\(\int (a-i b \arcsin (1+i d x^2))^{3/2} \, dx\) [336]

Optimal result

Integrand size = 22, antiderivative size = 310 \[ \int \left (a-i b \arcsin \left (1+i d x^2\right )\right )^{3/2} \, dx=-\frac {3 b \sqrt {-2 i d x^2+d^2 x^4} \sqrt {a-i b \arcsin \left (1+i d x^2\right )}}{d x}+x \left (a-i b \arcsin \left (1+i d x^2\right )\right )^{3/2}-\frac {3 b^2 \sqrt {\pi } x \operatorname {FresnelS}\left (\frac {\sqrt {a-i b \arcsin \left (1+i d x^2\right )}}{\sqrt {-i b} \sqrt {\pi }}\right ) \left (\cosh \left (\frac {a}{2 b}\right )+i \sinh \left (\frac {a}{2 b}\right )\right )}{\sqrt {-i b} \left (\cos \left (\frac {1}{2} \arcsin \left (1+i d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1+i d x^2\right )\right )\right )}-\frac {3 \sqrt {-i b} b \sqrt {\pi } x \operatorname {FresnelC}\left (\frac {\sqrt {a-i b \arcsin \left (1+i d x^2\right )}}{\sqrt {-i b} \sqrt {\pi }}\right ) \left (i \cosh \left (\frac {a}{2 b}\right )+\sinh \left (\frac {a}{2 b}\right )\right )}{\cos \left (\frac {1}{2} \arcsin \left (1+i d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1+i d x^2\right )\right )} \]

[Out]

x*(a-I*b*arcsin(1+I*d*x^2))^(3/2)-3*b^2*x*FresnelS((a-I*b*arcsin(1+I*d*x^2))^(1/2)/(-I*b)^(1/2)/Pi^(1/2))*(cos
h(1/2*a/b)+I*sinh(1/2*a/b))*Pi^(1/2)/(cos(1/2*arcsin(1+I*d*x^2))-sin(1/2*arcsin(1+I*d*x^2)))/(-I*b)^(1/2)-3*b*
x*FresnelC((a-I*b*arcsin(1+I*d*x^2))^(1/2)/(-I*b)^(1/2)/Pi^(1/2))*(I*cosh(1/2*a/b)+sinh(1/2*a/b))*(-I*b)^(1/2)
*Pi^(1/2)/(cos(1/2*arcsin(1+I*d*x^2))-sin(1/2*arcsin(1+I*d*x^2)))-3*b*(-2*I*d*x^2+d^2*x^4)^(1/2)*(a-I*b*arcsin
(1+I*d*x^2))^(1/2)/d/x

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {4898, 4903} \[ \int \left (a-i b \arcsin \left (1+i d x^2\right )\right )^{3/2} \, dx=-\frac {3 \sqrt {\pi } b^2 x \left (\cosh \left (\frac {a}{2 b}\right )+i \sinh \left (\frac {a}{2 b}\right )\right ) \operatorname {FresnelS}\left (\frac {\sqrt {a-i b \arcsin \left (i d x^2+1\right )}}{\sqrt {-i b} \sqrt {\pi }}\right )}{\sqrt {-i b} \left (\cos \left (\frac {1}{2} \arcsin \left (1+i d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1+i d x^2\right )\right )\right )}-\frac {3 b \sqrt {d^2 x^4-2 i d x^2} \sqrt {a-i b \arcsin \left (1+i d x^2\right )}}{d x}-\frac {3 \sqrt {\pi } \sqrt {-i b} b x \left (\sinh \left (\frac {a}{2 b}\right )+i \cosh \left (\frac {a}{2 b}\right )\right ) \operatorname {FresnelC}\left (\frac {\sqrt {a-i b \arcsin \left (i d x^2+1\right )}}{\sqrt {-i b} \sqrt {\pi }}\right )}{\cos \left (\frac {1}{2} \arcsin \left (1+i d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1+i d x^2\right )\right )}+x \left (a-i b \arcsin \left (1+i d x^2\right )\right )^{3/2} \]

