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

Optimal. Leaf size=14 \[ -\frac{\text{Si}\left (2 \cos ^{-1}(a x)\right )}{2 a^2} \]

[Out]

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

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Rubi [A]  time = 0.0348895, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4636, 4406, 12, 3299} \[ -\frac{\text{Si}\left (2 \cos ^{-1}(a x)\right )}{2 a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0187512, size = 14, normalized size = 1. \[ -\frac{\text{Si}\left (2 \cos ^{-1}(a x)\right )}{2 a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.043, size = 13, normalized size = 0.9 \begin{align*} -{\frac{{\it Si} \left ( 2\,\arccos \left ( ax \right ) \right ) }{2\,{a}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x/arccos(a*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x/arccos(a*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x/acos(a*x), x)

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Giac [A]  time = 1.16111, size = 16, normalized size = 1.14 \begin{align*} -\frac{\operatorname{Si}\left (2 \, \arccos \left (a x\right )\right )}{2 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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