3.43 \(\int \frac {1}{\sqrt {a+b \cos ^{-1}(c+d x)}} \, dx\)

Optimal. Leaf size=106 \[ \frac {\sqrt {2 \pi } \sin \left (\frac {a}{b}\right ) C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \cos ^{-1}(c+d x)}}{\sqrt {b}}\right )}{\sqrt {b} d}-\frac {\sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \cos ^{-1}(c+d x)}}{\sqrt {b}}\right )}{\sqrt {b} d} \]

[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]  time = 0.13, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4804, 4624, 3306, 3305, 3351, 3304, 3352} \[ \frac {\sqrt {2 \pi } \sin \left (\frac {a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \cos ^{-1}(c+d x)}}{\sqrt {b}}\right )}{\sqrt {b} d}-\frac {\sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \cos ^{-1}(c+d x)}}{\sqrt {b}}\right )}{\sqrt {b} d} \]

Antiderivative was successfully verified.

[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 3304

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 3305

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 3306

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 3351

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

Rule 3352

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

Rule 4624

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

Rule 4804

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*} \int \frac {1}{\sqrt {a+b \cos ^{-1}(c+d x)}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b \cos ^{-1}(x)}} \, dx,x,c+d x\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\sin \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \cos ^{-1}(c+d x)\right )}{b d}\\ &=-\frac {\cos \left (\frac {a}{b}\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \cos ^{-1}(c+d x)\right )}{b d}+\frac {\sin \left (\frac {a}{b}\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \cos ^{-1}(c+d x)\right )}{b d}\\ &=-\frac {\left (2 \cos \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \sin \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \cos ^{-1}(c+d x)}\right )}{b d}+\frac {\left (2 \sin \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \cos \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \cos ^{-1}(c+d x)}\right )}{b d}\\ &=-\frac {\sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \cos ^{-1}(c+d x)}}{\sqrt {b}}\right )}{\sqrt {b} d}+\frac {\sqrt {2 \pi } C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \cos ^{-1}(c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{\sqrt {b} d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.05, size = 128, normalized size = 1.21 \[ \frac {e^{-\frac {i a}{b}} \left (\sqrt {-\frac {i \left (a+b \cos ^{-1}(c+d x)\right )}{b}} \Gamma \left (\frac {1}{2},-\frac {i \left (a+b \cos ^{-1}(c+d x)\right )}{b}\right )+e^{\frac {2 i a}{b}} \sqrt {\frac {i \left (a+b \cos ^{-1}(c+d x)\right )}{b}} \Gamma \left (\frac {1}{2},\frac {i \left (a+b \cos ^{-1}(c+d x)\right )}{b}\right )\right )}{2 d \sqrt {a+b \cos ^{-1}(c+d x)}} \]

Warning: Unable to verify antiderivative.

[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]])

________________________________________________________________________________________

fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

giac [A]  time = 1.67, size = 171, normalized size = 1.61 \[ \frac {\sqrt {\pi } i \operatorname {erf}\left (-\frac {\sqrt {2} \sqrt {b \arccos \left (d x + c\right ) + a} i}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arccos \left (d x + c\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (\frac {a i}{b}\right )}}{{\left (\frac {\sqrt {2} b i}{\sqrt {{\left | b \right |}}} + \sqrt {2} \sqrt {{\left | b \right |}}\right )} d} + \frac {\sqrt {\pi } i \operatorname {erf}\left (\frac {\sqrt {2} \sqrt {b \arccos \left (d x + c\right ) + a} i}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arccos \left (d x + c\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (-\frac {a i}{b}\right )}}{{\left (\frac {\sqrt {2} b i}{\sqrt {{\left | b \right |}}} - \sqrt {2} \sqrt {{\left | b \right |}}\right )} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.10, size = 89, normalized size = 0.84 \[ -\frac {\sqrt {2}\, \sqrt {\pi }\, \sqrt {\frac {1}{b}}\, \left (\cos \left (\frac {a}{b}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {a +b \arccos \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right )-\sin \left (\frac {a}{b}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {a +b \arccos \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right )\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b \arccos \left (d x + c\right ) + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {a+b\,\mathrm {acos}\left (c+d\,x\right )}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*acos(c + d*x))^(1/2),x)

[Out]

int(1/(a + b*acos(c + d*x))^(1/2), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a + b \operatorname {acos}{\left (c + d x \right )}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________