Integrand size = 14, antiderivative size = 99 \[ \int \frac {x}{\sqrt {a+b \arccos (c x)}} \, dx=-\frac {\sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arccos (c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{2 \sqrt {b} c^2}+\frac {\sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arccos (c x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{2 \sqrt {b} c^2} \] Output:
-1/2*Pi^(1/2)*cos(2*a/b)*FresnelS(2*(a+b*arccos(c*x))^(1/2)/b^(1/2)/Pi^(1/ 2))/b^(1/2)/c^2+1/2*Pi^(1/2)*FresnelC(2*(a+b*arccos(c*x))^(1/2)/b^(1/2)/Pi ^(1/2))*sin(2*a/b)/b^(1/2)/c^2
Time = 0.18 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.86 \[ \int \frac {x}{\sqrt {a+b \arccos (c x)}} \, dx=\frac {\sqrt {\pi } \left (-\cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arccos (c x)}}{\sqrt {b} \sqrt {\pi }}\right )+\operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arccos (c x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )\right )}{2 \sqrt {b} c^2} \] Input:
Integrate[x/Sqrt[a + b*ArcCos[c*x]],x]
Output:
(Sqrt[Pi]*(-(Cos[(2*a)/b]*FresnelS[(2*Sqrt[a + b*ArcCos[c*x]])/(Sqrt[b]*Sq rt[Pi])]) + FresnelC[(2*Sqrt[a + b*ArcCos[c*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[( 2*a)/b]))/(2*Sqrt[b]*c^2)
Time = 0.58 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.99, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {5147, 25, 4906, 27, 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.
\(\displaystyle \int \frac {x}{\sqrt {a+b \arccos (c x)}} \, dx\) |
\(\Big \downarrow \) 5147 |
\(\displaystyle -\frac {\int -\frac {\cos \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right ) \sin \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{b c^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {\cos \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right ) \sin \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{b c^2}\) |
\(\Big \downarrow \) 4906 |
\(\displaystyle \frac {\int \frac {\sin \left (\frac {2 a}{b}-\frac {2 (a+b \arccos (c x))}{b}\right )}{2 \sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{b c^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {\sin \left (\frac {2 a}{b}-\frac {2 (a+b \arccos (c x))}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{2 b c^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {\sin \left (\frac {2 a}{b}-\frac {2 (a+b \arccos (c x))}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{2 b c^2}\) |
\(\Big \downarrow \) 3787 |
\(\displaystyle -\frac {-\sin \left (\frac {2 a}{b}\right ) \int \frac {\cos \left (\frac {2 (a+b \arccos (c x))}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))-\cos \left (\frac {2 a}{b}\right ) \int -\frac {\sin \left (\frac {2 (a+b \arccos (c x))}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{2 b c^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\cos \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 (a+b \arccos (c x))}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))-\sin \left (\frac {2 a}{b}\right ) \int \frac {\cos \left (\frac {2 (a+b \arccos (c x))}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{2 b c^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\cos \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 (a+b \arccos (c x))}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))-\sin \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 (a+b \arccos (c x))}{b}+\frac {\pi }{2}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{2 b c^2}\) |
\(\Big \downarrow \) 3785 |
\(\displaystyle -\frac {\cos \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 (a+b \arccos (c x))}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))-2 \sin \left (\frac {2 a}{b}\right ) \int \cos \left (\frac {2 (a+b \arccos (c x))}{b}\right )d\sqrt {a+b \arccos (c x)}}{2 b c^2}\) |
\(\Big \downarrow \) 3786 |
\(\displaystyle -\frac {2 \cos \left (\frac {2 a}{b}\right ) \int \sin \left (\frac {2 (a+b \arccos (c x))}{b}\right )d\sqrt {a+b \arccos (c x)}-2 \sin \left (\frac {2 a}{b}\right ) \int \cos \left (\frac {2 (a+b \arccos (c x))}{b}\right )d\sqrt {a+b \arccos (c x)}}{2 b c^2}\) |
