3.326 \(\int \frac {1}{(a-i b \sin ^{-1}(1+i d x^2))^2} \, dx\)

Optimal. Leaf size=244 \[ \frac {x \left (\cosh \left (\frac {a}{2 b}\right )+i \sinh \left (\frac {a}{2 b}\right )\right ) \text {Ci}\left (\frac {i \left (a-i b \sin ^{-1}\left (i d x^2+1\right )\right )}{2 b}\right )}{4 b^2 \left (\cos \left (\frac {1}{2} \sin ^{-1}\left (1+i d x^2\right )\right )-\sin \left (\frac {1}{2} \sin ^{-1}\left (1+i d x^2\right )\right )\right )}-\frac {x \left (\sinh \left (\frac {a}{2 b}\right )+i \cosh \left (\frac {a}{2 b}\right )\right ) \text {Shi}\left (\frac {a-i b \sin ^{-1}\left (i d x^2+1\right )}{2 b}\right )}{4 b^2 \left (\cos \left (\frac {1}{2} \sin ^{-1}\left (1+i d x^2\right )\right )-\sin \left (\frac {1}{2} \sin ^{-1}\left (1+i d x^2\right )\right )\right )}-\frac {\sqrt {d^2 x^4-2 i d x^2}}{2 b d x \left (a-i b \sin ^{-1}\left (1+i d x^2\right )\right )} \]

[Out]

1/4*x*Ci(1/2*I*(a-I*b*arcsin(1+I*d*x^2))/b)*(cosh(1/2*a/b)+I*sinh(1/2*a/b))/b^2/(cos(1/2*arcsin(1+I*d*x^2))-si
n(1/2*arcsin(1+I*d*x^2)))-1/4*x*Shi(1/2*(a-I*b*arcsin(1+I*d*x^2))/b)*(I*cosh(1/2*a/b)+sinh(1/2*a/b))/b^2/(cos(
1/2*arcsin(1+I*d*x^2))-sin(1/2*arcsin(1+I*d*x^2)))-1/2*(-2*I*d*x^2+d^2*x^4)^(1/2)/b/d/x/(a-I*b*arcsin(1+I*d*x^
2))

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {4825} \[ \frac {x \left (\cosh \left (\frac {a}{2 b}\right )+i \sinh \left (\frac {a}{2 b}\right )\right ) \text {CosIntegral}\left (\frac {i \left (a-i b \sin ^{-1}\left (1+i d x^2\right )\right )}{2 b}\right )}{4 b^2 \left (\cos \left (\frac {1}{2} \sin ^{-1}\left (1+i d x^2\right )\right )-\sin \left (\frac {1}{2} \sin ^{-1}\left (1+i d x^2\right )\right )\right )}-\frac {x \left (\sinh \left (\frac {a}{2 b}\right )+i \cosh \left (\frac {a}{2 b}\right )\right ) \text {Shi}\left (\frac {a-i b \sin ^{-1}\left (i d x^2+1\right )}{2 b}\right )}{4 b^2 \left (\cos \left (\frac {1}{2} \sin ^{-1}\left (1+i d x^2\right )\right )-\sin \left (\frac {1}{2} \sin ^{-1}\left (1+i d x^2\right )\right )\right )}-\frac {\sqrt {d^2 x^4-2 i d x^2}}{2 b d x \left (a-i b \sin ^{-1}\left (1+i d x^2\right )\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Sqrt[(-2*I)*d*x^2 + d^2*x^4]/(2*b*d*x*(a - I*b*ArcSin[1 + I*d*x^2])) + (x*CosIntegral[((I/2)*(a - I*b*ArcSin[
1 + I*d*x^2]))/b]*(Cosh[a/(2*b)] + I*Sinh[a/(2*b)]))/(4*b^2*(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)])*SinhIntegral[(a - I*b*ArcSin[1 + I*d*x^2])/(2*b)])/(4*b^2*(Cos
[ArcSin[1 + I*d*x^2]/2] - Sin[ArcSin[1 + I*d*x^2]/2]))

