\(\int \frac {1}{(a-b \arcsin (1-d x^2))^{7/2}} \, dx\) [305]

Optimal result

Integrand size = 18, antiderivative size = 339 \[ \int \frac {1}{\left (a-b \arcsin \left (1-d x^2\right )\right )^{7/2}} \, dx=-\frac {\sqrt {2 d x^2-d^2 x^4}}{5 b d x \left (a-b \arcsin \left (1-d x^2\right )\right )^{5/2}}+\frac {x}{15 b^2 \left (a-b \arcsin \left (1-d x^2\right )\right )^{3/2}}+\frac {\sqrt {2 d x^2-d^2 x^4}}{15 b^3 d x \sqrt {a-b \arcsin \left (1-d x^2\right )}}+\frac {\left (-\frac {1}{b}\right )^{7/2} \sqrt {\pi } x \operatorname {FresnelC}\left (\frac {\sqrt {-\frac {1}{b}} \sqrt {a-b \arcsin \left (1-d x^2\right )}}{\sqrt {\pi }}\right ) \left (\cos \left (\frac {a}{2 b}\right )-\sin \left (\frac {a}{2 b}\right )\right )}{15 \left (\cos \left (\frac {1}{2} \arcsin \left (1-d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1-d x^2\right )\right )\right )}-\frac {\left (-\frac {1}{b}\right )^{7/2} \sqrt {\pi } x \operatorname {FresnelS}\left (\frac {\sqrt {-\frac {1}{b}} \sqrt {a-b \arcsin \left (1-d x^2\right )}}{\sqrt {\pi }}\right ) \left (\cos \left (\frac {a}{2 b}\right )+\sin \left (\frac {a}{2 b}\right )\right )}{15 \left (\cos \left (\frac {1}{2} \arcsin \left (1-d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1-d x^2\right )\right )\right )} \] Output:

-1/5*(-d^2*x^4+2*d*x^2)^(1/2)/b/d/x/(a+b*arcsin(d*x^2-1))^(5/2)+1/15*x/b^2 
/(a+b*arcsin(d*x^2-1))^(3/2)+1/15*(-d^2*x^4+2*d*x^2)^(1/2)/b^3/d/x/(a+b*ar 
csin(d*x^2-1))^(1/2)+(-1/b)^(7/2)*Pi^(1/2)*x*FresnelC((-1/b)^(1/2)*(a+b*ar 
csin(d*x^2-1))^(1/2)/Pi^(1/2))*(cos(1/2*a/b)-sin(1/2*a/b))/(15*cos(1/2*arc 
sin(d*x^2-1))+15*sin(1/2*arcsin(d*x^2-1)))-(-1/b)^(7/2)*Pi^(1/2)*x*Fresnel 
S((-1/b)^(1/2)*(a+b*arcsin(d*x^2-1))^(1/2)/Pi^(1/2))*(cos(1/2*a/b)+sin(1/2 
*a/b))/(15*cos(1/2*arcsin(d*x^2-1))+15*sin(1/2*arcsin(d*x^2-1)))
 

Mathematica [A] (verified)

Time = 0.60 (sec) , antiderivative size = 319, normalized size of antiderivative = 0.94 \[ \int \frac {1}{\left (a-b \arcsin \left (1-d x^2\right )\right )^{7/2}} \, dx=\frac {\frac {-\frac {3 b \sqrt {d x^2 \left (2-d x^2\right )}}{d}+x^2 \left (a-b \arcsin \left (1-d x^2\right )\right )+\frac {\sqrt {d x^2 \left (2-d x^2\right )} \left (a-b \arcsin \left (1-d x^2\right )\right )^2}{b d}}{x \left (a-b \arcsin \left (1-d x^2\right )\right )^{5/2}}+\frac {\left (-\frac {1}{b}\right )^{3/2} \sqrt {\pi } x \operatorname {FresnelC}\left (\frac {\sqrt {-\frac {1}{b}} \sqrt {a-b \arcsin \left (1-d x^2\right )}}{\sqrt {\pi }}\right ) \left (\cos \left (\frac {a}{2 b}\right )-\sin \left (\frac {a}{2 b}\right )\right )}{\cos \left (\frac {1}{2} \arcsin \left (1-d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1-d x^2\right )\right )}+\frac {\left (-\frac {1}{b}\right )^{5/2} b \sqrt {\pi } x \operatorname {FresnelS}\left (\frac {\sqrt {-\frac {1}{b}} \sqrt {a-b \arcsin \left (1-d x^2\right )}}{\sqrt {\pi }}\right ) \left (\cos \left (\frac {a}{2 b}\right )+\sin \left (\frac {a}{2 b}\right )\right )}{\cos \left (\frac {1}{2} \arcsin \left (1-d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1-d x^2\right )\right )}}{15 b^2} \] Input:

