\(\int x^4 \arccos (a x)^{5/2} \, dx\) [86]

Optimal result

Integrand size = 12, antiderivative size = 298 \[ \int x^4 \arccos (a x)^{5/2} \, dx=-\frac {2 x \sqrt {\arccos (a x)}}{5 a^4}-\frac {x^3 \sqrt {\arccos (a x)}}{15 a^2}-\frac {3}{100} x^5 \sqrt {\arccos (a x)}-\frac {4 \sqrt {1-a^2 x^2} \arccos (a x)^{3/2}}{15 a^5}-\frac {2 x^2 \sqrt {1-a^2 x^2} \arccos (a x)^{3/2}}{15 a^3}-\frac {x^4 \sqrt {1-a^2 x^2} \arccos (a x)^{3/2}}{10 a}+\frac {1}{5} x^5 \arccos (a x)^{5/2}+\frac {15 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{32 a^5}+\frac {\sqrt {\frac {\pi }{6}} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{60 a^5}+\frac {\sqrt {\frac {3 \pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{320 a^5}+\frac {3 \sqrt {\frac {\pi }{10}} \operatorname {FresnelC}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arccos (a x)}\right )}{1600 a^5} \]

[Out]

1/5*x^5*arccos(a*x)^(5/2)+3/16000*FresnelC(10^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*10^(1/2)*Pi^(1/2)/a^5+5/1152*F
resnelC(6^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*6^(1/2)*Pi^(1/2)/a^5+15/64*FresnelC(2^(1/2)/Pi^(1/2)*arccos(a*x)^(
1/2))*2^(1/2)*Pi^(1/2)/a^5-4/15*arccos(a*x)^(3/2)*(-a^2*x^2+1)^(1/2)/a^5-2/15*x^2*arccos(a*x)^(3/2)*(-a^2*x^2+
1)^(1/2)/a^3-1/10*x^4*arccos(a*x)^(3/2)*(-a^2*x^2+1)^(1/2)/a-2/5*x*arccos(a*x)^(1/2)/a^4-1/15*x^3*arccos(a*x)^
(1/2)/a^2-3/100*x^5*arccos(a*x)^(1/2)

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {4726, 4796, 4768, 4716, 4810, 3385, 3433, 3393} \[ \int x^4 \arccos (a x)^{5/2} \, dx=\frac {15 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{32 a^5}+\frac {\sqrt {\frac {3 \pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{320 a^5}+\frac {\sqrt {\frac {\pi }{6}} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{60 a^5}+\frac {3 \sqrt {\frac {\pi }{10}} \operatorname {FresnelC}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arccos (a x)}\right )}{1600 a^5}-\frac {2 x \sqrt {\arccos (a x)}}{5 a^4}-\frac {x^3 \sqrt {\arccos (a x)}}{15 a^2}-\frac {x^4 \sqrt {1-a^2 x^2} \arccos (a x)^{3/2}}{10 a}-\frac {4 \sqrt {1-a^2 x^2} \arccos (a x)^{3/2}}{15 a^5}-\frac {2 x^2 \sqrt {1-a^2 x^2} \arccos (a x)^{3/2}}{15 a^3}+\frac {1}{5} x^5 \arccos (a x)^{5/2}-\frac {3}{100} x^5 \sqrt {\arccos (a x)} \]

[In]

Int[x^4*ArcCos[a*x]^(5/2),x]

[Out]

(-2*x*Sqrt[ArcCos[a*x]])/(5*a^4) - (x^3*Sqrt[ArcCos[a*x]])/(15*a^2) - (3*x^5*Sqrt[ArcCos[a*x]])/100 - (4*Sqrt[
1 - a^2*x^2]*ArcCos[a*x]^(3/2))/(15*a^5) - (2*x^2*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^(3/2))/(15*a^3) - (x^4*Sqrt[1
- a^2*x^2]*ArcCos[a*x]^(3/2))/(10*a) + (x^5*ArcCos[a*x]^(5/2))/5 + (15*Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Sqrt[Arc
Cos[a*x]]])/(32*a^5) + (Sqrt[Pi/6]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcCos[a*x]]])/(60*a^5) + (Sqrt[(3*Pi)/2]*FresnelC
[Sqrt[6/Pi]*Sqrt[ArcCos[a*x]]])/(320*a^5) + (3*Sqrt[Pi/10]*FresnelC[Sqrt[10/Pi]*Sqrt[ArcCos[a*x]]])/(1600*a^5)