Rule 4825

Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)^2]*(b_.))^(-2), x_Symbol] :> -Simp[Sqrt[-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*ArcSin[c + d*x^2])])/(4*b^2*(Cos[ArcSin[c + d*x^2]/2] - c*Sin[ArcSi
n[c + d*x^2]/2])), x]) /; FreeQ[{a, b, c, d}, x] && EqQ[c^2, 1]

Rubi steps

\begin {align*} \int \frac {1}{\left (a-i b \sin ^{-1}\left (1+i d x^2\right )\right )^2} \, dx &=-\frac {\sqrt {-2 i d x^2+d^2 x^4}}{2 b d x \left (a-i b \sin ^{-1}\left (1+i d x^2\right )\right )}+\frac {x \text {Ci}\left (\frac {i \left (a-i b \sin ^{-1}\left (1+i d x^2\right )\right )}{2 b}\right ) \left (\cosh \left (\frac {a}{2 b}\right )+i \sinh \left (\frac {a}{2 b}\right )\right )}{4 b^2 \left (\cos \left (\frac {1}{2} \sin ^{-1}\left (1+i d x^2\right )\right )-\sin \left (\frac {1}{2} \sin ^{-1}\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 {Shi}\left (\frac {a-i b \sin ^{-1}\left (1+i d x^2\right )}{2 b}\right )}{4 b^2 \left (\cos \left (\frac {1}{2} \sin ^{-1}\left (1+i d x^2\right )\right )-\sin \left (\frac {1}{2} \sin ^{-1}\left (1+i d x^2\right )\right )\right )}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 1.49, size = 196, normalized size = 0.80 \[ \frac {\frac {x^2 \left (\left (\cosh \left (\frac {a}{2 b}\right )+i \sinh \left (\frac {a}{2 b}\right )\right ) \text {Ci}\left (\frac {1}{2} \left (\frac {i a}{b}+\sin ^{-1}\left (i d x^2+1\right )\right )\right )-\left (\cosh \left (\frac {a}{2 b}\right )-i \sinh \left (\frac {a}{2 b}\right )\right ) \text {Si}\left (\frac {1}{2} \left (\frac {i a}{b}+\sin ^{-1}\left (i d x^2+1\right )\right )\right )\right )}{\cos \left (\frac {1}{2} \sin ^{-1}\left (1+i d x^2\right )\right )-\sin \left (\frac {1}{2} \sin ^{-1}\left (1+i d x^2\right )\right )}-\frac {2 b \sqrt {d x^2 \left (d x^2-2 i\right )}}{d \left (a-i b \sin ^{-1}\left (1+i d x^2\right )\right )}}{4 b^2 x} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-2*b*Sqrt[d*x^2*(-2*I + d*x^2)])/(d*(a - I*b*ArcSin[1 + I*d*x^2])) + (x^2*(CosIntegral[((I*a)/b + ArcSin[1 +
I*d*x^2])/2]*(Cosh[a/(2*b)] + I*Sinh[a/(2*b)]) - (Cosh[a/(2*b)] - I*Sinh[a/(2*b)])*SinIntegral[((I*a)/b + Arc
Sin[1 + I*d*x^2])/2]))/(Cos[ArcSin[1 + I*d*x^2]/2] - Sin[ArcSin[1 + I*d*x^2]/2]))/(4*b^2*x)

