Integrand size = 14, antiderivative size = 216 \[ \int x (a+b \arccos (c x))^{5/2} \, dx=\frac {15 b^2 \sqrt {a+b \arccos (c x)}}{64 c^2}-\frac {15}{32} b^2 x^2 \sqrt {a+b \arccos (c x)}-\frac {5 b x \sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{8 c}-\frac {(a+b \arccos (c x))^{5/2}}{4 c^2}+\frac {1}{2} x^2 (a+b \arccos (c x))^{5/2}+\frac {15 b^{5/2} \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arccos (c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{128 c^2}+\frac {15 b^{5/2} \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arccos (c x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{128 c^2} \]
-1/4*(a+b*arccos(c*x))^(5/2)/c^2+1/2*x^2*(a+b*arccos(c*x))^(5/2)+15/128*b^ (5/2)*cos(2*a/b)*FresnelC(2*(a+b*arccos(c*x))^(1/2)/b^(1/2)/Pi^(1/2))*Pi^( 1/2)/c^2+15/128*b^(5/2)*FresnelS(2*(a+b*arccos(c*x))^(1/2)/b^(1/2)/Pi^(1/2 ))*sin(2*a/b)*Pi^(1/2)/c^2-5/8*b*x*(a+b*arccos(c*x))^(3/2)*(-c^2*x^2+1)^(1 /2)/c+15/64*b^2*(a+b*arccos(c*x))^(1/2)/c^2-15/32*b^2*x^2*(a+b*arccos(c*x) )^(1/2)
Time = 0.94 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.87 \[ \int x (a+b \arccos (c x))^{5/2} \, dx=\frac {15 b^{5/2} \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arccos (c x)}}{\sqrt {b} \sqrt {\pi }}\right )+15 b^{5/2} \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arccos (c x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )+2 \sqrt {a+b \arccos (c x)} \left (\left (16 a^2-15 b^2\right ) \cos (2 \arccos (c x))+16 b^2 \arccos (c x)^2 \cos (2 \arccos (c x))-20 a b \sin (2 \arccos (c x))+4 b \arccos (c x) (8 a \cos (2 \arccos (c x))-5 b \sin (2 \arccos (c x)))\right )}{128 c^2} \]
(15*b^(5/2)*Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[a + b*ArcCos[c*x]])/(Sq rt[b]*Sqrt[Pi])] + 15*b^(5/2)*Sqrt[Pi]*FresnelS[(2*Sqrt[a + b*ArcCos[c*x]] )/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b] + 2*Sqrt[a + b*ArcCos[c*x]]*((16*a^2 - 15*b^2)*Cos[2*ArcCos[c*x]] + 16*b^2*ArcCos[c*x]^2*Cos[2*ArcCos[c*x]] - 20* a*b*Sin[2*ArcCos[c*x]] + 4*b*ArcCos[c*x]*(8*a*Cos[2*ArcCos[c*x]] - 5*b*Sin [2*ArcCos[c*x]])))/(128*c^2)
Time = 1.23 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.03, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {5141, 5211, 5141, 5153, 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.
| \(\displaystyle \int x (a+b \arccos (c x))^{5/2} \, dx\) |
| \(\Big \downarrow \) 5141 |
| \(\displaystyle \frac {5}{4} b c \int \frac {x^2 (a+b \arccos (c x))^{3/2}}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} x^2 (a+b \arccos (c x))^{5/2}\) |
| \(\Big \downarrow \) 5211 |
| \(\displaystyle \frac {5}{4} b c \left (\frac {\int \frac {(a+b \arccos (c x))^{3/2}}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {3 b \int x \sqrt {a+b \arccos (c x)}dx}{4 c}-\frac {x \sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{2 c^2}\right )+\frac {1}{2} x^2 (a+b \arccos (c x))^{5/2}\) |
| \(\Big \downarrow \) 5141 |
| \(\displaystyle \frac {5}{4} b c \left (-\frac {3 b \left (\frac {1}{4} b c \int \frac {x^2}{\sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}dx+\frac {1}{2} x^2 \sqrt {a+b \arccos (c x)}\right )}{4 c}+\frac {\int \frac {(a+b \arccos (c x))^{3/2}}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{2 c^2}\right )+\frac {1}{2} x^2 (a+b \arccos (c x))^{5/2}\) |
| \(\Big \downarrow \) 5153 |
