3.5.13 \(\int \frac {1}{(a-b \arcsin (1-d x^2))^2} \, dx\) [413]

3.5.13.1 Optimal result

Integrand size = 16, antiderivative size = 216 \[ \int \frac {1}{\left (a-b \arcsin \left (1-d x^2\right )\right )^2} \, dx=-\frac {\sqrt {2 d x^2-d^2 x^4}}{2 b d x \left (a-b \arcsin \left (1-d x^2\right )\right )}-\frac {x \operatorname {CosIntegral}\left (-\frac {a-b \arcsin \left (1-d x^2\right )}{2 b}\right ) \left (\cos \left (\frac {a}{2 b}\right )-\sin \left (\frac {a}{2 b}\right )\right )}{4 b^2 \left (\cos \left (\frac {1}{2} \arcsin \left (1-d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1-d x^2\right )\right )\right )}-\frac {x \left (\cos \left (\frac {a}{2 b}\right )+\sin \left (\frac {a}{2 b}\right )\right ) \text {Si}\left (\frac {a}{2 b}-\frac {1}{2} \arcsin \left (1-d x^2\right )\right )}{4 b^2 \left (\cos \left (\frac {1}{2} \arcsin \left (1-d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1-d x^2\right )\right )\right )} \]

output

-1/4*x*Ci(1/2*(-a-b*arcsin(d*x^2-1))/b)*(cos(1/2*a/b)-sin(1/2*a/b))/b^2/(c 
os(1/2*arcsin(d*x^2-1))+sin(1/2*arcsin(d*x^2-1)))-1/4*x*Si(1/2*a/b+1/2*arc 
sin(d*x^2-1))*(cos(1/2*a/b)+sin(1/2*a/b))/b^2/(cos(1/2*arcsin(d*x^2-1))+si 
n(1/2*arcsin(d*x^2-1)))-1/2*(-d^2*x^4+2*d*x^2)^(1/2)/b/d/x/(a+b*arcsin(d*x 
^2-1))
 

3.5.13.2 Mathematica [A] (verified)

Time = 0.53 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\left (a-b \arcsin \left (1-d x^2\right )\right )^2} \, dx=\frac {2 b \sqrt {d x^2 \left (2-d x^2\right )}+\left (a-b \arcsin \left (1-d x^2\right )\right ) \left (\cos \left (\frac {1}{2} \arcsin \left (1-d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1-d x^2\right )\right )\right ) \left (\operatorname {CosIntegral}\left (\frac {1}{2} \left (-\frac {a}{b}+\arcsin \left (1-d x^2\right )\right )\right ) \left (\cos \left (\frac {a}{2 b}\right )-\sin \left (\frac {a}{2 b}\right )\right )+\left (\cos \left (\frac {a}{2 b}\right )+\sin \left (\frac {a}{2 b}\right )\right ) \text {Si}\left (\frac {a-b \arcsin \left (1-d x^2\right )}{2 b}\right )\right )}{4 b^2 d x \left (-a+b \arcsin \left (1-d x^2\right )\right )} \]

input

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

output

(2*b*Sqrt[d*x^2*(2 - d*x^2)] + (a - b*ArcSin[1 - d*x^2])*(Cos[ArcSin[1 - d 
*x^2]/2] - Sin[ArcSin[1 - d*x^2]/2])*(CosIntegral[(-(a/b) + ArcSin[1 - d*x 
^2])/2]*(Cos[a/(2*b)] - Sin[a/(2*b)]) + (Cos[a/(2*b)] + Sin[a/(2*b)])*SinI 
ntegral[(a - b*ArcSin[1 - d*x^2])/(2*b)]))/(4*b^2*d*x*(-a + b*ArcSin[1 - d 
*x^2]))
 

3.5.13.3 Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {5324}

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.

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

\(\Big \downarrow \) 5324

\(\displaystyle -\frac {x \left (\cos \left (\frac {a}{2 b}\right )-\sin \left (\frac {a}{2 b}\right )\right ) \operatorname {CosIntegral}\left (-\frac {a-b \arcsin \left (1-d x^2\right )}{2 b}\right )}{4 b^2 \left (\cos \left (\frac {1}{2} \arcsin \left (1-d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1-d x^2\right )\right )\right )}-\frac {x \left (\sin \left (\frac {a}{2 b}\right )+\cos \left (\frac {a}{2 b}\right )\right ) \text {Si}\left (\frac {a}{2 b}-\frac {1}{2} \arcsin \left (1-d x^2\right )\right )}{4 b^2 \left (\cos \left (\frac {1}{2} \arcsin \left (1-d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1-d x^2\right )\right )\right )}-\frac {\sqrt {2 d x^2-d^2 x^4}}{2 b d x \left (a-b \arcsin \left (1-d x^2\right )\right )}\)

input

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

output

-1/2*Sqrt[2*d*x^2 - d^2*x^4]/(b*d*x*(a - b*ArcSin[1 - d*x^2])) - (x*CosInt 
egral[-1/2*(a - b*ArcSin[1 - d*x^2])/b]*(Cos[a/(2*b)] - Sin[a/(2*b)]))/(4* 
b^2*(Cos[ArcSin[1 - d*x^2]/2] - Sin[ArcSin[1 - d*x^2]/2])) - (x*(Cos[a/(2* 
b)] + Sin[a/(2*b)])*SinIntegral[a/(2*b) - ArcSin[1 - d*x^2]/2])/(4*b^2*(Co 
s[ArcSin[1 - d*x^2]/2] - Sin[ArcSin[1 - d*x^2]/2]))
 

3.5.13.3.1 Defintions of rubi rules used

rule 5324

Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)^2]*(b_.))^(-2), x_Symbol] :> Simp[-Sq 
rt[-2*c*d*x^2 - d^2*x^4]/(2*b*d*x*(a + b*ArcSin[c + d*x^2])), x] + (-Simp[x 
*(Cos[a/(2*b)] + c*Sin[a/(2*b)])*(CosIntegral[(c/(2*b))*(a + b*ArcSin[c + d 
*x^2])]/(4*b^2*(Cos[ArcSin[c + d*x^2]/2] - c*Sin[ArcSin[c + d*x^2]/2]))), x 
] + Simp[x*(Cos[a/(2*b)] - c*Sin[a/(2*b)])*(SinIntegral[(c/(2*b))*(a + b*Ar 
cSin[c + d*x^2])]/(4*b^2*(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]
 

3.5.13.4 Maple [F]

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

input

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

output

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

3.5.13.5 Fricas [F]

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

input

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

output

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

3.5.13.6 Sympy [F]

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

input

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

output

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

3.5.13.7 Maxima [F]

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

input

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

output

1/2*(2*(b^2*d*arctan2(d*x^2 - 1, sqrt(-d*x^2 + 2)*sqrt(d)*x) + a*b*d)*sqrt 
(d)*integrate(1/2*sqrt(-d*x^2 + 2)*x/(a*b*d*x^2 - 2*a*b + (b^2*d*x^2 - 2*b 
^2)*arctan2(d*x^2 - 1, sqrt(-d*x^2 + 2)*sqrt(d)*x)), x) - sqrt(-d*x^2 + 2) 
*sqrt(d))/(b^2*d*arctan2(d*x^2 - 1, sqrt(-d*x^2 + 2)*sqrt(d)*x) + a*b*d)
 

3.5.13.8 Giac [F]

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

input

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

output

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

3.5.13.9 Mupad [F(-1)]

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

input

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

output

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