3.109 \(\int \frac{x^2}{\cos ^{-1}(a x)^{5/2}} \, dx\)

Optimal. Leaf size=125 \[ \frac{\sqrt{2 \pi } S\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{3 a^3}+\frac{\sqrt{6 \pi } S\left (\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{a^3}+\frac{2 x^2 \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^{3/2}}-\frac{8 x}{3 a^2 \sqrt{\cos ^{-1}(a x)}}+\frac{4 x^3}{\sqrt{\cos ^{-1}(a x)}} \]

[Out]

(2*x^2*Sqrt[1 - a^2*x^2])/(3*a*ArcCos[a*x]^(3/2)) - (8*x)/(3*a^2*Sqrt[ArcCos[a*x]]) + (4*x^3)/Sqrt[ArcCos[a*x]
] + (Sqrt[2*Pi]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(3*a^3) + (Sqrt[6*Pi]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcCos[
a*x]]])/a^3

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Rubi [A]  time = 0.286432, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 7, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583, Rules used = {4634, 4720, 4636, 4406, 3305, 3351, 4624} \[ \frac{\sqrt{2 \pi } S\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{3 a^3}+\frac{\sqrt{6 \pi } S\left (\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{a^3}+\frac{2 x^2 \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^{3/2}}-\frac{8 x}{3 a^2 \sqrt{\cos ^{-1}(a x)}}+\frac{4 x^3}{\sqrt{\cos ^{-1}(a x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^2*Sqrt[1 - a^2*x^2])/(3*a*ArcCos[a*x]^(3/2)) - (8*x)/(3*a^2*Sqrt[ArcCos[a*x]]) + (4*x^3)/Sqrt[ArcCos[a*x]
] + (Sqrt[2*Pi]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(3*a^3) + (Sqrt[6*Pi]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcCos[
a*x]]])/a^3

Rule 4634

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> -Simp[(x^m*Sqrt[1 - c^2*x^2]*(a + b*ArcCo
s[c*x])^(n + 1))/(b*c*(n + 1)), x] + (-Dist[(c*(m + 1))/(b*(n + 1)), Int[(x^(m + 1)*(a + b*ArcCos[c*x])^(n + 1
))/Sqrt[1 - c^2*x^2], x], x] + Dist[m/(b*c*(n + 1)), Int[(x^(m - 1)*(a + b*ArcCos[c*x])^(n + 1))/Sqrt[1 - c^2*
x^2], x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]

Rule 4720

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

Rubi steps

\begin{align*} \int \frac{x^2}{\cos ^{-1}(a x)^{5/2}} \, dx &=\frac{2 x^2 \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^{3/2}}-\frac{4 \int \frac{x}{\sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}} \, dx}{3 a}+(2 a) \int \frac{x^3}{\sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}} \, dx\\ &=\frac{2 x^2 \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^{3/2}}-\frac{8 x}{3 a^2 \sqrt{\cos ^{-1}(a x)}}+\frac{4 x^3}{\sqrt{\cos ^{-1}(a x)}}-12 \int \frac{x^2}{\sqrt{\cos ^{-1}(a x)}} \, dx+\frac{8 \int \frac{1}{\sqrt{\cos ^{-1}(a x)}} \, dx}{3 a^2}\\ &=\frac{2 x^2 \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^{3/2}}-\frac{8 x}{3 a^2 \sqrt{\cos ^{-1}(a x)}}+\frac{4 x^3}{\sqrt{\cos ^{-1}(a x)}}-\frac{8 \operatorname{Subst}\left (\int \frac{\sin (x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{3 a^3}+\frac{12 \operatorname{Subst}\left (\int \frac{\cos ^2(x) \sin (x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{a^3}\\ &=\frac{2 x^2 \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^{3/2}}-\frac{8 x}{3 a^2 \sqrt{\cos ^{-1}(a x)}}+\frac{4 x^3}{\sqrt{\cos ^{-1}(a x)}}-\frac{16 \operatorname{Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{3 a^3}+\frac{12 \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 )}{a^3}\\ &=\frac{2 x^2 \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^{3/2}}-\frac{8 x}{3 a^2 \sqrt{\cos ^{-1}(a x)}}+\frac{4 x^3}{\sqrt{\cos ^{-1}(a x)}}-\frac{8 \sqrt{2 \pi } S\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{3 a^3}+\frac{3 \operatorname{Subst}\left (\int \frac{\sin (x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{a^3}+\frac{3 \operatorname{Subst}\left (\int \frac{\sin (3 x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{a^3}\\ &=\frac{2 x^2 \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^{3/2}}-\frac{8 x}{3 a^2 \sqrt{\cos ^{-1}(a x)}}+\frac{4 x^3}{\sqrt{\cos ^{-1}(a x)}}-\frac{8 \sqrt{2 \pi } S\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{3 a^3}+\frac{6 \operatorname{Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{a^3}+\frac{6 \operatorname{Subst}\left (\int \sin \left (3 x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{a^3}\\ &=\frac{2 x^2 \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^{3/2}}-\frac{8 x}{3 a^2 \sqrt{\cos ^{-1}(a x)}}+\frac{4 x^3}{\sqrt{\cos ^{-1}(a x)}}+\frac{\sqrt{2 \pi } S\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{3 a^3}+\frac{\sqrt{6 \pi } S\left (\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{a^3}\\ \end{align*}

