3.1.88 \(\int x^2 \text {ArcCos}(a x)^{5/2} \, dx\) [88]

Optimal. Leaf size=178 \[ -\frac {5 x \sqrt {\text {ArcCos}(a x)}}{6 a^2}-\frac {5}{36} x^3 \sqrt {\text {ArcCos}(a x)}-\frac {5 \sqrt {1-a^2 x^2} \text {ArcCos}(a x)^{3/2}}{9 a^3}-\frac {5 x^2 \sqrt {1-a^2 x^2} \text {ArcCos}(a x)^{3/2}}{18 a}+\frac {1}{3} x^3 \text {ArcCos}(a x)^{5/2}+\frac {15 \sqrt {\frac {\pi }{2}} \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{16 a^3}+\frac {5 \sqrt {\frac {\pi }{6}} \text {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{144 a^3} \]

[Out]

1/3*x^3*arccos(a*x)^(5/2)+5/864*FresnelC(6^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*6^(1/2)*Pi^(1/2)/a^3+15/32*Fresne
lC(2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a^3-5/9*arccos(a*x)^(3/2)*(-a^2*x^2+1)^(1/2)/a^3-5/18*
x^2*arccos(a*x)^(3/2)*(-a^2*x^2+1)^(1/2)/a-5/6*x*arccos(a*x)^(1/2)/a^2-5/36*x^3*arccos(a*x)^(1/2)

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Rubi [A]
time = 0.31, antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 15, 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 )}{16 a^3}+\frac {5 \sqrt {\frac {\pi }{6}} \text {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{144 a^3}-\frac {5 x^2 \sqrt {1-a^2 x^2} \text {ArcCos}(a x)^{3/2}}{18 a}-\frac {5 x \sqrt {\text {ArcCos}(a x)}}{6 a^2}-\frac {5 \sqrt {1-a^2 x^2} \text {ArcCos}(a x)^{3/2}}{9 a^3}+\frac {1}{3} x^3 \text {ArcCos}(a x)^{5/2}-\frac {5}{36} x^3 \sqrt {\text {ArcCos}(a x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*x*Sqrt[ArcCos[a*x]])/(6*a^2) - (5*x^3*Sqrt[ArcCos[a*x]])/36 - (5*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^(3/2))/(9*a
^3) - (5*x^2*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^(3/2))/(18*a) + (x^3*ArcCos[a*x]^(5/2))/3 + (15*Sqrt[Pi/2]*FresnelC
[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(16*a^3) + (5*Sqrt[Pi/6]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcCos[a*x]]])/(144*a^3)

