3.68 \(\int \frac{x^3}{\cos ^{-1}(a x)^4} \, dx\)

Optimal. Leaf size=143 \[ \frac{\text{CosIntegral}\left (2 \cos ^{-1}(a x)\right )}{3 a^4}+\frac{4 \text{CosIntegral}\left (4 \cos ^{-1}(a x)\right )}{3 a^4}-\frac{8 x^3 \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)}+\frac{x^3 \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^3}-\frac{x^2}{2 a^2 \cos ^{-1}(a x)^2}+\frac{x \sqrt{1-a^2 x^2}}{a^3 \cos ^{-1}(a x)}+\frac{2 x^4}{3 \cos ^{-1}(a x)^2} \]

[Out]

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

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Rubi [A]  time = 0.290715, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4634, 4720, 4632, 3302} \[ \frac{\text{CosIntegral}\left (2 \cos ^{-1}(a x)\right )}{3 a^4}+\frac{4 \text{CosIntegral}\left (4 \cos ^{-1}(a x)\right )}{3 a^4}-\frac{8 x^3 \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)}+\frac{x^3 \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^3}-\frac{x^2}{2 a^2 \cos ^{-1}(a x)^2}+\frac{x \sqrt{1-a^2 x^2}}{a^3 \cos ^{-1}(a x)}+\frac{2 x^4}{3 \cos ^{-1}(a x)^2} \]

Antiderivative was successfully verified.

[In]

Int[x^3/ArcCos[a*x]^4,x]

[Out]

(x^3*Sqrt[1 - a^2*x^2])/(3*a*ArcCos[a*x]^3) - x^2/(2*a^2*ArcCos[a*x]^2) + (2*x^4)/(3*ArcCos[a*x]^2) + (x*Sqrt[
1 - a^2*x^2])/(a^3*ArcCos[a*x]) - (8*x^3*Sqrt[1 - a^2*x^2])/(3*a*ArcCos[a*x]) + CosIntegral[2*ArcCos[a*x]]/(3*
a^4) + (4*CosIntegral[4*ArcCos[a*x]])/(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 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)^4} \, dx &=\frac{x^3 \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^3}-\frac{\int \frac{x^2}{\sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3} \, dx}{a}+\frac{1}{3} (4 a) \int \frac{x^4}{\sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3} \, dx\\ &=\frac{x^3 \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^3}-\frac{x^2}{2 a^2 \cos ^{-1}(a x)^2}+\frac{2 x^4}{3 \cos ^{-1}(a x)^2}-\frac{8}{3} \int \frac{x^3}{\cos ^{-1}(a x)^2} \, dx+\frac{\int \frac{x}{\cos ^{-1}(a x)^2} \, dx}{a^2}\\ &=\frac{x^3 \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^3}-\frac{x^2}{2 a^2 \cos ^{-1}(a x)^2}+\frac{2 x^4}{3 \cos ^{-1}(a x)^2}+\frac{x \sqrt{1-a^2 x^2}}{a^3 \cos ^{-1}(a x)}-\frac{8 x^3 \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \frac{\cos (2 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{a^4}-\frac{8 \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 )}{3 a^4}\\ &=\frac{x^3 \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^3}-\frac{x^2}{2 a^2 \cos ^{-1}(a x)^2}+\frac{2 x^4}{3 \cos ^{-1}(a x)^2}+\frac{x \sqrt{1-a^2 x^2}}{a^3 \cos ^{-1}(a x)}-\frac{8 x^3 \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)}-\frac{\text{Ci}\left (2 \cos ^{-1}(a x)\right )}{a^4}+\frac{4 \operatorname{Subst}\left (\int \frac{\cos (2 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{3 a^4}+\frac{4 \operatorname{Subst}\left (\int \frac{\cos (4 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{3 a^4}\\ &=\frac{x^3 \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^3}-\frac{x^2}{2 a^2 \cos ^{-1}(a x)^2}+\frac{2 x^4}{3 \cos ^{-1}(a x)^2}+\frac{x \sqrt{1-a^2 x^2}}{a^3 \cos ^{-1}(a x)}-\frac{8 x^3 \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)}+\frac{\text{Ci}\left (2 \cos ^{-1}(a x)\right )}{3 a^4}+\frac{4 \text{Ci}\left (4 \cos ^{-1}(a x)\right )}{3 a^4}\\ \end{align*}

Mathematica [A]  time = 0.254877, size = 107, normalized size = 0.75 \[ \frac{\frac{a x \left (2 a^2 x^2 \sqrt{1-a^2 x^2}+a x \left (4 a^2 x^2-3\right ) \cos ^{-1}(a x)-2 \sqrt{1-a^2 x^2} \left (8 a^2 x^2-3\right ) \cos ^{-1}(a x)^2\right )}{\cos ^{-1}(a x)^3}+2 \text{CosIntegral}\left (2 \cos ^{-1}(a x)\right )+8 \text{CosIntegral}\left (4 \cos ^{-1}(a x)\right )}{6 a^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/ArcCos[a*x]^4,x]

[Out]

((a*x*(2*a^2*x^2*Sqrt[1 - a^2*x^2] + a*x*(-3 + 4*a^2*x^2)*ArcCos[a*x] - 2*Sqrt[1 - a^2*x^2]*(-3 + 8*a^2*x^2)*A
rcCos[a*x]^2))/ArcCos[a*x]^3 + 2*CosIntegral[2*ArcCos[a*x]] + 8*CosIntegral[4*ArcCos[a*x]])/(6*a^4)

