Integrand size = 12, antiderivative size = 282 \[ \int x^4 \arccos (a x)^{3/2} \, dx=-\frac {4 \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{25 a^5}-\frac {2 x^2 \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{25 a^3}-\frac {3 x^4 \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{50 a}+\frac {1}{5} x^5 \arccos (a x)^{3/2}+\frac {11 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{400 a^5}+\frac {2 \sqrt {2 \pi } \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{25 a^5}+\frac {\sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{50 a^5}+\frac {3 \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{800 a^5}+\frac {3 \sqrt {\frac {\pi }{10}} \operatorname {FresnelS}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arccos (a x)}\right )}{800 a^5} \]
1/5*x^5*arccos(a*x)^(3/2)+3/8000*FresnelS(10^(1/2)/Pi^(1/2)*arccos(a*x)^(1 /2))*10^(1/2)*Pi^(1/2)/a^5+1/192*FresnelS(6^(1/2)/Pi^(1/2)*arccos(a*x)^(1/ 2))*6^(1/2)*Pi^(1/2)/a^5+3/32*FresnelS(2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2)) *2^(1/2)*Pi^(1/2)/a^5-4/25*(-a^2*x^2+1)^(1/2)*arccos(a*x)^(1/2)/a^5-2/25*x ^2*(-a^2*x^2+1)^(1/2)*arccos(a*x)^(1/2)/a^3-3/50*x^4*(-a^2*x^2+1)^(1/2)*ar ccos(a*x)^(1/2)/a
Result contains complex when optimal does not.
Time = 0.09 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.66 \[ \int x^4 \arccos (a x)^{3/2} \, dx=-\frac {2250 \left (\sqrt {-i \arccos (a x)} \Gamma \left (\frac {5}{2},-i \arccos (a x)\right )+\sqrt {i \arccos (a x)} \Gamma \left (\frac {5}{2},i \arccos (a x)\right )\right )+125 \sqrt {3} \left (\sqrt {-i \arccos (a x)} \Gamma \left (\frac {5}{2},-3 i \arccos (a x)\right )+\sqrt {i \arccos (a x)} \Gamma \left (\frac {5}{2},3 i \arccos (a x)\right )\right )+9 \sqrt {5} \left (\sqrt {-i \arccos (a x)} \Gamma \left (\frac {5}{2},-5 i \arccos (a x)\right )+\sqrt {i \arccos (a x)} \Gamma \left (\frac {5}{2},5 i \arccos (a x)\right )\right )}{36000 a^5 \sqrt {\arccos (a x)}} \]
-1/36000*(2250*(Sqrt[(-I)*ArcCos[a*x]]*Gamma[5/2, (-I)*ArcCos[a*x]] + Sqrt [I*ArcCos[a*x]]*Gamma[5/2, I*ArcCos[a*x]]) + 125*Sqrt[3]*(Sqrt[(-I)*ArcCos [a*x]]*Gamma[5/2, (-3*I)*ArcCos[a*x]] + Sqrt[I*ArcCos[a*x]]*Gamma[5/2, (3* I)*ArcCos[a*x]]) + 9*Sqrt[5]*(Sqrt[(-I)*ArcCos[a*x]]*Gamma[5/2, (-5*I)*Arc Cos[a*x]] + Sqrt[I*ArcCos[a*x]]*Gamma[5/2, (5*I)*ArcCos[a*x]]))/(a^5*Sqrt[ ArcCos[a*x]])
Time = 1.89 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.20, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.167, Rules used = {5141, 5211, 5147, 4906, 2009, 5211, 5147, 4906, 2009, 5183, 5135, 3042, 3786, 3832}
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.
