3.82 \(\int x^2 \cos ^{-1}(a x)^{3/2} \, dx\)

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

[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 = 0.30, 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 = {4630, 4708, 4678, 4624, 3305, 3351, 4636, 4406} \[ \frac {3 \sqrt {\frac {\pi }{2}} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{8 a^3}+\frac {\sqrt {\frac {\pi }{6}} S\left (\sqrt {\frac {6}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{24 a^3}-\frac {x^2 \sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}}{6 a}-\frac {\sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}}{3 a^3}+\frac {1}{3} x^3 \cos ^{-1}(a x)^{3/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[1 - a^2*x^2]*Sqrt[ArcCos[a*x]])/(3*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 3305

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 3351

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)])/
(f*Rt[d, 2]), x] /; FreeQ[{d, e, f}, x]

Rule 4406

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 4624

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

Rule 4630

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 4636

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> -Dist[(c^(m + 1))^(-1), Subst[Int[(a + b*
x)^n*Cos[x]^m*Sin[x], x], x, ArcCos[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]

Rule 4678

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*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p + 1
)*(1 - c^2*x^2)^FracPart[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 4708

Int[(((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[
(f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*(a + b*ArcCos[c*x])^n)/(e*m), x] + (Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m
- 2)*(a + b*ArcCos[c*x])^n)/Sqrt[d + e*x^2], x], x] - Dist[(b*f*n*Sqrt[1 - c^2*x^2])/(c*m*Sqrt[d + e*x^2]), I
nt[(f*x)^(m - 1)*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] &&
GtQ[n, 0] && GtQ[m, 1] && IntegerQ[m]

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 {\operatorname {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 {\operatorname {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 {\operatorname {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 {\operatorname {Subst}\left (\int \frac {\sin (x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{48 a^3}+\frac {\operatorname {Subst}\left (\int \frac {\sin (3 x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{48 a^3}+\frac {\operatorname {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 {\operatorname {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{24 a^3}+\frac {\operatorname {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]  time = 0.10, size = 125, normalized size = 0.85 \[ -\frac {27 \sqrt {-i \cos ^{-1}(a x)} \Gamma \left (\frac {5}{2},-i \cos ^{-1}(a x)\right )+27 \sqrt {i \cos ^{-1}(a x)} \Gamma \left (\frac {5}{2},i \cos ^{-1}(a x)\right )+\sqrt {3} \left (\sqrt {-i \cos ^{-1}(a x)} \Gamma \left (\frac {5}{2},-3 i \cos ^{-1}(a x)\right )+\sqrt {i \cos ^{-1}(a x)} \Gamma \left (\frac {5}{2},3 i \cos ^{-1}(a x)\right )\right )}{216 a^3 \sqrt {\cos ^{-1}(a x)}} \]

Warning: Unable to verify antiderivative.

