Integrand size = 22, antiderivative size = 231 \[ \int \frac {1}{\sqrt {a-i b \arcsin \left (1+i d x^2\right )}} \, dx=-\frac {\sqrt {\pi } x \operatorname {FresnelC}\left (\frac {\sqrt {a-i b \arcsin \left (1+i d x^2\right )}}{\sqrt {-i b} \sqrt {\pi }}\right ) \left (\cosh \left (\frac {a}{2 b}\right )-i \sinh \left (\frac {a}{2 b}\right )\right )}{\sqrt {-i b} \left (\cos \left (\frac {1}{2} \arcsin \left (1+i d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1+i d x^2\right )\right )\right )}-\frac {\sqrt {\pi } x \operatorname {FresnelS}\left (\frac {\sqrt {a-i b \arcsin \left (1+i d x^2\right )}}{\sqrt {-i b} \sqrt {\pi }}\right ) \left (\cosh \left (\frac {a}{2 b}\right )+i \sinh \left (\frac {a}{2 b}\right )\right )}{\sqrt {-i b} \left (\cos \left (\frac {1}{2} \arcsin \left (1+i d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1+i d x^2\right )\right )\right )} \] Output:
-Pi^(1/2)*x*FresnelC((a-I*b*arcsin(1+I*d*x^2))^(1/2)/(-I*b)^(1/2)/Pi^(1/2) )*(cosh(1/2*a/b)-I*sinh(1/2*a/b))/(-I*b)^(1/2)/(cos(1/2*arcsin(1+I*d*x^2)) -sin(1/2*arcsin(1+I*d*x^2)))-Pi^(1/2)*x*FresnelS((a-I*b*arcsin(1+I*d*x^2)) ^(1/2)/(-I*b)^(1/2)/Pi^(1/2))*(cosh(1/2*a/b)+I*sinh(1/2*a/b))/(-I*b)^(1/2) /(cos(1/2*arcsin(1+I*d*x^2))-sin(1/2*arcsin(1+I*d*x^2)))
Time = 0.01 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.78 \[ \int \frac {1}{\sqrt {a-i b \arcsin \left (1+i d x^2\right )}} \, dx=\frac {\sqrt {\pi } x \left (-\operatorname {FresnelC}\left (\frac {\sqrt {a-i b \arcsin \left (1+i d x^2\right )}}{\sqrt {-i b} \sqrt {\pi }}\right ) \left (\cosh \left (\frac {a}{2 b}\right )-i \sinh \left (\frac {a}{2 b}\right )\right )-\operatorname {FresnelS}\left (\frac {\sqrt {a-i b \arcsin \left (1+i d x^2\right )}}{\sqrt {-i b} \sqrt {\pi }}\right ) \left (\cosh \left (\frac {a}{2 b}\right )+i \sinh \left (\frac {a}{2 b}\right )\right )\right )}{\sqrt {-i b} \left (\cos \left (\frac {1}{2} \arcsin \left (1+i d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1+i d x^2\right )\right )\right )} \] Input:
Integrate[1/Sqrt[a - I*b*ArcSin[1 + I*d*x^2]],x]
Output:
(Sqrt[Pi]*x*(-(FresnelC[Sqrt[a - I*b*ArcSin[1 + I*d*x^2]]/(Sqrt[(-I)*b]*Sq rt[Pi])]*(Cosh[a/(2*b)] - I*Sinh[a/(2*b)])) - FresnelS[Sqrt[a - I*b*ArcSin [1 + I*d*x^2]]/(Sqrt[(-I)*b]*Sqrt[Pi])]*(Cosh[a/(2*b)] + I*Sinh[a/(2*b)])) )/(Sqrt[(-I)*b]*(Cos[ArcSin[1 + I*d*x^2]/2] - Sin[ArcSin[1 + I*d*x^2]/2]))
Time = 0.28 (sec) , antiderivative size = 231, 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 = {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.
