\(\int (a+i b \arcsin (1-i d x^2))^2 \, dx\) [316]

Optimal result

Integrand size = 20, antiderivative size = 76 \[ \int \left (a+i b \arcsin \left (1-i d x^2\right )\right )^2 \, dx=8 b^2 x-\frac {4 b \sqrt {2 i d x^2+d^2 x^4} \left (a+i b \arcsin \left (1-i d x^2\right )\right )}{d x}+x \left (a+i b \arcsin \left (1-i d x^2\right )\right )^2 \]

[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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {4898, 8} \[ \int \left (a+i b \arcsin \left (1-i d x^2\right )\right )^2 \, dx=-\frac {4 b \sqrt {d^2 x^4+2 i d x^2} \left (a+i b \arcsin \left (1-i d x^2\right )\right )}{d x}+x \left (a+i b \arcsin \left (1-i d x^2\right )\right )^2+8 b^2 x \]

[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 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*} \text {integral}& = -\frac {4 b \sqrt {2 i d x^2+d^2 x^4} \left (a+i b \arcsin \left (1-i d x^2\right )\right )}{d x}+x \left (a+i b \arcsin \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 \arcsin \left (1-i d x^2\right )\right )}{d x}+x \left (a+i b \arcsin \left (1-i d x^2\right )\right )^2 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00 \[ \int \left (a+i b \arcsin \left (1-i d x^2\right )\right )^2 \, dx=8 b^2 x-\frac {4 b \sqrt {2 i d x^2+d^2 x^4} \left (a+i b \arcsin \left (1-i d x^2\right )\right )}{d x}+x \left (a+i b \arcsin \left (1-i d x^2\right )\right )^2 \]

[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

Maple [F]

\[\int {\left (a +b \,\operatorname {arcsinh}\left (d \,x^{2}+i\right )\right )}^{2}d x\]

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.50 \[ \int \left (a+i b \arcsin \left (1-i d x^2\right )\right )^2 \, dx=\frac {b^{2} d x \log \left (d x^{2} + \sqrt {d^{2} x^{2} + 2 i \, d} x + i\right )^{2} + {\left (a^{2} + 8 \, b^{2}\right )} d x - 4 \, \sqrt {d^{2} x^{2} + 2 i \, d} a b + 2 \, {\left (a b d x - 2 \, \sqrt {d^{2} x^{2} + 2 i \, d} b^{2}\right )} \log \left (d x^{2} + \sqrt {d^{2} x^{2} + 2 i \, d} x + i\right )}{d} \]

[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

Sympy [F(-2)]

Exception generated. \[ \int \left (a+i b \arcsin \left (1-i d x^2\right )\right )^2 \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((a+b*asinh(I+d*x**2))**2,x)

[Out]

Exception raised: TypeError >> Invalid comparison of non-real I

Maxima [F]

\[ \int \left (a+i b \arcsin \left (1-i d x^2\right )\right )^2 \, dx=\int { {\left (b \operatorname {arsinh}\left (d x^{2} + i\right ) + a\right )}^{2} \,d x } \]

[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 + 2*I*d*x^2 + (d^(3/2)*x^3 + I*sqrt(d)*x)*sqrt(d*x^2 + 2*I))*l
og(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

Giac [F(-2)]

Exception generated. \[ \int \left (a+i b \arcsin \left (1-i d x^2\right )\right )^2 \, dx=\text {Exception raised: TypeError} \]

[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,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \left (a+i b \arcsin \left (1-i d x^2\right )\right )^2 \, dx=\int {\left (a+b\,\mathrm {asinh}\left (d\,x^2+1{}\mathrm {i}\right )\right )}^2 \,d x \]

[In]

int((a + b*asinh(d*x^2 + 1i))^2,x)

[Out]

int((a + b*asinh(d*x^2 + 1i))^2, x)