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Maple [A]  time = 0.056, size = 114, normalized size = 0.8 \begin{align*}{\frac{1}{{a}^{4}} \left ({\frac{\sin \left ( 2\,\arccos \left ( ax \right ) \right ) }{12\, \left ( \arccos \left ( ax \right ) \right ) ^{3}}}+{\frac{\cos \left ( 2\,\arccos \left ( ax \right ) \right ) }{12\, \left ( \arccos \left ( ax \right ) \right ) ^{2}}}-{\frac{\sin \left ( 2\,\arccos \left ( ax \right ) \right ) }{6\,\arccos \left ( ax \right ) }}+{\frac{{\it Ci} \left ( 2\,\arccos \left ( ax \right ) \right ) }{3}}+{\frac{\sin \left ( 4\,\arccos \left ( ax \right ) \right ) }{24\, \left ( \arccos \left ( ax \right ) \right ) ^{3}}}+{\frac{\cos \left ( 4\,\arccos \left ( ax \right ) \right ) }{12\, \left ( \arccos \left ( ax \right ) \right ) ^{2}}}-{\frac{\sin \left ( 4\,\arccos \left ( ax \right ) \right ) }{3\,\arccos \left ( ax \right ) }}+{\frac{4\,{\it Ci} \left ( 4\,\arccos \left ( ax \right ) \right ) }{3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/arccos(a*x)^4,x)

[Out]

1/a^4*(1/12/arccos(a*x)^3*sin(2*arccos(a*x))+1/12/arccos(a*x)^2*cos(2*arccos(a*x))-1/6/arccos(a*x)*sin(2*arcco
s(a*x))+1/3*Ci(2*arccos(a*x))+1/24/arccos(a*x)^3*sin(4*arccos(a*x))+1/12/arccos(a*x)^2*cos(4*arccos(a*x))-1/3*
sin(4*arccos(a*x))/arccos(a*x)+4/3*Ci(4*arccos(a*x)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{2 \, a^{3} \arctan \left (\sqrt{a x + 1} \sqrt{-a x + 1}, a x\right )^{3} \int \frac{{\left (32 \, a^{4} x^{4} - 30 \, a^{2} x^{2} + 3\right )} \sqrt{a x + 1} \sqrt{-a x + 1}}{{\left (a^{5} x^{2} - a^{3}\right )} \arctan \left (\sqrt{a x + 1} \sqrt{-a x + 1}, a x\right )}\,{d x} + 2 \,{\left (a^{2} x^{3} -{\left (8 \, a^{2} x^{3} - 3 \, x\right )} \arctan \left (\sqrt{a x + 1} \sqrt{-a x + 1}, a x\right )^{2}\right )} \sqrt{a x + 1} \sqrt{-a x + 1} +{\left (4 \, a^{3} x^{4} - 3 \, a x^{2}\right )} \arctan \left (\sqrt{a x + 1} \sqrt{-a x + 1}, a x\right )}{6 \, a^{3} \arctan \left (\sqrt{a x + 1} \sqrt{-a x + 1}, a x\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/arccos(a*x)^4,x, algorithm="maxima")

[Out]

1/6*(6*a^3*arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)^3*integrate(1/3*(32*a^4*x^4 - 30*a^2*x^2 + 3)*sqrt(a*x +
1)*sqrt(-a*x + 1)/((a^5*x^2 - a^3)*arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)), x) + 2*(a^2*x^3 - (8*a^2*x^3
- 3*x)*arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)^2)*sqrt(a*x + 1)*sqrt(-a*x + 1) + (4*a^3*x^4 - 3*a*x^2)*arct
an2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x))/(a^3*arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)^3)

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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 )^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/arccos(a*x)^4,x, algorithm="fricas")

[Out]

integral(x^3/arccos(a*x)^4, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\operatorname{acos}^{4}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/acos(a*x)**4,x)

[Out]

Integral(x**3/acos(a*x)**4, x)

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Giac [A]  time = 1.20239, size = 169, normalized size = 1.18 \begin{align*} \frac{2 \, x^{4}}{3 \, \arccos \left (a x\right )^{2}} - \frac{8 \, \sqrt{-a^{2} x^{2} + 1} x^{3}}{3 \, a \arccos \left (a x\right )} + \frac{\sqrt{-a^{2} x^{2} + 1} x^{3}}{3 \, a \arccos \left (a x\right )^{3}} - \frac{x^{2}}{2 \, a^{2} \arccos \left (a x\right )^{2}} + \frac{\sqrt{-a^{2} x^{2} + 1} x}{a^{3} \arccos \left (a x\right )} + \frac{4 \, \operatorname{Ci}\left (4 \, \arccos \left (a x\right )\right )}{3 \, a^{4}} + \frac{\operatorname{Ci}\left (2 \, \arccos \left (a x\right )\right )}{3 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/arccos(a*x)^4,x, algorithm="giac")

[Out]

2/3*x^4/arccos(a*x)^2 - 8/3*sqrt(-a^2*x^2 + 1)*x^3/(a*arccos(a*x)) + 1/3*sqrt(-a^2*x^2 + 1)*x^3/(a*arccos(a*x)
^3) - 1/2*x^2/(a^2*arccos(a*x)^2) + sqrt(-a^2*x^2 + 1)*x/(a^3*arccos(a*x)) + 4/3*cos_integral(4*arccos(a*x))/a
^4 + 1/3*cos_integral(2*arccos(a*x))/a^4