\(\int \frac {1}{\sqrt {\arcsin (a x)}} \, dx\) [96]

Optimal result

Integrand size = 8, antiderivative size = 30 \[ \int \frac {1}{\sqrt {\arcsin (a x)}} \, dx=\frac {\sqrt {2 \pi } \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{a} \]

[Out]

FresnelC(2^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4719, 3385, 3433} \[ \int \frac {1}{\sqrt {\arcsin (a x)}} \, dx=\frac {\sqrt {2 \pi } \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{a} \]

[In]

Int[1/Sqrt[ArcSin[a*x]],x]

[Out]

(Sqrt[2*Pi]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/a

Rule 3385

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Cos[f*(x^2/d)],
x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]

Rule 3433

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[d, 2]))*FresnelC[Sqrt[2/Pi]*Rt[d, 2
]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]

Rule 4719

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[1/(b*c), Subst[Int[x^n*Cos[-a/b + x/b], x], x,
a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, c, n}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\arcsin (a x)\right )}{a} \\ & = \frac {2 \text {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\arcsin (a x)}\right )}{a} \\ & = \frac {\sqrt {2 \pi } \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{a} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.02 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.30 \[ \int \frac {1}{\sqrt {\arcsin (a x)}} \, dx=-\frac {i \left (\sqrt {-i \arcsin (a x)} \Gamma \left (\frac {1}{2},-i \arcsin (a x)\right )-\sqrt {i \arcsin (a x)} \Gamma \left (\frac {1}{2},i \arcsin (a x)\right )\right )}{2 a \sqrt {\arcsin (a x)}} \]

[In]

Integrate[1/Sqrt[ArcSin[a*x]],x]

[Out]

((-1/2*I)*(Sqrt[(-I)*ArcSin[a*x]]*Gamma[1/2, (-I)*ArcSin[a*x]] - Sqrt[I*ArcSin[a*x]]*Gamma[1/2, I*ArcSin[a*x]]
))/(a*Sqrt[ArcSin[a*x]])

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83

method	result	size
default	\(\frac {\operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {2}\, \sqrt {\pi }}{a}\)	\(25\)

[In]

int(1/arcsin(a*x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

FresnelC(2^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a

Fricas [F(-2)]

Exception generated. \[ \int \frac {1}{\sqrt {\arcsin (a x)}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(1/arcsin(a*x)^(1/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {1}{\sqrt {\arcsin (a x)}} \, dx=\int \frac {1}{\sqrt {\operatorname {asin}{\left (a x \right )}}}\, dx \]

[In]

integrate(1/asin(a*x)**(1/2),x)

[Out]

Integral(1/sqrt(asin(a*x)), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{\sqrt {\arcsin (a x)}} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(1/arcsin(a*x)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.31 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.57 \[ \int \frac {1}{\sqrt {\arcsin (a x)}} \, dx=-\frac {\left (i + 1\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arcsin \left (a x\right )}\right )}{4 \, a} + \frac {\left (i - 1\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arcsin \left (a x\right )}\right )}{4 \, a} \]

[In]

integrate(1/arcsin(a*x)^(1/2),x, algorithm="giac")

[Out]

-(1/4*I + 1/4)*sqrt(2)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt(2)*sqrt(arcsin(a*x)))/a + (1/4*I - 1/4)*sqrt(2)*sqrt(pi
)*erf(-(1/2*I + 1/2)*sqrt(2)*sqrt(arcsin(a*x)))/a

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {\arcsin (a x)}} \, dx=\int \frac {1}{\sqrt {\mathrm {asin}\left (a\,x\right )}} \,d x \]

[In]

int(1/asin(a*x)^(1/2),x)

[Out]

int(1/asin(a*x)^(1/2), x)