| \(\displaystyle \int x^4 \arccos (a x)^{3/2} \, dx\) |
| \(\Big \downarrow \) 5141 |
| \(\displaystyle \frac {3}{10} a \int \frac {x^5 \sqrt {\arccos (a x)}}{\sqrt {1-a^2 x^2}}dx+\frac {1}{5} x^5 \arccos (a x)^{3/2}\) |
| \(\Big \downarrow \) 5211 |
| \(\displaystyle \frac {3}{10} a \left (\frac {4 \int \frac {x^3 \sqrt {\arccos (a x)}}{\sqrt {1-a^2 x^2}}dx}{5 a^2}-\frac {\int \frac {x^4}{\sqrt {\arccos (a x)}}dx}{10 a}-\frac {x^4 \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{5 a^2}\right )+\frac {1}{5} x^5 \arccos (a x)^{3/2}\) |
| \(\Big \downarrow \) 5147 |
| \(\displaystyle \frac {3}{10} a \left (\frac {4 \int \frac {x^3 \sqrt {\arccos (a x)}}{\sqrt {1-a^2 x^2}}dx}{5 a^2}+\frac {\int \frac {a^4 x^4 \sqrt {1-a^2 x^2}}{\sqrt {\arccos (a x)}}d\arccos (a x)}{10 a^6}-\frac {x^4 \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{5 a^2}\right )+\frac {1}{5} x^5 \arccos (a x)^{3/2}\) |
| \(\Big \downarrow \) 4906 |
| \(\displaystyle \frac {3}{10} a \left (\frac {4 \int \frac {x^3 \sqrt {\arccos (a x)}}{\sqrt {1-a^2 x^2}}dx}{5 a^2}+\frac {\int \left (\frac {3 \sin (3 \arccos (a x))}{16 \sqrt {\arccos (a x)}}+\frac {\sin (5 \arccos (a x))}{16 \sqrt {\arccos (a x)}}+\frac {\sqrt {1-a^2 x^2}}{8 \sqrt {\arccos (a x)}}\right )d\arccos (a x)}{10 a^6}-\frac {x^4 \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{5 a^2}\right )+\frac {1}{5} x^5 \arccos (a x)^{3/2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3}{10} a \left (\frac {4 \int \frac {x^3 \sqrt {\arccos (a x)}}{\sqrt {1-a^2 x^2}}dx}{5 a^2}+\frac {\frac {1}{4} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )+\frac {1}{8} \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{10}} \operatorname {FresnelS}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arccos (a x)}\right )}{10 a^6}-\frac {x^4 \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{5 a^2}\right )+\frac {1}{5} x^5 \arccos (a x)^{3/2}\) |
| \(\Big \downarrow \) 5211 |
| \(\displaystyle \frac {3}{10} a \left (\frac {4 \left (\frac {2 \int \frac {x \sqrt {\arccos (a x)}}{\sqrt {1-a^2 x^2}}dx}{3 a^2}-\frac {\int \frac {x^2}{\sqrt {\arccos (a x)}}dx}{6 a}-\frac {x^2 \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{3 a^2}\right )}{5 a^2}+\frac {\frac {1}{4} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )+\frac {1}{8} \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{10}} \operatorname {FresnelS}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arccos (a x)}\right )}{10 a^6}-\frac {x^4 \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{5 a^2}\right )+\frac {1}{5} x^5 \arccos (a x)^{3/2}\) |
| \(\Big \downarrow \) 5147 |
| \(\displaystyle \frac {3}{10} a \left (\frac {4 \left (\frac {2 \int \frac {x \sqrt {\arccos (a x)}}{\sqrt {1-a^2 x^2}}dx}{3 a^2}+\frac {\int \frac {a^2 x^2 \sqrt {1-a^2 x^2}}{\sqrt {\arccos (a x)}}d\arccos (a x)}{6 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{3 a^2}\right )}{5 a^2}+\frac {\frac {1}{4} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )+\frac {1}{8} \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{10}} \operatorname {FresnelS}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arccos (a x)}\right )}{10 a^6}-\frac {x^4 \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{5 a^2}\right )+\frac {1}{5} x^5 \arccos (a x)^{3/2}\) |
| \(\Big \downarrow \) 4906 |
| \(\displaystyle \frac {3}{10} a \left (\frac {4 \left (\frac {2 \int \frac {x \sqrt {\arccos (a x)}}{\sqrt {1-a^2 x^2}}dx}{3 a^2}+\frac {\int \left (\frac {\sin (3 \arccos (a x))}{4 \sqrt {\arccos (a x)}}+\frac {\sqrt {1-a^2 x^2}}{4 \sqrt {\arccos (a x)}}\right )d\arccos (a x)}{6 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{3 a^2}\right )}{5 a^2}+\frac {\frac {1}{4} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )+\frac {1}{8} \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{10}} \operatorname {FresnelS}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arccos (a x)}\right )}{10 a^6}-\frac {x^4 \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{5 a^2}\right )+\frac {1}{5} x^5 \arccos (a x)^{3/2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3}{10} a \left (\frac {4 \left (\frac {2 \int \frac {x \sqrt {\arccos (a x)}}{\sqrt {1-a^2 x^2}}dx}{3 a^2}+\frac {\frac {1}{2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )+\frac {1}{2} \sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{6 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{3 a^2}\right )}{5 a^2}+\frac {\frac {1}{4} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )+\frac {1}{8} \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{10}} \operatorname {FresnelS}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arccos (a x)}\right )}{10 a^6}-\frac {x^4 \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{5 a^2}\right )+\frac {1}{5} x^5 \arccos (a x)^{3/2}\) |
| \(\Big \downarrow \) 5183 |
| \(\displaystyle \frac {3}{10} a \left (\frac {4 \left (\frac {2 \left (-\frac {\int \frac {1}{\sqrt {\arccos (a x)}}dx}{2 a}-\frac {\sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{a^2}\right )}{3 a^2}+\frac {\frac {1}{2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )+\frac {1}{2} \sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{6 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{3 a^2}\right )}{5 a^2}+\frac {\frac {1}{4} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )+\frac {1}{8} \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{10}} \operatorname {FresnelS}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arccos (a x)}\right )}{10 a^6}-\frac {x^4 \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{5 a^2}\right )+\frac {1}{5} x^5 \arccos (a x)^{3/2}\) |
| \(\Big \downarrow \) 5135 |
| \(\displaystyle \frac {3}{10} a \left (\frac {4 \left (\frac {2 \left (\frac {\int \frac {\sqrt {1-a^2 x^2}}{\sqrt {\arccos (a x)}}d\arccos (a x)}{2 a^2}-\frac {\sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{a^2}\right )}{3 a^2}+\frac {\frac {1}{2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )+\frac {1}{2} \sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{6 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{3 a^2}\right )}{5 a^2}+\frac {\frac {1}{4} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )+\frac {1}{8} \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{10}} \operatorname {FresnelS}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arccos (a x)}\right )}{10 a^6}-\frac {x^4 \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{5 a^2}\right )+\frac {1}{5} x^5 \arccos (a x)^{3/2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{10} a \left (\frac {4 \left (\frac {2 \left (\frac {\int \frac {\sin (\arccos (a x))}{\sqrt {\arccos (a x)}}d\arccos (a x)}{2 a^2}-\frac {\sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{a^2}\right )}{3 a^2}+\frac {\frac {1}{2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )+\frac {1}{2} \sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{6 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{3 a^2}\right )}{5 a^2}+\frac {\frac {1}{4} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )+\frac {1}{8} \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{10}} \operatorname {FresnelS}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arccos (a x)}\right )}{10 a^6}-\frac {x^4 \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{5 a^2}\right )+\frac {1}{5} x^5 \arccos (a x)^{3/2}\) |
| \(\Big \downarrow \) 3786 |
| \(\displaystyle \frac {3}{10} a \left (\frac {4 \left (\frac {2 \left (\frac {\int \sqrt {1-a^2 x^2}d\sqrt {\arccos (a x)}}{a^2}-\frac {\sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{a^2}\right )}{3 a^2}+\frac {\frac {1}{2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )+\frac {1}{2} \sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{6 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{3 a^2}\right )}{5 a^2}+\frac {\frac {1}{4} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )+\frac {1}{8} \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{10}} \operatorname {FresnelS}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arccos (a x)}\right )}{10 a^6}-\frac {x^4 \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{5 a^2}\right )+\frac {1}{5} x^5 \arccos (a x)^{3/2}\) |
| \(\Big \downarrow \) 3832 |
