\(\int \frac {1}{\sqrt {a-b \arcsin (1-d x^2)}} \, dx\) [302]

Optimal result

Integrand size = 18, antiderivative size = 201 \[ \int \frac {1}{\sqrt {a-b \arcsin \left (1-d x^2\right )}} \, dx=-\frac {\sqrt {\pi } x \operatorname {FresnelS}\left (\frac {\sqrt {a-b \arcsin \left (1-d x^2\right )}}{\sqrt {-b} \sqrt {\pi }}\right ) \left (\cos \left (\frac {a}{2 b}\right )-\sin \left (\frac {a}{2 b}\right )\right )}{\sqrt {-b} \left (\cos \left (\frac {1}{2} \arcsin \left (1-d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1-d x^2\right )\right )\right )}-\frac {\sqrt {\pi } x \operatorname {FresnelC}\left (\frac {\sqrt {a-b \arcsin \left (1-d x^2\right )}}{\sqrt {-b} \sqrt {\pi }}\right ) \left (\cos \left (\frac {a}{2 b}\right )+\sin \left (\frac {a}{2 b}\right )\right )}{\sqrt {-b} \left (\cos \left (\frac {1}{2} \arcsin \left (1-d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1-d x^2\right )\right )\right )} \] Output:

-Pi^(1/2)*x*FresnelS((a+b*arcsin(d*x^2-1))^(1/2)/(-b)^(1/2)/Pi^(1/2))*(cos 
(1/2*a/b)-sin(1/2*a/b))/(-b)^(1/2)/(cos(1/2*arcsin(d*x^2-1))+sin(1/2*arcsi 
n(d*x^2-1)))-Pi^(1/2)*x*FresnelC((a+b*arcsin(d*x^2-1))^(1/2)/(-b)^(1/2)/Pi 
^(1/2))*(cos(1/2*a/b)+sin(1/2*a/b))/(-b)^(1/2)/(cos(1/2*arcsin(d*x^2-1))+s 
in(1/2*arcsin(d*x^2-1)))
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.77 \[ \int \frac {1}{\sqrt {a-b \arcsin \left (1-d x^2\right )}} \, dx=\frac {b \sqrt {\pi } x \left (\operatorname {FresnelS}\left (\frac {\sqrt {a-b \arcsin \left (1-d x^2\right )}}{\sqrt {-b} \sqrt {\pi }}\right ) \left (\cos \left (\frac {a}{2 b}\right )-\sin \left (\frac {a}{2 b}\right )\right )+\operatorname {FresnelC}\left (\frac {\sqrt {a-b \arcsin \left (1-d x^2\right )}}{\sqrt {-b} \sqrt {\pi }}\right ) \left (\cos \left (\frac {a}{2 b}\right )+\sin \left (\frac {a}{2 b}\right )\right )\right )}{(-b)^{3/2} \left (\cos \left (\frac {1}{2} \arcsin \left (1-d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1-d x^2\right )\right )\right )} \] Input:

Integrate[1/Sqrt[a - b*ArcSin[1 - d*x^2]],x]
 

Output:

(b*Sqrt[Pi]*x*(FresnelS[Sqrt[a - b*ArcSin[1 - d*x^2]]/(Sqrt[-b]*Sqrt[Pi])] 
*(Cos[a/(2*b)] - Sin[a/(2*b)]) + FresnelC[Sqrt[a - b*ArcSin[1 - d*x^2]]/(S 
qrt[-b]*Sqrt[Pi])]*(Cos[a/(2*b)] + Sin[a/(2*b)])))/((-b)^(3/2)*(Cos[ArcSin 
[1 - d*x^2]/2] - Sin[ArcSin[1 - d*x^2]/2]))
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {5318}

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.

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

\(\Big \downarrow \) 5318

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

Input:

Int[1/Sqrt[a - b*ArcSin[1 - d*x^2]],x]
 

Output:

-((Sqrt[Pi]*x*FresnelS[Sqrt[a - b*ArcSin[1 - d*x^2]]/(Sqrt[-b]*Sqrt[Pi])]* 
(Cos[a/(2*b)] - Sin[a/(2*b)]))/(Sqrt[-b]*(Cos[ArcSin[1 - d*x^2]/2] - Sin[A 
rcSin[1 - d*x^2]/2]))) - (Sqrt[Pi]*x*FresnelC[Sqrt[a - b*ArcSin[1 - d*x^2] 
]/(Sqrt[-b]*Sqrt[Pi])]*(Cos[a/(2*b)] + Sin[a/(2*b)]))/(Sqrt[-b]*(Cos[ArcSi 
n[1 - d*x^2]/2] - Sin[ArcSin[1 - d*x^2]/2]))
 

Defintions of rubi rules used

rule 5318

Int[1/Sqrt[(a_.) + ArcSin[(c_) + (d_.)*(x_)^2]*(b_.)], x_Symbol] :> Simp[(- 
Sqrt[Pi])*x*(Cos[a/(2*b)] - c*Sin[a/(2*b)])*(FresnelC[(1/(Sqrt[b*c]*Sqrt[Pi 
]))*Sqrt[a + b*ArcSin[c + d*x^2]]]/(Sqrt[b*c]*(Cos[ArcSin[c + d*x^2]/2] - c 
*Sin[ArcSin[c + d*x^2]/2]))), x] - Simp[Sqrt[Pi]*x*(Cos[a/(2*b)] + c*Sin[a/ 
(2*b)])*(FresnelS[(1/(Sqrt[b*c]*Sqrt[Pi]))*Sqrt[a + b*ArcSin[c + d*x^2]]]/( 
Sqrt[b*c]*(Cos[ArcSin[c + d*x^2]/2] - c*Sin[ArcSin[c + d*x^2]/2]))), x] /; 
FreeQ[{a, b, c, d}, x] && EqQ[c^2, 1]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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