3.11 \(\int \frac{\cos ^{-1}(a x)}{x^6} \, dx\)

Optimal. Leaf size=80 \[ \frac{3 a^3 \sqrt{1-a^2 x^2}}{40 x^2}+\frac{a \sqrt{1-a^2 x^2}}{20 x^4}+\frac{3}{40} a^5 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-\frac{\cos ^{-1}(a x)}{5 x^5} \]

[Out]

(a*Sqrt[1 - a^2*x^2])/(20*x^4) + (3*a^3*Sqrt[1 - a^2*x^2])/(40*x^2) - ArcCos[a*x]/(5*x^5) + (3*a^5*ArcTanh[Sqr
t[1 - a^2*x^2]])/40

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Rubi [A]  time = 0.0466105, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {4628, 266, 51, 63, 208} \[ \frac{3 a^3 \sqrt{1-a^2 x^2}}{40 x^2}+\frac{a \sqrt{1-a^2 x^2}}{20 x^4}+\frac{3}{40} a^5 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-\frac{\cos ^{-1}(a x)}{5 x^5} \]

Antiderivative was successfully verified.

[In]

Int[ArcCos[a*x]/x^6,x]

[Out]

(a*Sqrt[1 - a^2*x^2])/(20*x^4) + (3*a^3*Sqrt[1 - a^2*x^2])/(40*x^2) - ArcCos[a*x]/(5*x^5) + (3*a^5*ArcTanh[Sqr
t[1 - a^2*x^2]])/40

Rule 4628

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

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{\cos ^{-1}(a x)}{x^6} \, dx &=-\frac{\cos ^{-1}(a x)}{5 x^5}-\frac{1}{5} a \int \frac{1}{x^5 \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{\cos ^{-1}(a x)}{5 x^5}-\frac{1}{10} a \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{1-a^2 x}} \, dx,x,x^2\right )\\ &=\frac{a \sqrt{1-a^2 x^2}}{20 x^4}-\frac{\cos ^{-1}(a x)}{5 x^5}-\frac{1}{40} \left (3 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1-a^2 x}} \, dx,x,x^2\right )\\ &=\frac{a \sqrt{1-a^2 x^2}}{20 x^4}+\frac{3 a^3 \sqrt{1-a^2 x^2}}{40 x^2}-\frac{\cos ^{-1}(a x)}{5 x^5}-\frac{1}{80} \left (3 a^5\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )\\ &=\frac{a \sqrt{1-a^2 x^2}}{20 x^4}+\frac{3 a^3 \sqrt{1-a^2 x^2}}{40 x^2}-\frac{\cos ^{-1}(a x)}{5 x^5}+\frac{1}{40} \left (3 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{a^2}-\frac{x^2}{a^2}} \, dx,x,\sqrt{1-a^2 x^2}\right )\\ &=\frac{a \sqrt{1-a^2 x^2}}{20 x^4}+\frac{3 a^3 \sqrt{1-a^2 x^2}}{40 x^2}-\frac{\cos ^{-1}(a x)}{5 x^5}+\frac{3}{40} a^5 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )\\ \end{align*}

Mathematica [A]  time = 0.061937, size = 72, normalized size = 0.9 \[ \frac{1}{40} \left (\frac{a \sqrt{1-a^2 x^2} \left (3 a^2 x^2+2\right )}{x^4}+3 a^5 \log \left (\sqrt{1-a^2 x^2}+1\right )-3 a^5 \log (x)-\frac{8 \cos ^{-1}(a x)}{x^5}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCos[a*x]/x^6,x]

[Out]

((a*Sqrt[1 - a^2*x^2]*(2 + 3*a^2*x^2))/x^4 - (8*ArcCos[a*x])/x^5 - 3*a^5*Log[x] + 3*a^5*Log[1 + Sqrt[1 - a^2*x
^2]])/40

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Maple [A]  time = 0.003, size = 73, normalized size = 0.9 \begin{align*}{a}^{5} \left ( -{\frac{\arccos \left ( ax \right ) }{5\,{a}^{5}{x}^{5}}}+{\frac{1}{20\,{a}^{4}{x}^{4}}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{3}{40\,{a}^{2}{x}^{2}}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{3}{40}{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) } \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccos(a*x)/x^6,x)

[Out]

a^5*(-1/5*arccos(a*x)/a^5/x^5+1/20/a^4/x^4*(-a^2*x^2+1)^(1/2)+3/40/a^2/x^2*(-a^2*x^2+1)^(1/2)+3/40*arctanh(1/(
-a^2*x^2+1)^(1/2)))