[In]

Int[(a - I*b*ArcSin[1 + I*d*x^2])^(3/2),x]

[Out]

(-3*b*Sqrt[(-2*I)*d*x^2 + d^2*x^4]*Sqrt[a - I*b*ArcSin[1 + I*d*x^2]])/(d*x) + x*(a - I*b*ArcSin[1 + I*d*x^2])^
(3/2) - (3*b^2*Sqrt[Pi]*x*FresnelS[Sqrt[a - I*b*ArcSin[1 + I*d*x^2]]/(Sqrt[(-I)*b]*Sqrt[Pi])]*(Cosh[a/(2*b)] +
I*Sinh[a/(2*b)]))/(Sqrt[(-I)*b]*(Cos[ArcSin[1 + I*d*x^2]/2] - Sin[ArcSin[1 + I*d*x^2]/2])) - (3*Sqrt[(-I)*b]*
b*Sqrt[Pi]*x*FresnelC[Sqrt[a - I*b*ArcSin[1 + I*d*x^2]]/(Sqrt[(-I)*b]*Sqrt[Pi])]*(I*Cosh[a/(2*b)] + Sinh[a/(2*
b)]))/(Cos[ArcSin[1 + I*d*x^2]/2] - Sin[ArcSin[1 + I*d*x^2]/2])

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]

Rule 4903

Int[1/Sqrt[(a_.) + ArcSin[(c_) + (d_.)*(x_)^2]*(b_.)], x_Symbol] :> Simp[(-Sqrt[Pi])*x*(Cos[a/(2*b)] - c*Sin[a
/(2*b)])*(FresnelC[(1/(Sqrt[b*c]*Sqrt[Pi]))*Sqrt[a + b*ArcSin[c + d*x^2]]]/(Sqrt[b*c]*(Cos[ArcSin[c + d*x^2]/2
] - c*Sin[ArcSin[c + d*x^2]/2]))), x] - Simp[Sqrt[Pi]*x*(Cos[a/(2*b)] + c*Sin[a/(2*b)])*(FresnelS[(1/(Sqrt[b*c
]*Sqrt[Pi]))*Sqrt[a + b*ArcSin[c + d*x^2]]]/(Sqrt[b*c]*(Cos[ArcSin[c + d*x^2]/2] - c*Sin[ArcSin[c + d*x^2]/2])
)), x] /; FreeQ[{a, b, c, d}, x] && EqQ[c^2, 1]

Rubi steps \begin{align*} \text {integral}& = -\frac {3 b \sqrt {-2 i d x^2+d^2 x^4} \sqrt {a-i b \arcsin \left (1+i d x^2\right )}}{d x}+x \left (a-i b \arcsin \left (1+i d x^2\right )\right )^{3/2}+\left (3 b^2\right ) \int \frac {1}{\sqrt {a-i b \arcsin \left (1+i d x^2\right )}} \, dx \\ & = -\frac {3 b \sqrt {-2 i d x^2+d^2 x^4} \sqrt {a-i b \arcsin \left (1+i d x^2\right )}}{d x}+x \left (a-i b \arcsin \left (1+i d x^2\right )\right )^{3/2}-\frac {3 b^2 \sqrt {\pi } x \operatorname {FresnelS}\left (\frac {\sqrt {a-i b \arcsin \left (1+i d x^2\right )}}{\sqrt {-i b} \sqrt {\pi }}\right ) \left (\cosh \left (\frac {a}{2 b}\right )+i \sinh \left (\frac {a}{2 b}\right )\right )}{\sqrt {-i b} \left (\cos \left (\frac {1}{2} \arcsin \left (1+i d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1+i d x^2\right )\right )\right )}-\frac {3 \sqrt {-i b} b \sqrt {\pi } x \operatorname {FresnelC}\left (\frac {\sqrt {a-i b \arcsin \left (1+i d x^2\right )}}{\sqrt {-i b} \sqrt {\pi }}\right ) \left (i \cosh \left (\frac {a}{2 b}\right )+\sinh \left (\frac {a}{2 b}\right )\right )}{\cos \left (\frac {1}{2} \arcsin \left (1+i d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1+i d x^2\right )\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.82 \[ \int \left (a-i b \arcsin \left (1+i d x^2\right )\right )^{3/2} \, dx=-\frac {3 b \sqrt {d x^2 \left (-2 i+d x^2\right )} \sqrt {a-i b \arcsin \left (1+i d x^2\right )}}{d x}+x \left (a-i b \arcsin \left (1+i d x^2\right )\right )^{3/2}-\frac {3 (-i b)^{3/2} \sqrt {\pi } x \left (-\operatorname {FresnelC}\left (\frac {\sqrt {a-i b \arcsin \left (1+i d x^2\right )}}{\sqrt {-i b} \sqrt {\pi }}\right ) \left (\cosh \left (\frac {a}{2 b}\right )-i \sinh \left (\frac {a}{2 b}\right )\right )-\operatorname {FresnelS}\left (\frac {\sqrt {a-i b \arcsin \left (1+i d x^2\right )}}{\sqrt {-i b} \sqrt {\pi }}\right ) \left (\cosh \left (\frac {a}{2 b}\right )+i \sinh \left (\frac {a}{2 b}\right )\right )\right )}{\cos \left (\frac {1}{2} \arcsin \left (1+i d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1+i d x^2\right )\right )} \]

