\right )}{\cos \left (\frac {1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )-\sin \left (\frac {1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )}+\frac {-\left (x^2 \left (a+i b \sin ^{-1}\left (1-i d x^2\right )\right )\right )+\frac {\sqrt {d x^2 \left (d x^2+2 i\right )} \left (b \sin ^{-1}\left (1-i d x^2\right )-i a\right )^2}{b d}-\frac {3 b \sqrt {d x^2 \left (d x^2+2 i\right )}}{d}}{x \left (a+i b \sin ^{-1}\left (1-i d x^2\right )\right )^{5/2}}}{15 b^2} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + I*b*ArcSin[1 - I*d*x^2])^(-7/2),x]
[Out]
(((-3*b*Sqrt[d*x^2*(2*I + d*x^2)])/d - x^2*(a + I*b*ArcSin[1 - I*d*x^2]) + (Sqrt[d*x^2*(2*I + d*x^2)]*((-I)*a
+ b*ArcSin[1 - I*d*x^2])^2)/(b*d))/(x*(a + I*b*ArcSin[1 - I*d*x^2])^(5/2)) - (((-I)/b)^(3/2)*Sqrt[Pi]*x*Fresne
lC[(Sqrt[(-I)/b]*Sqrt[a + I*b*ArcSin[1 - I*d*x^2]])/Sqrt[Pi]]*(Cosh[a/(2*b)] - I*Sinh[a/(2*b)]))/(Cos[ArcSin[1
- I*d*x^2]/2] - Sin[ArcSin[1 - I*d*x^2]/2]) + (((-I)/b)^(3/2)*Sqrt[Pi]*x*FresnelS[(Sqrt[(-I)/b]*Sqrt[a + I*b*
ArcSin[1 - I*d*x^2]])/Sqrt[Pi]]*(Cosh[a/(2*b)] + I*Sinh[a/(2*b)]))/(Cos[ArcSin[1 - I*d*x^2]/2] - Sin[ArcSin[1
- I*d*x^2]/2]))/(15*b^2)
________________________________________________________________________________________
fricas [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+b*arcsinh(I+d*x^2))^(7/2),x, algorithm="fricas")
[Out]
Exception raised: TypeError >> Error detected within library code: integrate: implementation incomplete (co
nstant residues)
________________________________________________________________________________________
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+b*arcsinh(I+d*x^2))^(7/2),x, algorithm="giac")
[Out]
Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was
done assuming [d,x]=[89,77]schur row 1 1.22263e-09Francis algorithm not precise enough for[1.0,0.0,-1.67068342
657e+12,-1.17545053497e+18,-2.32598592657e+23]Bad conditionned root j= 2 value -527644.406418 ratio 8.33415008
204 mindist 33.7256546889Warning, need to choose a branch for the root of a polynomial with parameters. This m
ight be wrong.The choice was done assuming [d,x]=[-66,8]schur row 1 9.91586e-07Francis algorithm not precise e
nough for[1.0,0.0,-107053064.0,602922946560,-9.55030161457e+14]Bad conditionned root j= 0 value 4231.08593187
ratio 1.71555050372 mindist 7.08837726861schur row 1 9.85057e-10Francis algorithm not precise enough for[1.0,0
.0,-137282971022,-2.76877787308e+16,-1.57055117809e+21]Unable to isolate roots number Vector [0,1][-0.15126799
6566355e6,-0.151267183209932e6]Bad conditionned root j= 2 value -151253.820244 ratio 2.23329136282 mindist 13.
