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

Optimal. Leaf size=89 \[ \frac{8 \sqrt{\pi } S\left (\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right )}{3 a^2}+\frac{2 x \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^{3/2}}-\frac{4}{3 a^2 \sqrt{\cos ^{-1}(a x)}}+\frac{8 x^2}{3 \sqrt{\cos ^{-1}(a x)}} \]

[Out]

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

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Rubi [A]  time = 0.172752, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.8, Rules used = {4634, 4720, 4636, 4406, 12, 3305, 3351, 4642} \[ \frac{8 \sqrt{\pi } S\left (\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right )}{3 a^2}+\frac{2 x \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^{3/2}}-\frac{4}{3 a^2 \sqrt{\cos ^{-1}(a x)}}+\frac{8 x^2}{3 \sqrt{\cos ^{-1}(a x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 4642

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

Rubi steps

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

Mathematica [A]  time = 0.0832623, size = 61, normalized size = 0.69 \[ \frac{8 \sqrt{\pi } S\left (\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right )+\frac{4 \cos ^{-1}(a x) \cos \left (2 \cos ^{-1}(a x)\right )+\sin \left (2 \cos ^{-1}(a x)\right )}{\cos ^{-1}(a x)^{3/2}}}{3 a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(8*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcCos[a*x]])/Sqrt[Pi]] + (4*ArcCos[a*x]*Cos[2*ArcCos[a*x]] + Sin[2*ArcCos[a*x]])
/ArcCos[a*x]^(3/2))/(3*a^2)

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Maple [A]  time = 0.069, size = 56, normalized size = 0.6 \begin{align*}{\frac{1}{3\,{a}^{2}} \left ( 8\,\sqrt{\pi }{\it FresnelS} \left ( 2\,{\frac{\sqrt{\arccos \left ( ax \right ) }}{\sqrt{\pi }}} \right ) \left ( \arccos \left ( ax \right ) \right ) ^{3/2}+4\,\arccos \left ( ax \right ) \cos \left ( 2\,\arccos \left ( ax \right ) \right ) +\sin \left ( 2\,\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/arccos(a*x)^(5/2),x)

[Out]

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

[Out]

Integral(x/acos(a*x)**(5/2), x)

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

[Out]

integrate(x/arccos(a*x)^(5/2), x)