Rule 3385

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[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]

Rule 3393

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[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])
)

Rule 3433

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]

Rule 4716

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcCos[c*x])^n, x] + Dist[b*c*n, Int[
x*((a + b*ArcCos[c*x])^(n - 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 4726

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcCos[c*x])^n/(m
+ 1)), x] + Dist[b*c*(n/(m + 1)), Int[x^(m + 1)*((a + b*ArcCos[c*x])^(n - 1)/Sqrt[1 - c^2*x^2]), x], x] /; Fre
eQ[{a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]

Rule 4768

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] - Dist[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p
], Int[(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]

Rule 4796

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] + (Dist[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] - Dist[b*f*(n/(c*(m + 2*p + 1)))*S
imp[(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]

Rule 4810

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[(-(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] && IGt
Q[m, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} x^5 \arccos (a x)^{5/2}+\frac {1}{2} a \int \frac {x^5 \arccos (a x)^{3/2}}{\sqrt {1-a^2 x^2}} \, dx \\ & = -\frac {x^4 \sqrt {1-a^2 x^2} \arccos (a x)^{3/2}}{10 a}+\frac {1}{5} x^5 \arccos (a x)^{5/2}-\frac {3}{20} \int x^4 \sqrt {\arccos (a x)} \, dx+\frac {2 \int \frac {x^3 \arccos (a x)^{3/2}}{\sqrt {1-a^2 x^2}} \, dx}{5 a} \\ & = -\frac {3}{100} x^5 \sqrt {\arccos (a x)}-\frac {2 x^2 \sqrt {1-a^2 x^2} \arccos (a x)^{3/2}}{15 a^3}-\frac {x^4 \sqrt {1-a^2 x^2} \arccos (a x)^{3/2}}{10 a}+\frac {1}{5} x^5 \arccos (a x)^{5/2}+\frac {4 \int \frac {x \arccos (a x)^{3/2}}{\sqrt {1-a^2 x^2}} \, dx}{15 a^3}-\frac {\int x^2 \sqrt {\arccos (a x)} \, dx}{5 a^2}-\frac {1}{200} (3 a) \int \frac {x^5}{\sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}} \, dx \\ & = -\frac {x^3 \sqrt {\arccos (a x)}}{15 a^2}-\frac {3}{100} x^5 \sqrt {\arccos (a x)}-\frac {4 \sqrt {1-a^2 x^2} \arccos (a x)^{3/2}}{15 a^5}-\frac {2 x^2 \sqrt {1-a^2 x^2} \arccos (a x)^{3/2}}{15 a^3}-\frac {x^4 \sqrt {1-a^2 x^2} \arccos (a x)^{3/2}}{10 a}+\frac {1}{5} x^5 \arccos (a x)^{5/2}+\frac {3 \text {Subst}\left (\int \frac {\cos ^5(x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{200 a^5}-\frac {2 \int \sqrt {\arccos (a x)} \, dx}{5 a^4}-\frac {\int \frac {x^3}{\sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}} \, dx}{30 a} \\ & = -\frac {2 x \sqrt {\arccos (a x)}}{5 a^4}-\frac {x^3 \sqrt {\arccos (a x)}}{15 a^2}-\frac {3}{100} x^5 \sqrt {\arccos (a x)}-\frac {4 \sqrt {1-a^2 x^2} \arccos (a x)^{3/2}}{15 a^5}-\frac {2 x^2 \sqrt {1-a^2 x^2} \arccos (a x)^{3/2}}{15 a^3}-\frac {x^4 \sqrt {1-a^2 x^2} \arccos (a x)^{3/2}}{10 a}+\frac {1}{5} x^5 \arccos (a x)^{5/2}+\frac {3 \text {Subst}\left (\int \left (\frac {5 \cos (x)}{8 \sqrt {x}}+\frac {5 \cos (3 x)}{16 \sqrt {x}}+\frac {\cos (5 x)}{16 \sqrt {x}}\right ) \, dx,x,\arccos (a x)\right )}{200 a^5}+\frac {\text {Subst}\left (\int \frac {\cos ^3(x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{30 a^5}-\frac {\int \frac {x}{\sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}} \, dx}{5 a^3} \\ & = -\frac {2 x \sqrt {\arccos (a x)}}{5 a^4}-\frac {x^3 \sqrt {\arccos (a x)}}{15 a^2}-\frac {3}{100} x^5 \sqrt {\arccos (a x)}-\frac {4 \sqrt {1-a^2 x^2} \arccos (a x)^{3/2}}{15 a^5}-\frac {2 x^2 \sqrt {1-a^2 x^2} \arccos (a x)^{3/2}}{15 a^3}-\frac {x^4 \sqrt {1-a^2 x^2} \arccos (a x)^{3/2}}{10 a}+\frac {1}{5} x^5 \arccos (a x)^{5/2}+\frac {3 \text {Subst}\left (\int \frac {\cos (5 x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{3200 