Integrand size = 12, antiderivative size = 121 \[ \int x^4 \sqrt {\arccos (a x)} \, dx=\frac {1}{5} x^5 \sqrt {\arccos (a x)}-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{8 a^5}-\frac {\sqrt {\frac {\pi }{6}} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{16 a^5}-\frac {\sqrt {\frac {\pi }{10}} \operatorname {FresnelC}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arccos (a x)}\right )}{80 a^5} \]
-1/800*FresnelC(10^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*10^(1/2)*Pi^(1/2)/a^5 -1/96*FresnelC(6^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*6^(1/2)*Pi^(1/2)/a^5-1/ 16*FresnelC(2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a^5+1/5*x ^5*arccos(a*x)^(1/2)
Result contains complex when optimal does not.
Time = 0.08 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.60 \[ \int x^4 \sqrt {\arccos (a x)} \, dx=\frac {i \left (150 \sqrt {-i \arccos (a x)} \Gamma \left (\frac {3}{2},-i \arccos (a x)\right )-150 \sqrt {i \arccos (a x)} \Gamma \left (\frac {3}{2},i \arccos (a x)\right )+25 \sqrt {3} \sqrt {-i \arccos (a x)} \Gamma \left (\frac {3}{2},-3 i \arccos (a x)\right )-25 \sqrt {3} \sqrt {i \arccos (a x)} \Gamma \left (\frac {3}{2},3 i \arccos (a x)\right )+3 \sqrt {5} \sqrt {-i \arccos (a x)} \Gamma \left (\frac {3}{2},-5 i \arccos (a x)\right )-3 \sqrt {5} \sqrt {i \arccos (a x)} \Gamma \left (\frac {3}{2},5 i \arccos (a x)\right )\right )}{2400 a^5 \sqrt {\arccos (a x)}} \]
((I/2400)*(150*Sqrt[(-I)*ArcCos[a*x]]*Gamma[3/2, (-I)*ArcCos[a*x]] - 150*S qrt[I*ArcCos[a*x]]*Gamma[3/2, I*ArcCos[a*x]] + 25*Sqrt[3]*Sqrt[(-I)*ArcCos [a*x]]*Gamma[3/2, (-3*I)*ArcCos[a*x]] - 25*Sqrt[3]*Sqrt[I*ArcCos[a*x]]*Gam ma[3/2, (3*I)*ArcCos[a*x]] + 3*Sqrt[5]*Sqrt[(-I)*ArcCos[a*x]]*Gamma[3/2, ( -5*I)*ArcCos[a*x]] - 3*Sqrt[5]*Sqrt[I*ArcCos[a*x]]*Gamma[3/2, (5*I)*ArcCos [a*x]]))/(a^5*Sqrt[ArcCos[a*x]])
Time = 0.46 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.99, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, 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.
\(\displaystyle \int x^4 \sqrt {\arccos (a x)} \, dx\) |
\(\Big \downarrow \) 5141 |
\(\displaystyle \frac {1}{10} a \int \frac {x^5}{\sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}dx+\frac {1}{5} x^5 \sqrt {\arccos (a x)}\) |
\(\Big \downarrow \) 5225 |
\(\displaystyle \frac {1}{5} x^5 \sqrt {\arccos (a x)}-\frac {\int \frac {a^5 x^5}{\sqrt {\arccos (a x)}}d\arccos (a x)}{10 a^5}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{5} x^5 \sqrt {\arccos (a x)}-\frac {\int \frac {\sin \left (\arccos (a x)+\frac {\pi }{2}\right )^5}{\sqrt {\arccos (a x)}}d\arccos (a x)}{10 a^5}\) |
\(\Big \downarrow \) 3793 |
\(\displaystyle \frac {1}{5} x^5 \sqrt {\arccos (a x)}-\frac {\int \left (\frac {5 a x}{8 \sqrt {\arccos (a x)}}+\frac {5 \cos (3 \arccos (a x))}{16 \sqrt {\arccos (a x)}}+\frac {\cos (5 \arccos (a x))}{16 \sqrt {\arccos (a x)}}\right )d\arccos (a x)}{10 a^5}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{5} x^5 \sqrt {\arccos (a x)}-\frac {\frac {5}{4} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )+\frac {5}{8} \sqrt {\frac {\pi }{6}} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{10}} \operatorname {FresnelC}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arccos (a x)}\right )}{10 a^5}\) |
(x^5*Sqrt[ArcCos[a*x]])/5 - ((5*Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcCos [a*x]]])/4 + (5*Sqrt[Pi/6]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcCos[a*x]]])/8 + (Sq rt[Pi/10]*FresnelC[Sqrt[10/Pi]*Sqrt[ArcCos[a*x]]])/8)/(10*a^5)
3.1.74.3.1 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 = 0.94 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.18
method | result | size |
default | \(\frac {-3 \sqrt {5}\, \sqrt {2}\, \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {5}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )-25 \sqrt {3}\, \sqrt {2}\, \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )-150 \sqrt {2}\, \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )+300 \arccos \left (a x \right ) a x +150 \arccos \left (a x \right ) \cos \left (3 \arccos \left (a x \right )\right )+30 \arccos \left (a x \right ) \cos \left (5 \arccos \left (a x \right )\right )}{2400 a^{5} \sqrt {\arccos \left (a x \right )}}\) | \(143\) |
1/2400/a^5/arccos(a*x)^(1/2)*(-3*5^(1/2)*2^(1/2)*arccos(a*x)^(1/2)*Pi^(1/2 )*FresnelC(2^(1/2)/Pi^(1/2)*5^(1/2)*arccos(a*x)^(1/2))-25*3^(1/2)*2^(1/2)* arccos(a*x)^(1/2)*Pi^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*3^(1/2)*arccos(a*x)^( 1/2))-150*2^(1/2)*arccos(a*x)^(1/2)*Pi^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*arc cos(a*x)^(1/2))+300*arccos(a*x)*a*x+150*arccos(a*x)*cos(3*arccos(a*x))+30* arccos(a*x)*cos(5*arccos(a*x)))
Exception generated. \[ \int x^4 \sqrt {\arccos (a x)} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int x^4 \sqrt {\arccos (a x)} \, dx=\int x^{4} \sqrt {\operatorname {acos}{\left (a x \right )}}\, dx \]
Exception generated. \[ \int x^4 \sqrt {\arccos (a x)} \, dx=\text {Exception raised: RuntimeError} \]
Result contains complex when optimal does not.