Integrate[(a - b*ArcSin[1 - d*x^2])^(-7/2),x]
 

Output:

(((-3*b*Sqrt[d*x^2*(2 - d*x^2)])/d + x^2*(a - b*ArcSin[1 - d*x^2]) + (Sqrt 
[d*x^2*(2 - d*x^2)]*(a - b*ArcSin[1 - d*x^2])^2)/(b*d))/(x*(a - b*ArcSin[1 
 - d*x^2])^(5/2)) + ((-b^(-1))^(3/2)*Sqrt[Pi]*x*FresnelC[(Sqrt[-b^(-1)]*Sq 
rt[a - b*ArcSin[1 - d*x^2]])/Sqrt[Pi]]*(Cos[a/(2*b)] - Sin[a/(2*b)]))/(Cos 
[ArcSin[1 - d*x^2]/2] - Sin[ArcSin[1 - d*x^2]/2]) + ((-b^(-1))^(5/2)*b*Sqr 
t[Pi]*x*FresnelS[(Sqrt[-b^(-1)]*Sqrt[a - b*ArcSin[1 - d*x^2]])/Sqrt[Pi]]*( 
Cos[a/(2*b)] + Sin[a/(2*b)]))/(Cos[ArcSin[1 - d*x^2]/2] - Sin[ArcSin[1 - d 
*x^2]/2]))/(15*b^2)
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {5327, 5321}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {1}{\left (a-b \arcsin \left (1-d x^2\right )\right )^{7/2}} \, dx\)

\(\Big \downarrow \) 5327

\(\displaystyle -\frac {\int \frac {1}{\left (a-b \arcsin \left (1-d x^2\right )\right )^{3/2}}dx}{15 b^2}+\frac {x}{15 b^2 \left (a-b \arcsin \left (1-d x^2\right )\right )^{3/2}}-\frac {\sqrt {2 d x^2-d^2 x^4}}{5 b d x \left (a-b \arcsin \left (1-d x^2\right )\right )^{5/2}}\)

\(\Big \downarrow \) 5321

\(\displaystyle -\frac {-\frac {\sqrt {2 d x^2-d^2 x^4}}{b d x \sqrt {a-b \arcsin \left (1-d x^2\right )}}-\frac {\sqrt {\pi } \left (-\frac {1}{b}\right )^{3/2} x \left (\cos \left (\frac {a}{2 b}\right )-\sin \left (\frac {a}{2 b}\right )\right ) \operatorname {FresnelC}\left (\frac {\sqrt {-\frac {1}{b}} \sqrt {a-b \arcsin \left (1-d x^2\right )}}{\sqrt {\pi }}\right )}{\cos \left (\frac {1}{2} \arcsin \left (1-d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1-d x^2\right )\right )}+\frac {\sqrt {\pi } \left (-\frac {1}{b}\right )^{3/2} x \left (\sin \left (\frac {a}{2 b}\right )+\cos \left (\frac {a}{2 b}\right )\right ) \operatorname {FresnelS}\left (\frac {\sqrt {-\frac {1}{b}} \sqrt {a-b \arcsin \left (1-d x^2\right )}}{\sqrt {\pi }}\right )}{\cos \left (\frac {1}{2} \arcsin \left (1-d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1-d x^2\right )\right )}}{15 b^2}+\frac {x}{15 b^2 \left (a-b \arcsin \left (1-d x^2\right )\right )^{3/2}}-\frac {\sqrt {2 d x^2-d^2 x^4}}{5 b d x \left (a-b \arcsin \left (1-d x^2\right )\right )^{5/2}}\)

