Integrand size = 14, antiderivative size = 137 \[ \int x \sqrt {a+b \arccos (c x)} \, dx=-\frac {\sqrt {a+b \arccos (c x)}}{4 c^2}+\frac {1}{2} x^2 \sqrt {a+b \arccos (c x)}-\frac {\sqrt {b} \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arccos (c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{8 c^2}-\frac {\sqrt {b} \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arccos (c x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{8 c^2} \] Output:
-1/4*(a+b*arccos(c*x))^(1/2)/c^2+1/2*x^2*(a+b*arccos(c*x))^(1/2)-1/8*b^(1/ 2)*Pi^(1/2)*cos(2*a/b)*FresnelC(2*(a+b*arccos(c*x))^(1/2)/b^(1/2)/Pi^(1/2) )/c^2-1/8*b^(1/2)*Pi^(1/2)*FresnelS(2*(a+b*arccos(c*x))^(1/2)/b^(1/2)/Pi^( 1/2))*sin(2*a/b)/c^2
Time = 0.19 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.85 \[ \int x \sqrt {a+b \arccos (c x)} \, dx=\frac {2 \sqrt {a+b \arccos (c x)} \cos (2 \arccos (c x))-\sqrt {b} \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arccos (c x)}}{\sqrt {b} \sqrt {\pi }}\right )-\sqrt {b} \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arccos (c x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{8 c^2} \] Input:
Integrate[x*Sqrt[a + b*ArcCos[c*x]],x]
Output:
(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 = 0.67 (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 |
\(\displaystyle \frac {1}{2} x^2 \sqrt {a+b \arccos (c x)}-\frac {\frac {1}{2} \sqrt {\pi } \sqrt {b} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arccos (c x)}}{\sqrt {b} \sqrt {\pi }}\right )+\frac {1}{2} \sqrt {\pi } \sqrt {b} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arccos (c x)}}{\sqrt {b} \sqrt {\pi }}\right )+\sqrt {a+b \arccos (c x)}}{4 c^2}\) |
Input:
Int[x*Sqrt[a + b*ArcCos[c*x]],x]
Output:
(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)
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 = 0.14 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.36
method | result | size |
default | \(\frac {-\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \sqrt {a +b \arccos \left (c x \right )}\, \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) \cos \left (\frac {2 a}{b}\right ) b +\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \sqrt {a +b \arccos \left (c x \right )}\, \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) \sin \left (\frac {2 a}{b}\right ) b +2 \arccos \left (c x \right ) \cos \left (-\frac {2 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {2 a}{b}\right ) b +2 \cos \left (-\frac {2 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {2 a}{b}\right ) a}{8 c^{2} \sqrt {a +b \arccos \left (c x \right )}}\) | \(186\) |
Input:
int(x*(a+b*arccos(c*x))^(1/2),x,method=_RETURNVERBOSE)
Output:
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)*FresnelC(2*2^(1/2)/Pi^(1/2)/(-2/b)^(1/2)*(a+b*arccos(c*x))^(1/2)/b)* cos(2*a/b)*b+Pi^(1/2)*(-1/b)^(1/2)*(a+b*arccos(c*x))^(1/2)*FresnelS(2*2^(1 /2)/Pi^(1/2)/(-2/b)^(1/2)*(a+b*arccos(c*x))^(1/2)/b)*sin(2*a/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. \[ \int x \sqrt {a+b \arccos (c x)} \, dx=\text {Exception raised: TypeError} \] 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)
\[ \int x \sqrt {a+b \arccos (c x)} \, dx=\int x \sqrt {a + b \operatorname {acos}{\left (c x \right )}}\, dx \] Input:
integrate(x*(a+b*acos(c*x))**(1/2),x)
Output:
Integral(x*sqrt(a + b*acos(c*x)), x)
\[ \int x \sqrt {a+b \arccos (c x)} \, dx=\int { \sqrt {b \arccos \left (c x\right ) + a} x \,d x } \] Input:
integrate(x*(a+b*arccos(c*x))^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(b*arccos(c*x) + a)*x, x)
Result contains complex when optimal does not.
Time = 0.56 (sec) , antiderivative size = 448, normalized size of antiderivative = 3.27 \[ \int x \sqrt {a+b \arccos (c x)} \, dx =\text {Too large to display} \] Input:
integrate(x*(a+b*arccos(c*x))^(1/2),x, algorithm="giac")
Output:
-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. \[ \int x \sqrt {a+b \arccos (c x)} \, dx=\int x\,\sqrt {a+b\,\mathrm {acos}\left (c\,x\right )} \,d x \] Input:
int(x*(a + b*acos(c*x))^(1/2),x)
Output:
int(x*(a + b*acos(c*x))^(1/2), x)
\[ \int x \sqrt {a+b \arccos (c x)} \, dx=\int \sqrt {\mathit {acos} \left (c x \right ) b +a}\, x d x \] Input:
int(x*(a+b*acos(c*x))^(1/2),x)
Output:
int(sqrt(acos(c*x)*b + a)*x,x)