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

Optimal. Leaf size=76 \[ -\frac {4 b \sqrt {d^2 x^4-2 i d x^2} \left (a-i b \sin ^{-1}\left (1+i d x^2\right )\right )}{d x}+x \left (a-i b \sin ^{-1}\left (1+i d x^2\right )\right )^2+8 b^2 x \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {4814, 8} \[ -\frac {4 b \sqrt {d^2 x^4-2 i d x^2} \left (a-i b \sin ^{-1}\left (1+i d x^2\right )\right )}{d x}+x \left (a-i b \sin ^{-1}\left (1+i d x^2\right )\right )^2+8 b^2 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 4814

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

Rubi steps

\begin {align*} \int \left (a-i b \sin ^{-1}\left (1+i d x^2\right )\right )^2 \, dx &=-\frac {4 b \sqrt {-2 i d x^2+d^2 x^4} \left (a-i b \sin ^{-1}\left (1+i d x^2\right )\right )}{d x}+x \left (a-i b \sin ^{-1}\left (1+i d x^2\right )\right )^2+\left (8 b^2\right ) \int 1 \, dx\\ &=8 b^2 x-\frac {4 b \sqrt {-2 i d x^2+d^2 x^4} \left (a-i b \sin ^{-1}\left (1+i d x^2\right )\right )}{d x}+x \left (a-i b \sin ^{-1}\left (1+i d x^2\right )\right )^2\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 76, normalized size = 1.00 \[ -\frac {4 b \sqrt {d^2 x^4-2 i d x^2} \left (a-i b \sin ^{-1}\left (1+i d x^2\right )\right )}{d x}+x \left (a-i b \sin ^{-1}\left (1+i d x^2\right )\right )^2+8 b^2 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.51, size = 114, normalized size = 1.50 \[ \frac {b^{2} d x \log \left (d x^{2} + \sqrt {d^{2} x^{2} - 2 i \, d} x - i\right )^{2} + {\left (a^{2} + 8 \, b^{2}\right )} d x - 4 \, \sqrt {d^{2} x^{2} - 2 i \, d} a b + 2 \, {\left (a b d x - 2 \, \sqrt {d^{2} x^{2} - 2 i \, d} b^{2}\right )} \log \left (d x^{2} + \sqrt {d^{2} x^{2} - 2 i \, d} x - i\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b^2*d*x*log(d*x^2 + sqrt(d^2*x^2 - 2*I*d)*x - I)^2 + (a^2 + 8*b^2)*d*x - 4*sqrt(d^2*x^2 - 2*I*d)*a*b + 2*(a*b
*d*x - 2*sqrt(d^2*x^2 - 2*I*d)*b^2)*log(d*x^2 + sqrt(d^2*x^2 - 2*I*d)*x - I))/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))^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: 6.85sym2poly/r2sym(const gen &
e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [F]  time = 0.18, size = 0, normalized size = 0.00 \[ \int \left (a +b \arcsinh \left (d \,x^{2}-i\right )\right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asinh(d*x^2 - 1i))^2,x)

[Out]

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

[Out]

Exception raised: TypeError

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