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]

-(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)

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Rubi [A]  time = 0.302481, antiderivative size = 147, normalized size of antiderivative = 1., 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 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 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]

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 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 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 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 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]

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*}

Mathematica [C]  time = 0.0862844, size = 125, normalized size = 0.85 \[ -\frac{27 \sqrt{-i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{5}{2},-i \cos ^{-1}(a x)\right )+27 \sqrt{i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{5}{2},i \cos ^{-1}(a x)\right )+\sqrt{3} \left (\sqrt{-i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{5}{2},-3 i \cos ^{-1}(a x)\right )+\sqrt{i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{5}{2},3 i \cos ^{-1}(a x)\right )\right )}{216 a^3 \sqrt{\cos ^{-1}(a x)}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(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)*ArcCos[a
*x]]))/(216*a^3*Sqrt[ArcCos[a*x]])

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Maple [A]  time = 0.087, size = 130, normalized size = 0.9 \begin{align*}{\frac{1}{144\,{a}^{3}} \left ( 36\,ax \left ( \arccos \left ( ax \right ) \right ) ^{2}+\sqrt{3}\sqrt{2}\sqrt{\arccos \left ( ax \right ) }\sqrt{\pi }{\it FresnelS} \left ({\frac{\sqrt{3}\sqrt{2}}{\sqrt{\pi }}\sqrt{\arccos \left ( ax \right ) }} \right ) +12\, \left ( \arccos \left ( ax \right ) \right ) ^{2}\cos \left ( 3\,\arccos \left ( ax \right ) \right ) +27\,\sqrt{2}\sqrt{\arccos \left ( ax \right ) }\sqrt{\pi }{\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{\arccos \left ( ax \right ) }}{\sqrt{\pi }}} \right ) -54\,\arccos \left ( ax \right ) \sqrt{-{a}^{2}{x}^{2}+1}-6\,\arccos \left ( ax \right ) \sin \left ( 3\,\arccos \left ( ax \right ) \right ) \right ){\frac{1}{\sqrt{\arccos \left ( ax \right ) }}}} \end{align*}

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*(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., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

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

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

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: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \operatorname{acos}^{\frac{3}{2}}{\left (a x \right )}\, dx \end{align*}

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 [B]  time = 1.29681, size = 390, normalized size = 2.65 \begin{align*} \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 )}} \end{align*}

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))