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

Optimal. Leaf size=38 \[ \frac{x \sqrt{1-a^2 x^2}}{a \cos ^{-1}(a x)}-\frac{\text{CosIntegral}\left (2 \cos ^{-1}(a x)\right )}{a^2} \]

[Out]

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

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Rubi [A]  time = 0.0244747, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4632, 3302} \[ \frac{x \sqrt{1-a^2 x^2}}{a \cos ^{-1}(a x)}-\frac{\text{CosIntegral}\left (2 \cos ^{-1}(a x)\right )}{a^2} \]

Antiderivative was successfully verified.

[In]

Int[x/ArcCos[a*x]^2,x]

[Out]

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

Rule 4632

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> -Simp[(x^m*Sqrt[1 - c^2*x^2]*(a + b*ArcCo
s[c*x])^(n + 1))/(b*c*(n + 1)), x] - Dist[1/(b*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[(a + b*x)^(n + 1
), Cos[x]^(m - 1)*(m - (m + 1)*Cos[x]^2), x], x], x, ArcCos[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] &&
GeQ[n, -2] && LtQ[n, -1]

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

Mathematica [A]  time = 0.0854411, size = 37, normalized size = 0.97 \[ \frac{\frac{a x \sqrt{1-a^2 x^2}}{\cos ^{-1}(a x)}-\text{CosIntegral}\left (2 \cos ^{-1}(a x)\right )}{a^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x/ArcCos[a*x]^2,x]

[Out]

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

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Maple [A]  time = 0.045, size = 30, normalized size = 0.8 \begin{align*}{\frac{1}{{a}^{2}} \left ({\frac{\sin \left ( 2\,\arccos \left ( ax \right ) \right ) }{2\,\arccos \left ( ax \right ) }}-{\it Ci} \left ( 2\,\arccos \left ( ax \right ) \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x}{\arccos \left (a x\right )^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x/arccos(a*x)^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x/acos(a*x)**2, x)

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Giac [A]  time = 1.2191, size = 49, normalized size = 1.29 \begin{align*} \frac{\sqrt{-a^{2} x^{2} + 1} x}{a \arccos \left (a x\right )} - \frac{\operatorname{Ci}\left (2 \, \arccos \left (a x\right )\right )}{a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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