Integrand size = 15, antiderivative size = 108 \[ \int \frac {1}{\sqrt {a-b \arccos (c+d x)}} \, dx=-\frac {\sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a-b \arccos (c+d x)}}{\sqrt {b}}\right )}{\sqrt {b} d}+\frac {\sqrt {2 \pi } \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a-b \arccos (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{\sqrt {b} d} \] Output:
-2^(1/2)*Pi^(1/2)*cos(a/b)*FresnelS(2^(1/2)/Pi^(1/2)*(a-b*arccos(d*x+c))^( 1/2)/b^(1/2))/b^(1/2)/d+2^(1/2)*Pi^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*(a-b*ar ccos(d*x+c))^(1/2)/b^(1/2))*sin(a/b)/b^(1/2)/d
Result contains complex when optimal does not.
Time = 0.09 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.23 \[ \int \frac {1}{\sqrt {a-b \arccos (c+d x)}} \, dx=\frac {e^{-\frac {i a}{b}} \left (\sqrt {-\frac {i (a-b \arccos (c+d x))}{b}} \Gamma \left (\frac {1}{2},-\frac {i (a-b \arccos (c+d x))}{b}\right )+e^{\frac {2 i a}{b}} \sqrt {\frac {i (a-b \arccos (c+d x))}{b}} \Gamma \left (\frac {1}{2},\frac {i (a-b \arccos (c+d x))}{b}\right )\right )}{2 d \sqrt {a-b \arccos (c+d x)}} \] Input:
Integrate[1/Sqrt[a - b*ArcCos[c + d*x]],x]
Output:
(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]])
Time = 0.57 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.01, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {5303, 5135, 3042, 3787, 25, 3042, 3785, 3786, 3832, 3833}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{\sqrt {a-b \arccos (c+d x)}} \, dx\) |
\(\Big \downarrow \) 5303 |
\(\displaystyle \frac {\int \frac {1}{\sqrt {a-b \arccos (c+d x)}}d(c+d x)}{d}\) |
\(\Big \downarrow \) 5135 |
\(\displaystyle \frac {\int \frac {\sin \left (\frac {a}{b}-\frac {a-b \arccos (c+d x)}{b}\right )}{\sqrt {a-b \arccos (c+d x)}}d(a-b \arccos (c+d x))}{b d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {\sin \left (\frac {a}{b}-\frac {a-b \arccos (c+d x)}{b}\right )}{\sqrt {a-b \arccos (c+d x)}}d(a-b \arccos (c+d x))}{b d}\) |
\(\Big \downarrow \) 3787 |
\(\displaystyle \frac {\sin \left (\frac {a}{b}\right ) \int \frac {\cos \left (\frac {a-b \arccos (c+d x)}{b}\right )}{\sqrt {a-b \arccos (c+d x)}}d(a-b \arccos (c+d x))+\cos \left (\frac {a}{b}\right ) \int -\frac {\sin \left (\frac {a-b \arccos (c+d x)}{b}\right )}{\sqrt {a-b \arccos (c+d x)}}d(a-b \arccos (c+d x))}{b d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\sin \left (\frac {a}{b}\right ) \int \frac {\cos \left (\frac {a-b \arccos (c+d x)}{b}\right )}{\sqrt {a-b \arccos (c+d x)}}d(a-b \arccos (c+d x))-\cos \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a-b \arccos (c+d x)}{b}\right )}{\sqrt {a-b \arccos (c+d x)}}d(a-b \arccos (c+d x))}{b d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sin \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a-b \arccos (c+d x)}{b}+\frac {\pi }{2}\right )}{\sqrt {a-b \arccos (c+d x)}}d(a-b \arccos (c+d x))-\cos \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a-b \arccos (c+d x)}{b}\right )}{\sqrt {a-b \arccos (c+d x)}}d(a-b \arccos (c+d x))}{b d}\) |
\(\Big \downarrow \) 3785 |
\(\displaystyle \frac {2 \sin \left (\frac {a}{b}\right ) \int \cos \left (\frac {a-b \arccos (c+d x)}{b}\right )d\sqrt {a-b \arccos (c+d x)}-\cos \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a-b \arccos (c+d x)}{b}\right )}{\sqrt {a-b \arccos (c+d x)}}d(a-b \arccos (c+d x))}{b d}\) |
\(\Big \downarrow \) 3786 |
\(\displaystyle \frac {2 \sin \left (\frac {a}{b}\right ) \int \cos \left (\frac {a-b \arccos (c+d x)}{b}\right )d\sqrt {a-b \arccos (c+d x)}-2 \cos \left (\frac {a}{b}\right ) \int \sin \left (\frac {a-b \arccos (c+d x)}{b}\right )d\sqrt {a-b \arccos (c+d x)}}{b d}\) |
\(\Big \downarrow \) 3832 |
\(\displaystyle \frac {2 \sin \left (\frac {a}{b}\right ) \int \cos \left (\frac {a-b \arccos (c+d x)}{b}\right )d\sqrt {a-b \arccos (c+d x)}-\sqrt {2 \pi } \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a-b \arccos (c+d x)}}{\sqrt {b}}\right )}{b d}\) |
\(\Big \downarrow \) 3833 |
\(\displaystyle \frac {\sqrt {2 \pi } \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a-b \arccos (c+d x)}}{\sqrt {b}}\right )-\sqrt {2 \pi } \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a-b \arccos (c+d x)}}{\sqrt {b}}\right )}{b d}\) |
