Integrand size = 14, antiderivative size = 137 \[ \int x \sqrt {a+b \arccos (c x)} \, dx=-\frac {\sqrt {a+b \arccos (c x)}}{4 c^2}+\frac {1}{2} x^2 \sqrt {a+b \arccos (c x)}-\frac {\sqrt {b} \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arccos (c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{8 c^2}-\frac {\sqrt {b} \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arccos (c x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{8 c^2} \]
-1/8*cos(2*a/b)*FresnelC(2*(a+b*arccos(c*x))^(1/2)/b^(1/2)/Pi^(1/2))*b^(1/ 2)*Pi^(1/2)/c^2-1/8*FresnelS(2*(a+b*arccos(c*x))^(1/2)/b^(1/2)/Pi^(1/2))*s in(2*a/b)*b^(1/2)*Pi^(1/2)/c^2-1/4*(a+b*arccos(c*x))^(1/2)/c^2+1/2*x^2*(a+ b*arccos(c*x))^(1/2)
Time = 0.22 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.85 \[ \int x \sqrt {a+b \arccos (c x)} \, dx=\frac {2 \sqrt {a+b \arccos (c x)} \cos (2 \arccos (c x))-\sqrt {b} \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arccos (c x)}}{\sqrt {b} \sqrt {\pi }}\right )-\sqrt {b} \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arccos (c x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{8 c^2} \]
(2*Sqrt[a + b*ArcCos[c*x]]*Cos[2*ArcCos[c*x]] - Sqrt[b]*Sqrt[Pi]*Cos[(2*a) /b]*FresnelC[(2*Sqrt[a + b*ArcCos[c*x]])/(Sqrt[b]*Sqrt[Pi])] - Sqrt[b]*Sqr t[Pi]*FresnelS[(2*Sqrt[a + b*ArcCos[c*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b ])/(8*c^2)
Time = 0.57 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.96, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {5141, 5225, 3042, 3793, 2009}
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.
| \(\displaystyle \int x \sqrt {a+b \arccos (c x)} \, dx\) |
| \(\Big \downarrow \) 5141 |
| \(\displaystyle \frac {1}{4} b c \int \frac {x^2}{\sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}dx+\frac {1}{2} x^2 \sqrt {a+b \arccos (c x)}\) |
| \(\Big \downarrow \) 5225 |
| \(\displaystyle \frac {1}{2} x^2 \sqrt {a+b \arccos (c x)}-\frac {\int \frac {\cos ^2\left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{4 c^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} x^2 \sqrt {a+b \arccos (c x)}-\frac {\int \frac {\sin \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}+\frac {\pi }{2}\right )^2}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{4 c^2}\) |
| \(\Big \downarrow \) 3793 |
| \(\displaystyle \frac {1}{2} x^2 \sqrt {a+b \arccos (c x)}-\frac {\int \left (\frac {\cos \left (\frac {2 a}{b}-\frac {2 (a+b \arccos (c x))}{b}\right )}{2 \sqrt {a+b \arccos (c x)}}+\frac {1}{2 \sqrt {a+b \arccos (c x)}}\right )d(a+b \arccos (c x))}{4 c^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} x^2 \sqrt {a+b \arccos (c x)}-\frac {\frac {1}{2} \sqrt {\pi } \sqrt {b} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arccos (c x)}}{\sqrt {b} \sqrt {\pi }}\right )+\frac {1}{2} \sqrt {\pi } \sqrt {b} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arccos (c x)}}{\sqrt {b} \sqrt {\pi }}\right )+\sqrt {a+b \arccos (c x)}}{4 c^2}\) |
