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

Optimal result

Integrand size = 8, antiderivative size = 31 \[ \int \frac {1}{\sqrt {\arccos (a x)}} \, dx=-\frac {\sqrt {2 \pi } \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{a} \]

[Out]

-FresnelS(2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 31, 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 = {4720, 3386, 3432} \[ \int \frac {1}{\sqrt {\arccos (a x)}} \, dx=-\frac {\sqrt {2 \pi } \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{a} \]

[In]

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

[Out]

-((Sqrt[2*Pi]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/a)

Rule 3386

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Sin[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 3432

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

Rule 4720

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

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.02 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.19 \[ \int \frac {1}{\sqrt {\arccos (a x)}} \, dx=-\frac {-\sqrt {-i \arccos (a x)} \Gamma \left (\frac {1}{2},-i \arccos (a x)\right )-\sqrt {i \arccos (a x)} \Gamma \left (\frac {1}{2},i \arccos (a x)\right )}{2 a \sqrt {\arccos (a x)}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.70 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84

method	result	size
default	\(-\frac {\operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {2}\, \sqrt {\pi }}{a}\)	\(26\)

[In]

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

[Out]

-FresnelS(2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a

Fricas [F(-2)]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral(1/sqrt(acos(a*x)), x)

Maxima [F(-2)]

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

[In]

integrate(1/arccos(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 = 0.29 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.52 \[ \int \frac {1}{\sqrt {\arccos (a x)}} \, dx=-\frac {\left (i - 1\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arccos \left (a x\right )}\right )}{4 \, a} + \frac {\left (i + 1\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arccos \left (a x\right )}\right )}{4 \, a} \]

[In]

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

Mupad [F(-1)]

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

[In]

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

[Out]

int(1/acos(a*x)^(1/2), x)