3.4.21 \(\int (a-i b \text {ArcSin}(1+i d x^2))^4 \, dx\) [321]

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

[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 = 0.02, 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 = {4898, 8} \begin {gather*} -\frac {192 b^3 \sqrt {d^2 x^4-2 i d x^2} \left (a-i b \text {ArcSin}\left (1+i d x^2\right )\right )}{d x}+48 b^2 x \left (a-i b \text {ArcSin}\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 \text {ArcSin}\left (1+i d x^2\right )\right )^3}{d x}+x \left (a-i b \text {ArcSin}\left (1+i d x^2\right )\right )^4+384 b^4 x \end {gather*}

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*Arc
Sin[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*Ar
cSin[1 + I*d*x^2])^4

Rule 8

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

Rule 4898

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 = 0.09, size = 149, normalized size = 0.97 \begin {gather*} -\frac {8 b \sqrt {d x^2 \left (-2 i+d x^2\right )} \left (a-i b \text {ArcSin}\left (1+i d x^2\right )\right )^3}{d x}+x \left (a-i b \text {ArcSin}\left (1+i d x^2\right )\right )^4+48 b^2 \left (8 b^2 x-\frac {4 b \sqrt {d x^2 \left (-2 i+d x^2\right )} \left (a-i b \text {ArcSin}\left (1+i d x^2\right )\right )}{d x}+x \left (a-i b \text {ArcSin}\left (1+i d x^2\right )\right )^2\right ) \end {gather*}

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 +
48*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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Maple [F]
time = 0.93, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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(2*(2*((a*b^3*d^2 - 2*b^4*d^2)*x^4 - 2*a*b^3 - (3*I*a*b^3*d -
4*I*b^4*d)*x^2 + ((a*b^3*d^(3/2) - 2*b^4*d^(3/2))*x^3 - 2*(I*a*b^3*sqrt(d) - 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 + 3*(a^2*b^2*d^2*x^4 - 3*I*a^2*b^2*d*x^2 - 2*a^2*b^2 + (a^
2*b^2*d^(3/2)*x^3 - 2*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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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 269 vs. \(2 (129) = 258\).
time = 0.39, size = 269, normalized size = 1.76 \begin {gather*} \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} \end {gather*}

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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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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 >> Invalid comparison of non-real -I

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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,sageVARx):;OUTP
UT:Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choi
ce was done

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

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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