3.108 \(\int \frac{x^3}{\cos ^{-1}(a x)^{5/2}} \, dx\)

Optimal. Leaf size=126 \[ \frac{4 \sqrt{2 \pi } S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{3 a^4}+\frac{4 \sqrt{\pi } S\left (\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right )}{3 a^4}+\frac{2 x^3 \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^{3/2}}-\frac{4 x^2}{a^2 \sqrt{\cos ^{-1}(a x)}}+\frac{16 x^4}{3 \sqrt{\cos ^{-1}(a x)}} \]

[Out]

(2*x^3*Sqrt[1 - a^2*x^2])/(3*a*ArcCos[a*x]^(3/2)) - (4*x^2)/(a^2*Sqrt[ArcCos[a*x]]) + (16*x^4)/(3*Sqrt[ArcCos[
a*x]]) + (4*Sqrt[2*Pi]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(3*a^4) + (4*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcCos
[a*x]])/Sqrt[Pi]])/(3*a^4)

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Rubi [A]  time = 0.325012, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 15, 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, 12} \[ \frac{4 \sqrt{2 \pi } S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{3 a^4}+\frac{4 \sqrt{\pi } S\left (\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right )}{3 a^4}+\frac{2 x^3 \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^{3/2}}-\frac{4 x^2}{a^2 \sqrt{\cos ^{-1}(a x)}}+\frac{16 x^4}{3 \sqrt{\cos ^{-1}(a x)}} \]

Antiderivative was successfully verified.

[In]

Int[x^3/ArcCos[a*x]^(5/2),x]

[Out]

(2*x^3*Sqrt[1 - a^2*x^2])/(3*a*ArcCos[a*x]^(3/2)) - (4*x^2)/(a^2*Sqrt[ArcCos[a*x]]) + (16*x^4)/(3*Sqrt[ArcCos[
a*x]]) + (4*Sqrt[2*Pi]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(3*a^4) + (4*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcCos
[a*x]])/Sqrt[Pi]])/(3*a^4)

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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rubi steps

\begin{align*} \int \frac{x^3}{\cos ^{-1}(a x)^{5/2}} \, dx &=\frac{2 x^3 \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^{3/2}}-\frac{2 \int \frac{x^2}{\sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}} \, dx}{a}+\frac{1}{3} (8 a) \int \frac{x^4}{\sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}} \, dx\\ &=\frac{2 x^3 \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^{3/2}}-\frac{4 x^2}{a^2 \sqrt{\cos ^{-1}(a x)}}+\frac{16 x^4}{3 \sqrt{\cos ^{-1}(a x)}}-\frac{64}{3} \int \frac{x^3}{\sqrt{\cos ^{-1}(a x)}} \, dx+\frac{8 \int \frac{x}{\sqrt{\cos ^{-1}(a x)}} \, dx}{a^2}\\ &=\frac{2 x^3 \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^{3/2}}-\frac{4 x^2}{a^2 \sqrt{\cos ^{-1}(a x)}}+\frac{16 x^4}{3 \sqrt{\cos ^{-1}(a x)}}-\frac{8 \operatorname{Subst}\left (\int \frac{\cos (x) \sin (x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{a^4}+\frac{64 \operatorname{Subst}\left (\int \frac{\cos ^3(x) \sin (x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{3 a^4}\\ &=\frac{2 x^3 \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^{3/2}}-\frac{4 x^2}{a^2 \sqrt{\cos ^{-1}(a x)}}+\frac{16 x^4}{3 \sqrt{\cos ^{-1}(a x)}}-\frac{8 \operatorname{Subst}\left (\int \frac{\sin (2 x)}{2 \sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{a^4}+\frac{64 \operatorname{Subst}\left (\int \left (\frac{\sin (2 x)}{4 \sqrt{x}}+\frac{\sin (4 x)}{8 \sqrt{x}}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{3 a^4}\\ &=\frac{2 x^3 \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^{3/2}}-\frac{4 x^2}{a^2 \sqrt{\cos ^{-1}(a x)}}+\frac{16 x^4}{3 \sqrt{\cos ^{-1}(a x)}}+\frac{8 \operatorname{Subst}\left (\int \frac{\sin (4 x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{3 a^4}-\frac{4 \operatorname{Subst}\left (\int \frac{\sin (2 x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{a^4}+\frac{16 \operatorname{Subst}\left (\int \frac{\sin (2 x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{3 a^4}\\ &=\frac{2 x^3 \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^{3/2}}-\frac{4 x^2}{a^2 \sqrt{\cos ^{-1}(a x)}}+\frac{16 x^4}{3 \sqrt{\cos ^{-1}(a x)}}+\frac{16 \operatorname{Subst}\left (\int \sin \left (4 x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{3 a^4}-\frac{8 \operatorname{Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{a^4}+\frac{32 \operatorname{Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{3 a^4}\\ &=\frac{2 x^3 \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^{3/2}}-\frac{4 x^2}{a^2 \sqrt{\cos ^{-1}(a x)}}+\frac{16 x^4}{3 \sqrt{\cos ^{-1}(a x)}}+\frac{4 \sqrt{2 \pi } S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{3 a^4}+\frac{4 \sqrt{\pi } S\left (\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right )}{3 a^4}\\ \end{align*}

