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

Optimal. Leaf size=10 \[ -\frac{\text{Si}\left (\cos ^{-1}(a x)\right )}{a} \]

[Out]

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

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Rubi [A]  time = 0.0153486, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4624, 3299} \[ -\frac{\text{Si}\left (\cos ^{-1}(a x)\right )}{a} \]

Antiderivative was successfully verified.

[In]

Int[ArcCos[a*x]^(-1),x]

[Out]

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

Rule 4624

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[1/(b*c), Subst[Int[x^n*Sin[a/b - x/b], x], x, a
+ b*ArcCos[c*x]], x] /; FreeQ[{a, b, c, n}, 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{1}{\cos ^{-1}(a x)} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\sin (x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{a}\\ &=-\frac{\text{Si}\left (\cos ^{-1}(a x)\right )}{a}\\ \end{align*}

Mathematica [A]  time = 0.0235791, size = 10, normalized size = 1. \[ -\frac{\text{Si}\left (\cos ^{-1}(a x)\right )}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCos[a*x]^(-1),x]

[Out]

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

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Maple [A]  time = 0.041, size = 11, normalized size = 1.1 \begin{align*} -{\frac{{\it Si} \left ( \arccos \left ( ax \right ) \right ) }{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/arccos(a*x),x)

[Out]

-Si(arccos(a*x))/a

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\arccos \left (a x\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arccos(a*x),x, algorithm="maxima")

[Out]

integrate(1/arccos(a*x), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{\arccos \left (a x\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arccos(a*x),x, algorithm="fricas")

[Out]

integral(1/arccos(a*x), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\operatorname{acos}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/acos(a*x),x)

[Out]

Integral(1/acos(a*x), x)

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Giac [A]  time = 1.11767, size = 14, normalized size = 1.4 \begin{align*} -\frac{\operatorname{Si}\left (\arccos \left (a x\right )\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arccos(a*x),x, algorithm="giac")

[Out]

-sin_integral(arccos(a*x))/a