3629663866Warning, choosing root of [1,0,%%%{-6,[2,4]%%%}+%%%{-8,[0,0]%%%},%%%{-8,[3,6]%%%}+%%%{-32,[1,2]%%%},
%%%{-3,[4,8]%%%}+%%%{-24,[2,4]%%%}+%%%{16,[0,0]%%%}] at parameters values [63,-49]Warning, need to choose a br
anch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [d,x]=[84,
-65]Bad conditionned root j= 1 value -354917.321547 ratio 3.58199229828 mindist 17.6876820158Bad conditionned
root j= 2 value -354899.633865 ratio 36.3553454927 mindist 16.5892679524Bad conditionned root j= 1 value -3388
0.7669046 ratio 0.612379387673 mindist 0.825253607938Warning, choosing root of [1,0,%%%{-6,[2,4]%%%}+%%%{-8,[0
,0]%%%},%%%{-8,[3,6]%%%}+%%%{-32,[1,2]%%%},%%%{-3,[4,8]%%%}+%%%{-24,[2,4]%%%}+%%%{16,[0,0]%%%}] at parameters
values [70,22]Warning, need to choose a branch for the root of a polynomial with parameters. This might be wro
ng.The choice was done assuming [d,x]=[30,-21]Warning, need to choose a branch for the root of a polynomial wi
th parameters. This might be wrong.The choice was done assuming [d,x]=[-24,-63]Warning, need to choose a branc
h for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [d,x]=[11,52]
schur row 3 -3.22193e-07Warning, choosing root of [1,0,%%%{-4,[2,4]%%%}+%%%{-8,[0,0]%%%},%%%{-4,[3,6]%%%}+%%%{
-16,[1,2]%%%},%%%{-1,[4,8]%%%}+%%%{-4,[2,4]%%%}+%%%{16,[0,0]%%%}] at parameters values [18,-49]Warning, choosi
ng root of [1,0,%%%{-4,[2,4]%%%}+%%%{-8,[0,0]%%%},%%%{-4,[3,6]%%%}+%%%{-16,[1,2]%%%},%%%{-1,[4,8]%%%}+%%%{-4,[
2,4]%%%}+%%%{16,[0,0]%%%}] at parameters values [-33,-70]schur row 3 1.93169e-07Warning, choosing root of [1,0
,%%%{-4,[2,4]%%%}+%%%{-8,[0,0]%%%},%%%{-4,[3,6]%%%}+%%%{-16,[1,2]%%%},%%%{-1,[4,8]%%%}+%%%{-4,[2,4]%%%}+%%%{16
,[0,0]%%%}] at parameters values [8,63]Warning, need to choose a branch for the root of a polynomial with para
meters. This might be wrong.The choice was done assuming [d,x]=[-8,-94]schur row 1 1.15782e-08Francis algorith
m not precise enough for[1.0,0.0,-29980760072,2.82570662547e+15,-7.49038312879e+19]Bad conditionned root j= 0
value 70702.5530055 ratio 3.30491028239 mindist 18.5135468412Warning, need to choose a branch for the root of
a polynomial with parameters. This might be wrong.The choice was done assuming [d,x]=[-8,-38]schur row 1 3.472
75e-08Francis algorithm not precise enough for[1.0,0.0,-800692232.0,1.23327957985e+13,-5.34256730006e+16]Bad c
onditionned root j= 0 value 11555.1783957 ratio 1.30663425153 mindist 3.17957631341schur row 1 5.69583e-10Fran
cis algorithm not precise enough for[1.0,0.0,-1.32328584376e+12,-8.28597485843e+17,-1.45923785361e+23]Unable t
o isolate roots number Vector [0,2][-0.469612557480389e6,-0.469615331440520e6]Bad conditionned root j= 1 value
-469647.111085 ratio 3.72291056632 mindist 31.7796449475Warning, choosing root of [1,0,%%%{-6,[2,4]%%%}+%%%{-
8,[0,0]%%%},%%%{-8,[3,6]%%%}+%%%{-32,[1,2]%%%},%%%{-3,[4,8]%%%}+%%%{-24,[2,4]%%%}+%%%{16,[0,0]%%%}] at paramet
ers values [65,-85]Warning, need to choose a branch for the root of a polynomial with parameters. This might b
e wrong.The choice was done assuming [d,x]=[-86,73]Bad conditionned root j= 0 value 458272.621508 ratio 152.16
9961926 mindist 255.569113305Bad conditionned root j= 2 value 458592.326104 ratio 66.2919566381 mindist 302.96
7080007Bad conditionned root j= 1 value -770132.243981 ratio 141.661540265 mindist 52.7628472447Bad conditionn
ed root j= 2 value -770187.27489 ratio 12.2575973716 mindist 55.0309094898Unable to isolate roots number Vecto
r [1,3][-0.770132243980564e6,-0.770079481133319e6]Warning, choosing root of [1,0,%%%{-6,[2,4]%%%}+%%%{-8,[0,0]
%%%},%%%{-8,[3,6]%%%}+%%%{-32,[1,2]%%%},%%%{-3,[4,8]%%%}+%%%{-24,[2,4]%%%}+%%%{16,[0,0]%%%}] at parameters val
ues [93,91]Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.
