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

Optimal. Leaf size=35 \[ \frac{\sqrt{1-a^2 x^2}}{a \cos ^{-1}(a x)}-\frac{\text{CosIntegral}\left (\cos ^{-1}(a x)\right )}{a} \]

[Out]

Sqrt[1 - a^2*x^2]/(a*ArcCos[a*x]) - CosIntegral[ArcCos[a*x]]/a

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Rubi [A]  time = 0.0811583, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4622, 4724, 3302} \[ \frac{\sqrt{1-a^2 x^2}}{a \cos ^{-1}(a x)}-\frac{\text{CosIntegral}\left (\cos ^{-1}(a x)\right )}{a} \]

Antiderivative was successfully verified.

[In]

Int[ArcCos[a*x]^(-2),x]

[Out]

Sqrt[1 - a^2*x^2]/(a*ArcCos[a*x]) - CosIntegral[ArcCos[a*x]]/a

Rule 4622

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x])^(n + 1)
)/(b*c*(n + 1)), x] - Dist[c/(b*(n + 1)), Int[(x*(a + b*ArcCos[c*x])^(n + 1))/Sqrt[1 - c^2*x^2], x], x] /; Fre
eQ[{a, b, c}, x] && LtQ[n, -1]

Rule 4724

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> -Dist[d^p/c^
(m + 1), Subst[Int[(a + b*x)^n*Cos[x]^m*Sin[x]^(2*p + 1), x], x, ArcCos[c*x]], x] /; FreeQ[{a, b, c, d, e, n},
x] && EqQ[c^2*d + e, 0] && IntegerQ[2*p] && GtQ[p, -1] && IGtQ[m, 0] && (IntegerQ[p] || GtQ[d, 0])

Rule 3302

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rubi steps

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

Mathematica [A]  time = 0.041016, size = 35, normalized size = 1. \[ \frac{\sqrt{1-a^2 x^2}}{a \cos ^{-1}(a x)}-\frac{\text{CosIntegral}\left (\cos ^{-1}(a x)\right )}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCos[a*x]^(-2),x]

[Out]

Sqrt[1 - a^2*x^2]/(a*ArcCos[a*x]) - CosIntegral[ArcCos[a*x]]/a

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Maple [A]  time = 0.046, size = 32, normalized size = 0.9 \begin{align*}{\frac{1}{a} \left ({\frac{1}{\arccos \left ( ax \right ) }\sqrt{-{a}^{2}{x}^{2}+1}}-{\it Ci} \left ( \arccos \left ( ax \right ) \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/arccos(a*x)^2,x)

[Out]

1/a*((-a^2*x^2+1)^(1/2)/arccos(a*x)-Ci(arccos(a*x)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{a^{2} \arctan \left (\sqrt{a x + 1} \sqrt{-a x + 1}, a x\right ) \int \frac{\sqrt{-a x + 1} x}{\sqrt{a x + 1}{\left (a x - 1\right )} \arctan \left (\sqrt{a x + 1} \sqrt{-a x + 1}, a x\right )}\,{d x} - \sqrt{a x + 1} \sqrt{-a x + 1}}{a \arctan \left (\sqrt{a x + 1} \sqrt{-a x + 1}, a x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arccos(a*x)^2,x, algorithm="maxima")

[Out]

-(a^2*arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)*integrate(sqrt(a*x + 1)*sqrt(-a*x + 1)*x/((a^2*x^2 - 1)*arcta
n2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)), x) - sqrt(a*x + 1)*sqrt(-a*x + 1))/(a*arctan2(sqrt(a*x + 1)*sqrt(-a*x
+ 1), a*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 )^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arccos(a*x)^2,x, algorithm="fricas")

[Out]

integral(arccos(a*x)^(-2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\operatorname{acos}^{2}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/acos(a*x)**2,x)

[Out]

Integral(acos(a*x)**(-2), x)

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Giac [A]  time = 1.14229, size = 45, normalized size = 1.29 \begin{align*} -\frac{\operatorname{Ci}\left (\arccos \left (a x\right )\right )}{a} + \frac{\sqrt{-a^{2} x^{2} + 1}}{a \arccos \left (a x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arccos(a*x)^2,x, algorithm="giac")

[Out]

-cos_integral(arccos(a*x))/a + sqrt(-a^2*x^2 + 1)/(a*arccos(a*x))