3.314 \(\int (a+i b \sin ^{-1}(1-i d x^2))^4 \, dx\)

Optimal. Leaf size=153 \[ -\frac {192 b^3 \sqrt {d^2 x^4+2 i d x^2} \left (a+i b \sin ^{-1}\left (1-i d x^2\right )\right )}{d x}+48 b^2 x \left (a+i b \sin ^{-1}\left (1-i d x^2\right )\right )^2-\frac {8 b \sqrt {d^2 x^4+2 i d x^2} \left (a+i b \sin ^{-1}\left (1-i d x^2\right )\right )^3}{d x}+x \left (a+i b \sin ^{-1}\left (1-i d x^2\right )\right )^4+384 b^4 x \]

[Out]

384*b^4*x+48*b^2*x*(a-I*b*arcsin(-1+I*d*x^2))^2+x*(a-I*b*arcsin(-1+I*d*x^2))^4-192*b^3*(a-I*b*arcsin(-1+I*d*x^
2))*(2*I*d*x^2+d^2*x^4)^(1/2)/d/x-8*b*(a-I*b*arcsin(-1+I*d*x^2))^3*(2*I*d*x^2+d^2*x^4)^(1/2)/d/x

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Rubi [A]  time = 0.04, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {4814, 8} \[ -\frac {192 b^3 \sqrt {d^2 x^4+2 i d x^2} \left (a+i b \sin ^{-1}\left (1-i d x^2\right )\right )}{d x}+48 b^2 x \left (a+i b \sin ^{-1}\left (1-i d x^2\right )\right )^2-\frac {8 b \sqrt {d^2 x^4+2 i d x^2} \left (a+i b \sin ^{-1}\left (1-i d x^2\right )\right )^3}{d x}+x \left (a+i b \sin ^{-1}\left (1-i d x^2\right )\right )^4+384 b^4 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

384*b^4*x - (192*b^3*Sqrt[(2*I)*d*x^2 + d^2*x^4]*(a + I*b*ArcSin[1 - I*d*x^2]))/(d*x) + 48*b^2*x*(a + I*b*ArcS
in[1 - I*d*x^2])^2 - (8*b*Sqrt[(2*I)*d*x^2 + d^2*x^4]*(a + I*b*ArcSin[1 - I*d*x^2])^3)/(d*x) + x*(a + I*b*ArcS
in[1 - I*d*x^2])^4

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 )^4 \, dx &=-\frac {8 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 )^3}{d x}+x \left (a+i b \sin ^{-1}\left (1-i d x^2\right )\right )^4+\left (48 b^2\right ) \int \left (a+i b \sin ^{-1}\left (1-i d x^2\right )\right )^2 \, dx\\ &=-\frac {192 b^3 \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}+48 b^2 x \left (a+i b \sin ^{-1}\left (1-i d x^2\right )\right )^2-\frac {8 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 )^3}{d x}+x \left (a+i b \sin ^{-1}\left (1-i d x^2\right )\right )^4+\left (384 b^4\right ) \int 1 \, dx\\ &=384 b^4 x-\frac {192 b^3 \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}+48 b^2 x \left (a+i b \sin ^{-1}\left (1-i d x^2\right )\right )^2-\frac {8 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 )^3}{d x}+x \left (a+i b \sin ^{-1}\left (1-i d x^2\right )\right )^4\\ \end {align*}

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Mathematica [A]  time = 0.13, size = 149, normalized size = 0.97 \[ 48 b^2 \left (-\frac {4 b \sqrt {d x^2 \left (d x^2+2 i\right )} \left (a+i b \sin ^{-1}\left (1-i d x^2\right )\right )}{d x}+x \left (a+i b \sin ^{-1}\left (1-i d x^2\right )\right )^2+8 b^2 x\right )+x \left (a+i b \sin ^{-1}\left (1-i d x^2\right )\right )^4-\frac {8 b \sqrt {d x^2 \left (d x^2+2 i\right )} \left (a+i b \sin ^{-1}\left (1-i d x^2\right )\right )^3}{d x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*b*Sqrt[d*x^2*(2*I + d*x^2)]*(a + I*b*ArcSin[1 - I*d*x^2])^3)/(d*x) + x*(a + I*b*ArcSin[1 - I*d*x^2])^4 + 4
8*b^2*(8*b^2*x - (4*b*Sqrt[d*x^2*(2*I + d*x^2)]*(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 [B]  time = 0.52, size = 269, normalized size = 1.76 \[ \frac {b^{4} d x \log \left (d x^{2} + \sqrt {d^{2} x^{2} + 2 i \, d} x + i\right )^{4} + 4 \, {\left (a b^{3} d x - 2 \, \sqrt {d^{2} x^{2} + 2 i \, d} b^{4}\right )} \log \left (d x^{2} + \sqrt {d^{2} x^{2} + 2 i \, d} x + i\right )^{3} + {\left (a^{4} + 48 \, a^{2} b^{2} + 384 \, b^{4}\right )} d x - 6 \, {\left (4 \, \sqrt {d^{2} x^{2} + 2 i \, d} a b^{3} - {\left (a^{2} b^{2} + 8 \, b^{4}\right )} d x\right )} \log \left (d x^{2} + \sqrt {d^{2} x^{2} + 2 i \, d} x + i\right )^{2} + 4 \, {\left ({\left (a^{3} b + 24 \, a b^{3}\right )} d x - 6 \, {\left (a^{2} b^{2} + 8 \, b^{4}\right )} \sqrt {d^{2} x^{2} + 2 i \, d}\right )} \log \left (d x^{2} + \sqrt {d^{2} x^{2} + 2 i \, d} x + i\right ) - 8 \, {\left (a^{3} b + 24 \, a b^{3}\right )} \sqrt {d^{2} x^{2} + 2 i \, d}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsinh(I+d*x^2))^4,x, algorithm="fricas")