The choice was done assuming [d,x]=[-82,36]Warning, need to choose a branch for the root of a polynomial with
parameters. This might be wrong.The choice was done assuming [d,x]=[9,15]Warning, need to choose a branch for
the root of a polynomial with parameters. This might be wrong.The choice was done assuming [d,x]=[-11,52]schur
row 3 1.00361e-07Warning, choosing root of [1,0,%%%{-4,[2,4]%%%}+%%%{-8,[0,0]%%%},%%%{-4,[3,6]%%%}+%%%{-16,[1
,2]%%%},%%%{-1,[4,8]%%%}+%%%{-4,[2,4]%%%}+%%%{16,[0,0]%%%}] at parameters values [79,-88]Warning, choosing roo
t of [1,0,%%%{-4,[2,4]%%%}+%%%{-8,[0,0]%%%},%%%{-4,[3,6]%%%}+%%%{-16,[1,2]%%%},%%%{-1,[4,8]%%%}+%%%{-4,[2,4]%%
%}+%%%{16,[0,0]%%%}] at parameters values [9,6]Warning, choosing root of [1,0,%%%{-4,[2,4]%%%}+%%%{-8,[0,0]%%%
},%%%{-4,[3,6]%%%}+%%%{-16,[1,2]%%%},%%%{-1,[4,8]%%%}+%%%{-4,[2,4]%%%}+%%%{16,[0,0]%%%}] at parameters values
[-69,-8]Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The
choice was done assuming [d,t_nostep]=[91,52]Bad conditionned root j= 1 value -246064.602658 ratio 4.13844710
897 mindist 2.78272759262schur row 1 9.14435e-09Francis algorithm not precise enough for[1.0,0.0,-2124702752,-
5.33102089176e+13,-3.76196821029e+17]Bad conditionned root j= 1 value -18820.5074185 ratio 0.838796175516 mind
ist 3.20710914307Warning, choosing root of [1,0,%%%{-6,[2,4]%%%}+%%%{-8,[0,0]%%%},%%%{-8,[3,6]%%%}+%%%{-32,[1,
2]%%%},%%%{-3,[4,8]%%%}+%%%{-24,[2,4]%%%}+%%%{16,[0,0]%%%}] at parameters values [2,97]Warning, need to choose
a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [d,t_
nostep]=[-82,75]schur row 1 1.2297e-10Francis algorithm not precise enough for[1.0,0.0,-1.27650937501e+12,7.85
05326564e+17,-1.35789682044e+23]Bad conditionned root j= 2 value 461240.783946 ratio 2.37092177178 mindist 11.
7745041251schur row 1 1.96129e-07Francis algorithm not precise enough for[1.0,0.0,-288648584.0,2.6694222528e+1
2,-6.94316785683e+15]Bad conditionned root j= 0 value 6940.98554667 ratio 1.29997372862 mindist 4.99589011288W
arning, choosing root of [1,0,%%%{-6,[2,4]%%%}+%%%{-8,[0,0]%%%},%%%{-8,[3,6]%%%}+%%%{-32,[1,2]%%%},%%%{-3,[4,8
]%%%}+%%%{-24,[2,4]%%%}+%%%{16,[0,0]%%%}] at parameters values [-24,-17]Evaluation time: 1.97sym2poly/r2sym(co
nst gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
________________________________________________________________________________________
maple [F] time = 0.13, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a +b \arcsinh \left (d \,x^{2}+i\right )\right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(a+b*arcsinh(I+d*x^2))^(7/2),x)
[Out]
int(1/(a+b*arcsinh(I+d*x^2))^(7/2),x)
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b \operatorname {arsinh}\left (d x^{2} + i\right ) + a\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b*arcsinh(I+d*x^2))^(7/2),x, algorithm="maxima")
[Out]
integrate((b*arcsinh(d*x^2 + I) + a)^(-7/2), x)
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (a+b\,\mathrm {asinh}\left (d\,x^2+1{}\mathrm {i}\right )\right )}^{7/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(a + b*asinh(d*x^2 + 1i))^(7/2),x)
[Out]
int(1/(a + b*asinh(d*x^2 + 1i))^(7/2), x)
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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+b*asinh(I+d*x**2))**(7/2),x)
[Out]
Timed out
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