| \(\displaystyle \frac {3}{10} a \left (\frac {\frac {1}{4} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )+\frac {1}{8} \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{10}} \operatorname {FresnelS}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arccos (a x)}\right )}{10 a^6}-\frac {x^4 \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{5 a^2}+\frac {4 \left (\frac {\frac {1}{2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )+\frac {1}{2} \sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{6 a^4}+\frac {2 \left (\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{a^2}-\frac {\sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{a^2}\right )}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{3 a^2}\right )}{5 a^2}\right )+\frac {1}{5} x^5 \arccos (a x)^{3/2}\) |
(x^5*ArcCos[a*x]^(3/2))/5 + (3*a*(-1/5*(x^4*Sqrt[1 - a^2*x^2]*Sqrt[ArcCos[ a*x]])/a^2 + (4*(-1/3*(x^2*Sqrt[1 - a^2*x^2]*Sqrt[ArcCos[a*x]])/a^2 + (2*( -((Sqrt[1 - a^2*x^2]*Sqrt[ArcCos[a*x]])/a^2) + (Sqrt[Pi/2]*FresnelS[Sqrt[2 /Pi]*Sqrt[ArcCos[a*x]]])/a^2))/(3*a^2) + ((Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]* Sqrt[ArcCos[a*x]]])/2 + (Sqrt[Pi/6]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcCos[a*x]]] )/2)/(6*a^4)))/(5*a^2) + ((Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcCos[a*x] ]])/4 + (Sqrt[(3*Pi)/2]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcCos[a*x]]])/8 + (Sqrt[ Pi/10]*FresnelS[Sqrt[10/Pi]*Sqrt[ArcCos[a*x]]])/8)/(10*a^6)))/10
3.1.80.3.1 Defintions of rubi rules used
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[(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[(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_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]
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_.), 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]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_ .), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcCos[c*x])^n/(2*e*(p + 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] I nt[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -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]
Time = 0.99 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.68
| method | result | size |
| default | \(\frac {9 \,\operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {5}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {5}\, \sqrt {2}\, \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }+125 \,\operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {3}\, \sqrt {2}\, \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }+3000 \arccos \left (a x \right )^{2} a x +2250 \,\operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {2}\, \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }+1500 \arccos \left (a x \right )^{2} \cos \left (3 \arccos \left (a x \right )\right )+300 \arccos \left (a x \right )^{2} \cos \left (5 \arccos \left (a x \right )\right )-4500 \arccos \left (a x \right ) \sqrt {-a^{2} x^{2}+1}-750 \arccos \left (a x \right ) \sin \left (3 \arccos \left (a x \right )\right )-90 \arccos \left (a x \right ) \sin \left (5 \arccos \left (a x \right )\right )}{24000 a^{5} \sqrt {\arccos \left (a x \right )}}\) | \(193\) |
1/24000/a^5*(9*FresnelS(2^(1/2)/Pi^(1/2)*5^(1/2)*arccos(a*x)^(1/2))*5^(1/2 )*2^(1/2)*arccos(a*x)^(1/2)*Pi^(1/2)+125*FresnelS(2^(1/2)/Pi^(1/2)*3^(1/2) *arccos(a*x)^(1/2))*3^(1/2)*2^(1/2)*arccos(a*x)^(1/2)*Pi^(1/2)+3000*arccos (a*x)^2*a*x+2250*FresnelS(2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*2^(1/2)*arcc os(a*x)^(1/2)*Pi^(1/2)+1500*arccos(a*x)^2*cos(3*arccos(a*x))+300*arccos(a* x)^2*cos(5*arccos(a*x))-4500*arccos(a*x)*(-a^2*x^2+1)^(1/2)-750*arccos(a*x )*sin(3*arccos(a*x))-90*arccos(a*x)*sin(5*arccos(a*x)))/arccos(a*x)^(1/2)
Exception generated. \[ \int x^4 \arccos (a x)^{3/2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int x^4 \arccos (a x)^{3/2} \, dx=\int x^{4} \operatorname {acos}^{\frac {3}{2}}{\left (a x \right )}\, dx \]
Exception generated. \[ \int x^4 \arccos (a x)^{3/2} \, dx=\text {Exception raised: RuntimeError} \]
Result contains complex when optimal does not.