a^5}+\frac {3 \text {Subst}\left (\int \frac {\cos (3 x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{640 a^5}+\frac {3 \text {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{320 a^5}+\frac {\text {Subst}\left (\int \left (\frac {3 \cos (x)}{4 \sqrt {x}}+\frac {\cos (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\arccos (a x)\right )}{30 a^5}+\frac {\text {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{5 a^5} \\ & = -\frac {2 x \sqrt {\arccos (a x)}}{5 a^4}-\frac {x^3 \sqrt {\arccos (a x)}}{15 a^2}-\frac {3}{100} x^5 \sqrt {\arccos (a x)}-\frac {4 \sqrt {1-a^2 x^2} \arccos (a x)^{3/2}}{15 a^5}-\frac {2 x^2 \sqrt {1-a^2 x^2} \arccos (a x)^{3/2}}{15 a^3}-\frac {x^4 \sqrt {1-a^2 x^2} \arccos (a x)^{3/2}}{10 a}+\frac {1}{5} x^5 \arccos (a x)^{5/2}+\frac {3 \text {Subst}\left (\int \cos \left (5 x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{1600 a^5}+\frac {\text {Subst}\left (\int \frac {\cos (3 x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{120 a^5}+\frac {3 \text {Subst}\left (\int \cos \left (3 x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{320 a^5}+\frac {3 \text {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{160 a^5}+\frac {\text {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{40 a^5}+\frac {2 \text {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{5 a^5} \\ & = -\frac {2 x \sqrt {\arccos (a x)}}{5 a^4}-\frac {x^3 \sqrt {\arccos (a x)}}{15 a^2}-\frac {3}{100} x^5 \sqrt {\arccos (a x)}-\frac {4 \sqrt {1-a^2 x^2} \arccos (a x)^{3/2}}{15 a^5}-\frac {2 x^2 \sqrt {1-a^2 x^2} \arccos (a x)^{3/2}}{15 a^3}-\frac {x^4 \sqrt {1-a^2 x^2} \arccos (a x)^{3/2}}{10 a}+\frac {1}{5} x^5 \arccos (a x)^{5/2}+\frac {3 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{160 a^5}+\frac {\sqrt {2 \pi } \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{5 a^5}+\frac {\sqrt {\frac {3 \pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{320 a^5}+\frac {3 \sqrt {\frac {\pi }{10}} \operatorname {FresnelC}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arccos (a x)}\right )}{1600 a^5}+\frac {\text {Subst}\left (\int \cos \left (3 x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{60 a^5}+\frac {\text {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{20 a^5} \\ & = -\frac {2 x \sqrt {\arccos (a x)}}{5 a^4}-\frac {x^3 \sqrt {\arccos (a x)}}{15 a^2}-\frac {3}{100} x^5 \sqrt {\arccos (a x)}-\frac {4 \sqrt {1-a^2 x^2} \arccos (a x)^{3/2}}{15 a^5}-\frac {2 x^2 \sqrt {1-a^2 x^2} \arccos (a x)^{3/2}}{15 a^3}-\frac {x^4 \sqrt {1-a^2 x^2} \arccos (a x)^{3/2}}{10 a}+\frac {1}{5} x^5 \arccos (a x)^{5/2}+\frac {11 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{160 a^5}+\frac {\sqrt {2 \pi } \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{5 a^5}+\frac {\sqrt {\frac {\pi }{6}} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{60 a^5}+\frac {\sqrt {\frac {3 \pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{320 a^5}+\frac {3 \sqrt {\frac {\pi }{10}} \operatorname {FresnelC}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arccos (a x)}\right )}{1600 a^5} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.08 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.65 \[ \int x^4 \arccos (a x)^{5/2} \, dx=-\frac {i \left (33750 \sqrt {-i \arccos (a x)} \Gamma \left (\frac {7}{2},-i \arccos (a x)\right )-33750 \sqrt {i \arccos (a x)} \Gamma \left (\frac {7}{2},i \arccos (a x)\right )+625 \sqrt {3} \sqrt {-i \arccos (a x)} \Gamma \left (\frac {7}{2},-3 i \arccos (a x)\right )-625 \sqrt {3} \sqrt {i \arccos (a x)} \Gamma \left (\frac {7}{2},3 i \arccos (a x)\right )+27 \sqrt {5} \sqrt {-i \arccos (a x)} \Gamma \left (\frac {7}{2},-5 i \arccos (a x)\right )-27 \sqrt {5} \sqrt {i \arccos (a x)} \Gamma \left (\frac {7}{2},5 i \arccos (a x)\right )\right )}{540000 a^5 \sqrt {\arccos (a x)}} \]

