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

Optimal. Leaf size=194 \[ \frac {x \left (-\sinh \left (\frac {a}{2 b}\right )+i \cosh \left (\frac {a}{2 b}\right )\right ) \text {Ci}\left (-\frac {i \left (a+i b \sin ^{-1}\left (1-i d x^2\right )\right )}{2 b}\right )}{2 b \left (\cos \left (\frac {1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )-\sin \left (\frac {1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )\right )}-\frac {x \left (\sinh \left (\frac {a}{2 b}\right )+i \cosh \left (\frac {a}{2 b}\right )\right ) \text {Si}\left (\frac {i a}{2 b}-\frac {1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )}{2 b \left (\cos \left (\frac {1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )-\sin \left (\frac {1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )\right )} \]

[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)))

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Rubi [A]  time = 0.05, antiderivative size = 194, 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 = {4816} \[ \frac {x \left (-\sinh \left (\frac {a}{2 b}\right )+i \cosh \left (\frac {a}{2 b}\right )\right ) \text {CosIntegral}\left (-\frac {i \left (a+i b \sin ^{-1}\left (1-i d x^2\right )\right )}{2 b}\right )}{2 b \left (\cos \left (\frac {1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )-\sin \left (\frac {1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )\right )}-\frac {x \left (\sinh \left (\frac {a}{2 b}\right )+i \cosh \left (\frac {a}{2 b}\right )\right ) \text {Si}\left (\frac {i a}{2 b}-\frac {1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )}{2 b \left (\cos \left (\frac {1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )-\sin \left (\frac {1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*CosIntegral[((-I/2)*(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 4816

Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)^2]*(b_.))^(-1), x_Symbol] :> -Simp[(x*(c*Cos[a/(2*b)] - Sin[a/(2*b)])*Co
sIntegral[(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[ArcS
in[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*} \int \frac {1}{a+i b \sin ^{-1}\left (1-i d x^2\right )} \, dx &=\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 (i \cosh \left (\frac {a}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right )\right )}{2 b \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 {Si}\left (\frac {i a}{2 b}-\frac {1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )}{2 b \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*}

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Mathematica [A]  time = 0.79, size = 150, normalized size = 0.77 \[ \frac {x \left (\left (-\sinh \left (\frac {a}{2 b}\right )+i \cosh \left (\frac {a}{2 b}\right )\right ) \text {Ci}\left (\frac {1}{2} \left (\sin ^{-1}\left (1-i d x^2\right )-\frac {i a}{b}\right )\right )+\left (-\sinh \left (\frac {a}{2 b}\right )-i \cosh \left (\frac {a}{2 b}\right )\right ) \text {Si}\left (\frac {i a}{2 b}-\frac {1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )\right )}{2 b \left (\cos \left (\frac {1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )-\sin \left (\frac {1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )\right )} \]

Antiderivative was successfully verified.

[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]))

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fricas [F]  time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{b \log \left (d x^{2} + \sqrt {d^{2} x^{2} + 2 i \, d} x + i\right ) + a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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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)),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]=[60,97]schur row 1 7
.48504e-11Francis algorithm not precise enough for[1.0,0.0,-1.91223246961e+12,-1.43937562454e+18,-3.0471941815
7e+23]Bad conditionned root j= 2 value -564549.069246 ratio 4.13534933689 mindist 8.26009499958schur row 3 8.3
2254e-09schur row 3 8.32254e-09Francis algorithm not precise enough for[1.0,0.0,-137282971022,-2.76877787308e+
16,-1.57055117809e+21]Bad conditionned root j= 3 value -151279.357647 ratio 3.67253338015 mindist 17.337314081
1Warning, 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 [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]=[-20,1
4]schur row 1 6.86402e-10Francis algorithm not precise enough for[1.0,0.0,-129654000008,2.54121840047e+16,-1.4
0084664352e+21]Bad conditionned root j= 2 value 146992.858887 ratio 1.597707895 mindist 8.30647902455Warning,
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: 3.81sym2poly/r2sym(const gen &
e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [F]  time = 0.16, size = 0, normalized size = 0.00 \[ \int \frac {1}{a +b \arcsinh \left (d \,x^{2}+i\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{b \operatorname {arsinh}\left (d x^{2} + i\right ) + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{a+b\,\mathrm {asinh}\left (d\,x^2+1{}\mathrm {i}\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*asinh(d*x^2 + 1i)),x)

[Out]

int(1/(a + b*asinh(d*x^2 + 1i)), x)

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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)),x)

[Out]

Exception raised: TypeError

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