Time = 0.33 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.26 \[ \int x^4 \arccos (a x)^{3/2} \, dx=\frac {\arccos \left (a x\right )^{\frac {3}{2}} e^{\left (5 i \, \arccos \left (a x\right )\right )}}{160 \, a^{5}} + \frac {\arccos \left (a x\right )^{\frac {3}{2}} e^{\left (3 i \, \arccos \left (a x\right )\right )}}{32 \, a^{5}} + \frac {\arccos \left (a x\right )^{\frac {3}{2}} e^{\left (i \, \arccos \left (a x\right )\right )}}{16 \, a^{5}} + \frac {\arccos \left (a x\right )^{\frac {3}{2}} e^{\left (-i \, \arccos \left (a x\right )\right )}}{16 \, a^{5}} + \frac {\arccos \left (a x\right )^{\frac {3}{2}} e^{\left (-3 i \, \arccos \left (a x\right )\right )}}{32 \, a^{5}} + \frac {\arccos \left (a x\right )^{\frac {3}{2}} e^{\left (-5 i \, \arccos \left (a x\right )\right )}}{160 \, a^{5}} + \frac {\left (3 i - 3\right ) \, \sqrt {10} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {10} \sqrt {\arccos \left (a x\right )}\right )}{32000 \, a^{5}} - \frac {\left (3 i + 3\right ) \, \sqrt {10} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {10} \sqrt {\arccos \left (a x\right )}\right )}{32000 \, a^{5}} + \frac {\left (i - 1\right ) \, \sqrt {6} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {6} \sqrt {\arccos \left (a x\right )}\right )}{768 \, a^{5}} - \frac {\left (i + 1\right ) \, \sqrt {6} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {6} \sqrt {\arccos \left (a x\right )}\right )}{768 \, a^{5}} + \frac {\left (3 i - 3\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arccos \left (a x\right )}\right )}{128 \, a^{5}} - \frac {\left (3 i + 3\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arccos \left (a x\right )}\right )}{128 \, a^{5}} + \frac {3 i \, \sqrt {\arccos \left (a x\right )} e^{\left (5 i \, \arccos \left (a x\right )\right )}}{1600 \, a^{5}} + \frac {i \, \sqrt {\arccos \left (a x\right )} e^{\left (3 i \, \arccos \left (a x\right )\right )}}{64 \, a^{5}} + \frac {3 i \, \sqrt {\arccos \left (a x\right )} e^{\left (i \, \arccos \left (a x\right )\right )}}{32 \, a^{5}} - \frac {3 i \, \sqrt {\arccos \left (a x\right )} e^{\left (-i \, \arccos \left (a x\right )\right )}}{32 \, a^{5}} - \frac {i \, \sqrt {\arccos \left (a x\right )} e^{\left (-3 i \, \arccos \left (a x\right )\right )}}{64 \, a^{5}} - \frac {3 i \, \sqrt {\arccos \left (a x\right )} e^{\left (-5 i \, \arccos \left (a x\right )\right )}}{1600 \, a^{5}} \]
1/160*arccos(a*x)^(3/2)*e^(5*I*arccos(a*x))/a^5 + 1/32*arccos(a*x)^(3/2)*e ^(3*I*arccos(a*x))/a^5 + 1/16*arccos(a*x)^(3/2)*e^(I*arccos(a*x))/a^5 + 1/ 16*arccos(a*x)^(3/2)*e^(-I*arccos(a*x))/a^5 + 1/32*arccos(a*x)^(3/2)*e^(-3 *I*arccos(a*x))/a^5 + 1/160*arccos(a*x)^(3/2)*e^(-5*I*arccos(a*x))/a^5 + ( 3/32000*I - 3/32000)*sqrt(10)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt(10)*sqrt(arc cos(a*x)))/a^5 - (3/32000*I + 3/32000)*sqrt(10)*sqrt(pi)*erf(-(1/2*I + 1/2 )*sqrt(10)*sqrt(arccos(a*x)))/a^5 + (1/768*I - 1/768)*sqrt(6)*sqrt(pi)*erf ((1/2*I - 1/2)*sqrt(6)*sqrt(arccos(a*x)))/a^5 - (1/768*I + 1/768)*sqrt(6)* sqrt(pi)*erf(-(1/2*I + 1/2)*sqrt(6)*sqrt(arccos(a*x)))/a^5 + (3/128*I - 3/ 128)*sqrt(2)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt(2)*sqrt(arccos(a*x)))/a^5 - ( 3/128*I + 3/128)*sqrt(2)*sqrt(pi)*erf(-(1/2*I + 1/2)*sqrt(2)*sqrt(arccos(a *x)))/a^5 + 3/1600*I*sqrt(arccos(a*x))*e^(5*I*arccos(a*x))/a^5 + 1/64*I*sq rt(arccos(a*x))*e^(3*I*arccos(a*x))/a^5 + 3/32*I*sqrt(arccos(a*x))*e^(I*ar ccos(a*x))/a^5 - 3/32*I*sqrt(arccos(a*x))*e^(-I*arccos(a*x))/a^5 - 1/64*I* sqrt(arccos(a*x))*e^(-3*I*arccos(a*x))/a^5 - 3/1600*I*sqrt(arccos(a*x))*e^ (-5*I*arccos(a*x))/a^5
Timed out. \[ \int x^4 \arccos (a x)^{3/2} \, dx=\int x^4\,{\mathrm {acos}\left (a\,x\right )}^{3/2} \,d x \]