[In]

Integrate[(a - I*b*ArcSin[1 + I*d*x^2])^(3/2),x]

[Out]

(-3*b*Sqrt[d*x^2*(-2*I + d*x^2)]*Sqrt[a - I*b*ArcSin[1 + I*d*x^2]])/(d*x) + x*(a - I*b*ArcSin[1 + I*d*x^2])^(3
/2) - (3*((-I)*b)^(3/2)*Sqrt[Pi]*x*(-(FresnelC[Sqrt[a - I*b*ArcSin[1 + I*d*x^2]]/(Sqrt[(-I)*b]*Sqrt[Pi])]*(Cos
h[a/(2*b)] - I*Sinh[a/(2*b)])) - FresnelS[Sqrt[a - I*b*ArcSin[1 + I*d*x^2]]/(Sqrt[(-I)*b]*Sqrt[Pi])]*(Cosh[a/(
2*b)] + I*Sinh[a/(2*b)])))/(Cos[ArcSin[1 + I*d*x^2]/2] - Sin[ArcSin[1 + I*d*x^2]/2])

Maple [F]

\[\int {\left (a +b \,\operatorname {arcsinh}\left (d \,x^{2}-i\right )\right )}^{\frac {3}{2}}d x\]

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \left (a-i b \arcsin \left (1+i d x^2\right )\right )^{3/2} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((a+b*arcsinh(-I+d*x^2))^(3/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [F(-2)]

Exception generated. \[ \int \left (a-i b \arcsin \left (1+i d x^2\right )\right )^{3/2} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((a+b*asinh(-I+d*x**2))**(3/2),x)

[Out]

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

Maxima [F]

\[ \int \left (a-i b \arcsin \left (1+i d x^2\right )\right )^{3/2} \, dx=\int { {\left (b \operatorname {arsinh}\left (d x^{2} - i\right ) + a\right )}^{\frac {3}{2}} \,d x } \]

[In]

integrate((a+b*arcsinh(-I+d*x^2))^(3/2),x, algorithm="maxima")

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \left (a-i b \arcsin \left (1+i d x^2\right )\right )^{3/2} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((a+b*arcsinh(-I+d*x^2))^(3/2),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. \[ \int \left (a-i b \arcsin \left (1+i d x^2\right )\right )^{3/2} \, dx=\int {\left (a+b\,\mathrm {asinh}\left (d\,x^2-\mathrm {i}\right )\right )}^{3/2} \,d x \]

[In]

int((a + b*asinh(d*x^2 - 1i))^(3/2),x)

[Out]

int((a + b*asinh(d*x^2 - 1i))^(3/2), x)