Time = 0.34 (sec) , antiderivative size = 247, normalized size of antiderivative = 2.04 \[ \int x^4 \sqrt {\arccos (a x)} \, dx=\frac {\left (i + 1\right ) \, \sqrt {10} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {10} \sqrt {\arccos \left (a x\right )}\right )}{3200 \, a^{5}} - \frac {\left (i - 1\right ) \, \sqrt {10} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {10} \sqrt {\arccos \left (a x\right )}\right )}{3200 \, a^{5}} + \frac {\left (i + 1\right ) \, \sqrt {6} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {6} \sqrt {\arccos \left (a x\right )}\right )}{384 \, a^{5}} - \frac {\left (i - 1\right ) \, \sqrt {6} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {6} \sqrt {\arccos \left (a x\right )}\right )}{384 \, a^{5}} + \frac {\left (i + 1\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arccos \left (a x\right )}\right )}{64 \, a^{5}} - \frac {\left (i - 1\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arccos \left (a x\right )}\right )}{64 \, a^{5}} + \frac {\sqrt {\arccos \left (a x\right )} e^{\left (5 i \, \arccos \left (a x\right )\right )}}{160 \, a^{5}} + \frac {\sqrt {\arccos \left (a x\right )} e^{\left (3 i \, \arccos \left (a x\right )\right )}}{32 \, a^{5}} + \frac {\sqrt {\arccos \left (a x\right )} e^{\left (i \, \arccos \left (a x\right )\right )}}{16 \, a^{5}} + \frac {\sqrt {\arccos \left (a x\right )} e^{\left (-i \, \arccos \left (a x\right )\right )}}{16 \, a^{5}} + \frac {\sqrt {\arccos \left (a x\right )} e^{\left (-3 i \, \arccos \left (a x\right )\right )}}{32 \, a^{5}} + \frac {\sqrt {\arccos \left (a x\right )} e^{\left (-5 i \, \arccos \left (a x\right )\right )}}{160 \, a^{5}} \]
(1/3200*I + 1/3200)*sqrt(10)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt(10)*sqrt(arcc os(a*x)))/a^5 - (1/3200*I - 1/3200)*sqrt(10)*sqrt(pi)*erf(-(1/2*I + 1/2)*s qrt(10)*sqrt(arccos(a*x)))/a^5 + (1/384*I + 1/384)*sqrt(6)*sqrt(pi)*erf((1 /2*I - 1/2)*sqrt(6)*sqrt(arccos(a*x)))/a^5 - (1/384*I - 1/384)*sqrt(6)*sqr t(pi)*erf(-(1/2*I + 1/2)*sqrt(6)*sqrt(arccos(a*x)))/a^5 + (1/64*I + 1/64)* sqrt(2)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt(2)*sqrt(arccos(a*x)))/a^5 - (1/64* I - 1/64)*sqrt(2)*sqrt(pi)*erf(-(1/2*I + 1/2)*sqrt(2)*sqrt(arccos(a*x)))/a ^5 + 1/160*sqrt(arccos(a*x))*e^(5*I*arccos(a*x))/a^5 + 1/32*sqrt(arccos(a* x))*e^(3*I*arccos(a*x))/a^5 + 1/16*sqrt(arccos(a*x))*e^(I*arccos(a*x))/a^5 + 1/16*sqrt(arccos(a*x))*e^(-I*arccos(a*x))/a^5 + 1/32*sqrt(arccos(a*x))* e^(-3*I*arccos(a*x))/a^5 + 1/160*sqrt(arccos(a*x))*e^(-5*I*arccos(a*x))/a^ 5
Timed out. \[ \int x^4 \sqrt {\arccos (a x)} \, dx=\int x^4\,\sqrt {\mathrm {acos}\left (a\,x\right )} \,d x \]