\(\displaystyle \int \frac {1}{\sqrt {a-i b \arcsin \left (1+i d x^2\right )}} \, dx\) |
\(\Big \downarrow \) 5318 |
\(\displaystyle -\frac {\sqrt {\pi } x \left (\cosh \left (\frac {a}{2 b}\right )-i \sinh \left (\frac {a}{2 b}\right )\right ) \operatorname {FresnelC}\left (\frac {\sqrt {a-i b \arcsin \left (i d x^2+1\right )}}{\sqrt {-i b} \sqrt {\pi }}\right )}{\sqrt {-i b} \left (\cos \left (\frac {1}{2} \arcsin \left (1+i d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1+i d x^2\right )\right )\right )}-\frac {\sqrt {\pi } x \left (\cosh \left (\frac {a}{2 b}\right )+i \sinh \left (\frac {a}{2 b}\right )\right ) \operatorname {FresnelS}\left (\frac {\sqrt {a-i b \arcsin \left (i d x^2+1\right )}}{\sqrt {-i b} \sqrt {\pi }}\right )}{\sqrt {-i b} \left (\cos \left (\frac {1}{2} \arcsin \left (1+i d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1+i d x^2\right )\right )\right )}\) |
Input:
Int[1/Sqrt[a - I*b*ArcSin[1 + I*d*x^2]],x]
Output:
-((Sqrt[Pi]*x*FresnelC[Sqrt[a - I*b*ArcSin[1 + I*d*x^2]]/(Sqrt[(-I)*b]*Sqr t[Pi])]*(Cosh[a/(2*b)] - I*Sinh[a/(2*b)]))/(Sqrt[(-I)*b]*(Cos[ArcSin[1 + I *d*x^2]/2] - Sin[ArcSin[1 + I*d*x^2]/2]))) - (Sqrt[Pi]*x*FresnelS[Sqrt[a - I*b*ArcSin[1 + I*d*x^2]]/(Sqrt[(-I)*b]*Sqrt[Pi])]*(Cosh[a/(2*b)] + I*Sinh [a/(2*b)]))/(Sqrt[(-I)*b]*(Cos[ArcSin[1 + I*d*x^2]/2] - Sin[ArcSin[1 + I*d *x^2]/2]))
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]
\[\int \frac {1}{\sqrt {a -i b \arcsin \left (i d \,x^{2}+1\right )}}d x\]
Input:
int(1/(a-I*b*arcsin(1+I*d*x^2))^(1/2),x)
Output:
int(1/(a-I*b*arcsin(1+I*d*x^2))^(1/2),x)
Exception generated. \[ \int \frac {1}{\sqrt {a-i b \arcsin \left (1+i d x^2\right )}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(1/(a-I*b*arcsin(1+I*d*x^2))^(1/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {1}{\sqrt {a-i b \arcsin \left (1+i d x^2\right )}} \, dx=\int \frac {1}{\sqrt {a - i b \operatorname {asin}{\left (i d x^{2} + 1 \right )}}}\, dx \] Input:
integrate(1/(a-I*b*asin(1+I*d*x**2))**(1/2),x)
Output:
Integral(1/sqrt(a - I*b*asin(I*d*x**2 + 1)), x)
\[ \int \frac {1}{\sqrt {a-i b \arcsin \left (1+i d x^2\right )}} \, dx=\int { \frac {1}{\sqrt {-i \, b \arcsin \left (i \, d x^{2} + 1\right ) + a}} \,d x } \] Input:
integrate(1/(a-I*b*arcsin(1+I*d*x^2))^(1/2),x, algorithm="maxima")
Output:
integrate(1/sqrt(-I*b*arcsin(I*d*x^2 + 1) + a), x)
Exception generated. \[ \int \frac {1}{\sqrt {a-i b \arcsin \left (1+i d x^2\right )}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(1/(a-I*b*arcsin(1+I*d*x^2))^(1/2),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
Timed out. \[ \int \frac {1}{\sqrt {a-i b \arcsin \left (1+i d x^2\right )}} \, dx=\int \frac {1}{\sqrt {a-b\,\mathrm {asin}\left (1{}\mathrm {i}\,d\,x^2+1\right )\,1{}\mathrm {i}}} \,d x \] Input:
int(1/(a - b*asin(d*x^2*1i + 1)*1i)^(1/2),x)
Output:
int(1/(a - b*asin(d*x^2*1i + 1)*1i)^(1/2), x)
\[ \int \frac {1}{\sqrt {a-i b \arcsin \left (1+i d x^2\right )}} \, dx=\left (\int \frac {\sqrt {-\mathit {asin} \left (d i \,x^{2}+1\right ) b i +a}}{\mathit {asin} \left (d i \,x^{2}+1\right )^{2} b^{2}+a^{2}}d x \right ) a +\left (\int \frac {\sqrt {-\mathit {asin} \left (d i \,x^{2}+1\right ) b i +a}\, \mathit {asin} \left (d i \,x^{2}+1\right )}{\mathit {asin} \left (d i \,x^{2}+1\right )^{2} b^{2}+a^{2}}d x \right ) b i \] Input:
int(1/(a-I*b*asin(1+I*d*x^2))^(1/2),x)
Output:
int(sqrt( - asin(d*i*x**2 + 1)*b*i + a)/(asin(d*i*x**2 + 1)**2*b**2 + a**2 ),x)*a + int((sqrt( - asin(d*i*x**2 + 1)*b*i + a)*asin(d*i*x**2 + 1))/(asi n(d*i*x**2 + 1)**2*b**2 + a**2),x)*b*i