\(\int \frac {\arccos (a x)}{x^6} \, dx\) [11]

Optimal result

Integrand size = 8, antiderivative size = 80 \[ \int \frac {\arccos (a x)}{x^6} \, dx=\frac {a \sqrt {1-a^2 x^2}}{20 x^4}+\frac {3 a^3 \sqrt {1-a^2 x^2}}{40 x^2}-\frac {\arccos (a x)}{5 x^5}+\frac {3}{40} a^5 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {4724, 272, 44, 65, 214} \[ \int \frac {\arccos (a x)}{x^6} \, dx=\frac {a \sqrt {1-a^2 x^2}}{20 x^4}+\frac {3}{40} a^5 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )+\frac {3 a^3 \sqrt {1-a^2 x^2}}{40 x^2}-\frac {\arccos (a x)}{5 x^5} \]

[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 44

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] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

Rule 65

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 214

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

Rule 272

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 4724

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]

Rubi steps \begin{align*} \text {integral}& = -\frac {\arccos (a x)}{5 x^5}-\frac {1}{5} a \int \frac {1}{x^5 \sqrt {1-a^2 x^2}} \, dx \\ & = -\frac {\arccos (a x)}{5 x^5}-\frac {1}{10} a \text {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 {\arccos (a x)}{5 x^5}-\frac {1}{40} \left (3 a^3\right ) \text {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 {\arccos (a x)}{5 x^5}-\frac {1}{80} \left (3 a^5\right ) \text {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 {\arccos (a x)}{5 x^5}+\frac {1}{40} \left (3 a^3\right ) \text {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 {\arccos (a x)}{5 x^5}+\frac {3}{40} a^5 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.90 \[ \int \frac {\arccos (a x)}{x^6} \, dx=\frac {1}{40} \left (\frac {a \sqrt {1-a^2 x^2} \left (2+3 a^2 x^2\right )}{x^4}-\frac {8 \arccos (a x)}{x^5}-3 a^5 \log (x)+3 a^5 \log \left (1+\sqrt {1-a^2 x^2}\right )\right ) \]

[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

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.91

method	result	size
derivativedivides	\(a^{5} \left (-\frac {\arccos \left (a x \right )}{5 a^{5} x^{5}}+\frac {\sqrt {-a^{2} x^{2}+1}}{20 a^{4} x^{4}}+\frac {3 \sqrt {-a^{2} x^{2}+1}}{40 a^{2} x^{2}}+\frac {3 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{40}\right )\)	\(73\)
default	\(a^{5} \left (-\frac {\arccos \left (a x \right )}{5 a^{5} x^{5}}+\frac {\sqrt {-a^{2} x^{2}+1}}{20 a^{4} x^{4}}+\frac {3 \sqrt {-a^{2} x^{2}+1}}{40 a^{2} x^{2}}+\frac {3 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{40}\right )\)	\(73\)
parts	\(-\frac {\arccos \left (a x \right )}{5 x^{5}}-\frac {a \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{4 x^{4}}+\frac {3 a^{2} \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{2 x^{2}}-\frac {a^{2} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}\right )}{4}\right )}{5}\)	\(73\)

[In]

int(arccos(a*x)/x^6,x,method=_RETURNVERBOSE)

[Out]

a^5*(-1/5/a^5/x^5*arccos(a*x)+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)))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.52 \[ \int \frac {\arccos (a x)}{x^6} \, dx=\frac {3 \, a^{5} x^{5} \log \left (\sqrt {-a^{2} x^{2} + 1} + 1\right ) - 3 \, a^{5} x^{5} \log \left (\sqrt {-a^{2} x^{2} + 1} - 1\right ) - 16 \, x^{5} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} a x}{a^{2} x^{2} - 1}\right ) + 16 \, {\left (x^{5} - 1\right )} \arccos \left (a x\right ) + 2 \, {\left (3 \, a^{3} x^{3} + 2 \, a x\right )} \sqrt {-a^{2} x^{2} + 1}}{80 \, x^{5}} \]

[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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 3.32 (sec) , antiderivative size = 184, normalized size of antiderivative = 2.30 \[ \int \frac {\arccos (a x)}{x^6} \, dx=- \frac {a \left (\begin {cases} - \frac {3 a^{4} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{8} + \frac {3 a^{3}}{8 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {a}{8 x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {1}{4 a x^{5} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {3 i a^{4} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{8} - \frac {3 i a^{3}}{8 x \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {i a}{8 x^{3} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {i}{4 a x^{5} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} & \text {otherwise} \end {cases}\right )}{5} - \frac {\operatorname {acos}{\left (a x \right )}}{5 x^{5}} \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.02 \[ \int \frac {\arccos (a x)}{x^6} \, dx=\frac {1}{40} \, {\left (3 \, a^{4} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \frac {3 \, \sqrt {-a^{2} x^{2} + 1} a^{2}}{x^{2}} + \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{x^{4}}\right )} a - \frac {\arccos \left (a x\right )}{5 \, x^{5}} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.26 \[ \int \frac {\arccos (a x)}{x^6} \, dx=\frac {3 \, a^{6} \log \left (\sqrt {-a^{2} x^{2} + 1} + 1\right ) - 3 \, a^{6} \log \left (-\sqrt {-a^{2} x^{2} + 1} + 1\right ) - \frac {2 \, {\left (3 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{6} - 5 \, \sqrt {-a^{2} x^{2} + 1} a^{6}\right )}}{a^{4} x^{4}}}{80 \, a} - \frac {\arccos \left (a x\right )}{5 \, x^{5}} \]

[In]

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

[Out]

1/80*(3*a^6*log(sqrt(-a^2*x^2 + 1) + 1) - 3*a^6*log(-sqrt(-a^2*x^2 + 1) + 1) - 2*(3*(-a^2*x^2 + 1)^(3/2)*a^6 -
5*sqrt(-a^2*x^2 + 1)*a^6)/(a^4*x^4))/a - 1/5*arccos(a*x)/x^5

Mupad [F(-1)]

Timed out. \[ \int \frac {\arccos (a x)}{x^6} \, dx=\int \frac {\mathrm {acos}\left (a\,x\right )}{x^6} \,d x \]

[In]

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

[Out]

int(acos(a*x)/x^6, x)