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^2 \cos ^{-1}(a x)^{5/2} \, dx &=\frac {1}{3} x^3 \cos ^{-1}(a x)^{5/2}+\frac {1}{6} (5 a) \int \frac {x^3 \cos ^{-1}(a x)^{3/2}}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {5 x^2 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{18 a}+\frac {1}{3} x^3 \cos ^{-1}(a x)^{5/2}-\frac {5}{12} \int x^2 \sqrt {\cos ^{-1}(a x)} \, dx+\frac {5 \int \frac {x \cos ^{-1}(a x)^{3/2}}{\sqrt {1-a^2 x^2}} \, dx}{9 a}\\ &=-\frac {5}{36} x^3 \sqrt {\cos ^{-1}(a x)}-\frac {5 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{9 a^3}-\frac {5 x^2 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{18 a}+\frac {1}{3} x^3 \cos ^{-1}(a x)^{5/2}-\frac {5 \int \sqrt {\cos ^{-1}(a x)} \, dx}{6 a^2}-\frac {1}{72} (5 a) \int \frac {x^3}{\sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}} \, dx\\ &=-\frac {5 x \sqrt {\cos ^{-1}(a x)}}{6 a^2}-\frac {5}{36} x^3 \sqrt {\cos ^{-1}(a x)}-\frac {5 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{9 a^3}-\frac {5 x^2 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{18 a}+\frac {1}{3} x^3 \cos ^{-1}(a x)^{5/2}+\frac {5 \text {Subst}\left (\int \frac {\cos ^3(x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{72 a^3}-\frac {5 \int \frac {x}{\sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}} \, dx}{12 a}\\ &=-\frac {5 x \sqrt {\cos ^{-1}(a x)}}{6 a^2}-\frac {5}{36} x^3 \sqrt {\cos ^{-1}(a x)}-\frac {5 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{9 a^3}-\frac {5 x^2 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{18 a}+\frac {1}{3} x^3 \cos ^{-1}(a x)^{5/2}+\frac {5 \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 )}{72 a^3}+\frac {5 \text {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{12 a^3}\\ &=-\frac {5 x \sqrt {\cos ^{-1}(a x)}}{6 a^2}-\frac {5}{36} x^3 \sqrt {\cos ^{-1}(a x)}-\frac {5 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{9 a^3}-\frac {5 x^2 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{18 a}+\frac {1}{3} x^3 \cos ^{-1}(a x)^{5/2}+\frac {5 \text {Subst}\left (\int \frac {\cos (3 x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{288 a^3}+\frac {5 \text {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{96 a^3}+\frac {5 \text {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{6 a^3}\\ &=-\frac {5 x \sqrt {\cos ^{-1}(a x)}}{6 a^2}-\frac {5}{36} x^3 \sqrt {\cos ^{-1}(a x)}-\frac {5 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{9 a^3}-\frac {5 x^2 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{18 a}+\frac {1}{3} x^3 \cos ^{-1}(a x)^{5/2}+\frac {5 \sqrt {\frac {\pi }{2}} C\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{6 a^3}+\frac {5 \text {Subst}\left (\int \cos \left (3 x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{144 a^3}+\frac {5 \text {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{48 a^3}\\ &=-\frac {5 x \sqrt {\cos ^{-1}(a x)}}{6 a^2}-\frac {5}{36} x^3 \sqrt {\cos ^{-1}(a x)}-\frac {5 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{9 a^3}-\frac {5 x^2 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{18 a}+\frac {1}{3} x^3 \cos ^{-1}(a x)^{5/2}+\frac {15 \sqrt {\frac {\pi }{2}} C\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{16 a^3}+\frac {5 \sqrt {\frac {\pi }{6}} C\left (\sqrt {\frac {6}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{144 a^3}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.09, size = 122, normalized size = 0.69 \begin {gather*} -\frac {81 i \sqrt {\text {ArcCos}(a x)^2} \text {Gamma}\left (\frac {7}{2},-i \text {ArcCos}(a x)\right )+81 \text {ArcCos}(a x) \text {Gamma}\left (\frac {7}{2},i \text {ArcCos}(a x)\right )+\sqrt {3} \left (i \sqrt {\text {ArcCos}(a x)^2} \text {Gamma}\left (\frac {7}{2},-3 i \text {ArcCos}(a x)\right )+\text {ArcCos}(a x) \text {Gamma}\left (\frac {7}{2},3 i \text {ArcCos}(a x)\right )\right )}{648 a^3 \sqrt {i \text {ArcCos}(a x)} \sqrt {\text {ArcCos}(a x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/648*((81*I)*Sqrt[ArcCos[a*x]^2]*Gamma[7/2, (-I)*ArcCos[a*x]] + 81*ArcCos[a*x]*Gamma[7/2, I*ArcCos[a*x]] + S
qrt[3]*(I*Sqrt[ArcCos[a*x]^2]*Gamma[7/2, (-3*I)*ArcCos[a*x]] + ArcCos[a*x]*Gamma[7/2, (3*I)*ArcCos[a*x]]))/(a^
3*Sqrt[I*ArcCos[a*x]]*Sqrt[ArcCos[a*x]])

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Maple [A]
time = 0.23, size = 156, normalized size = 0.88

method	result	size
default	\(\frac {216 a x \arccos \left (a x \right )^{3}+72 \arccos \left (a x \right )^{3} \cos \left (3 \arccos \left (a x \right )\right )+5 \sqrt {3}\, \sqrt {2}\, \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }\, \FresnelC \left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )-540 \arccos \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}-60 \arccos \left (a x \right )^{2} \sin \left (3 \arccos \left (a x \right )\right )+405 \FresnelC \left (\frac {\sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {2}\, \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }-810 a x \arccos \left (a x \right )-30 \arccos \left (a x \right ) \cos \left (3 \arccos \left (a x \right )\right )}{864 a^{3} \sqrt {\arccos \left (a x \right )}}\)	\(156\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/864/a^3*(216*a*x*arccos(a*x)^3+72*arccos(a*x)^3*cos(3*arccos(a*x))+5*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))-540*arccos(a*x)^2*(-a^2*x^2+1)^(1/2)-60*arccos(a*x)^2
*sin(3*arccos(a*x))+405*FresnelC(2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*2^(1/2)*arccos(a*x)^(1/2)*Pi^(1/2)-810*a*
x*arccos(a*x)-30*arccos(a*x)*cos(3*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^2*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^2*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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \operatorname {acos}^{\frac {5}{2}}{\left (a x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2*acos(a*x)**(5/2), x)