[In]

Integrate[x^4*ArcCos[a*x]^(5/2),x]

[Out]

((-1/540000*I)*(33750*Sqrt[(-I)*ArcCos[a*x]]*Gamma[7/2, (-I)*ArcCos[a*x]] - 33750*Sqrt[I*ArcCos[a*x]]*Gamma[7/
2, I*ArcCos[a*x]] + 625*Sqrt[3]*Sqrt[(-I)*ArcCos[a*x]]*Gamma[7/2, (-3*I)*ArcCos[a*x]] - 625*Sqrt[3]*Sqrt[I*Arc
Cos[a*x]]*Gamma[7/2, (3*I)*ArcCos[a*x]] + 27*Sqrt[5]*Sqrt[(-I)*ArcCos[a*x]]*Gamma[7/2, (-5*I)*ArcCos[a*x]] - 2
7*Sqrt[5]*Sqrt[I*ArcCos[a*x]]*Gamma[7/2, (5*I)*ArcCos[a*x]]))/(a^5*Sqrt[ArcCos[a*x]])

Maple [A] (verified)

Time = 1.05 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.78

method	result	size
default	\(\frac {18000 \arccos \left (a x \right )^{3} a x +9000 \arccos \left (a x \right )^{3} \cos \left (3 \arccos \left (a x \right )\right )+1800 \arccos \left (a x \right )^{3} \cos \left (5 \arccos \left (a x \right )\right )+27 \sqrt {5}\, \sqrt {2}\, \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {5}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )+625 \sqrt {3}\, \sqrt {2}\, \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )-45000 \arccos \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}-7500 \arccos \left (a x \right )^{2} \sin \left (3 \arccos \left (a x \right )\right )-900 \arccos \left (a x \right )^{2} \sin \left (5 \arccos \left (a x \right )\right )+33750 \sqrt {2}\, \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )-67500 \arccos \left (a x \right ) a x -3750 \arccos \left (a x \right ) \cos \left (3 \arccos \left (a x \right )\right )-270 \arccos \left (a x \right ) \cos \left (5 \arccos \left (a x \right )\right )}{144000 a^{5} \sqrt {\arccos \left (a x \right )}}\)	\(233\)

[In]

int(x^4*arccos(a*x)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/144000/a^5*(18000*arccos(a*x)^3*a*x+9000*arccos(a*x)^3*cos(3*arccos(a*x))+1800*arccos(a*x)^3*cos(5*arccos(a*
x))+27*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))+625*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))-45000*arccos(a*x)
^2*(-a^2*x^2+1)^(1/2)-7500*arccos(a*x)^2*sin(3*arccos(a*x))-900*arccos(a*x)^2*sin(5*arccos(a*x))+33750*2^(1/2)
*arccos(a*x)^(1/2)*Pi^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))-67500*arccos(a*x)*a*x-3750*arccos(a*x
)*cos(3*arccos(a*x))-270*arccos(a*x)*cos(5*arccos(a*x)))/arccos(a*x)^(1/2)

