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

Optimal. Leaf size=298 \[ -\frac {2 x \sqrt {\text {ArcCos}(a x)}}{5 a^4}-\frac {x^3 \sqrt {\text {ArcCos}(a x)}}{15 a^2}-\frac {3}{100} x^5 \sqrt {\text {ArcCos}(a x)}-\frac {4 \sqrt {1-a^2 x^2} \text {ArcCos}(a x)^{3/2}}{15 a^5}-\frac {2 x^2 \sqrt {1-a^2 x^2} \text {ArcCos}(a x)^{3/2}}{15 a^3}-\frac {x^4 \sqrt {1-a^2 x^2} \text {ArcCos}(a x)^{3/2}}{10 a}+\frac {1}{5} x^5 \text {ArcCos}(a x)^{5/2}+\frac {15 \sqrt {\frac {\pi }{2}} \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{32 a^5}+\frac {\sqrt {\frac {\pi }{6}} \text {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{60 a^5}+\frac {\sqrt {\frac {3 \pi }{2}} \text {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{320 a^5}+\frac {3 \sqrt {\frac {\pi }{10}} \text {FresnelC}\left (\sqrt {\frac {10}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{1600 a^5} \]

[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)

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Rubi [A]
time = 0.53, antiderivative size = 298, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 8, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {4726, 4796, 4768, 4716, 4810, 3385, 3433, 3393} \begin {gather*} \frac {15 \sqrt {\frac {\pi }{2}} \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{32 a^5}+\frac {\sqrt {\frac {3 \pi }{2}} \text {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{320 a^5}+\frac {\sqrt {\frac {\pi }{6}} \text {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{60 a^5}+\frac {3 \sqrt {\frac {\pi }{10}} \text {FresnelC}\left (\sqrt {\frac {10}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{1600 a^5}-\frac {2 x \sqrt {\text {ArcCos}(a x)}}{5 a^4}-\frac {x^3 \sqrt {\text {ArcCos}(a x)}}{15 a^2}-\frac {x^4 \sqrt {1-a^2 x^2} \text {ArcCos}(a x)^{3/2}}{10 a}-\frac {4 \sqrt {1-a^2 x^2} \text {ArcCos}(a x)^{3/2}}{15 a^5}-\frac {2 x^2 \sqrt {1-a^2 x^2} \text {ArcCos}(a x)^{3/2}}{15 a^3}+\frac {1}{5} x^5 \text {ArcCos}(a x)^{5/2}-\frac {3}{100} x^5 \sqrt {\text {ArcCos}(a x)} \end {gather*}

Antiderivative was successfully verified.

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

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Mathematica [C] Result contains complex when optimal does not.
time = 0.14, size = 212, normalized size = 0.71 \begin {gather*} -\frac {33750 \sqrt {i \text {ArcCos}(a x)} \sqrt {\text {ArcCos}(a x)^2} \text {Gamma}\left (\frac {7}{2},-i \text {ArcCos}(a x)\right )+33750 \sqrt {-i \text {ArcCos}(a x)} \sqrt {\text {ArcCos}(a x)^2} \text {Gamma}\left (\frac {7}{2},i \text {ArcCos}(a x)\right )-625 \sqrt {3} (-i \text {ArcCos}(a x))^{3/2} \text {Gamma}\left (\frac {7}{2},-3 i \text {ArcCos}(a x)\right )-625 \sqrt {3} (i \text {ArcCos}(a x))^{3/2} \text {Gamma}\left (\frac {7}{2},3 i \text {ArcCos}(a x)\right )-27 \sqrt {5} (-i \text {ArcCos}(a x))^{3/2} \text {Gamma}\left (\frac {7}{2},-5 i \text {ArcCos}(a x)\right )-27 \sqrt {5} (i \text {ArcCos}(a x))^{3/2} \text {Gamma}\left (\frac {7}{2},5 i \text {ArcCos}(a x)\right )}{540000 a^5 \text {ArcCos}(a x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.31, size = 233, normalized size = 0.78

