\(\int \frac {1}{(a+i b \arcsin (1-i d x^2))^{3/2}} \, dx\) [267]

Optimal result

Integrand size = 22, antiderivative size = 291 \[ \int \frac {1}{\left (a+i b \arcsin \left (1-i d x^2\right )\right )^{3/2}} \, dx=-\frac {\sqrt {2 i d x^2+d^2 x^4}}{b d x \sqrt {a+i b \arcsin \left (1-i d x^2\right )}}-\frac {\left (-\frac {i}{b}\right )^{3/2} \sqrt {\pi } x \operatorname {FresnelC}\left (\frac {\sqrt {-\frac {i}{b}} \sqrt {a+i b \arcsin \left (1-i d x^2\right )}}{\sqrt {\pi }}\right ) \left (\cosh \left (\frac {a}{2 b}\right )-i \sinh \left (\frac {a}{2 b}\right )\right )}{\cos \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )}+\frac {\left (-\frac {i}{b}\right )^{3/2} \sqrt {\pi } x \operatorname {FresnelS}\left (\frac {\sqrt {-\frac {i}{b}} \sqrt {a+i b \arcsin \left (1-i d x^2\right )}}{\sqrt {\pi }}\right ) \left (\cosh \left (\frac {a}{2 b}\right )+i \sinh \left (\frac {a}{2 b}\right )\right )}{\cos \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )} \] Output:

-(2*I*d*x^2+d^2*x^4)^(1/2)/b/d/x/(a-I*b*arcsin(-1+I*d*x^2))^(1/2)-(-I/b)^( 
3/2)*Pi^(1/2)*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))/(cos(1/2*arcsin(-1+I*d*x^2))+sin(1/ 
2*arcsin(-1+I*d*x^2)))+(-I/b)^(3/2)*Pi^(1/2)*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))/(cos 
(1/2*arcsin(-1+I*d*x^2))+sin(1/2*arcsin(-1+I*d*x^2)))
 

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (a+i b \arcsin \left (1-i d x^2\right )\right )^{3/2}} \, dx=-\frac {\sqrt {2 i d x^2+d^2 x^4}}{b d x \sqrt {a+i b \arcsin \left (1-i d x^2\right )}}-\frac {\left (-\frac {i}{b}\right )^{3/2} \sqrt {\pi } x \operatorname {FresnelC}\left (\frac {\sqrt {-\frac {i}{b}} \sqrt {a+i b \arcsin \left (1-i d x^2\right )}}{\sqrt {\pi }}\right ) \left (\cosh \left (\frac {a}{2 b}\right )-i \sinh \left (\frac {a}{2 b}\right )\right )}{\cos \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )}+\frac {\left (-\frac {i}{b}\right )^{3/2} \sqrt {\pi } x \operatorname {FresnelS}\left (\frac {\sqrt {-\frac {i}{b}} \sqrt {a+i b \arcsin \left (1-i d x^2\right )}}{\sqrt {\pi }}\right ) \left (\cosh \left (\frac {a}{2 b}\right )+i \sinh \left (\frac {a}{2 b}\right )\right )}{\cos \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )} \] Input:

Integrate[(a + I*b*ArcSin[1 - I*d*x^2])^(-3/2),x]
 

Output:

-(Sqrt[(2*I)*d*x^2 + d^2*x^4]/(b*d*x*Sqrt[a + I*b*ArcSin[1 - I*d*x^2]])) - 
 (((-I)/b)^(3/2)*Sqrt[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)]))/(Cos[ArcSin[1 - I 
*d*x^2]/2] - Sin[ArcSin[1 - I*d*x^2]/2]) + (((-I)/b)^(3/2)*Sqrt[Pi]*x*Fres 
nelS[(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])
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {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.