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Giac [C] Result contains complex when optimal does not.
time = 0.48, size = 309, normalized size = 1.74 \begin {gather*} \frac {\arccos \left (a x\right )^{\frac {5}{2}} e^{\left (3 i \, \arccos \left (a x\right )\right )}}{24 \, a^{3}} + \frac {\arccos \left (a x\right )^{\frac {5}{2}} e^{\left (i \, \arccos \left (a x\right )\right )}}{8 \, a^{3}} + \frac {\arccos \left (a x\right )^{\frac {5}{2}} e^{\left (-i \, \arccos \left (a x\right )\right )}}{8 \, a^{3}} + \frac {\arccos \left (a x\right )^{\frac {5}{2}} e^{\left (-3 i \, \arccos \left (a x\right )\right )}}{24 \, a^{3}} + \frac {5 i \, \arccos \left (a x\right )^{\frac {3}{2}} e^{\left (3 i \, \arccos \left (a x\right )\right )}}{144 \, a^{3}} + \frac {5 i \, \arccos \left (a x\right )^{\frac {3}{2}} e^{\left (i \, \arccos \left (a x\right )\right )}}{16 \, a^{3}} - \frac {5 i \, \arccos \left (a x\right )^{\frac {3}{2}} e^{\left (-i \, \arccos \left (a x\right )\right )}}{16 \, a^{3}} - \frac {5 i \, \arccos \left (a x\right )^{\frac {3}{2}} e^{\left (-3 i \, \arccos \left (a x\right )\right )}}{144 \, a^{3}} - \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 )}{3456 \, a^{3}} + \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 )}{3456 \, a^{3}} - \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 )}{128 \, a^{3}} + \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 )}{128 \, a^{3}} - \frac {5 \, \sqrt {\arccos \left (a x\right )} e^{\left (3 i \, \arccos \left (a x\right )\right )}}{288 \, a^{3}} - \frac {15 \, \sqrt {\arccos \left (a x\right )} e^{\left (i \, \arccos \left (a x\right )\right )}}{32 \, a^{3}} - \frac {15 \, \sqrt {\arccos \left (a x\right )} e^{\left (-i \, \arccos \left (a x\right )\right )}}{32 \, a^{3}} - \frac {5 \, \sqrt {\arccos \left (a x\right )} e^{\left (-3 i \, \arccos \left (a x\right )\right )}}{288 \, a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*arccos(a*x)^(5/2)*e^(3*I*arccos(a*x))/a^3 + 1/8*arccos(a*x)^(5/2)*e^(I*arccos(a*x))/a^3 + 1/8*arccos(a*x)
^(5/2)*e^(-I*arccos(a*x))/a^3 + 1/24*arccos(a*x)^(5/2)*e^(-3*I*arccos(a*x))/a^3 + 5/144*I*arccos(a*x)^(3/2)*e^
(3*I*arccos(a*x))/a^3 + 5/16*I*arccos(a*x)^(3/2)*e^(I*arccos(a*x))/a^3 - 5/16*I*arccos(a*x)^(3/2)*e^(-I*arccos
(a*x))/a^3 - 5/144*I*arccos(a*x)^(3/2)*e^(-3*I*arccos(a*x))/a^3 - (5/3456*I + 5/3456)*sqrt(6)*sqrt(pi)*erf((1/
2*I - 1/2)*sqrt(6)*sqrt(arccos(a*x)))/a^3 + (5/3456*I - 5/3456)*sqrt(6)*sqrt(pi)*erf(-(1/2*I + 1/2)*sqrt(6)*sq
rt(arccos(a*x)))/a^3 - (15/128*I + 15/128)*sqrt(2)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt(2)*sqrt(arccos(a*x)))/a^3 +
(15/128*I - 15/128)*sqrt(2)*sqrt(pi)*erf(-(1/2*I + 1/2)*sqrt(2)*sqrt(arccos(a*x)))/a^3 - 5/288*sqrt(arccos(a*
x))*e^(3*I*arccos(a*x))/a^3 - 15/32*sqrt(arccos(a*x))*e^(I*arccos(a*x))/a^3 - 15/32*sqrt(arccos(a*x))*e^(-I*ar
ccos(a*x))/a^3 - 5/288*sqrt(arccos(a*x))*e^(-3*I*arccos(a*x))/a^3

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^2\,{\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^2*acos(a*x)^(5/2),x)

[Out]

int(x^2*acos(a*x)^(5/2), x)

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