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

Optimal. Leaf size=29 \[ -\frac{\text{Si}\left (2 \cos ^{-1}(a x)\right )}{4 a^4}-\frac{\text{Si}\left (4 \cos ^{-1}(a x)\right )}{8 a^4} \]

[Out]

-SinIntegral[2*ArcCos[a*x]]/(4*a^4) - SinIntegral[4*ArcCos[a*x]]/(8*a^4)

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Rubi [A]  time = 0.06382, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {4636, 4406, 3299} \[ -\frac{\text{Si}\left (2 \cos ^{-1}(a x)\right )}{4 a^4}-\frac{\text{Si}\left (4 \cos ^{-1}(a x)\right )}{8 a^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-SinIntegral[2*ArcCos[a*x]]/(4*a^4) - SinIntegral[4*ArcCos[a*x]]/(8*a^4)

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 3299

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

Rubi steps

\begin{align*} \int \frac{x^3}{\cos ^{-1}(a x)} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\cos ^3(x) \sin (x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{a^4}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{\sin (2 x)}{4 x}+\frac{\sin (4 x)}{8 x}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{a^4}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\sin (4 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{8 a^4}-\frac{\operatorname{Subst}\left (\int \frac{\sin (2 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{4 a^4}\\ &=-\frac{\text{Si}\left (2 \cos ^{-1}(a x)\right )}{4 a^4}-\frac{\text{Si}\left (4 \cos ^{-1}(a x)\right )}{8 a^4}\\ \end{align*}

Mathematica [A]  time = 0.0623818, size = 24, normalized size = 0.83 \[ -\frac{2 \text{Si}\left (2 \cos ^{-1}(a x)\right )+\text{Si}\left (4 \cos ^{-1}(a x)\right )}{8 a^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(2*SinIntegral[2*ArcCos[a*x]] + SinIntegral[4*ArcCos[a*x]])/(8*a^4)

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Maple [A]  time = 0.046, size = 24, normalized size = 0.8 \begin{align*}{\frac{1}{{a}^{4}} \left ( -{\frac{{\it Si} \left ( 2\,\arccos \left ( ax \right ) \right ) }{4}}-{\frac{{\it Si} \left ( 4\,\arccos \left ( ax \right ) \right ) }{8}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a^4*(-1/4*Si(2*arccos(a*x))-1/8*Si(4*arccos(a*x)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\arccos \left (a x\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^3/arccos(a*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^3/arccos(a*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**3/acos(a*x), x)

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Giac [A]  time = 1.17723, size = 34, normalized size = 1.17 \begin{align*} -\frac{\operatorname{Si}\left (4 \, \arccos \left (a x\right )\right )}{8 \, a^{4}} - \frac{\operatorname{Si}\left (2 \, \arccos \left (a x\right )\right )}{4 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*sin_integral(4*arccos(a*x))/a^4 - 1/4*sin_integral(2*arccos(a*x))/a^4