[Out]

(b^4*d*x*log(d*x^2 + sqrt(d^2*x^2 + 2*I*d)*x + I)^4 + 4*(a*b^3*d*x - 2*sqrt(d^2*x^2 + 2*I*d)*b^4)*log(d*x^2 +
sqrt(d^2*x^2 + 2*I*d)*x + I)^3 + (a^4 + 48*a^2*b^2 + 384*b^4)*d*x - 6*(4*sqrt(d^2*x^2 + 2*I*d)*a*b^3 - (a^2*b^
2 + 8*b^4)*d*x)*log(d*x^2 + sqrt(d^2*x^2 + 2*I*d)*x + I)^2 + 4*((a^3*b + 24*a*b^3)*d*x - 6*(a^2*b^2 + 8*b^4)*s
qrt(d^2*x^2 + 2*I*d))*log(d*x^2 + sqrt(d^2*x^2 + 2*I*d)*x + I) - 8*(a^3*b + 24*a*b^3)*sqrt(d^2*x^2 + 2*I*d))/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))^4,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: 29.4sym2poly/r2sym(const gen & e,const index_m & i,const
vecteur & l) Error: Bad Argument Value

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maple [F]  time = 0.23, size = 0, normalized size = 0.00 \[ \int \left (a +b \arcsinh \left (d \,x^{2}+i\right )\right )^{4}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsinh(I+d*x^2))^4,x)

[Out]

int((a+b*arcsinh(I+d*x^2))^4,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ b^{4} x \log \left (d x^{2} + \sqrt {d x^{2} + 2 i} \sqrt {d} x + i\right )^{4} + 4 \, {\left (x \operatorname {arsinh}\left (d x^{2} + i\right ) - \frac {2 \, {\left (d^{\frac {3}{2}} x^{2} + 2 i \, \sqrt {d}\right )}}{\sqrt {d x^{2} + 2 i} d}\right )} a^{3} b + a^{4} x + \int \frac {{\left (4 \, {\left (a b^{3} d^{2} - 2 \, b^{4} d^{2}\right )} x^{4} - 8 \, a b^{3} + {\left (12 i \, a b^{3} d - 16 i \, b^{4} d\right )} x^{2} + {\left (4 \, {\left (a b^{3} d^{\frac {3}{2}} - 2 \, b^{4} d^{\frac {3}{2}}\right )} x^{3} + {\left (8 i \, a b^{3} \sqrt {d} - 8 i \, b^{4} \sqrt {d}\right )} x\right )} \sqrt {d x^{2} + 2 i}\right )} \log \left (d x^{2} + \sqrt {d x^{2} + 2 i} \sqrt {d} x + i\right )^{3} + {\left (6 \, a^{2} b^{2} d^{2} x^{4} + 18 i \, a^{2} b^{2} d x^{2} - 12 \, a^{2} b^{2} + {\left (6 \, a^{2} b^{2} d^{\frac {3}{2}} x^{3} + 12 i \, a^{2} b^{2} \sqrt {d} x\right )} \sqrt {d x^{2} + 2 i}\right )} \log \left (d x^{2} + \sqrt {d x^{2} + 2 i} \sqrt {d} x + i\right )^{2}}{d^{2} x^{4} + 3 i \, d x^{2} + {\left (d^{\frac {3}{2}} x^{3} + 2 i \, \sqrt {d} x\right )} \sqrt {d x^{2} + 2 i} - 2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsinh(I+d*x^2))^4,x, algorithm="maxima")

[Out]

b^4*x*log(d*x^2 + sqrt(d*x^2 + 2*I)*sqrt(d)*x + I)^4 + 4*(x*arcsinh(d*x^2 + I) - 2*(d^(3/2)*x^2 + 2*I*sqrt(d))
/(sqrt(d*x^2 + 2*I)*d))*a^3*b + a^4*x + integrate(((4*(a*b^3*d^2 - 2*b^4*d^2)*x^4 - 8*a*b^3 + (12*I*a*b^3*d -
16*I*b^4*d)*x^2 + (4*(a*b^3*d^(3/2) - 2*b^4*d^(3/2))*x^3 + (8*I*a*b^3*sqrt(d) - 8*I*b^4*sqrt(d))*x)*sqrt(d*x^2
+ 2*I))*log(d*x^2 + sqrt(d*x^2 + 2*I)*sqrt(d)*x + I)^3 + (6*a^2*b^2*d^2*x^4 + 18*I*a^2*b^2*d*x^2 - 12*a^2*b^2
+ (6*a^2*b^2*d^(3/2)*x^3 + 12*I*a^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)^2)/(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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a+b\,\mathrm {asinh}\left (d\,x^2+1{}\mathrm {i}\right )\right )}^4 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asinh(d*x^2 + 1i))^4,x)

[Out]

int((a + b*asinh(d*x^2 + 1i))^4, 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))**4,x)

[Out]

Exception raised: TypeError

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