Input:

Int[(a - b*ArcSin[1 - d*x^2])^(-7/2),x]
 

Output:

-1/5*Sqrt[2*d*x^2 - d^2*x^4]/(b*d*x*(a - b*ArcSin[1 - d*x^2])^(5/2)) + x/( 
15*b^2*(a - b*ArcSin[1 - d*x^2])^(3/2)) - (-(Sqrt[2*d*x^2 - d^2*x^4]/(b*d* 
x*Sqrt[a - b*ArcSin[1 - d*x^2]])) - ((-b^(-1))^(3/2)*Sqrt[Pi]*x*FresnelC[( 
Sqrt[-b^(-1)]*Sqrt[a - b*ArcSin[1 - d*x^2]])/Sqrt[Pi]]*(Cos[a/(2*b)] - Sin 
[a/(2*b)]))/(Cos[ArcSin[1 - d*x^2]/2] - Sin[ArcSin[1 - d*x^2]/2]) + ((-b^( 
-1))^(3/2)*Sqrt[Pi]*x*FresnelS[(Sqrt[-b^(-1)]*Sqrt[a - b*ArcSin[1 - d*x^2] 
])/Sqrt[Pi]]*(Cos[a/(2*b)] + Sin[a/(2*b)]))/(Cos[ArcSin[1 - d*x^2]/2] - Si 
n[ArcSin[1 - d*x^2]/2]))/(15*b^2)
 

Defintions of rubi rules used

rule 5321

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*Sqrt[a + b*ArcSin[c + d*x^2]]), x] + (-Si 
mp[(c/b)^(3/2)*Sqrt[Pi]*x*(Cos[a/(2*b)] + c*Sin[a/(2*b)])*(FresnelC[Sqrt[c/ 
(Pi*b)]*Sqrt[a + b*ArcSin[c + d*x^2]]]/(Cos[(1/2)*ArcSin[c + d*x^2]] - c*Si 
n[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]]]/(Co 
s[(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 5327

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] + (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] - Simp[1/(4*b^2*(n + 1)*(n + 2))   Int[(a + b*ArcSin[c + d*x^2])^(n + 
 2), x], x]) /; FreeQ[{a, b, c, d}, x] && EqQ[c^2, 1] && LtQ[n, -1] && NeQ[ 
n, -2]
 

Maple [F]

\[\int \frac {1}{{\left (a +b \arcsin \left (d \,x^{2}-1\right )\right )}^{\frac {7}{2}}}d x\]

Input:

int(1/(a+b*arcsin(d*x^2-1))^(7/2),x)
 

Output:

int(1/(a+b*arcsin(d*x^2-1))^(7/2),x)
 

Fricas [F(-2)]

Exception generated. \[ \int \frac {1}{\left (a-b \arcsin \left (1-d x^2\right )\right )^{7/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(1/(a+b*arcsin(d*x^2-1))^(7/2),x, algorithm="fricas")
 

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

Sympy [F]

\[ \int \frac {1}{\left (a-b \arcsin \left (1-d x^2\right )\right )^{7/2}} \, dx=\int \frac {1}{\left (a + b \operatorname {asin}{\left (d x^{2} - 1 \right )}\right )^{\frac {7}{2}}}\, dx \] Input:

integrate(1/(a+b*asin(d*x**2-1))**(7/2),x)
 

Output:

Integral((a + b*asin(d*x**2 - 1))**(-7/2), x)
 

Maxima [F]

\[ \int \frac {1}{\left (a-b \arcsin \left (1-d x^2\right )\right )^{7/2}} \, dx=\int { \frac {1}{{\left (b \arcsin \left (d x^{2} - 1\right ) + a\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate(1/(a+b*arcsin(d*x^2-1))^(7/2),x, algorithm="maxima")
 

Output:

integrate((b*arcsin(d*x^2 - 1) + a)^(-7/2), x)
 

