\(\int \frac {\arccos (a x)}{x^4} \, dx\) [9]

Optimal result

Integrand size = 8, antiderivative size = 56 \[ \int \frac {\arccos (a x)}{x^4} \, dx=\frac {a \sqrt {1-a^2 x^2}}{6 x^2}-\frac {\arccos (a x)}{3 x^3}+\frac {1}{6} a^3 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right ) \]

[Out]

-1/3*arccos(a*x)/x^3+1/6*a^3*arctanh((-a^2*x^2+1)^(1/2))+1/6*a*(-a^2*x^2+1)^(1/2)/x^2

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {4724, 272, 44, 65, 214} \[ \int \frac {\arccos (a x)}{x^4} \, dx=\frac {a \sqrt {1-a^2 x^2}}{6 x^2}+\frac {1}{6} a^3 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {\arccos (a x)}{3 x^3} \]

[In]

Int[ArcCos[a*x]/x^4,x]

[Out]

(a*Sqrt[1 - a^2*x^2])/(6*x^2) - ArcCos[a*x]/(3*x^3) + (a^3*ArcTanh[Sqrt[1 - a^2*x^2]])/6

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)}{3 x^3}-\frac {1}{3} a \int \frac {1}{x^3 \sqrt {1-a^2 x^2}} \, dx \\ & = -\frac {\arccos (a x)}{3 x^3}-\frac {1}{6} a \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}}{6 x^2}-\frac {\arccos (a x)}{3 x^3}-\frac {1}{12} a^3 \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}}{6 x^2}-\frac {\arccos (a x)}{3 x^3}+\frac {1}{6} a \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}}{6 x^2}-\frac {\arccos (a x)}{3 x^3}+\frac {1}{6} a^3 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.20 \[ \int \frac {\arccos (a x)}{x^4} \, dx=\frac {a \sqrt {1-a^2 x^2}}{6 x^2}-\frac {\arccos (a x)}{3 x^3}-\frac {1}{6} a^3 \log (x)+\frac {1}{6} a^3 \log \left (1+\sqrt {1-a^2 x^2}\right ) \]

[In]

Integrate[ArcCos[a*x]/x^4,x]

[Out]

(a*Sqrt[1 - a^2*x^2])/(6*x^2) - ArcCos[a*x]/(3*x^3) - (a^3*Log[x])/6 + (a^3*Log[1 + Sqrt[1 - a^2*x^2]])/6

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.89

method	result	size
parts	\(-\frac {\arccos \left (a x \right )}{3 x^{3}}-\frac {a \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{2 x^{2}}-\frac {a^{2} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}\right )}{3}\)	\(50\)
derivativedivides	\(a^{3} \left (-\frac {\arccos \left (a x \right )}{3 a^{3} x^{3}}+\frac {\sqrt {-a^{2} x^{2}+1}}{6 a^{2} x^{2}}+\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{6}\right )\)	\(53\)
default	\(a^{3} \left (-\frac {\arccos \left (a x \right )}{3 a^{3} x^{3}}+\frac {\sqrt {-a^{2} x^{2}+1}}{6 a^{2} x^{2}}+\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{6}\right )\)	\(53\)

[In]

int(arccos(a*x)/x^4,x,method=_RETURNVERBOSE)

[Out]

-1/3*arccos(a*x)/x^3-1/3*a*(-1/2/x^2*(-a^2*x^2+1)^(1/2)-1/2*a^2*arctanh(1/(-a^2*x^2+1)^(1/2)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 110 vs. \(2 (46) = 92\).

Time = 0.27 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.96 \[ \int \frac {\arccos (a x)}{x^4} \, dx=\frac {a^{3} x^{3} \log \left (\sqrt {-a^{2} x^{2} + 1} + 1\right ) - a^{3} x^{3} \log \left (\sqrt {-a^{2} x^{2} + 1} - 1\right ) - 4 \, x^{3} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} a x}{a^{2} x^{2} - 1}\right ) + 2 \, \sqrt {-a^{2} x^{2} + 1} a x + 4 \, {\left (x^{3} - 1\right )} \arccos \left (a x\right )}{12 \, x^{3}} \]

[In]

integrate(arccos(a*x)/x^4,x, algorithm="fricas")

[Out]

1/12*(a^3*x^3*log(sqrt(-a^2*x^2 + 1) + 1) - a^3*x^3*log(sqrt(-a^2*x^2 + 1) - 1) - 4*x^3*arctan(sqrt(-a^2*x^2 +
1)*a*x/(a^2*x^2 - 1)) + 2*sqrt(-a^2*x^2 + 1)*a*x + 4*(x^3 - 1)*arccos(a*x))/x^3

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.68 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.95 \[ \int \frac {\arccos (a x)}{x^4} \, dx=- \frac {a \left (\begin {cases} - \frac {a^{2} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{2} + \frac {a}{2 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {1}{2 a x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {i a^{2} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{2} - \frac {i a \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{2 x} & \text {otherwise} \end {cases}\right )}{3} - \frac {\operatorname {acos}{\left (a x \right )}}{3 x^{3}} \]

[In]

integrate(acos(a*x)/x**4,x)

[Out]

-a*Piecewise((-a**2*acosh(1/(a*x))/2 + a/(2*x*sqrt(-1 + 1/(a**2*x**2))) - 1/(2*a*x**3*sqrt(-1 + 1/(a**2*x**2))
), 1/Abs(a**2*x**2) > 1), (I*a**2*asin(1/(a*x))/2 - I*a*sqrt(1 - 1/(a**2*x**2))/(2*x), True))/3 - acos(a*x)/(3
*x**3)

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.07 \[ \int \frac {\arccos (a x)}{x^4} \, dx=\frac {1}{6} \, {\left (a^{2} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \frac {\sqrt {-a^{2} x^{2} + 1}}{x^{2}}\right )} a - \frac {\arccos \left (a x\right )}{3 \, x^{3}} \]

[In]

integrate(arccos(a*x)/x^4,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.38 \[ \int \frac {\arccos (a x)}{x^4} \, dx=\frac {a^{4} \log \left (\sqrt {-a^{2} x^{2} + 1} + 1\right ) - a^{4} \log \left (-\sqrt {-a^{2} x^{2} + 1} + 1\right ) + \frac {2 \, \sqrt {-a^{2} x^{2} + 1} a^{2}}{x^{2}}}{12 \, a} - \frac {\arccos \left (a x\right )}{3 \, x^{3}} \]

[In]

integrate(arccos(a*x)/x^4,x, algorithm="giac")

[Out]

1/12*(a^4*log(sqrt(-a^2*x^2 + 1) + 1) - a^4*log(-sqrt(-a^2*x^2 + 1) + 1) + 2*sqrt(-a^2*x^2 + 1)*a^2/x^2)/a - 1
/3*arccos(a*x)/x^3

Mupad [F(-1)]

Timed out. \[ \int \frac {\arccos (a x)}{x^4} \, dx=\int \frac {\mathrm {acos}\left (a\,x\right )}{x^4} \,d x \]

[In]

int(acos(a*x)/x^4,x)

[Out]

int(acos(a*x)/x^4, x)