\(\int \frac {1}{a-i b \arcsin (1+i d x^2)} \, dx\) [325]

Optimal result

Integrand size = 20, antiderivative size = 191 \[ \int \frac {1}{a-i b \arcsin \left (1+i d x^2\right )} \, dx=-\frac {x \operatorname {CosIntegral}\left (\frac {i \left (a-i b \arcsin \left (1+i d x^2\right )\right )}{2 b}\right ) \left (i \cosh \left (\frac {a}{2 b}\right )+\sinh \left (\frac {a}{2 b}\right )\right )}{2 b \left (\cos \left (\frac {1}{2} \arcsin \left (1+i d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1+i d x^2\right )\right )\right )}+\frac {x \left (\cosh \left (\frac {a}{2 b}\right )+i \sinh \left (\frac {a}{2 b}\right )\right ) \text {Shi}\left (\frac {a-i b \arcsin \left (1+i d x^2\right )}{2 b}\right )}{2 b \left (\cos \left (\frac {1}{2} \arcsin \left (1+i d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1+i d x^2\right )\right )\right )} \]

[Out]

1/2*x*Shi(1/2*(a-I*b*arcsin(1+I*d*x^2))/b)*(cosh(1/2*a/b)+I*sinh(1/2*a/b))/b/(cos(1/2*arcsin(1+I*d*x^2))-sin(1
/2*arcsin(1+I*d*x^2)))-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))-sin(1/2*arcsin(1+I*d*x^2)))

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 191, 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} \[ \int \frac {1}{a-i b \arcsin \left (1+i d x^2\right )} \, dx=\frac {x \left (\cosh \left (\frac {a}{2 b}\right )+i \sinh \left (\frac {a}{2 b}\right )\right ) \text {Shi}\left (\frac {a-i b \arcsin \left (i d x^2+1\right )}{2 b}\right )}{2 b \left (\cos \left (\frac {1}{2} \arcsin \left (1+i d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1+i d x^2\right )\right )\right )}-\frac {x \left (\sinh \left (\frac {a}{2 b}\right )+i \cosh \left (\frac {a}{2 b}\right )\right ) \operatorname {CosIntegral}\left (\frac {i \left (a-i b \arcsin \left (i d x^2+1\right )\right )}{2 b}\right )}{2 b \left (\cos \left (\frac {1}{2} \arcsin \left (1+i d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1+i d x^2\right )\right )\right )} \]

[In]

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

[Out]

-1/2*(x*CosIntegral[((I/2)*(a - I*b*ArcSin[1 + I*d*x^2]))/b]*(I*Cosh[a/(2*b)] + Sinh[a/(2*b)]))/(b*(Cos[ArcSin
[1 + I*d*x^2]/2] - Sin[ArcSin[1 + I*d*x^2]/2])) + (x*(Cosh[a/(2*b)] + I*Sinh[a/(2*b)])*SinhIntegral[(a - I*b*A
rcSin[1 + I*d*x^2])/(2*b)])/(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 (\cosh \left (\frac {a}{2 b}\right )+i \sinh \left (\frac {a}{2 b}\right )\right ) \text {Shi}\left (\frac {a-i b \arcsin \left (1+i d x^2\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 )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.57 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.76 \[ \int \frac {1}{a-i b \arcsin \left (1+i d x^2\right )} \, dx=\frac {x \left (\operatorname {CosIntegral}\left (\frac {1}{2} \left (\frac {i a}{b}+\arcsin \left (1+i d x^2\right )\right )\right ) \left (-i \cosh \left (\frac {a}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right )\right )+\left (-i \cosh \left (\frac {a}{2 b}\right )+\sinh \left (\frac {a}{2 b}\right )\right ) \text {Si}\left (\frac {1}{2} \left (\frac {i a}{b}+\arcsin \left (1+i d x^2\right )\right )\right )\right )}{2 b \left (\cos \left (\frac {1}{2} \arcsin \left (1+i d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1+i d x^2\right )\right )\right )} \]

[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*a)/b + ArcSin[1 + I*d*x^2])/2]))/(2*b*(Cos[ArcSin[1 + I*d*x^2]/2] - Sin[ArcSi
n[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. \[ \int \frac {1}{a-i b \arcsin \left (1+i d x^2\right )} \, dx=\int \frac {1}{a+b\,\mathrm {asinh}\left (d\,x^2-\mathrm {i}\right )} \,d x \]

[In]

int(1/(a + b*asinh(d*x^2 - 1i)),x)

[Out]

int(1/(a + b*asinh(d*x^2 - 1i)), x)