________________________________________________________________________________________

fricas [F]  time = 0.68, size = 0, normalized size = 0.00 \[ \frac {2 \, {\left (b^{2} d \log \left (d x^{2} + \sqrt {d^{2} x^{2} - 2 i \, d} x - i\right ) + a b d\right )} {\rm integral}\left (\frac {\sqrt {d^{2} x^{2} - 2 i \, d} x}{2 \, a b d x^{2} - 4 i \, a b + {\left (2 \, b^{2} d x^{2} - 4 i \, b^{2}\right )} \log \left (d x^{2} + \sqrt {d^{2} x^{2} - 2 i \, d} x - i\right )}, x\right ) - \sqrt {d^{2} x^{2} - 2 i \, d}}{2 \, {\left (b^{2} d \log \left (d x^{2} + \sqrt {d^{2} x^{2} - 2 i \, d} x - i\right ) + a b d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*(b^2*d*log(d*x^2 + sqrt(d^2*x^2 - 2*I*d)*x - I) + a*b*d)*integral(sqrt(d^2*x^2 - 2*I*d)*x/(2*a*b*d*x^2
- 4*I*a*b + (2*b^2*d*x^2 - 4*I*b^2)*log(d*x^2 + sqrt(d^2*x^2 - 2*I*d)*x - I)), x) - sqrt(d^2*x^2 - 2*I*d))/(b^
2*d*log(d*x^2 + sqrt(d^2*x^2 - 2*I*d)*x - I) + a*b*d)

________________________________________________________________________________________

giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was
done assuming [d,x]=[45,-28]Bad conditionned root j= 1 value -35280.3655931 ratio 0.379661795171 mindist 0.443
514529616Bad conditionned root j= 1 value -5105.29327315 ratio 1.07361778233 mindist 2.29350729132Warning, cho
osing root of [1,0,%%%{-6,[2,4]%%%}+%%%{-8,[0,0]%%%},%%%{-8,[3,6]%%%}+%%%{-32,[1,2]%%%},%%%{-3,[4,8]%%%}+%%%{-
24,[2,4]%%%}+%%%{16,[0,0]%%%}] at parameters values [7,-27]Warning, need to choose a branch for the root of a
polynomial with parameters. This might be wrong.The choice was done assuming [d,t_nostep]=[79,3]schur row 3 8.
74347e-08Bad conditionned root j= 2 value -151313.412862 ratio 11.2206791301 mindist 48.7986537395Warning, cho
osing root of [1,0,%%%{-6,[2,4]%%%}+%%%{-8,[0,0]%%%},%%%{-8,[3,6]%%%}+%%%{-32,[1,2]%%%},%%%{-3,[4,8]%%%}+%%%{-
24,[2,4]%%%}+%%%{16,[0,0]%%%}] at parameters values [63,-49]Warning, need to choose a branch for the root of a
polynomial with parameters. This might be wrong.The choice was done assuming [d,t_nostep]=[-27,9]Bad conditio
nned root j= 2 value 147025.62453 ratio 5.74493624992 mindist 24.9427695529Warning, choosing root of [1,0,%%%{
-6,[2,4]%%%}+%%%{-8,[0,0]%%%},%%%{-8,[3,6]%%%}+%%%{-32,[1,2]%%%},%%%{-3,[4,8]%%%}+%%%{-24,[2,4]%%%}+%%%{16,[0,
0]%%%}] at parameters values [-30,70]Evaluation time: 12.5sym2poly/r2sym(const gen & e,const index_m & i,const
vecteur & l) Error: Bad Argument Value