Mathematica [C]  time = 0.890657, size = 203, normalized size = 1.61 \[ -\frac{-4 \cos ^{-1}(a x) \left (-2 \sqrt{-i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-4 i \cos ^{-1}(a x)\right )-2 \sqrt{i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},4 i \cos ^{-1}(a x)\right )+e^{-4 i \cos ^{-1}(a x)}+e^{4 i \cos ^{-1}(a x)}\right )-2 \left (\sin \left (2 \cos ^{-1}(a x)\right )+2 \cos ^{-1}(a x) \left (-\sqrt{2} \sqrt{-i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-2 i \cos ^{-1}(a x)\right )-\sqrt{2} \sqrt{i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},2 i \cos ^{-1}(a x)\right )+e^{-2 i \cos ^{-1}(a x)}+e^{2 i \cos ^{-1}(a x)}\right )\right )-\sin \left (4 \cos ^{-1}(a x)\right )}{12 a^4 \cos ^{-1}(a x)^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^3/ArcCos[a*x]^(5/2),x]

[Out]

-(-4*ArcCos[a*x]*(E^((-4*I)*ArcCos[a*x]) + E^((4*I)*ArcCos[a*x]) - 2*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-4*I)*
ArcCos[a*x]] - 2*Sqrt[I*ArcCos[a*x]]*Gamma[1/2, (4*I)*ArcCos[a*x]]) - 2*(2*ArcCos[a*x]*(E^((-2*I)*ArcCos[a*x])
+ E^((2*I)*ArcCos[a*x]) - Sqrt[2]*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-2*I)*ArcCos[a*x]] - Sqrt[2]*Sqrt[I*ArcC
os[a*x]]*Gamma[1/2, (2*I)*ArcCos[a*x]]) + Sin[2*ArcCos[a*x]]) - Sin[4*ArcCos[a*x]])/(12*a^4*ArcCos[a*x]^(3/2))

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Maple [A]  time = 0.085, size = 107, normalized size = 0.9 \begin{align*}{\frac{1}{12\,{a}^{4}} \left ( 16\,\sqrt{2}\sqrt{\pi }{\it FresnelS} \left ( 2\,{\frac{\sqrt{2}\sqrt{\arccos \left ( ax \right ) }}{\sqrt{\pi }}} \right ) \left ( \arccos \left ( ax \right ) \right ) ^{3/2}+16\,\sqrt{\pi }{\it FresnelS} \left ( 2\,{\frac{\sqrt{\arccos \left ( ax \right ) }}{\sqrt{\pi }}} \right ) \left ( \arccos \left ( ax \right ) \right ) ^{3/2}+8\,\arccos \left ( ax \right ) \cos \left ( 2\,\arccos \left ( ax \right ) \right ) +8\,\arccos \left ( ax \right ) \cos \left ( 4\,\arccos \left ( ax \right ) \right ) +2\,\sin \left ( 2\,\arccos \left ( ax \right ) \right ) +\sin \left ( 4\,\arccos \left ( ax \right ) \right ) \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^3/arccos(a*x)^(5/2),x)

[Out]

1/12/a^4*(16*2^(1/2)*Pi^(1/2)*FresnelS(2*2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*arccos(a*x)^(3/2)+16*Pi^(1/2)*Fre
snelS(2*arccos(a*x)^(1/2)/Pi^(1/2))*arccos(a*x)^(3/2)+8*arccos(a*x)*cos(2*arccos(a*x))+8*arccos(a*x)*cos(4*arc
cos(a*x))+2*sin(2*arccos(a*x))+sin(4*arccos(a*x)))/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^3/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^3/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^{3}}{\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**3/acos(a*x)**(5/2),x)

[Out]

Integral(x**3/acos(a*x)**(5/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\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^3/arccos(a*x)^(5/2),x, algorithm="giac")

[Out]

integrate(x^3/arccos(a*x)^(5/2), x)