\(\int \frac {1}{\sqrt {a-b \arccos (c+d x)}} \, dx\) [44]

Optimal result

Integrand size = 15, antiderivative size = 108 \[ \int \frac {1}{\sqrt {a-b \arccos (c+d x)}} \, dx=-\frac {\sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a-b \arccos (c+d x)}}{\sqrt {b}}\right )}{\sqrt {b} d}+\frac {\sqrt {2 \pi } \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a-b \arccos (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{\sqrt {b} d} \]

[Out]

-cos(a/b)*FresnelS(2^(1/2)/Pi^(1/2)*(a-b*arccos(d*x+c))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/d/b^(1/2)+FresnelC(2^(
1/2)/Pi^(1/2)*(a-b*arccos(d*x+c))^(1/2)/b^(1/2))*sin(a/b)*2^(1/2)*Pi^(1/2)/d/b^(1/2)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {4888, 4720, 3387, 3386, 3432, 3385, 3433} \[ \int \frac {1}{\sqrt {a-b \arccos (c+d x)}} \, dx=\frac {\sqrt {2 \pi } \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a-b \arccos (c+d x)}}{\sqrt {b}}\right )}{\sqrt {b} d}-\frac {\sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a-b \arccos (c+d x)}}{\sqrt {b}}\right )}{\sqrt {b} d} \]

[In]

Int[1/Sqrt[a - b*ArcCos[c + d*x]],x]

[Out]

-((Sqrt[2*Pi]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a - b*ArcCos[c + d*x]])/Sqrt[b]])/(Sqrt[b]*d)) + (Sqrt[2*Pi]*
FresnelC[(Sqrt[2/Pi]*Sqrt[a - b*ArcCos[c + d*x]])/Sqrt[b]]*Sin[a/b])/(Sqrt[b]*d)

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 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 3387

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) +
f*x]/Sqrt[c + d*x], x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[c*(f/d) + f*x]/Sqrt[c + d*x], x], x] /; FreeQ[{c
, d, e, f}, x] && ComplexFreeQ[f] && NeQ[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 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 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 {a-b \arccos (x)}} \, dx,x,c+d x\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {\sin \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a-b \arccos (c+d x)\right )}{b d} \\ & = -\frac {\cos \left (\frac {a}{b}\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a-b \arccos (c+d x)\right )}{b d}+\frac {\sin \left (\frac {a}{b}\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a-b \arccos (c+d x)\right )}{b d} \\ & = -\frac {\left (2 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a-b \arccos (c+d x)}\right )}{b d}+\frac {\left (2 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a-b \arccos (c+d x)}\right )}{b d} \\ & = -\frac {\sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a-b \arccos (c+d x)}}{\sqrt {b}}\right )}{\sqrt {b} d}+\frac {\sqrt {2 \pi } \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a-b \arccos (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{\sqrt {b} d} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.17 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.23 \[ \int \frac {1}{\sqrt {a-b \arccos (c+d x)}} \, dx=\frac {e^{-\frac {i a}{b}} \left (\sqrt {-\frac {i (a-b \arccos (c+d x))}{b}} \Gamma \left (\frac {1}{2},-\frac {i (a-b \arccos (c+d x))}{b}\right )+e^{\frac {2 i a}{b}} \sqrt {\frac {i (a-b \arccos (c+d x))}{b}} \Gamma \left (\frac {1}{2},\frac {i (a-b \arccos (c+d x))}{b}\right )\right )}{2 d \sqrt {a-b \arccos (c+d x)}} \]

[In]

Integrate[1/Sqrt[a - b*ArcCos[c + d*x]],x]

[Out]

(Sqrt[((-I)*(a - b*ArcCos[c + d*x]))/b]*Gamma[1/2, ((-I)*(a - b*ArcCos[c + d*x]))/b] + E^(((2*I)*a)/b)*Sqrt[(I
*(a - b*ArcCos[c + d*x]))/b]*Gamma[1/2, (I*(a - b*ArcCos[c + d*x]))/b])/(2*d*E^((I*a)/b)*Sqrt[a - b*ArcCos[c +
d*x]])

Maple [A] (verified)

Time = 1.82 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.88

method	result	size
default	\(\frac {\sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \left (\cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {a -b \arccos \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right )+\sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {a -b \arccos \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right )\right )}{d}\)	\(95\)

[In]

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

[Out]

2^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*(cos(a/b)*FresnelS(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)/b*(a-b*arccos(d*x+c))^(1/2))+si
n(a/b)*FresnelC(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)/b*(a-b*arccos(d*x+c))^(1/2)))/d

Fricas [F(-2)]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral(1/sqrt(a - b*acos(c + d*x)), x)

Maxima [F]

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

[In]

integrate(1/(a-b*arccos(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate(1/sqrt(-b*arccos(d*x + c) + a), x)

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.42 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.58 \[ \int \frac {1}{\sqrt {a-b \arccos (c+d x)}} \, dx=\frac {i \, \sqrt {\pi } \operatorname {erf}\left (-\frac {i \, \sqrt {2} \sqrt {-b \arccos \left (d x + c\right ) + a}}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {-b \arccos \left (d x + c\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (\frac {i \, a}{b}\right )}}{d {\left (\frac {i \, \sqrt {2} b}{\sqrt {{\left | b \right |}}} + \sqrt {2} \sqrt {{\left | b \right |}}\right )}} - \frac {i \, \sqrt {\pi } \operatorname {erf}\left (\frac {i \, \sqrt {2} \sqrt {-b \arccos \left (d x + c\right ) + a}}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {-b \arccos \left (d x + c\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (-\frac {i \, a}{b}\right )}}{d {\left (-\frac {i \, \sqrt {2} b}{\sqrt {{\left | b \right |}}} + \sqrt {2} \sqrt {{\left | b \right |}}\right )}} \]

[In]

integrate(1/(a-b*arccos(d*x+c))^(1/2),x, algorithm="giac")

[Out]

I*sqrt(pi)*erf(-1/2*I*sqrt(2)*sqrt(-b*arccos(d*x + c) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(-b*arccos(d*x + c)
+ a)*sqrt(abs(b))/b)*e^(I*a/b)/(d*(I*sqrt(2)*b/sqrt(abs(b)) + sqrt(2)*sqrt(abs(b)))) - I*sqrt(pi)*erf(1/2*I*sq
rt(2)*sqrt(-b*arccos(d*x + c) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(-b*arccos(d*x + c) + a)*sqrt(abs(b))/b)*e^(
-I*a/b)/(d*(-I*sqrt(2)*b/sqrt(abs(b)) + sqrt(2)*sqrt(abs(b))))

Mupad [F(-1)]

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

[In]

int(1/(a - b*acos(c + d*x))^(1/2),x)

[Out]

int(1/(a - b*acos(c + d*x))^(1/2), x)