Integrand size = 12, antiderivative size = 102 \[ \int \frac {1}{\sqrt {a+b \arccos (c x)}} \, dx=-\frac {\sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{\sqrt {b} c}+\frac {\sqrt {2 \pi } \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{\sqrt {b} c} \] Output:
-2^(1/2)*Pi^(1/2)*cos(a/b)*FresnelS(2^(1/2)/Pi^(1/2)*(a+b*arccos(c*x))^(1/ 2)/b^(1/2))/b^(1/2)/c+2^(1/2)*Pi^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*(a+b*arcc os(c*x))^(1/2)/b^(1/2))*sin(a/b)/b^(1/2)/c
Result contains complex when optimal does not.
Time = 0.05 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.16 \[ \int \frac {1}{\sqrt {a+b \arccos (c x)}} \, dx=\frac {e^{-\frac {i a}{b}} \left (\sqrt {-\frac {i (a+b \arccos (c x))}{b}} \Gamma \left (\frac {1}{2},-\frac {i (a+b \arccos (c x))}{b}\right )+e^{\frac {2 i a}{b}} \sqrt {\frac {i (a+b \arccos (c x))}{b}} \Gamma \left (\frac {1}{2},\frac {i (a+b \arccos (c x))}{b}\right )\right )}{2 c \sqrt {a+b \arccos (c x)}} \] Input:
Integrate[1/Sqrt[a + b*ArcCos[c*x]],x]
Output:
(Sqrt[((-I)*(a + b*ArcCos[c*x]))/b]*Gamma[1/2, ((-I)*(a + b*ArcCos[c*x]))/ b] + E^(((2*I)*a)/b)*Sqrt[(I*(a + b*ArcCos[c*x]))/b]*Gamma[1/2, (I*(a + b* ArcCos[c*x]))/b])/(2*c*E^((I*a)/b)*Sqrt[a + b*ArcCos[c*x]])
Time = 0.51 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.02, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {5135, 25, 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.
\(\displaystyle \int \frac {1}{\sqrt {a+b \arccos (c x)}} \, dx\) |
\(\Big \downarrow \) 5135 |
\(\displaystyle -\frac {\int -\frac {\sin \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{b c}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {\sin \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{b c}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {\sin \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{b c}\) |
\(\Big \downarrow \) 3787 |
\(\displaystyle -\frac {-\sin \left (\frac {a}{b}\right ) \int \frac {\cos \left (\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))-\cos \left (\frac {a}{b}\right ) \int -\frac {\sin \left (\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{b c}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\cos \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))-\sin \left (\frac {a}{b}\right ) \int \frac {\cos \left (\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{b c}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\cos \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))-\sin \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arccos (c x)}{b}+\frac {\pi }{2}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{b c}\) |
\(\Big \downarrow \) 3785 |
\(\displaystyle -\frac {\cos \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))-2 \sin \left (\frac {a}{b}\right ) \int \cos \left (\frac {a+b \arccos (c x)}{b}\right )d\sqrt {a+b \arccos (c x)}}{b c}\) |
\(\Big \downarrow \) 3786 |
\(\displaystyle -\frac {2 \cos \left (\frac {a}{b}\right ) \int \sin \left (\frac {a+b \arccos (c x)}{b}\right )d\sqrt {a+b \arccos (c x)}-2 \sin \left (\frac {a}{b}\right ) \int \cos \left (\frac {a+b \arccos (c x)}{b}\right )d\sqrt {a+b \arccos (c x)}}{b c}\) |
\(\Big \downarrow \) 3832 |
\(\displaystyle -\frac {\sqrt {2 \pi } \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )-2 \sin \left (\frac {a}{b}\right ) \int \cos \left (\frac {a+b \arccos (c x)}{b}\right )d\sqrt {a+b \arccos (c x)}}{b c}\) |
\(\Big \downarrow \) 3833 |