(x^2*Sqrt[a + b*ArcCos[c*x]])/2 - (Sqrt[a + b*ArcCos[c*x]] + (Sqrt[b]*Sqrt [Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[a + b*ArcCos[c*x]])/(Sqrt[b]*Sqrt[Pi])] )/2 + (Sqrt[b]*Sqrt[Pi]*FresnelS[(2*Sqrt[a + b*ArcCos[c*x]])/(Sqrt[b]*Sqrt [Pi])]*Sin[(2*a)/b])/2)/(4*c^2)
3.2.74.3.1 Defintions of rubi rules used
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f , m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x ^(m + 1)*((a + b*ArcCos[c*x])^n/(m + 1)), x] + Simp[b*c*(n/(m + 1)) Int[x ^(m + 1)*((a + b*ArcCos[c*x])^(n - 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{ a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^ 2)^(p_.), x_Symbol] :> Simp[(-(b*c^(m + 1))^(-1))*Simp[(d + e*x^2)^p/(1 - c ^2*x^2)^p] Subst[Int[x^n*Cos[-a/b + x/b]^m*Sin[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCos[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e , 0] && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
Time = 1.84 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.36
| method | result | size |
| default | \(\frac {-\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \sqrt {a +b \arccos \left (c x \right )}\, \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) b +\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \sqrt {a +b \arccos \left (c x \right )}\, \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) b +2 \arccos \left (c x \right ) \cos \left (-\frac {2 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {2 a}{b}\right ) b +2 \cos \left (-\frac {2 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {2 a}{b}\right ) a}{8 c^{2} \sqrt {a +b \arccos \left (c x \right )}}\) | \(186\) |
1/8/c^2/(a+b*arccos(c*x))^(1/2)*(-Pi^(1/2)*(-1/b)^(1/2)*(a+b*arccos(c*x))^ (1/2)*cos(2*a/b)*FresnelC(2*2^(1/2)/Pi^(1/2)/(-2/b)^(1/2)*(a+b*arccos(c*x) )^(1/2)/b)*b+Pi^(1/2)*(-1/b)^(1/2)*(a+b*arccos(c*x))^(1/2)*sin(2*a/b)*Fres nelS(2*2^(1/2)/Pi^(1/2)/(-2/b)^(1/2)*(a+b*arccos(c*x))^(1/2)/b)*b+2*arccos (c*x)*cos(-2*(a+b*arccos(c*x))/b+2*a/b)*b+2*cos(-2*(a+b*arccos(c*x))/b+2*a /b)*a)
Exception generated. \[ \int x \sqrt {a+b \arccos (c x)} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int x \sqrt {a+b \arccos (c x)} \, dx=\int x \sqrt {a + b \operatorname {acos}{\left (c x \right )}}\, dx \]
\[ \int x \sqrt {a+b \arccos (c x)} \, dx=\int { \sqrt {b \arccos \left (c x\right ) + a} x \,d x } \]
Result contains complex when optimal does not.
Time = 0.69 (sec) , antiderivative size = 448, normalized size of antiderivative = 3.27 \[ \int x \sqrt {a+b \arccos (c x)} \, dx=-\frac {i \, \sqrt {\pi } a \sqrt {b} \operatorname {erf}\left (-\frac {\sqrt {b \arccos \left (c x\right ) + a}}{\sqrt {b}} - \frac {i \, \sqrt {b \arccos \left (c x\right ) + a} \sqrt {b}}{{\left | b \right |}}\right ) e^{\left (\frac {2 i \, a}{b}\right )}}{4 \, {\left (b + \frac {i \, b^{2}}{{\left | b \right |}}\right )} c^{2}} + \frac {\sqrt {\pi } b^{\frac {3}{2}} \operatorname {erf}\left (-\frac {\sqrt {b \arccos \left (c x\right ) + a}}{\sqrt {b}} - \frac {i \, \sqrt {b \arccos \left (c x\right ) + a} \sqrt {b}}{{\left | b \right |}}\right ) e^{\left (\frac {2 i \, a}{b}\right )}}{16 \, {\left (b + \frac {i \, b^{2}}{{\left | b \right |}}\right )} c^{2}} + \frac {i \, \sqrt {\pi } a \sqrt {b} \operatorname {erf}\left (-\frac {\sqrt {b \arccos \left (c x\right ) + a}}{\sqrt {b}} + \frac {i \, \sqrt {b \arccos \left (c x\right ) + a} \sqrt {b}}{{\left | b \right |}}\right ) e^{\left (-\frac {2 i \, a}{b}\right )}}{4 \, {\left (b - \frac {i \, b^{2}}{{\left | b \right |}}\right )} c^{2}} + \frac {\sqrt {\pi } b^{\frac {3}{2}} \operatorname {erf}\left (-\frac {\sqrt {b \arccos \left (c x\right ) + a}}{\sqrt {b}} + \frac {i \, \sqrt {b \arccos \left (c x\right ) + a} \sqrt {b}}{{\left | b \right |}}\right ) e^{\left (-\frac {2 i \, a}{b}\right )}}{16 \, {\left (b - \frac {i \, b^{2}}{{\left | b \right |}}\right )} c^{2}} - \frac {i \, \sqrt {\pi } a \operatorname {erf}\left (-\frac {\sqrt {b \arccos \left (c x\right ) + a}}{\sqrt {b}} + \frac {i \, \sqrt {b \arccos \left (c x\right ) + a} \sqrt {b}}{{\left | b \right |}}\right ) e^{\left (-\frac {2 i \, a}{b}\right )}}{4 \, c^{2} {\left (\sqrt {b} - \frac {i \, b^{\frac {3}{2}}}{{\left | b \right |}}\right )}} + \frac {i \, \sqrt {\pi } a \operatorname {erf}\left (-\frac {\sqrt {b \arccos \left (c x\right ) + a}}{\sqrt {b}} - \frac {i \, \sqrt {b \arccos \left (c x\right ) + a} \sqrt {b}}{{\left | b \right |}}\right ) e^{\left (\frac {2 i \, a}{b}\right )}}{4 \, \sqrt {b} c^{2} {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )}} + \frac {\sqrt {b \arccos \left (c x\right ) + a} e^{\left (2 i \, \arccos \left (c x\right )\right )}}{8 \, c^{2}} + \frac {\sqrt {b \arccos \left (c x\right ) + a} e^{\left (-2 i \, \arccos \left (c x\right )\right )}}{8 \, c^{2}} \]
-1/4*I*sqrt(pi)*a*sqrt(b)*erf(-sqrt(b*arccos(c*x) + a)/sqrt(b) - I*sqrt(b* arccos(c*x) + a)*sqrt(b)/abs(b))*e^(2*I*a/b)/((b + I*b^2/abs(b))*c^2) + 1/ 16*sqrt(pi)*b^(3/2)*erf(-sqrt(b*arccos(c*x) + a)/sqrt(b) - I*sqrt(b*arccos (c*x) + a)*sqrt(b)/abs(b))*e^(2*I*a/b)/((b + I*b^2/abs(b))*c^2) + 1/4*I*sq rt(pi)*a*sqrt(b)*erf(-sqrt(b*arccos(c*x) + a)/sqrt(b) + I*sqrt(b*arccos(c* x) + a)*sqrt(b)/abs(b))*e^(-2*I*a/b)/((b - I*b^2/abs(b))*c^2) + 1/16*sqrt( pi)*b^(3/2)*erf(-sqrt(b*arccos(c*x) + a)/sqrt(b) + I*sqrt(b*arccos(c*x) + a)*sqrt(b)/abs(b))*e^(-2*I*a/b)/((b - I*b^2/abs(b))*c^2) - 1/4*I*sqrt(pi)* a*erf(-sqrt(b*arccos(c*x) + a)/sqrt(b) + I*sqrt(b*arccos(c*x) + a)*sqrt(b) /abs(b))*e^(-2*I*a/b)/(c^2*(sqrt(b) - I*b^(3/2)/abs(b))) + 1/4*I*sqrt(pi)* a*erf(-sqrt(b*arccos(c*x) + a)/sqrt(b) - I*sqrt(b*arccos(c*x) + a)*sqrt(b) /abs(b))*e^(2*I*a/b)/(sqrt(b)*c^2*(I*b/abs(b) + 1)) + 1/8*sqrt(b*arccos(c* x) + a)*e^(2*I*arccos(c*x))/c^2 + 1/8*sqrt(b*arccos(c*x) + a)*e^(-2*I*arcc os(c*x))/c^2
Timed out. \[ \int x \sqrt {a+b \arccos (c x)} \, dx=\int x\,\sqrt {a+b\,\mathrm {acos}\left (c\,x\right )} \,d x \]