\(\displaystyle \int \frac {1}{\left (a+i b \arcsin \left (1-i d x^2\right )\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 5321

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

Input:

Int[(a + I*b*ArcSin[1 - I*d*x^2])^(-3/2),x]
 

Output:

-(Sqrt[(2*I)*d*x^2 + d^2*x^4]/(b*d*x*Sqrt[a + I*b*ArcSin[1 - I*d*x^2]])) - 
 (((-I)/b)^(3/2)*Sqrt[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)]))/(Cos[ArcSin[1 - I 
*d*x^2]/2] - Sin[ArcSin[1 - I*d*x^2]/2]) + (((-I)/b)^(3/2)*Sqrt[Pi]*x*Fres 
nelS[(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])
 

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]
 

Maple [F]

\[\int \frac {1}{{\left (a -i b \arcsin \left (i d \,x^{2}-1\right )\right )}^{\frac {3}{2}}}d x\]

Input:

int(1/(a-I*b*arcsin(-1+I*d*x^2))^(3/2),x)
 

Output:

int(1/(a-I*b*arcsin(-1+I*d*x^2))^(3/2),x)
 

Fricas [F(-2)]

Exception generated. \[ \int \frac {1}{\left (a+i b \arcsin \left (1-i d x^2\right )\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(1/(a-I*b*arcsin(-1+I*d*x^2))^(3/2),x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {1}{\left (a+i b \arcsin \left (1-i d x^2\right )\right )^{3/2}} \, dx=\int \frac {1}{\left (a - i b \operatorname {asin}{\left (i d x^{2} - 1 \right )}\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(1/(a-I*b*asin(-1+I*d*x**2))**(3/2),x)
 

Output:

Integral((a - I*b*asin(I*d*x**2 - 1))**(-3/2), x)
 

Maxima [F]

\[ \int \frac {1}{\left (a+i b \arcsin \left (1-i d x^2\right )\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-i \, b \arcsin \left (i \, d x^{2} - 1\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(a-I*b*arcsin(-1+I*d*x^2))^(3/2),x, algorithm="maxima")
 

Output:

integrate((-I*b*arcsin(I*d*x^2 - 1) + a)^(-3/2), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {1}{\left (a+i b \arcsin \left (1-i d x^2\right )\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(1/(a-I*b*arcsin(-1+I*d*x^2))^(3/2),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+i b \arcsin \left (1-i d x^2\right )\right )^{3/2}} \, dx=\int \frac {1}{{\left (a-b\,\mathrm {asin}\left (-1+d\,x^2\,1{}\mathrm {i}\right )\,1{}\mathrm {i}\right )}^{3/2}} \,d x \] Input:

int(1/(a - b*asin(d*x^2*1i - 1)*1i)^(3/2),x)
 

Output:

int(1/(a - b*asin(d*x^2*1i - 1)*1i)^(3/2), x)
 

Reduce [F]

\[ \int \frac {1}{\left (a+i b \arcsin \left (1-i d x^2\right )\right )^{3/2}} \, dx=-\left (\int \frac {\sqrt {-\mathit {asin} \left (d i \,x^{2}-1\right ) b i +a}}{\mathit {asin} \left (d i \,x^{2}-1\right )^{3} b^{3} i -\mathit {asin} \left (d i \,x^{2}-1\right )^{2} a \,b^{2}+\mathit {asin} \left (d i \,x^{2}-1\right ) a^{2} b i -a^{3}}d x \right ) a -\left (\int \frac {\sqrt {-\mathit {asin} \left (d i \,x^{2}-1\right ) b i +a}\, \mathit {asin} \left (d i \,x^{2}-1\right )}{\mathit {asin} \left (d i \,x^{2}-1\right )^{3} b^{3} i -\mathit {asin} \left (d i \,x^{2}-1\right )^{2} a \,b^{2}+\mathit {asin} \left (d i \,x^{2}-1\right ) a^{2} b i -a^{3}}d x \right ) b i \] Input:

int(1/(a-I*b*asin(-1+I*d*x^2))^(3/2),x)
 

Output:

 - (int(sqrt( - asin(d*i*x**2 - 1)*b*i + a)/(asin(d*i*x**2 - 1)**3*b**3*i 
- asin(d*i*x**2 - 1)**2*a*b**2 + asin(d*i*x**2 - 1)*a**2*b*i - a**3),x)*a 
+ int((sqrt( - asin(d*i*x**2 - 1)*b*i + a)*asin(d*i*x**2 - 1))/(asin(d*i*x 
**2 - 1)**3*b**3*i - asin(d*i*x**2 - 1)**2*a*b**2 + asin(d*i*x**2 - 1)*a** 
2*b*i - a**3),x)*b*i)