\(\displaystyle -\frac {\sqrt {2 \pi } \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )-\sqrt {2 \pi } \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{b c}\) |
Input:
Int[1/Sqrt[a + b*ArcCos[c*x]],x]
Output:
-((Sqrt[b]*Sqrt[2*Pi]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x] ])/Sqrt[b]] - Sqrt[b]*Sqrt[2*Pi]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c* x]])/Sqrt[b]]*Sin[a/b])/(b*c))
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]
Time = 0.00 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.87
method | result | size |
default | \(\frac {\sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \left (\cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right )+\sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right )\right )}{c}\) | \(89\) |
Input:
int(1/(a+b*arccos(c*x))^(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(c*x))^(1/2)/b)+sin(a/b)*FresnelC(2^(1/2)/Pi^(1/2)/(-1/b)^ (1/2)*(a+b*arccos(c*x))^(1/2)/b))/c
Exception generated. \[ \int \frac {1}{\sqrt {a+b \arccos (c x)}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(1/(a+b*arccos(c*x))^(1/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {1}{\sqrt {a+b \arccos (c x)}} \, dx=\int \frac {1}{\sqrt {a + b \operatorname {acos}{\left (c x \right )}}}\, dx \] Input:
integrate(1/(a+b*acos(c*x))**(1/2),x)
Output:
Integral(1/sqrt(a + b*acos(c*x)), x)
\[ \int \frac {1}{\sqrt {a+b \arccos (c x)}} \, dx=\int { \frac {1}{\sqrt {b \arccos \left (c x\right ) + a}} \,d x } \] Input:
integrate(1/(a+b*arccos(c*x))^(1/2),x, algorithm="maxima")
Output:
integrate(1/sqrt(b*arccos(c*x) + a), x)
Result contains complex when optimal does not.
Time = 0.20 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.56 \[ \int \frac {1}{\sqrt {a+b \arccos (c x)}} \, dx=\frac {i \, \sqrt {\pi } \operatorname {erf}\left (-\frac {i \, \sqrt {2} \sqrt {b \arccos \left (c x\right ) + a}}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arccos \left (c x\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (\frac {i \, a}{b}\right )}}{c {\left (\frac {i \, \sqrt {2} b}{\sqrt {{\left | b \right |}}} + \sqrt {2} \sqrt {{\left | b \right |}}\right )}} - \frac {i \, \sqrt {\pi } \operatorname {erf}\left (\frac {i \, \sqrt {2} \sqrt {b \arccos \left (c x\right ) + a}}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arccos \left (c x\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (-\frac {i \, a}{b}\right )}}{c {\left (-\frac {i \, \sqrt {2} b}{\sqrt {{\left | b \right |}}} + \sqrt {2} \sqrt {{\left | b \right |}}\right )}} \] Input:
integrate(1/(a+b*arccos(c*x))^(1/2),x, algorithm="giac")
Output:
I*sqrt(pi)*erf(-1/2*I*sqrt(2)*sqrt(b*arccos(c*x) + a)/sqrt(abs(b)) - 1/2*s qrt(2)*sqrt(b*arccos(c*x) + a)*sqrt(abs(b))/b)*e^(I*a/b)/(c*(I*sqrt(2)*b/s qrt(abs(b)) + sqrt(2)*sqrt(abs(b)))) - I*sqrt(pi)*erf(1/2*I*sqrt(2)*sqrt(b *arccos(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*sqrt( abs(b))/b)*e^(-I*a/b)/(c*(-I*sqrt(2)*b/sqrt(abs(b)) + sqrt(2)*sqrt(abs(b)) ))
Timed out. \[ \int \frac {1}{\sqrt {a+b \arccos (c x)}} \, dx=\int \frac {1}{\sqrt {a+b\,\mathrm {acos}\left (c\,x\right )}} \,d x \] Input:
int(1/(a + b*acos(c*x))^(1/2),x)
Output:
int(1/(a + b*acos(c*x))^(1/2), x)
\[ \int \frac {1}{\sqrt {a+b \arccos (c x)}} \, dx=\int \frac {\sqrt {\mathit {acos} \left (c x \right ) b +a}}{\mathit {acos} \left (c x \right ) b +a}d x \] Input:
int(1/(a+b*acos(c*x))^(1/2),x)
Output:
int(sqrt(acos(c*x)*b + a)/(acos(c*x)*b + a),x)