3.1.82 \(\int x^2 \text {ArcCos}(a x)^{3/2} \, dx\) [82]

Optimal. Leaf size=147 \[ -\frac {\sqrt {1-a^2 x^2} \sqrt {\text {ArcCos}(a x)}}{3 a^3}-\frac {x^2 \sqrt {1-a^2 x^2} \sqrt {\text {ArcCos}(a x)}}{6 a}+\frac {1}{3} x^3 \text {ArcCos}(a x)^{3/2}+\frac {3 \sqrt {\frac {\pi }{2}} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{8 a^3}+\frac {\sqrt {\frac {\pi }{6}} S\left (\sqrt {\frac {6}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{24 a^3} \]

[Out]

1/3*x^3*arccos(a*x)^(3/2)+1/144*FresnelS(6^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*6^(1/2)*Pi^(1/2)/a^3+3/16*Fresnel
S(2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a^3-1/3*(-a^2*x^2+1)^(1/2)*arccos(a*x)^(1/2)/a^3-1/6*x^
2*(-a^2*x^2+1)^(1/2)*arccos(a*x)^(1/2)/a

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Rubi [A]
time = 0.21, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {4726, 4796, 4768, 4720, 3386, 3432, 4732, 4491} \begin {gather*} \frac {3 \sqrt {\frac {\pi }{2}} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{8 a^3}+\frac {\sqrt {\frac {\pi }{6}} S\left (\sqrt {\frac {6}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{24 a^3}-\frac {x^2 \sqrt {1-a^2 x^2} \sqrt {\text {ArcCos}(a x)}}{6 a}-\frac {\sqrt {1-a^2 x^2} \sqrt {\text {ArcCos}(a x)}}{3 a^3}+\frac {1}{3} x^3 \text {ArcCos}(a x)^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*ArcCos[a*x]^(3/2),x]

[Out]

-1/3*(Sqrt[1 - a^2*x^2]*Sqrt[ArcCos[a*x]])/a^3 - (x^2*Sqrt[1 - a^2*x^2]*Sqrt[ArcCos[a*x]])/(6*a) + (x^3*ArcCos
[a*x]^(3/2))/3 + (3*Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(8*a^3) + (Sqrt[Pi/6]*FresnelS[Sqrt[6/P
i]*Sqrt[ArcCos[a*x]]])/(24*a^3)

Rule 3386

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

Rule 3432

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]

Rule 4491

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(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]
&& IGtQ[p, 0]

Rule 4720

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[-(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]

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 4732

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[-(b*c^(m + 1))^(-1), Subst[Int[x^n*C
os[-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]

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]

Rubi steps

\begin {align*} \int x^2 \cos ^{-1}(a x)^{3/2} \, dx &=\frac {1}{3} x^3 \cos ^{-1}(a x)^{3/2}+\frac {1}{2} a \int \frac {x^3 \sqrt {\cos ^{-1}(a x)}}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {x^2 \sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}}{6 a}+\frac {1}{3} x^3 \cos ^{-1}(a x)^{3/2}-\frac {1}{12} \int \frac {x^2}{\sqrt {\cos ^{-1}(a x)}} \, dx+\frac {\int \frac {x \sqrt {\cos ^{-1}(a x)}}{\sqrt {1-a^2 x^2}} \, dx}{3 a}\\ &=-\frac {\sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}}{3 a^3}-\frac {x^2 \sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}}{6 a}+\frac {1}{3} x^3 \cos ^{-1}(a x)^{3/2}+\frac {\text {Subst}\left (\int \frac {\cos ^2(x) \sin (x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{12 a^3}-\frac {\int \frac {1}{\sqrt {\cos ^{-1}(a x)}} \, dx}{6 a^2}\\ &=-\frac {\sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}}{3 a^3}-\frac {x^2 \sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}}{6 a}+\frac {1}{3} x^3 \cos ^{-1}(a x)^{3/2}+\frac {\text {Subst}\left (\int \left (\frac {\sin (x)}{4 \sqrt {x}}+\frac {\sin (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{12 a^3}+\frac {\text {Subst}\left (\int \frac {\sin (x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{6 a^3}\\ &=-\frac {\sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}}{3 a^3}-\frac {x^2 \sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}}{6 a}+\frac {1}{3} x^3 \cos ^{-1}(a x)^{3/2}+\frac {\text {Subst}\left (\int \frac {\sin (x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{48 a^3}+\frac {\text {Subst}\left (\int \frac {\sin (3 x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{48 a^3}+\frac {\text {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{3 a^3}\\ &=-\frac {\sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}}{3 a^3}-\frac {x^2 \sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}}{6 a}+\frac {1}{3} x^3 \cos ^{-1}(a x)^{3/2}+\frac {\sqrt {\frac {\pi }{2}} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{3 a^3}+\frac {\text {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{24 a^3}+\frac {\text {Subst}\left (\int \sin \left (3 x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{24 a^3}\\ &=-\frac {\sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}}{3 a^3}-\frac {x^2 \sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}}{6 a}+\frac {1}{3} x^3 \cos ^{-1}(a x)^{3/2}+\frac {3 \sqrt {\frac {\pi }{2}} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{8 a^3}+\frac {\sqrt {\frac {\pi }{6}} S\left (\sqrt {\frac {6}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{24 a^3}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.07, size = 125, normalized size = 0.85 \begin {gather*} -\frac {27 \sqrt {-i \text {ArcCos}(a x)} \text {Gamma}\left (\frac {5}{2},-i \text {ArcCos}(a x)\right )+27 \sqrt {i \text {ArcCos}(a x)} \text {Gamma}\left (\frac {5}{2},i \text {ArcCos}(a x)\right )+\sqrt {3} \left (\sqrt {-i \text {ArcCos}(a x)} \text {Gamma}\left (\frac {5}{2},-3 i \text {ArcCos}(a x)\right )+\sqrt {i \text {ArcCos}(a x)} \text {Gamma}\left (\frac {5}{2},3 i \text {ArcCos}(a x)\right )\right )}{216 a^3 \sqrt {\text {ArcCos}(a x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2*ArcCos[a*x]^(3/2),x]