Fricas [F(-2)]

Exception generated. \[ \int x^4 \arccos (a x)^{5/2} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x^4*arccos(a*x)^(5/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [F(-1)]

Timed out. \[ \int x^4 \arccos (a x)^{5/2} \, dx=\text {Timed out} \]

[In]

integrate(x**4*acos(a*x)**(5/2),x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int x^4 \arccos (a x)^{5/2} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(x^4*arccos(a*x)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.37 (sec) , antiderivative size = 463, normalized size of antiderivative = 1.55 \[ \int x^4 \arccos (a x)^{5/2} \, dx=\text {Too large to display} \]

[In]

integrate(x^4*arccos(a*x)^(5/2),x, algorithm="giac")

[Out]

1/160*arccos(a*x)^(5/2)*e^(5*I*arccos(a*x))/a^5 + 1/32*arccos(a*x)^(5/2)*e^(3*I*arccos(a*x))/a^5 + 1/16*arccos
(a*x)^(5/2)*e^(I*arccos(a*x))/a^5 + 1/16*arccos(a*x)^(5/2)*e^(-I*arccos(a*x))/a^5 + 1/32*arccos(a*x)^(5/2)*e^(
-3*I*arccos(a*x))/a^5 + 1/160*arccos(a*x)^(5/2)*e^(-5*I*arccos(a*x))/a^5 + 1/320*I*arccos(a*x)^(3/2)*e^(5*I*ar
ccos(a*x))/a^5 + 5/192*I*arccos(a*x)^(3/2)*e^(3*I*arccos(a*x))/a^5 + 5/32*I*arccos(a*x)^(3/2)*e^(I*arccos(a*x)
)/a^5 - 5/32*I*arccos(a*x)^(3/2)*e^(-I*arccos(a*x))/a^5 - 5/192*I*arccos(a*x)^(3/2)*e^(-3*I*arccos(a*x))/a^5 -
1/320*I*arccos(a*x)^(3/2)*e^(-5*I*arccos(a*x))/a^5 - (3/64000*I + 3/64000)*sqrt(10)*sqrt(pi)*erf((1/2*I - 1/2
)*sqrt(10)*sqrt(arccos(a*x)))/a^5 + (3/64000*I - 3/64000)*sqrt(10)*sqrt(pi)*erf(-(1/2*I + 1/2)*sqrt(10)*sqrt(a
rccos(a*x)))/a^5 - (5/4608*I + 5/4608)*sqrt(6)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt(6)*sqrt(arccos(a*x)))/a^5 + (5/
4608*I - 5/4608)*sqrt(6)*sqrt(pi)*erf(-(1/2*I + 1/2)*sqrt(6)*sqrt(arccos(a*x)))/a^5 - (15/256*I + 15/256)*sqrt
(2)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt(2)*sqrt(arccos(a*x)))/a^5 + (15/256*I - 15/256)*sqrt(2)*sqrt(pi)*erf(-(1/2
*I + 1/2)*sqrt(2)*sqrt(arccos(a*x)))/a^5 - 3/3200*sqrt(arccos(a*x))*e^(5*I*arccos(a*x))/a^5 - 5/384*sqrt(arcco
s(a*x))*e^(3*I*arccos(a*x))/a^5 - 15/64*sqrt(arccos(a*x))*e^(I*arccos(a*x))/a^5 - 15/64*sqrt(arccos(a*x))*e^(-
I*arccos(a*x))/a^5 - 5/384*sqrt(arccos(a*x))*e^(-3*I*arccos(a*x))/a^5 - 3/3200*sqrt(arccos(a*x))*e^(-5*I*arcco
s(a*x))/a^5

Mupad [F(-1)]

Timed out. \[ \int x^4 \arccos (a x)^{5/2} \, dx=\int x^4\,{\mathrm {acos}\left (a\,x\right )}^{5/2} \,d x \]

[In]

int(x^4*acos(a*x)^(5/2),x)

[Out]

int(x^4*acos(a*x)^(5/2), x)