| \(\displaystyle \frac {5}{4} b c \left (-\frac {3 b \left (\frac {1}{4} b c \int \frac {x^2}{\sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}dx+\frac {1}{2} x^2 \sqrt {a+b \arccos (c x)}\right )}{4 c}-\frac {(a+b \arccos (c x))^{5/2}}{5 b c^3}-\frac {x \sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{2 c^2}\right )+\frac {1}{2} x^2 (a+b \arccos (c x))^{5/2}\) |
| \(\Big \downarrow \) 5225 |
| \(\displaystyle \frac {5}{4} b c \left (-\frac {3 b \left (\frac {1}{2} x^2 \sqrt {a+b \arccos (c x)}-\frac {\int \frac {\cos ^2\left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{4 c^2}\right )}{4 c}-\frac {(a+b \arccos (c x))^{5/2}}{5 b c^3}-\frac {x \sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{2 c^2}\right )+\frac {1}{2} x^2 (a+b \arccos (c x))^{5/2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5}{4} b c \left (-\frac {3 b \left (\frac {1}{2} x^2 \sqrt {a+b \arccos (c x)}-\frac {\int \frac {\sin \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}+\frac {\pi }{2}\right )^2}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{4 c^2}\right )}{4 c}-\frac {(a+b \arccos (c x))^{5/2}}{5 b c^3}-\frac {x \sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{2 c^2}\right )+\frac {1}{2} x^2 (a+b \arccos (c x))^{5/2}\) |
| \(\Big \downarrow \) 3793 |
| \(\displaystyle \frac {5}{4} b c \left (-\frac {3 b \left (\frac {1}{2} x^2 \sqrt {a+b \arccos (c x)}-\frac {\int \left (\frac {\cos \left (\frac {2 a}{b}-\frac {2 (a+b \arccos (c x))}{b}\right )}{2 \sqrt {a+b \arccos (c x)}}+\frac {1}{2 \sqrt {a+b \arccos (c x)}}\right )d(a+b \arccos (c x))}{4 c^2}\right )}{4 c}-\frac {(a+b \arccos (c x))^{5/2}}{5 b c^3}-\frac {x \sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{2 c^2}\right )+\frac {1}{2} x^2 (a+b \arccos (c x))^{5/2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {5}{4} b c \left (-\frac {(a+b \arccos (c x))^{5/2}}{5 b c^3}-\frac {3 b \left (\frac {1}{2} x^2 \sqrt {a+b \arccos (c x)}-\frac {\frac {1}{2} \sqrt {\pi } \sqrt {b} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arccos (c x)}}{\sqrt {b} \sqrt {\pi }}\right )+\frac {1}{2} \sqrt {\pi } \sqrt {b} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arccos (c x)}}{\sqrt {b} \sqrt {\pi }}\right )+\sqrt {a+b \arccos (c x)}}{4 c^2}\right )}{4 c}-\frac {x \sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{2 c^2}\right )+\frac {1}{2} x^2 (a+b \arccos (c x))^{5/2}\) |
(x^2*(a + b*ArcCos[c*x])^(5/2))/2 + (5*b*c*(-1/2*(x*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x])^(3/2))/c^2 - (a + b*ArcCos[c*x])^(5/2)/(5*b*c^3) - (3*b*(( 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)))/(4*c)))/4
3.2.84.3.1 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_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S ymbol] :> Simp[(-(b*c*(n + 1))^(-1))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2] ]*(a + b*ArcCos[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^ 2*d + e, 0] && NeQ[n, -1]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcCos[c*x])^n/(e*(m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcCos[c*x])^n, x], x] - S imp[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] Int[(f* x)^(m - 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m , 1] && NeQ[m + 2*p + 1, 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]
Leaf count of result is larger than twice the leaf count of optimal. \(407\) vs. \(2(170)=340\).
