3.1.57 \(\int \frac {1}{\text {ArcCos}(a x)^2} \, dx\) [57]

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

[Out]

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

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Rubi [A]
time = 0.05, 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 = {4718, 4810, 3383} \begin {gather*} \frac {\sqrt {1-a^2 x^2}}{a \text {ArcCos}(a x)}-\frac {\text {CosIntegral}(\text {ArcCos}(a x))}{a} \end {gather*}

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 3383

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 4718

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] /; F
reeQ[{a, b, c}, x] && LtQ[n, -1]

Rule 4810

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[(-(b*c^
(m + 1))^(-1))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p], Subst[Int[x^n*Cos[-a/b + x/b]^m*Sin[-a/b + x/b]^(2*p + 1),
x], x, a + b*ArcCos[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && IGtQ[2*p + 2, 0] && IGt
Q[m, 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 {\text {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 = 0.03, size = 35, normalized size = 1.00 \begin {gather*} \frac {\sqrt {1-a^2 x^2}}{a \text {ArcCos}(a x)}-\frac {\text {CosIntegral}(\text {ArcCos}(a x))}{a} \end {gather*}

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 = 0.07, size = 32, normalized size = 0.91

method	result	size
derivativedivides	\(\frac {\frac {\sqrt {-a^{2} x^{2}+1}}{\arccos \left (a x \right )}-\cosineIntegral \left (\arccos \left (a x \right )\right )}{a}\)	\(32\)
default	\(\frac {\frac {\sqrt {-a^{2} x^{2}+1}}{\arccos \left (a x \right )}-\cosineIntegral \left (\arccos \left (a x \right )\right )}{a}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\operatorname {acos}^{2}{\left (a x \right )}}\, dx \end {gather*}

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 = 0.43, size = 33, normalized size = 0.94 \begin {gather*} -\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 {gather*}

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

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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