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Maxima [A]  time = 1.45141, size = 111, normalized size = 1.39 \begin{align*} \frac{1}{40} \,{\left (3 \, a^{4} \log \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) + \frac{3 \, \sqrt{-a^{2} x^{2} + 1} a^{2}}{x^{2}} + \frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{x^{4}}\right )} a - \frac{\arccos \left (a x\right )}{5 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccos(a*x)/x^6,x, algorithm="maxima")

[Out]

1/40*(3*a^4*log(2*sqrt(-a^2*x^2 + 1)/abs(x) + 2/abs(x)) + 3*sqrt(-a^2*x^2 + 1)*a^2/x^2 + 2*sqrt(-a^2*x^2 + 1)/
x^4)*a - 1/5*arccos(a*x)/x^5

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Fricas [A]  time = 2.19651, size = 289, normalized size = 3.61 \begin{align*} \frac{3 \, a^{5} x^{5} \log \left (\sqrt{-a^{2} x^{2} + 1} + 1\right ) - 3 \, a^{5} x^{5} \log \left (\sqrt{-a^{2} x^{2} + 1} - 1\right ) - 16 \, x^{5} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} a x}{a^{2} x^{2} - 1}\right ) + 16 \,{\left (x^{5} - 1\right )} \arccos \left (a x\right ) + 2 \,{\left (3 \, a^{3} x^{3} + 2 \, a x\right )} \sqrt{-a^{2} x^{2} + 1}}{80 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccos(a*x)/x^6,x, algorithm="fricas")

[Out]

1/80*(3*a^5*x^5*log(sqrt(-a^2*x^2 + 1) + 1) - 3*a^5*x^5*log(sqrt(-a^2*x^2 + 1) - 1) - 16*x^5*arctan(sqrt(-a^2*
x^2 + 1)*a*x/(a^2*x^2 - 1)) + 16*(x^5 - 1)*arccos(a*x) + 2*(3*a^3*x^3 + 2*a*x)*sqrt(-a^2*x^2 + 1))/x^5

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Sympy [C]  time = 6.43766, size = 184, normalized size = 2.3 \begin{align*} - \frac{a \left (\begin{cases} - \frac{3 a^{4} \operatorname{acosh}{\left (\frac{1}{a x} \right )}}{8} + \frac{3 a^{3}}{8 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{a}{8 x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{1}{4 a x^{5} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac{3 i a^{4} \operatorname{asin}{\left (\frac{1}{a x} \right )}}{8} - \frac{3 i a^{3}}{8 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i a}{8 x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{4 a x^{5} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right )}{5} - \frac{\operatorname{acos}{\left (a x \right )}}{5 x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acos(a*x)/x**6,x)

[Out]

-a*Piecewise((-3*a**4*acosh(1/(a*x))/8 + 3*a**3/(8*x*sqrt(-1 + 1/(a**2*x**2))) - a/(8*x**3*sqrt(-1 + 1/(a**2*x
**2))) - 1/(4*a*x**5*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (3*I*a**4*asin(1/(a*x))/8 - 3*I*a**3/(8
*x*sqrt(1 - 1/(a**2*x**2))) + I*a/(8*x**3*sqrt(1 - 1/(a**2*x**2))) + I/(4*a*x**5*sqrt(1 - 1/(a**2*x**2))), Tru
e))/5 - acos(a*x)/(5*x**5)

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Giac [A]  time = 1.15694, size = 120, normalized size = 1.5 \begin{align*} -\frac{1}{80} \, a^{5}{\left (\frac{2 \,{\left (3 \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} - 5 \, \sqrt{-a^{2} x^{2} + 1}\right )}}{a^{4} x^{4}} - 3 \, \log \left (\sqrt{-a^{2} x^{2} + 1} + 1\right ) + 3 \, \log \left (-\sqrt{-a^{2} x^{2} + 1} + 1\right )\right )} - \frac{\arccos \left (a x\right )}{5 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccos(a*x)/x^6,x, algorithm="giac")

[Out]

-1/80*a^5*(2*(3*(-a^2*x^2 + 1)^(3/2) - 5*sqrt(-a^2*x^2 + 1))/(a^4*x^4) - 3*log(sqrt(-a^2*x^2 + 1) + 1) + 3*log
(-sqrt(-a^2*x^2 + 1) + 1)) - 1/5*arccos(a*x)/x^5