Giac [F]

\[ \int \frac {1}{\left (a-b \arcsin \left (1-d x^2\right )\right )^{7/2}} \, dx=\int { \frac {1}{{\left (b \arcsin \left (d x^{2} - 1\right ) + a\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate(1/(a+b*arcsin(d*x^2-1))^(7/2),x, algorithm="giac")
 

Output:

integrate((b*arcsin(d*x^2 - 1) + a)^(-7/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a-b \arcsin \left (1-d x^2\right )\right )^{7/2}} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {asin}\left (d\,x^2-1\right )\right )}^{7/2}} \,d x \] Input:

int(1/(a + b*asin(d*x^2 - 1))^(7/2),x)
 

Output:

int(1/(a + b*asin(d*x^2 - 1))^(7/2), x)
 

Reduce [F]

\[ \int \frac {1}{\left (a-b \arcsin \left (1-d x^2\right )\right )^{7/2}} \, dx=\text {too large to display} \] Input:

int(1/(a+b*asin(d*x^2-1))^(7/2),x)
 

Output:

( - sqrt(d)*asin(d*x**2 - 1)**3*int((sqrt(asin(d*x**2 - 1)*b + a)*sqrt( - 
d*x**2 + 2)*asin(d*x**2 - 1)*x)/(asin(d*x**2 - 1)**4*b**4*d*x**2 - 2*asin( 
d*x**2 - 1)**4*b**4 + 4*asin(d*x**2 - 1)**3*a*b**3*d*x**2 - 8*asin(d*x**2 
- 1)**3*a*b**3 + 6*asin(d*x**2 - 1)**2*a**2*b**2*d*x**2 - 12*asin(d*x**2 - 
 1)**2*a**2*b**2 + 4*asin(d*x**2 - 1)*a**3*b*d*x**2 - 8*asin(d*x**2 - 1)*a 
**3*b + a**4*d*x**2 - 2*a**4),x)*a*b**3*d - sqrt(d)*asin(d*x**2 - 1)**3*in 
t((sqrt(asin(d*x**2 - 1)*b + a)*sqrt( - d*x**2 + 2)*asin(d*x**2 - 1)**2*x) 
/(asin(d*x**2 - 1)**4*b**4*d*x**2 - 2*asin(d*x**2 - 1)**4*b**4 + 4*asin(d* 
x**2 - 1)**3*a*b**3*d*x**2 - 8*asin(d*x**2 - 1)**3*a*b**3 + 6*asin(d*x**2 
- 1)**2*a**2*b**2*d*x**2 - 12*asin(d*x**2 - 1)**2*a**2*b**2 + 4*asin(d*x** 
2 - 1)*a**3*b*d*x**2 - 8*asin(d*x**2 - 1)*a**3*b + a**4*d*x**2 - 2*a**4),x 
)*b**4*d + 3*asin(d*x**2 - 1)**3*int((sqrt(asin(d*x**2 - 1)*b + a)*asin(d* 
x**2 - 1)*x**2)/(asin(d*x**2 - 1)**4*b**4*d*x**2 - 2*asin(d*x**2 - 1)**4*b 
**4 + 4*asin(d*x**2 - 1)**3*a*b**3*d*x**2 - 8*asin(d*x**2 - 1)**3*a*b**3 + 
 6*asin(d*x**2 - 1)**2*a**2*b**2*d*x**2 - 12*asin(d*x**2 - 1)**2*a**2*b**2 
 + 4*asin(d*x**2 - 1)*a**3*b*d*x**2 - 8*asin(d*x**2 - 1)*a**3*b + a**4*d*x 
**2 - 2*a**4),x)*b**4*d**2 - 6*asin(d*x**2 - 1)**3*int((sqrt(asin(d*x**2 - 
 1)*b + a)*asin(d*x**2 - 1))/(asin(d*x**2 - 1)**4*b**4*d*x**2 - 2*asin(d*x 
**2 - 1)**4*b**4 + 4*asin(d*x**2 - 1)**3*a*b**3*d*x**2 - 8*asin(d*x**2 - 1 
)**3*a*b**3 + 6*asin(d*x**2 - 1)**2*a**2*b**2*d*x**2 - 12*asin(d*x**2 -...