Mathematica [C]  time = 0.821265, size = 220, normalized size = 1.76 \[ -\frac{\sqrt{-i \cos ^{-1}(a x)} \cos ^{-1}(a x) \text{Gamma}\left (\frac{1}{2},-i \cos ^{-1}(a x)\right )+\sqrt{i \cos ^{-1}(a x)} \cos ^{-1}(a x) \text{Gamma}\left (\frac{1}{2},i \cos ^{-1}(a x)\right )-3 \cos ^{-1}(a x) \left (-\sqrt{3} \sqrt{-i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-3 i \cos ^{-1}(a x)\right )-\sqrt{3} \sqrt{i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},3 i \cos ^{-1}(a x)\right )+e^{-3 i \cos ^{-1}(a x)}+e^{3 i \cos ^{-1}(a x)}\right )-\sqrt{1-a^2 x^2}-e^{-i \cos ^{-1}(a x)} \cos ^{-1}(a x)-e^{i \cos ^{-1}(a x)} \cos ^{-1}(a x)-\sin \left (3 \cos ^{-1}(a x)\right )}{6 a^3 \cos ^{-1}(a x)^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(-Sqrt[1 - a^2*x^2] - ArcCos[a*x]/E^(I*ArcCos[a*x]) - E^(I*ArcCos[a*x])*ArcCos[a*x] + Sqrt[(-I)*ArcCos[a*x]]*
ArcCos[a*x]*Gamma[1/2, (-I)*ArcCos[a*x]] + Sqrt[I*ArcCos[a*x]]*ArcCos[a*x]*Gamma[1/2, I*ArcCos[a*x]] - 3*ArcCo
s[a*x]*(E^((-3*I)*ArcCos[a*x]) + E^((3*I)*ArcCos[a*x]) - Sqrt[3]*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-3*I)*ArcC
os[a*x]] - Sqrt[3]*Sqrt[I*ArcCos[a*x]]*Gamma[1/2, (3*I)*ArcCos[a*x]]) - Sin[3*ArcCos[a*x]])/(6*a^3*ArcCos[a*x]
^(3/2))

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Maple [A]  time = 0.085, size = 115, normalized size = 0.9 \begin{align*}{\frac{1}{6\,{a}^{3}} \left ( 6\,\sqrt{2}\sqrt{\pi }\sqrt{3}{\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{3}\sqrt{\arccos \left ( ax \right ) }}{\sqrt{\pi }}} \right ) \left ( \arccos \left ( ax \right ) \right ) ^{3/2}+2\,\sqrt{2}\sqrt{\pi }{\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{\arccos \left ( ax \right ) }}{\sqrt{\pi }}} \right ) \left ( \arccos \left ( ax \right ) \right ) ^{3/2}+2\,ax\arccos \left ( ax \right ) +6\,\arccos \left ( ax \right ) \cos \left ( 3\,\arccos \left ( ax \right ) \right ) +\sin \left ( 3\,\arccos \left ( ax \right ) \right ) +\sqrt{-{a}^{2}{x}^{2}+1} \right ) \left ( \arccos \left ( ax \right ) \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6/a^3*(6*2^(1/2)*Pi^(1/2)*3^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*3^(1/2)*arccos(a*x)^(1/2))*arccos(a*x)^(3/2)+2*2
^(1/2)*Pi^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*arccos(a*x)^(3/2)+2*a*x*arccos(a*x)+6*arccos(a*x)
*cos(3*arccos(a*x))+sin(3*arccos(a*x))+(-a^2*x^2+1)^(1/2))/arccos(a*x)^(3/2)

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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)^(5/2),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)^(5/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\operatorname{acos}^{\frac{5}{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)**(5/2),x)

[Out]

Integral(x**2/acos(a*x)**(5/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\arccos \left (a x\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2/arccos(a*x)^(5/2), x)