3.34 \(\int \frac{1}{\cos ^{-1}(a+b x)} \, dx\)

Optimal. Leaf size=12 \[ -\frac{\text{Si}\left (\cos ^{-1}(a+b x)\right )}{b} \]

[Out]

-(SinIntegral[ArcCos[a + b*x]]/b)

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Rubi [A]  time = 0.0215678, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {4804, 4624, 3299} \[ -\frac{\text{Si}\left (\cos ^{-1}(a+b x)\right )}{b} \]

Antiderivative was successfully verified.

[In]

Int[ArcCos[a + b*x]^(-1),x]

[Out]

-(SinIntegral[ArcCos[a + b*x]]/b)

Rule 4804

Int[((a_.) + ArcCos[(c_) + (d_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Dist[1/d, Subst[Int[(a + b*ArcCos[x])^n, x],
x, c + d*x], x] /; FreeQ[{a, b, c, d, n}, x]

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

Mathematica [A]  time = 0.0259629, size = 12, normalized size = 1. \[ -\frac{\text{Si}\left (\cos ^{-1}(a+b x)\right )}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCos[a + b*x]^(-1),x]

[Out]

-(SinIntegral[ArcCos[a + b*x]]/b)

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Maple [A]  time = 0.046, size = 13, normalized size = 1.1 \begin{align*} -{\frac{{\it Si} \left ( \arccos \left ( bx+a \right ) \right ) }{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/arccos(b*x + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/arccos(b*x + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/acos(a + b*x), x)

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Giac [A]  time = 1.34309, size = 16, normalized size = 1.33 \begin{align*} -\frac{\operatorname{Si}\left (\arccos \left (b x + a\right )\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sin_integral(arccos(b*x + a))/b