\(\int \frac {1}{\arccos (a x)^4} \, dx\) [71]

Optimal result

Integrand size = 6, antiderivative size = 78 \[ \int \frac {1}{\arccos (a x)^4} \, dx=\frac {\sqrt {1-a^2 x^2}}{3 a \arccos (a x)^3}+\frac {x}{6 \arccos (a x)^2}-\frac {\sqrt {1-a^2 x^2}}{6 a \arccos (a x)}+\frac {\operatorname {CosIntegral}(\arccos (a x))}{6 a} \]

[Out]

1/6*x/arccos(a*x)^2+1/6*Ci(arccos(a*x))/a+1/3*(-a^2*x^2+1)^(1/2)/a/arccos(a*x)^3-1/6*(-a^2*x^2+1)^(1/2)/a/arcc
os(a*x)

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {4718, 4808, 4810, 3383} \[ \int \frac {1}{\arccos (a x)^4} \, dx=-\frac {\sqrt {1-a^2 x^2}}{6 a \arccos (a x)}+\frac {\sqrt {1-a^2 x^2}}{3 a \arccos (a x)^3}+\frac {\operatorname {CosIntegral}(\arccos (a x))}{6 a}+\frac {x}{6 \arccos (a x)^2} \]

[In]

Int[ArcCos[a*x]^(-4),x]

[Out]

Sqrt[1 - a^2*x^2]/(3*a*ArcCos[a*x]^3) + x/(6*ArcCos[a*x]^2) - Sqrt[1 - a^2*x^2]/(6*a*ArcCos[a*x]) + CosIntegra
l[ArcCos[a*x]]/(6*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 4808

Int[(((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[
(-(f*x)^m/(b*c*(n + 1)))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcCos[c*x])^(n + 1), x] + Dist[f*(m/(
b*c*(n + 1)))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]], Int[(f*x)^(m - 1)*(a + b*ArcCos[c*x])^(n + 1), x], x] /
; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && 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*} \text {integral}& = \frac {\sqrt {1-a^2 x^2}}{3 a \arccos (a x)^3}+\frac {1}{3} a \int \frac {x}{\sqrt {1-a^2 x^2} \arccos (a x)^3} \, dx \\ & = \frac {\sqrt {1-a^2 x^2}}{3 a \arccos (a x)^3}+\frac {x}{6 \arccos (a x)^2}-\frac {1}{6} \int \frac {1}{\arccos (a x)^2} \, dx \\ & = \frac {\sqrt {1-a^2 x^2}}{3 a \arccos (a x)^3}+\frac {x}{6 \arccos (a x)^2}-\frac {\sqrt {1-a^2 x^2}}{6 a \arccos (a x)}-\frac {1}{6} a \int \frac {x}{\sqrt {1-a^2 x^2} \arccos (a x)} \, dx \\ & = \frac {\sqrt {1-a^2 x^2}}{3 a \arccos (a x)^3}+\frac {x}{6 \arccos (a x)^2}-\frac {\sqrt {1-a^2 x^2}}{6 a \arccos (a x)}+\frac {\text {Subst}\left (\int \frac {\cos (x)}{x} \, dx,x,\arccos (a x)\right )}{6 a} \\ & = \frac {\sqrt {1-a^2 x^2}}{3 a \arccos (a x)^3}+\frac {x}{6 \arccos (a x)^2}-\frac {\sqrt {1-a^2 x^2}}{6 a \arccos (a x)}+\frac {\operatorname {CosIntegral}(\arccos (a x))}{6 a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\arccos (a x)^4} \, dx=\frac {2 \sqrt {1-a^2 x^2}+a x \arccos (a x)-\sqrt {1-a^2 x^2} \arccos (a x)^2+\arccos (a x)^3 \operatorname {CosIntegral}(\arccos (a x))}{6 a \arccos (a x)^3} \]

[In]

Integrate[ArcCos[a*x]^(-4),x]

[Out]

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

Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.81

method	result	size
derivativedivides	\(\frac {\frac {\sqrt {-a^{2} x^{2}+1}}{3 \arccos \left (a x \right )^{3}}+\frac {a x}{6 \arccos \left (a x \right )^{2}}-\frac {\sqrt {-a^{2} x^{2}+1}}{6 \arccos \left (a x \right )}+\frac {\operatorname {Ci}\left (\arccos \left (a x \right )\right )}{6}}{a}\)	\(63\)
default	\(\frac {\frac {\sqrt {-a^{2} x^{2}+1}}{3 \arccos \left (a x \right )^{3}}+\frac {a x}{6 \arccos \left (a x \right )^{2}}-\frac {\sqrt {-a^{2} x^{2}+1}}{6 \arccos \left (a x \right )}+\frac {\operatorname {Ci}\left (\arccos \left (a x \right )\right )}{6}}{a}\)	\(63\)

[In]

int(1/arccos(a*x)^4,x,method=_RETURNVERBOSE)

[Out]

1/a*(1/3*(-a^2*x^2+1)^(1/2)/arccos(a*x)^3+1/6/arccos(a*x)^2*a*x-1/6*(-a^2*x^2+1)^(1/2)/arccos(a*x)+1/6*Ci(arcc
os(a*x)))

Fricas [F]

\[ \int \frac {1}{\arccos (a x)^4} \, dx=\int { \frac {1}{\arccos \left (a x\right )^{4}} \,d x } \]

[In]

integrate(1/arccos(a*x)^4,x, algorithm="fricas")

[Out]

integral(arccos(a*x)^(-4), x)

Sympy [F]

\[ \int \frac {1}{\arccos (a x)^4} \, dx=\int \frac {1}{\operatorname {acos}^{4}{\left (a x \right )}}\, dx \]

[In]

integrate(1/acos(a*x)**4,x)

[Out]

Integral(acos(a*x)**(-4), x)

Maxima [F]

\[ \int \frac {1}{\arccos (a x)^4} \, dx=\int { \frac {1}{\arccos \left (a x\right )^{4}} \,d x } \]

[In]

integrate(1/arccos(a*x)^4,x, algorithm="maxima")

[Out]

1/6*(6*a^2*arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)^3*integrate(1/6*sqrt(a*x + 1)*sqrt(-a*x + 1)*x/((a^2*x^2
- 1)*arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)), x) + a*x*arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x) - sqrt(
a*x + 1)*sqrt(-a*x + 1)*(arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)^2 - 2))/(a*arctan2(sqrt(a*x + 1)*sqrt(-a*x
+ 1), a*x)^3)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\arccos (a x)^4} \, dx=\frac {\operatorname {Ci}\left (\arccos \left (a x\right )\right )}{6 \, a} + \frac {x}{6 \, \arccos \left (a x\right )^{2}} - \frac {\sqrt {-a^{2} x^{2} + 1}}{6 \, a \arccos \left (a x\right )} + \frac {\sqrt {-a^{2} x^{2} + 1}}{3 \, a \arccos \left (a x\right )^{3}} \]

[In]

integrate(1/arccos(a*x)^4,x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\arccos (a x)^4} \, dx=\int \frac {1}{{\mathrm {acos}\left (a\,x\right )}^4} \,d x \]

[In]

int(1/acos(a*x)^4,x)

[Out]

int(1/acos(a*x)^4, x)