Input:
Int[1/Sqrt[a - b*ArcCos[c + d*x]],x]
Output:
(-(Sqrt[b]*Sqrt[2*Pi]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a - b*ArcCos[c + d*x]])/Sqrt[b]]) + Sqrt[b]*Sqrt[2*Pi]*FresnelC[(Sqrt[2/Pi]*Sqrt[a - b*ArcC os[c + d*x]])/Sqrt[b]]*Sin[a/b])/(b*d)
Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> S imp[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]
Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[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]
Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Cos [(d*e - c*f)/d] Int[Sin[c*(f/d) + f*x]/Sqrt[c + d*x], x], x] + Simp[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]
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]
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]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[-(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]
Int[((a_.) + ArcCos[(c_) + (d_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[1/d Subst[Int[(a + b*ArcCos[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, n}, x]
Time = 0.06 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.88
method | result | size |
default | \(\frac {\sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \left (\cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {a -b \arccos \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right )+\sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {a -b \arccos \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right )\right )}{d}\) | \(95\) |
Input:
int(1/(a-b*arccos(d*x+c))^(1/2),x,method=_RETURNVERBOSE)
Output:
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
Exception generated. \[ \int \frac {1}{\sqrt {a-b \arccos (c+d x)}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(1/(a-b*arccos(d*x+c))^(1/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {1}{\sqrt {a-b \arccos (c+d x)}} \, dx=\int \frac {1}{\sqrt {a - b \operatorname {acos}{\left (c + d x \right )}}}\, dx \] Input:
integrate(1/(a-b*acos(d*x+c))**(1/2),x)
Output:
Integral(1/sqrt(a - b*acos(c + d*x)), x)
\[ \int \frac {1}{\sqrt {a-b \arccos (c+d x)}} \, dx=\int { \frac {1}{\sqrt {-b \arccos \left (d x + c\right ) + a}} \,d x } \] Input:
integrate(1/(a-b*arccos(d*x+c))^(1/2),x, algorithm="maxima")
Output:
integrate(1/sqrt(-b*arccos(d*x + c) + a), x)
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.58 \[ \int \frac {1}{\sqrt {a-b \arccos (c+d x)}} \, dx=\frac {i \, \sqrt {\pi } \operatorname {erf}\left (-\frac {i \, \sqrt {2} \sqrt {-b \arccos \left (d x + c\right ) + a}}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {-b \arccos \left (d x + c\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (\frac {i \, a}{b}\right )}}{d {\left (\frac {i \, \sqrt {2} b}{\sqrt {{\left | b \right |}}} + \sqrt {2} \sqrt {{\left | b \right |}}\right )}} - \frac {i \, \sqrt {\pi } \operatorname {erf}\left (\frac {i \, \sqrt {2} \sqrt {-b \arccos \left (d x + c\right ) + a}}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {-b \arccos \left (d x + c\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (-\frac {i \, a}{b}\right )}}{d {\left (-\frac {i \, \sqrt {2} b}{\sqrt {{\left | b \right |}}} + \sqrt {2} \sqrt {{\left | b \right |}}\right )}} \] Input:
integrate(1/(a-b*arccos(d*x+c))^(1/2),x, algorithm="giac")
Output:
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*s qrt(2)*b/sqrt(abs(b)) + sqrt(2)*sqrt(abs(b)))) - 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))))
Timed out. \[ \int \frac {1}{\sqrt {a-b \arccos (c+d x)}} \, dx=\int \frac {1}{\sqrt {a-b\,\mathrm {acos}\left (c+d\,x\right )}} \,d x \] Input:
int(1/(a - b*acos(c + d*x))^(1/2),x)
Output:
int(1/(a - b*acos(c + d*x))^(1/2), x)
\[ \int \frac {1}{\sqrt {a-b \arccos (c+d x)}} \, dx=-\left (\int \frac {\sqrt {-\mathit {acos} \left (d x +c \right ) b +a}}{\mathit {acos} \left (d x +c \right ) b -a}d x \right ) \] Input:
int(1/(a-b*acos(d*x+c))^(1/2),x)
Output:
- int(sqrt( - acos(c + d*x)*b + a)/(acos(c + d*x)*b - a),x)