________________________________________________________________________________________

maple [F]  time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a +b \arcsinh \left (d \,x^{2}-i\right )\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {d^{2} x^{4} - 3 i \, d x^{2} + {\left (d^{\frac {3}{2}} x^{3} - 2 i \, \sqrt {d} x\right )} \sqrt {d x^{2} - 2 i} - 2}{2 \, a b d^{2} x^{3} - 4 i \, a b d x + {\left (2 \, b^{2} d^{2} x^{3} - 4 i \, b^{2} d x + {\left (2 \, b^{2} d^{\frac {3}{2}} x^{2} - 2 i \, b^{2} \sqrt {d}\right )} \sqrt {d x^{2} - 2 i}\right )} \log \left (d x^{2} + \sqrt {d x^{2} - 2 i} \sqrt {d} x - i\right ) + 2 \, {\left (a b d^{\frac {3}{2}} x^{2} - i \, a b \sqrt {d}\right )} \sqrt {d x^{2} - 2 i}} + \int \frac {2 \, d^{3} x^{6} - 6 i \, d^{2} x^{4} + {\left (2 \, d^{2} x^{4} - 2 i \, d x^{2} - 4\right )} {\left (d x^{2} - 2 i\right )} + 2 \, {\left (2 \, d^{\frac {5}{2}} x^{5} - 4 i \, d^{\frac {3}{2}} x^{3} - \sqrt {d} x\right )} \sqrt {d x^{2} - 2 i} - 8 i}{4 \, a b d^{3} x^{6} - 16 i \, a b d^{2} x^{4} - 16 \, a b d x^{2} + {\left (4 \, a b d^{2} x^{4} - 8 i \, a b d x^{2} - 4 \, a b\right )} {\left (d x^{2} - 2 i\right )} + {\left (4 \, b^{2} d^{3} x^{6} - 16 i \, b^{2} d^{2} x^{4} - 16 \, b^{2} d x^{2} + 4 \, {\left (b^{2} d^{2} x^{4} - 2 i \, b^{2} d x^{2} - b^{2}\right )} {\left (d x^{2} - 2 i\right )} + 8 \, {\left (b^{2} d^{\frac {5}{2}} x^{5} - 3 i \, b^{2} d^{\frac {3}{2}} x^{3} - 2 \, b^{2} \sqrt {d} x\right )} \sqrt {d x^{2} - 2 i}\right )} \log \left (d x^{2} + \sqrt {d x^{2} - 2 i} \sqrt {d} x - i\right ) + {\left (8 \, a b d^{\frac {5}{2}} x^{5} - 24 i \, a b d^{\frac {3}{2}} x^{3} - 16 \, a b \sqrt {d} x\right )} \sqrt {d x^{2} - 2 i}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(d^2*x^4 - 3*I*d*x^2 + (d^(3/2)*x^3 - 2*I*sqrt(d)*x)*sqrt(d*x^2 - 2*I) - 2)/(2*a*b*d^2*x^3 - 4*I*a*b*d*x + (2
*b^2*d^2*x^3 - 4*I*b^2*d*x + (2*b^2*d^(3/2)*x^2 - 2*I*b^2*sqrt(d))*sqrt(d*x^2 - 2*I))*log(d*x^2 + sqrt(d*x^2 -
2*I)*sqrt(d)*x - I) + 2*(a*b*d^(3/2)*x^2 - I*a*b*sqrt(d))*sqrt(d*x^2 - 2*I)) + integrate((2*d^3*x^6 - 6*I*d^2
*x^4 + (2*d^2*x^4 - 2*I*d*x^2 - 4)*(d*x^2 - 2*I) + 2*(2*d^(5/2)*x^5 - 4*I*d^(3/2)*x^3 - sqrt(d)*x)*sqrt(d*x^2
- 2*I) - 8*I)/(4*a*b*d^3*x^6 - 16*I*a*b*d^2*x^4 - 16*a*b*d*x^2 + (4*a*b*d^2*x^4 - 8*I*a*b*d*x^2 - 4*a*b)*(d*x^
2 - 2*I) + (4*b^2*d^3*x^6 - 16*I*b^2*d^2*x^4 - 16*b^2*d*x^2 + 4*(b^2*d^2*x^4 - 2*I*b^2*d*x^2 - b^2)*(d*x^2 - 2
*I) + 8*(b^2*d^(5/2)*x^5 - 3*I*b^2*d^(3/2)*x^3 - 2*b^2*sqrt(d)*x)*sqrt(d*x^2 - 2*I))*log(d*x^2 + sqrt(d*x^2 -
2*I)*sqrt(d)*x - I) + (8*a*b*d^(5/2)*x^5 - 24*I*a*b*d^(3/2)*x^3 - 16*a*b*sqrt(d)*x)*sqrt(d*x^2 - 2*I)), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (a+b\,\mathrm {asinh}\left (d\,x^2-\mathrm {i}\right )\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*asinh(d*x^2 - 1i))^2,x)

[Out]

int(1/(a + b*asinh(d*x^2 - 1i))^2, x)

________________________________________________________________________________________

sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError

________________________________________________________________________________________