\(\int \frac {1}{a+i b \arcsin (1-i d x^2)} \, dx\) [318]

Optimal result

Integrand size = 20, antiderivative size = 194 \[ \int \frac {1}{a+i b \arcsin \left (1-i d x^2\right )} \, dx=\frac {x \operatorname {CosIntegral}\left (-\frac {i \left (a+i b \arcsin \left (1-i d x^2\right )\right )}{2 b}\right ) \left (i \cosh \left (\frac {a}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right )\right )}{2 b \left (\cos \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )\right )}-\frac {x \left (i \cosh \left (\frac {a}{2 b}\right )+\sinh \left (\frac {a}{2 b}\right )\right ) \text {Si}\left (\frac {i a}{2 b}-\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )}{2 b \left (\cos \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )\right )} \]

[Out]

1/2*x*Ci(-1/2*I*(a-I*b*arcsin(-1+I*d*x^2))/b)*(I*cosh(1/2*a/b)-sinh(1/2*a/b))/b/(cos(1/2*arcsin(-1+I*d*x^2))+s
in(1/2*arcsin(-1+I*d*x^2)))-1/2*x*Si(1/2*I*a/b+1/2*arcsin(-1+I*d*x^2))*(I*cosh(1/2*a/b)+sinh(1/2*a/b))/b/(cos(
1/2*arcsin(-1+I*d*x^2))+sin(1/2*arcsin(-1+I*d*x^2)))

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {4900} \[ \int \frac {1}{a+i b \arcsin \left (1-i d x^2\right )} \, dx=\frac {x \left (-\sinh \left (\frac {a}{2 b}\right )+i \cosh \left (\frac {a}{2 b}\right )\right ) \operatorname {CosIntegral}\left (-\frac {i \left (a+i b \arcsin \left (1-i d x^2\right )\right )}{2 b}\right )}{2 b \left (\cos \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )\right )}-\frac {x \left (\sinh \left (\frac {a}{2 b}\right )+i \cosh \left (\frac {a}{2 b}\right )\right ) \text {Si}\left (\frac {i a}{2 b}-\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )}{2 b \left (\cos \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )\right )} \]

[In]

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

[Out]

(x*CosIntegral[((-1/2*I)*(a + I*b*ArcSin[1 - I*d*x^2]))/b]*(I*Cosh[a/(2*b)] - Sinh[a/(2*b)]))/(2*b*(Cos[ArcSin
[1 - I*d*x^2]/2] - Sin[ArcSin[1 - I*d*x^2]/2])) - (x*(I*Cosh[a/(2*b)] + Sinh[a/(2*b)])*SinIntegral[((I/2)*a)/b
- ArcSin[1 - I*d*x^2]/2])/(2*b*(Cos[ArcSin[1 - I*d*x^2]/2] - Sin[ArcSin[1 - I*d*x^2]/2]))

Rule 4900

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

Rubi steps \begin{align*} \text {integral}& = \frac {x \operatorname {CosIntegral}\left (-\frac {i \left (a+i b \arcsin \left (1-i d x^2\right )\right )}{2 b}\right ) \left (i \cosh \left (\frac {a}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right )\right )}{2 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 {x \left (i \cosh \left (\frac {a}{2 b}\right )+\sinh \left (\frac {a}{2 b}\right )\right ) \text {Si}\left (\frac {i a}{2 b}-\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )}{2 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 )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.62 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.77 \[ \int \frac {1}{a+i b \arcsin \left (1-i d x^2\right )} \, dx=\frac {x \left (\operatorname {CosIntegral}\left (\frac {1}{2} \left (-\frac {i a}{b}+\arcsin \left (1-i d x^2\right )\right )\right ) \left (i \cosh \left (\frac {a}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right )\right )+\left (-i \cosh \left (\frac {a}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right )\right ) \text {Si}\left (\frac {i a}{2 b}-\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )\right )}{2 b \left (\cos \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )\right )} \]

[In]

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

[Out]

(x*(CosIntegral[(((-I)*a)/b + ArcSin[1 - I*d*x^2])/2]*(I*Cosh[a/(2*b)] - Sinh[a/(2*b)]) + ((-I)*Cosh[a/(2*b)]
- Sinh[a/(2*b)])*SinIntegral[((I/2)*a)/b - ArcSin[1 - I*d*x^2]/2]))/(2*b*(Cos[ArcSin[1 - I*d*x^2]/2] - Sin[Arc
Sin[1 - I*d*x^2]/2]))

Maple [F]

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

[In]

int(1/(a+b*arcsinh(I+d*x^2)),x)

[Out]

int(1/(a+b*arcsinh(I+d*x^2)),x)

Fricas [F]

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

[In]

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

[Out]

integral(1/(b*log(d*x^2 + sqrt(d^2*x^2 + 2*I*d)*x + I) + a), x)

Sympy [F(-2)]

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

[In]

integrate(1/(a+b*asinh(I+d*x**2)),x)

[Out]

Exception raised: TypeError >> Invalid comparison of non-real I

Maxima [F]

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

[In]

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

[Out]

integrate(1/(b*arcsinh(d*x^2 + I) + a), x)

Giac [F(-2)]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(1/(a + b*asinh(d*x^2 + 1i)),x)

[Out]

int(1/(a + b*asinh(d*x^2 + 1i)), x)