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

Optimal. Leaf size=50 \[ a x-\frac {2 b \sqrt {d^2 x^4+2 i d x^2}}{d x}+i b x \sin ^{-1}\left (1-i d x^2\right ) \]

[Out]

a*x-I*b*x*arcsin(-1+I*d*x^2)-2*b*(2*I*d*x^2+d^2*x^4)^(1/2)/d/x

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Rubi [A]  time = 0.04, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4840, 12, 1588} \[ a x-\frac {2 b \sqrt {d^2 x^4+2 i d x^2}}{d x}+i b x \sin ^{-1}\left (1-i d x^2\right ) \]

Antiderivative was successfully verified.

[In]

Int[a + I*b*ArcSin[1 - I*d*x^2],x]

[Out]

a*x - (2*b*Sqrt[(2*I)*d*x^2 + d^2*x^4])/(d*x) + I*b*x*ArcSin[1 - I*d*x^2]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1588

Int[(Pp_)*(Qq_)^(m_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*x^(p - q
+ 1)*Qq^(m + 1))/((p + m*q + 1)*Coeff[Qq, x, q]), x] /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x,
q]*Pp, Coeff[Pp, x, p]*x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]] /; FreeQ[m, x] && PolyQ[Pp, x] && Pol
yQ[Qq, x] && NeQ[m, -1]

Rule 4840

Int[ArcSin[u_], x_Symbol] :> Simp[x*ArcSin[u], x] - Int[SimplifyIntegrand[(x*D[u, x])/Sqrt[1 - u^2], x], x] /;
InverseFunctionFreeQ[u, x] &&  !FunctionOfExponentialQ[u, x]

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 48, normalized size = 0.96 \[ a x-\frac {2 b \sqrt {d x^2 \left (d x^2+2 i\right )}}{d x}+i b x \sin ^{-1}\left (1-i d x^2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.45, size = 52, normalized size = 1.04 \[ \frac {b d x \log \left (d x^{2} + \sqrt {d^{2} x^{2} + 2 i \, d} x + i\right ) + a d x - 2 \, \sqrt {d^{2} x^{2} + 2 i \, d} b}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*d*x*log(d*x^2 + sqrt(d^2*x^2 + 2*I*d)*x + I) + a*d*x - 2*sqrt(d^2*x^2 + 2*I*d)*b)/d

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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(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,x]=[29,3]schur row 1 3.19975e-
11schur row 2 -2.1116e-07Francis algorithm not precise enough for[1.0,0.0,-137282971022,-2.76877787308e+16,-1.
57055117809e+21]Bad conditionned root j= 1 value -151344.383904 ratio 18.0968847681 mindist 79.7696956145Warni
ng, 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]Evaluation time: 1.26sym2poly/r2sym(const g
en & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [A]  time = 0.02, size = 47, normalized size = 0.94 \[ a x +b \left (x \arcsinh \left (d \,x^{2}+i\right )-\frac {2 x \left (d \,x^{2}+2 i\right )}{\sqrt {d^{2} x^{4}+2 i d \,x^{2}}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x+b*(x*arcsinh(I+d*x^2)-2/(2*I*d*x^2+d^2*x^4)^(1/2)*x*(d*x^2+2*I))

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maxima [A]  time = 0.73, size = 44, normalized size = 0.88 \[ {\left (x \operatorname {arsinh}\left (d x^{2} + i\right ) - \frac {2 \, {\left (d^{\frac {3}{2}} x^{2} + 2 i \, \sqrt {d}\right )}}{\sqrt {d x^{2} + 2 i} d}\right )} b + a x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x*arcsinh(d*x^2 + I) - 2*(d^(3/2)*x^2 + 2*I*sqrt(d))/(sqrt(d*x^2 + 2*I)*d))*b + a*x

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mupad [B]  time = 0.53, size = 39, normalized size = 0.78 \[ a\,x+b\,x\,\mathrm {asinh}\left (d\,x^2+1{}\mathrm {i}\right )-\frac {2\,b\,\sqrt {{\left (d\,x^2+1{}\mathrm {i}\right )}^2+1}}{d\,x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a + b*asinh(d*x^2 + 1i),x)

[Out]

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

[Out]

Exception raised: TypeError

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