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

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

[Out]

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

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Rubi [A]  time = 0.08, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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 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]

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])

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*}

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Mathematica [A]  time = 0.04, size = 35, normalized size = 1.00 \[ \frac {\sqrt {1-a^2 x^2}}{a \cos ^{-1}(a x)}-\frac {\text {Ci}\left (\cos ^{-1}(a x)\right )}{a} \]

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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fricas [F]  time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\arccos \left (a x\right )^{2}}, x\right ) \]

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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giac [A]  time = 1.72, size = 33, normalized size = 0.94 \[ -\frac {\operatorname {Ci}\left (\arccos \left (a x\right )\right )}{a} + \frac {\sqrt {-a^{2} x^{2} + 1}}{a \arccos \left (a x\right )} \]

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))

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maple [A]  time = 0.04, size = 32, normalized size = 0.91 \[ \frac {\frac {\sqrt {-a^{2} x^{2}+1}}{\arccos \left (a x \right )}-\Ci \left (\arccos \left (a x \right )\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {a^{2} \arctan \left (\sqrt {a x + 1} \sqrt {-a x + 1}, a x\right ) \int \frac {\sqrt {-a x + 1} x}{\sqrt {a x + 1} {\left (a x - 1\right )} \arctan \left (\sqrt {a x + 1} \sqrt {-a x + 1}, a x\right )}\,{d x} - \sqrt {a x + 1} \sqrt {-a x + 1}}{a \arctan \left (\sqrt {a x + 1} \sqrt {-a x + 1}, a x\right )} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{{\mathrm {acos}\left (a\,x\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/acos(a*x)^2, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\operatorname {acos}^{2}{\left (a x \right )}}\, dx \]

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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