\(\int \frac {1}{\sqrt {\arccos (a+b x)}} \, dx\) [40]

Optimal result

Integrand size = 10, antiderivative size = 33 \[ \int \frac {1}{\sqrt {\arccos (a+b x)}} \, dx=-\frac {\sqrt {2 \pi } \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a+b x)}\right )}{b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4888, 4720, 3386, 3432} \[ \int \frac {1}{\sqrt {\arccos (a+b x)}} \, dx=-\frac {\sqrt {2 \pi } \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a+b x)}\right )}{b} \]

[In]

Int[1/Sqrt[ArcCos[a + b*x]],x]

[Out]

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

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]

Rule 4888

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

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\sqrt {\arccos (x)}} \, dx,x,a+b x\right )}{b} \\ & = -\frac {\text {Subst}\left (\int \frac {\sin (x)}{\sqrt {x}} \, dx,x,\arccos (a+b x)\right )}{b} \\ & = -\frac {2 \text {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {\arccos (a+b x)}\right )}{b} \\ & = -\frac {\sqrt {2 \pi } \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a+b x)}\right )}{b} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.03 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.36 \[ \int \frac {1}{\sqrt {\arccos (a+b x)}} \, dx=-\frac {-\sqrt {-i \arccos (a+b x)} \Gamma \left (\frac {1}{2},-i \arccos (a+b x)\right )-\sqrt {i \arccos (a+b x)} \Gamma \left (\frac {1}{2},i \arccos (a+b x)\right )}{2 b \sqrt {\arccos (a+b x)}} \]

[In]

Integrate[1/Sqrt[ArcCos[a + b*x]],x]

[Out]

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

Maple [A] (verified)

Time = 0.80 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.85

method	result	size
default	\(-\frac {\operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {\arccos \left (b x +a \right )}}{\sqrt {\pi }}\right ) \sqrt {2}\, \sqrt {\pi }}{b}\)	\(28\)

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral(1/sqrt(acos(a + b*x)), x)

Maxima [F(-2)]

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

[In]

integrate(1/arccos(b*x+a)^(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 = 0.32 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.55 \[ \int \frac {1}{\sqrt {\arccos (a+b x)}} \, dx=-\frac {\left (i - 1\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arccos \left (b x + a\right )}\right )}{4 \, b} + \frac {\left (i + 1\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arccos \left (b x + a\right )}\right )}{4 \, b} \]

[In]

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

Mupad [F(-1)]

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

[In]

int(1/acos(a + b*x)^(1/2),x)

[Out]

int(1/acos(a + b*x)^(1/2), x)