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

Optimal. Leaf size=56 \[ -\frac{\text{CosIntegral}\left (2 \cos ^{-1}(a x)\right )}{2 a^4}-\frac{\text{CosIntegral}\left (4 \cos ^{-1}(a x)\right )}{2 a^4}+\frac{x^3 \sqrt{1-a^2 x^2}}{a \cos ^{-1}(a x)} \]

[Out]

(x^3*Sqrt[1 - a^2*x^2])/(a*ArcCos[a*x]) - CosIntegral[2*ArcCos[a*x]]/(2*a^4) - CosIntegral[4*ArcCos[a*x]]/(2*a
^4)

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Rubi [A]  time = 0.0502264, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4632, 3302} \[ -\frac{\text{CosIntegral}\left (2 \cos ^{-1}(a x)\right )}{2 a^4}-\frac{\text{CosIntegral}\left (4 \cos ^{-1}(a x)\right )}{2 a^4}+\frac{x^3 \sqrt{1-a^2 x^2}}{a \cos ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^3*Sqrt[1 - a^2*x^2])/(a*ArcCos[a*x]) - CosIntegral[2*ArcCos[a*x]]/(2*a^4) - CosIntegral[4*ArcCos[a*x]]/(2*a
^4)

Rule 4632

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[1/(b*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[(a + b*x)^(n + 1
), Cos[x]^(m - 1)*(m - (m + 1)*Cos[x]^2), x], x], x, ArcCos[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] &&
GeQ[n, -2] && LtQ[n, -1]

Rule 3302

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rubi steps

\begin{align*} \int \frac{x^3}{\cos ^{-1}(a x)^2} \, dx &=\frac{x^3 \sqrt{1-a^2 x^2}}{a \cos ^{-1}(a x)}+\frac{\operatorname{Subst}\left (\int \left (-\frac{\cos (2 x)}{2 x}-\frac{\cos (4 x)}{2 x}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{a^4}\\ &=\frac{x^3 \sqrt{1-a^2 x^2}}{a \cos ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \frac{\cos (2 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{2 a^4}-\frac{\operatorname{Subst}\left (\int \frac{\cos (4 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{2 a^4}\\ &=\frac{x^3 \sqrt{1-a^2 x^2}}{a \cos ^{-1}(a x)}-\frac{\text{Ci}\left (2 \cos ^{-1}(a x)\right )}{2 a^4}-\frac{\text{Ci}\left (4 \cos ^{-1}(a x)\right )}{2 a^4}\\ \end{align*}

Mathematica [A]  time = 0.138831, size = 50, normalized size = 0.89 \[ -\frac{-\frac{2 a^3 x^3 \sqrt{1-a^2 x^2}}{\cos ^{-1}(a x)}+\text{CosIntegral}\left (2 \cos ^{-1}(a x)\right )+\text{CosIntegral}\left (4 \cos ^{-1}(a x)\right )}{2 a^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.045, size = 54, normalized size = 1. \begin{align*}{\frac{1}{{a}^{4}} \left ({\frac{\sin \left ( 2\,\arccos \left ( ax \right ) \right ) }{4\,\arccos \left ( ax \right ) }}-{\frac{{\it Ci} \left ( 2\,\arccos \left ( ax \right ) \right ) }{2}}+{\frac{\sin \left ( 4\,\arccos \left ( ax \right ) \right ) }{8\,\arccos \left ( ax \right ) }}-{\frac{{\it Ci} \left ( 4\,\arccos \left ( ax \right ) \right ) }{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a^4*(1/4/arccos(a*x)*sin(2*arccos(a*x))-1/2*Ci(2*arccos(a*x))+1/8*sin(4*arccos(a*x))/arccos(a*x)-1/2*Ci(4*ar
ccos(a*x)))

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{3}}{\arccos \left (a x\right )^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^3/arccos(a*x)^2, x)

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

[Out]

Integral(x**3/acos(a*x)**2, x)

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Giac [A]  time = 1.18811, size = 68, normalized size = 1.21 \begin{align*} \frac{\sqrt{-a^{2} x^{2} + 1} x^{3}}{a \arccos \left (a x\right )} - \frac{\operatorname{Ci}\left (4 \, \arccos \left (a x\right )\right )}{2 \, a^{4}} - \frac{\operatorname{Ci}\left (2 \, \arccos \left (a x\right )\right )}{2 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(-a^2*x^2 + 1)*x^3/(a*arccos(a*x)) - 1/2*cos_integral(4*arccos(a*x))/a^4 - 1/2*cos_integral(2*arccos(a*x))
/a^4