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/144000/a^5*(18000*a*x*arccos(a*x)^3+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*Fresnel
C(2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*2^(1/2)*arccos(a*x)^(1/2)*Pi^(1/2)-67500*a*x*arccos(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)

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3060 deep

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Giac [C] Result contains complex when optimal does not.
time = 0.50, size = 463, normalized size = 1.55 \begin {gather*} \frac {\arccos \left (a x\right )^{\frac {5}{2}} e^{\left (5 i \, \arccos \left (a x\right )\right )}}{160 \, a^{5}} + \frac {\arccos \left (a x\right )^{\frac {5}{2}} e^{\left (3 i \, \arccos \left (a x\right )\right )}}{32 \, a^{5}} + \frac {\arccos \left (a x\right )^{\frac {5}{2}} e^{\left (i \, \arccos \left (a x\right )\right )}}{16 \, a^{5}} + \frac {\arccos \left (a x\right )^{\frac {5}{2}} e^{\left (-i \, \arccos \left (a x\right )\right )}}{16 \, a^{5}} + \frac {\arccos \left (a x\right )^{\frac {5}{2}} e^{\left (-3 i \, \arccos \left (a x\right )\right )}}{32 \, a^{5}} + \frac {\arccos \left (a x\right )^{\frac {5}{2}} e^{\left (-5 i \, \arccos \left (a x\right )\right )}}{160 \, a^{5}} + \frac {i \, \arccos \left (a x\right )^{\frac {3}{2}} e^{\left (5 i \, \arccos \left (a x\right )\right )}}{320 \, a^{5}} + \frac {5 i \, \arccos \left (a x\right )^{\frac {3}{2}} e^{\left (3 i \, \arccos \left (a x\right )\right )}}{192 \, a^{5}} + \frac {5 i \, \arccos \left (a x\right )^{\frac {3}{2}} e^{\left (i \, \arccos \left (a x\right )\right )}}{32 \, a^{5}} - \frac {5 i \, \arccos \left (a x\right )^{\frac {3}{2}} e^{\left (-i \, \arccos \left (a x\right )\right )}}{32 \, a^{5}} - \frac {5 i \, \arccos \left (a x\right )^{\frac {3}{2}} e^{\left (-3 i \, \arccos \left (a x\right )\right )}}{192 \, a^{5}} - \frac {i \, \arccos \left (a x\right )^{\frac {3}{2}} e^{\left (-5 i \, \arccos \left (a x\right )\right )}}{320 \, 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 )}{64000 \, 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 )}{64000 \, a^{5}} - \frac {\left (5 i + 5\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 )}{4608 \, a^{5}} + \frac {\left (5 i - 5\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 )}{4608 \, a^{5}} - \frac {\left (15 i + 15\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 )}{256 \, a^{5}} + \frac {\left (15 i - 15\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 )}{256 \, a^{5}} - \frac {3 \, \sqrt {\arccos \left (a x\right )} e^{\left (5 i \, \arccos \left (a x\right )\right )}}{3200 \, a^{5}} - \frac {5 \, \sqrt {\arccos \left (a x\right )} e^{\left (3 i \, \arccos \left (a x\right )\right )}}{384 \, a^{5}} - \frac {15 \, \sqrt {\arccos \left (a x\right )} e^{\left (i \, \arccos \left (a x\right )\right )}}{64 \, a^{5}} - \frac {15 \, \sqrt {\arccos \left (a x\right )} e^{\left (-i \, \arccos \left (a x\right )\right )}}{64 \, a^{5}} - \frac {5 \, \sqrt {\arccos \left (a x\right )} e^{\left (-3 i \, \arccos \left (a x\right )\right )}}{384 \, a^{5}} - \frac {3 \, \sqrt {\arccos \left (a x\right )} e^{\left (-5 i \, \arccos \left (a x\right )\right )}}{3200 \, a^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^4\,{\mathrm {acos}\left (a\,x\right )}^{5/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^4*acos(a*x)^(5/2), x)

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