\(\Big \downarrow \) 3832 |
\(\displaystyle -\frac {\sqrt {\pi } \sqrt {b} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arccos (c x)}}{\sqrt {b} \sqrt {\pi }}\right )-2 \sin \left (\frac {2 a}{b}\right ) \int \cos \left (\frac {2 (a+b \arccos (c x))}{b}\right )d\sqrt {a+b \arccos (c x)}}{2 b c^2}\) |
\(\Big \downarrow \) 3833 |
\(\displaystyle -\frac {\sqrt {\pi } \sqrt {b} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arccos (c x)}}{\sqrt {b} \sqrt {\pi }}\right )-\sqrt {\pi } \sqrt {b} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arccos (c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{2 b c^2}\) |
Input:
Int[x/Sqrt[a + b*ArcCos[c*x]],x]
Output:
-1/2*(Sqrt[b]*Sqrt[Pi]*Cos[(2*a)/b]*FresnelS[(2*Sqrt[a + b*ArcCos[c*x]])/( Sqrt[b]*Sqrt[Pi])] - Sqrt[b]*Sqrt[Pi]*FresnelC[(2*Sqrt[a + b*ArcCos[c*x]]) /(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(b*c^2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b _.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin[a + b*x ]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IG tQ[p, 0]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[- (b*c^(m + 1))^(-1) Subst[Int[x^n*Cos[-a/b + x/b]^m*Sin[-a/b + x/b], x], x , a + b*ArcCos[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]
Time = 0.12 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.92
method | result | size |
default | \(\frac {\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \left (\cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right )+\sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right )\right )}{2 c^{2}}\) | \(91\) |
Input:
int(x/(a+b*arccos(c*x))^(1/2),x,method=_RETURNVERBOSE)
Output:
1/2*Pi^(1/2)*(-1/b)^(1/2)*(cos(2*a/b)*FresnelS(2*2^(1/2)/Pi^(1/2)/(-2/b)^( 1/2)*(a+b*arccos(c*x))^(1/2)/b)+sin(2*a/b)*FresnelC(2*2^(1/2)/Pi^(1/2)/(-2 /b)^(1/2)*(a+b*arccos(c*x))^(1/2)/b))/c^2
Exception generated. \[ \int \frac {x}{\sqrt {a+b \arccos (c x)}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x/(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)
\[ \int \frac {x}{\sqrt {a+b \arccos (c x)}} \, dx=\int \frac {x}{\sqrt {a + b \operatorname {acos}{\left (c x \right )}}}\, dx \] Input:
integrate(x/(a+b*acos(c*x))**(1/2),x)
Output:
Integral(x/sqrt(a + b*acos(c*x)), x)
\[ \int \frac {x}{\sqrt {a+b \arccos (c x)}} \, dx=\int { \frac {x}{\sqrt {b \arccos \left (c x\right ) + a}} \,d x } \] Input:
integrate(x/(a+b*arccos(c*x))^(1/2),x, algorithm="maxima")
Output:
integrate(x/sqrt(b*arccos(c*x) + a), x)
Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.33 \[ \int \frac {x}{\sqrt {a+b \arccos (c x)}} \, dx=-\frac {i \, \sqrt {\pi } \operatorname {erf}\left (-\frac {\sqrt {b \arccos \left (c x\right ) + a}}{\sqrt {b}} + \frac {i \, \sqrt {b \arccos \left (c x\right ) + a} \sqrt {b}}{{\left | b \right |}}\right ) e^{\left (-\frac {2 i \, a}{b}\right )}}{4 \, c^{2} {\left (\sqrt {b} - \frac {i \, b^{\frac {3}{2}}}{{\left | b \right |}}\right )}} + \frac {i \, \sqrt {\pi } \operatorname {erf}\left (-\frac {\sqrt {b \arccos \left (c x\right ) + a}}{\sqrt {b}} - \frac {i \, \sqrt {b \arccos \left (c x\right ) + a} \sqrt {b}}{{\left | b \right |}}\right ) e^{\left (\frac {2 i \, a}{b}\right )}}{4 \, \sqrt {b} c^{2} {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )}} \] Input:
integrate(x/(a+b*arccos(c*x))^(1/2),x, algorithm="giac")
Output:
-1/4*I*sqrt(pi)*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)*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))
Timed out. \[ \int \frac {x}{\sqrt {a+b \arccos (c x)}} \, dx=\int \frac {x}{\sqrt {a+b\,\mathrm {acos}\left (c\,x\right )}} \,d x \] Input:
int(x/(a + b*acos(c*x))^(1/2),x)
Output:
int(x/(a + b*acos(c*x))^(1/2), x)
\[ \int \frac {x}{\sqrt {a+b \arccos (c x)}} \, dx=\int \frac {\sqrt {\mathit {acos} \left (c x \right ) b +a}\, x}{\mathit {acos} \left (c x \right ) b +a}d x \] Input:
int(x/(a+b*acos(c*x))^(1/2),x)
Output:
int((sqrt(acos(c*x)*b + a)*x)/(acos(c*x)*b + a),x)