[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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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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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giac [B]  time = 0.31, size = 289, normalized size = 1.97 \[ \frac {i \sqrt {\arccos \left (a x\right )} e^{\left (3 \, i \arccos \left (a x\right )\right )}}{48 \, a^{3}} + \frac {\arccos \left (a x\right )^{\frac {3}{2}} e^{\left (3 \, i \arccos \left (a x\right )\right )}}{24 \, a^{3}} + \frac {3 \, i \sqrt {\arccos \left (a x\right )} e^{\left (i \arccos \left (a x\right )\right )}}{16 \, a^{3}} + \frac {\arccos \left (a x\right )^{\frac {3}{2}} e^{\left (i \arccos \left (a x\right )\right )}}{8 \, a^{3}} - \frac {3 \, i \sqrt {\arccos \left (a x\right )} e^{\left (-i \arccos \left (a x\right )\right )}}{16 \, a^{3}} + \frac {\arccos \left (a x\right )^{\frac {3}{2}} e^{\left (-i \arccos \left (a x\right )\right )}}{8 \, a^{3}} - \frac {i \sqrt {\arccos \left (a x\right )} e^{\left (-3 \, i \arccos \left (a x\right )\right )}}{48 \, a^{3}} + \frac {\arccos \left (a x\right )^{\frac {3}{2}} e^{\left (-3 \, i \arccos \left (a x\right )\right )}}{24 \, a^{3}} - \frac {\sqrt {6} \sqrt {\pi } i \operatorname {erf}\left (-\frac {\sqrt {6} i \sqrt {\arccos \left (a x\right )}}{i - 1}\right )}{288 \, a^{3} {\left (i - 1\right )}} - \frac {3 \, \sqrt {2} \sqrt {\pi } i \operatorname {erf}\left (-\frac {\sqrt {2} i \sqrt {\arccos \left (a x\right )}}{i - 1}\right )}{32 \, a^{3} {\left (i - 1\right )}} + \frac {\sqrt {6} \sqrt {\pi } \operatorname {erf}\left (\frac {\sqrt {6} \sqrt {\arccos \left (a x\right )}}{i - 1}\right )}{288 \, a^{3} {\left (i - 1\right )}} + \frac {3 \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\frac {\sqrt {2} \sqrt {\arccos \left (a x\right )}}{i - 1}\right )}{32 \, a^{3} {\left (i - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*i*sqrt(arccos(a*x))*e^(3*i*arccos(a*x))/a^3 + 1/24*arccos(a*x)^(3/2)*e^(3*i*arccos(a*x))/a^3 + 3/16*i*sqr
t(arccos(a*x))*e^(i*arccos(a*x))/a^3 + 1/8*arccos(a*x)^(3/2)*e^(i*arccos(a*x))/a^3 - 3/16*i*sqrt(arccos(a*x))*
e^(-i*arccos(a*x))/a^3 + 1/8*arccos(a*x)^(3/2)*e^(-i*arccos(a*x))/a^3 - 1/48*i*sqrt(arccos(a*x))*e^(-3*i*arcco
s(a*x))/a^3 + 1/24*arccos(a*x)^(3/2)*e^(-3*i*arccos(a*x))/a^3 - 1/288*sqrt(6)*sqrt(pi)*i*erf(-sqrt(6)*i*sqrt(a
rccos(a*x))/(i - 1))/(a^3*(i - 1)) - 3/32*sqrt(2)*sqrt(pi)*i*erf(-sqrt(2)*i*sqrt(arccos(a*x))/(i - 1))/(a^3*(i
- 1)) + 1/288*sqrt(6)*sqrt(pi)*erf(sqrt(6)*sqrt(arccos(a*x))/(i - 1))/(a^3*(i - 1)) + 3/32*sqrt(2)*sqrt(pi)*e
rf(sqrt(2)*sqrt(arccos(a*x))/(i - 1))/(a^3*(i - 1))

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maple [A]  time = 0.24, size = 130, normalized size = 0.88 \[ \frac {\mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {3}\, \sqrt {2}\, \sqrt {\pi }\, \sqrt {\arccos \left (a x \right )}+36 a x \arccos \left (a x \right )^{2}+27 \,\mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {2}\, \sqrt {\pi }\, \sqrt {\arccos \left (a x \right )}+12 \arccos \left (a x \right )^{2} \cos \left (3 \arccos \left (a x \right )\right )-54 \arccos \left (a x \right ) \sqrt {-a^{2} x^{2}+1}-6 \arccos \left (a x \right ) \sin \left (3 \arccos \left (a x \right )\right )}{144 a^{3} \sqrt {\arccos \left (a x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/144/a^3*(FresnelS(2^(1/2)/Pi^(1/2)*3^(1/2)*arccos(a*x)^(1/2))*3^(1/2)*2^(1/2)*Pi^(1/2)*arccos(a*x)^(1/2)+36*
a*x*arccos(a*x)^2+27*FresnelS(2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*2^(1/2)*Pi^(1/2)*arccos(a*x)^(1/2)+12*arccos
(a*x)^2*cos(3*arccos(a*x))-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 \[ \text {Exception raised: RuntimeError} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x^2\,{\mathrm {acos}\left (a\,x\right )}^{3/2} \,d x \]

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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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \operatorname {acos}^{\frac {3}{2}}{\left (a x \right )}\, dx \]

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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