[Out]

-1/216*(27*Sqrt[(-I)*ArcCos[a*x]]*Gamma[5/2, (-I)*ArcCos[a*x]] + 27*Sqrt[I*ArcCos[a*x]]*Gamma[5/2, I*ArcCos[a*
x]] + 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)*Ar
cCos[a*x]]))/(a^3*Sqrt[ArcCos[a*x]])

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Maple [A]
time = 0.22, size = 130, normalized size = 0.88

method	result	size
default	\(\frac {36 a x \arccos \left (a x \right )^{2}+\sqrt {3}\, \sqrt {2}\, \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }\, \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )+12 \arccos \left (a x \right )^{2} \cos \left (3 \arccos \left (a x \right )\right )+27 \sqrt {2}\, \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }\, \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )-54 \arccos \left (a x \right ) \sqrt {-a^{2} x^{2}+1}-6 \arccos \left (a x \right ) \sin \left (3 \arccos \left (a x \right )\right )}{144 a^{3} \sqrt {\arccos \left (a x \right )}}\)	\(130\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*arccos(a*x)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/144/a^3*(36*a*x*arccos(a*x)^2+3^(1/2)*2^(1/2)*arccos(a*x)^(1/2)*Pi^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*3^(1/2)*a
rccos(a*x)^(1/2))+12*arccos(a*x)^2*cos(3*arccos(a*x))+27*2^(1/2)*arccos(a*x)^(1/2)*Pi^(1/2)*FresnelS(2^(1/2)/P
i^(1/2)*arccos(a*x)^(1/2))-54*arccos(a*x)*(-a^2*x^2+1)^(1/2)-6*arccos(a*x)*sin(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)^(3/2),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)^(3/2),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 {3}{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)**(3/2),x)

[Out]

Integral(x**2*acos(a*x)**(3/2), x)

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Giac [C] Result contains complex when optimal does not.
time = 0.47, size = 237, normalized size = 1.61 \begin {gather*} \frac {\arccos \left (a x\right )^{\frac {3}{2}} e^{\left (3 i \, \arccos \left (a x\right )\right )}}{24 \, a^{3}} + \frac {\arccos \left (a x\right )^{\frac {3}{2}} e^{\left (i \, \arccos \left (a x\right )\right )}}{8 \, a^{3}} + \frac {\arccos \left (a x\right )^{\frac {3}{2}} e^{\left (-i \, \arccos \left (a x\right )\right )}}{8 \, a^{3}} + \frac {\arccos \left (a x\right )^{\frac {3}{2}} e^{\left (-3 i \, \arccos \left (a x\right )\right )}}{24 \, a^{3}} + \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 )}{576 \, a^{3}} - \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 )}{576 \, a^{3}} + \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 )}{64 \, a^{3}} - \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 )}{64 \, a^{3}} + \frac {i \, \sqrt {\arccos \left (a x\right )} e^{\left (3 i \, \arccos \left (a x\right )\right )}}{48 \, a^{3}} + \frac {3 i \, \sqrt {\arccos \left (a x\right )} e^{\left (i \, \arccos \left (a x\right )\right )}}{16 \, a^{3}} - \frac {3 i \, \sqrt {\arccos \left (a x\right )} e^{\left (-i \, \arccos \left (a x\right )\right )}}{16 \, a^{3}} - \frac {i \, \sqrt {\arccos \left (a x\right )} e^{\left (-3 i \, \arccos \left (a x\right )\right )}}{48 \, a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arccos(a*x)^(3/2),x, algorithm="giac")

[Out]

1/24*arccos(a*x)^(3/2)*e^(3*I*arccos(a*x))/a^3 + 1/8*arccos(a*x)^(3/2)*e^(I*arccos(a*x))/a^3 + 1/8*arccos(a*x)
^(3/2)*e^(-I*arccos(a*x))/a^3 + 1/24*arccos(a*x)^(3/2)*e^(-3*I*arccos(a*x))/a^3 + (1/576*I - 1/576)*sqrt(6)*sq
rt(pi)*erf((1/2*I - 1/2)*sqrt(6)*sqrt(arccos(a*x)))/a^3 - (1/576*I + 1/576)*sqrt(6)*sqrt(pi)*erf(-(1/2*I + 1/2
)*sqrt(6)*sqrt(arccos(a*x)))/a^3 + (3/64*I - 3/64)*sqrt(2)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt(2)*sqrt(arccos(a*x)
))/a^3 - (3/64*I + 3/64)*sqrt(2)*sqrt(pi)*erf(-(1/2*I + 1/2)*sqrt(2)*sqrt(arccos(a*x)))/a^3 + 1/48*I*sqrt(arcc
os(a*x))*e^(3*I*arccos(a*x))/a^3 + 3/16*I*sqrt(arccos(a*x))*e^(I*arccos(a*x))/a^3 - 3/16*I*sqrt(arccos(a*x))*e
^(-I*arccos(a*x))/a^3 - 1/48*I*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 )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*acos(a*x)^(3/2),x)

[Out]

int(x^2*acos(a*x)^(3/2), x)

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