Time = 2.01 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.89
| method | result | size |
| default | \(\frac {15 \sqrt {a +b \arccos \left (c x \right )}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) b^{3}-15 \sqrt {a +b \arccos \left (c x \right )}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) b^{3}+32 \arccos \left (c x \right )^{3} \cos \left (-\frac {2 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {2 a}{b}\right ) b^{3}+96 \arccos \left (c x \right )^{2} \cos \left (-\frac {2 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {2 a}{b}\right ) a \,b^{2}+40 \arccos \left (c x \right )^{2} \sin \left (-\frac {2 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {2 a}{b}\right ) b^{3}+96 \arccos \left (c x \right ) \cos \left (-\frac {2 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {2 a}{b}\right ) a^{2} b -30 \arccos \left (c x \right ) \cos \left (-\frac {2 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {2 a}{b}\right ) b^{3}+80 \arccos \left (c x \right ) \sin \left (-\frac {2 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {2 a}{b}\right ) a \,b^{2}+32 \cos \left (-\frac {2 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {2 a}{b}\right ) a^{3}-30 \cos \left (-\frac {2 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {2 a}{b}\right ) a \,b^{2}+40 \sin \left (-\frac {2 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {2 a}{b}\right ) a^{2} b}{128 c^{2} \sqrt {a +b \arccos \left (c x \right )}}\) | \(408\) |
1/128/c^2/(a+b*arccos(c*x))^(1/2)*(15*(a+b*arccos(c*x))^(1/2)*Pi^(1/2)*(-1 /b)^(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^3-15*(a+b*arccos(c*x))^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*sin(2* a/b)*FresnelS(2*2^(1/2)/Pi^(1/2)/(-2/b)^(1/2)*(a+b*arccos(c*x))^(1/2)/b)*b ^3+32*arccos(c*x)^3*cos(-2*(a+b*arccos(c*x))/b+2*a/b)*b^3+96*arccos(c*x)^2 *cos(-2*(a+b*arccos(c*x))/b+2*a/b)*a*b^2+40*arccos(c*x)^2*sin(-2*(a+b*arcc os(c*x))/b+2*a/b)*b^3+96*arccos(c*x)*cos(-2*(a+b*arccos(c*x))/b+2*a/b)*a^2 *b-30*arccos(c*x)*cos(-2*(a+b*arccos(c*x))/b+2*a/b)*b^3+80*arccos(c*x)*sin (-2*(a+b*arccos(c*x))/b+2*a/b)*a*b^2+32*cos(-2*(a+b*arccos(c*x))/b+2*a/b)* a^3-30*cos(-2*(a+b*arccos(c*x))/b+2*a/b)*a*b^2+40*sin(-2*(a+b*arccos(c*x)) /b+2*a/b)*a^2*b)
Exception generated. \[ \int x (a+b \arccos (c x))^{5/2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int x (a+b \arccos (c x))^{5/2} \, dx=\int x \left (a + b \operatorname {acos}{\left (c x \right )}\right )^{\frac {5}{2}}\, dx \]
\[ \int x (a+b \arccos (c x))^{5/2} \, dx=\int { {\left (b \arccos \left (c x\right ) + a\right )}^{\frac {5}{2}} x \,d x } \]
Result contains complex when optimal does not.
Time = 1.29 (sec) , antiderivative size = 1307, normalized size of antiderivative = 6.05 \[ \int x (a+b \arccos (c x))^{5/2} \, dx=\text {Too large to display} \]
-1/4*I*sqrt(pi)*a^3*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^2 + I*b^3/abs(b))*c^2) + 3/8*sqrt(pi)*a^2*b^(5/2)*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^2 + I*b^3/abs(b))*c^2) + 1/4*I*sqrt(pi)*a^3*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^2 - I*b^3/abs(b))*c^2) + 3/8*sqrt(pi)*a^2*b^(5/2)*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^2 - I*b^3/abs(b))*c^2) + 1/8*sqrt(b*arccos(c*x) + a)*b^2*arccos(c*x)^2*e^(2*I*arccos(c*x))/c^2 + 1/8*sqrt(b*arccos(c*x) + a)*b^2*arccos(c*x)^2*e^(-2*I*arccos(c*x))/c^2 - 3/8*sqrt(pi)*a^2*b^2*erf(-sqrt(b*arccos(c*x) + a)/sqrt(b) - I*sqrt(b*arcco s(c*x) + a)*sqrt(b)/abs(b))*e^(2*I*a/b)/((b^(3/2) + I*b^(5/2)/abs(b))*c^2) + 9/64*I*sqrt(pi)*a*b^3*erf(-sqrt(b*arccos(c*x) + a)/sqrt(b) - I*sqrt(b*a rccos(c*x) + a)*sqrt(b)/abs(b))*e^(2*I*a/b)/((b^(3/2) + I*b^(5/2)/abs(b))* c^2) - 1/4*I*sqrt(pi)*a^3*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^(3/2) - I*b^(5/2)/abs( b))*c^2) - 3/8*sqrt(pi)*a^2*b^2*erf(-sqrt(b*arccos(c*x) + a)/sqrt(b) + I*s qrt(b*arccos(c*x) + a)*sqrt(b)/abs(b))*e^(-2*I*a/b)/((b^(3/2) - I*b^(5/2)/ abs(b))*c^2) - 9/64*I*sqrt(pi)*a*b^3*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^(3/2) - I*...
Timed out. \[ \int x (a+b \arccos (c x))^{5/2} \, dx=\int x\,{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^{5/2} \,d x \]