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

Optimal. Leaf size=27 \[ -\frac{\text{Si}\left (\cos ^{-1}(a x)\right )}{4 a^3}-\frac{\text{Si}\left (3 \cos ^{-1}(a x)\right )}{4 a^3} \]

[Out]

-SinIntegral[ArcCos[a*x]]/(4*a^3) - SinIntegral[3*ArcCos[a*x]]/(4*a^3)

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Rubi [A]  time = 0.0600118, antiderivative size = 27, 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} \[ -\frac{\text{Si}\left (\cos ^{-1}(a x)\right )}{4 a^3}-\frac{\text{Si}\left (3 \cos ^{-1}(a x)\right )}{4 a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-SinIntegral[ArcCos[a*x]]/(4*a^3) - SinIntegral[3*ArcCos[a*x]]/(4*a^3)

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

Mathematica [A]  time = 0.0527136, size = 20, normalized size = 0.74 \[ -\frac{\text{Si}\left (\cos ^{-1}(a x)\right )+\text{Si}\left (3 \cos ^{-1}(a x)\right )}{4 a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(SinIntegral[ArcCos[a*x]] + SinIntegral[3*ArcCos[a*x]])/(4*a^3)

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Maple [A]  time = 0.045, size = 22, normalized size = 0.8 \begin{align*}{\frac{1}{{a}^{3}} \left ( -{\frac{{\it Si} \left ( 3\,\arccos \left ( ax \right ) \right ) }{4}}-{\frac{{\it Si} \left ( \arccos \left ( ax \right ) \right ) }{4}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a^3*(-1/4*Si(3*arccos(a*x))-1/4*Si(arccos(a*x)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\arccos \left (a x\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2/arccos(a*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^2/arccos(a*x), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\operatorname{acos}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/acos(a*x),x)

[Out]

Integral(x**2/acos(a*x), x)

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Giac [A]  time = 1.17207, size = 31, normalized size = 1.15 \begin{align*} -\frac{\operatorname{Si}\left (3 \, \arccos \left (a x\right )\right )}{4 \, a^{3}} - \frac{\operatorname{Si}\left (\arccos \left (a x\right )\right )}{4 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*sin_integral(3*arccos(a*x))/a^3 - 1/4*sin_integral(arccos(a*x))/a^3