Integrand size = 6, antiderivative size = 78 \[ \int \frac {1}{\arccos (a x)^4} \, dx=\frac {\sqrt {1-a^2 x^2}}{3 a \arccos (a x)^3}+\frac {x}{6 \arccos (a x)^2}-\frac {\sqrt {1-a^2 x^2}}{6 a \arccos (a x)}+\frac {\operatorname {CosIntegral}(\arccos (a x))}{6 a} \] Output:
1/3*(-a^2*x^2+1)^(1/2)/a/arccos(a*x)^3+1/6*x/arccos(a*x)^2-1/6*(-a^2*x^2+1 )^(1/2)/a/arccos(a*x)+1/6*Ci(arccos(a*x))/a
Time = 0.04 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\arccos (a x)^4} \, dx=\frac {2 \sqrt {1-a^2 x^2}+a x \arccos (a x)-\sqrt {1-a^2 x^2} \arccos (a x)^2+\arccos (a x)^3 \operatorname {CosIntegral}(\arccos (a x))}{6 a \arccos (a x)^3} \] Input:
Integrate[ArcCos[a*x]^(-4),x]
Output:
(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)
Time = 0.55 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.15, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5133, 5223, 5133, 5225, 3042, 3783}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{\arccos (a x)^4} \, dx\) |
\(\Big \downarrow \) 5133 |
\(\displaystyle \frac {1}{3} a \int \frac {x}{\sqrt {1-a^2 x^2} \arccos (a x)^3}dx+\frac {\sqrt {1-a^2 x^2}}{3 a \arccos (a x)^3}\) |
\(\Big \downarrow \) 5223 |
\(\displaystyle \frac {1}{3} a \left (\frac {x}{2 a \arccos (a x)^2}-\frac {\int \frac {1}{\arccos (a x)^2}dx}{2 a}\right )+\frac {\sqrt {1-a^2 x^2}}{3 a \arccos (a x)^3}\) |
\(\Big \downarrow \) 5133 |
\(\displaystyle \frac {1}{3} a \left (\frac {x}{2 a \arccos (a x)^2}-\frac {a \int \frac {x}{\sqrt {1-a^2 x^2} \arccos (a x)}dx+\frac {\sqrt {1-a^2 x^2}}{a \arccos (a x)}}{2 a}\right )+\frac {\sqrt {1-a^2 x^2}}{3 a \arccos (a x)^3}\) |
\(\Big \downarrow \) 5225 |
\(\displaystyle \frac {1}{3} a \left (\frac {x}{2 a \arccos (a x)^2}-\frac {\frac {\sqrt {1-a^2 x^2}}{a \arccos (a x)}-\frac {\int \frac {a x}{\arccos (a x)}d\arccos (a x)}{a}}{2 a}\right )+\frac {\sqrt {1-a^2 x^2}}{3 a \arccos (a x)^3}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{3} a \left (\frac {x}{2 a \arccos (a x)^2}-\frac {\frac {\sqrt {1-a^2 x^2}}{a \arccos (a x)}-\frac {\int \frac {\sin \left (\arccos (a x)+\frac {\pi }{2}\right )}{\arccos (a x)}d\arccos (a x)}{a}}{2 a}\right )+\frac {\sqrt {1-a^2 x^2}}{3 a \arccos (a x)^3}\) |
\(\Big \downarrow \) 3783 |
\(\displaystyle \frac {1}{3} a \left (\frac {x}{2 a \arccos (a x)^2}-\frac {\frac {\sqrt {1-a^2 x^2}}{a \arccos (a x)}-\frac {\operatorname {CosIntegral}(\arccos (a x))}{a}}{2 a}\right )+\frac {\sqrt {1-a^2 x^2}}{3 a \arccos (a x)^3}\) |
Input:
Int[ArcCos[a*x]^(-4),x]
Output:
Sqrt[1 - a^2*x^2]/(3*a*ArcCos[a*x]^3) + (a*(x/(2*a*ArcCos[a*x]^2) - (Sqrt[ 1 - a^2*x^2]/(a*ArcCos[a*x]) - CosIntegral[ArcCos[a*x]]/a)/(2*a)))/3
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosInte gral[e - Pi/2 + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]
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] - Simp[c/(b*(n + 1 )) Int[x*((a + b*ArcCos[c*x])^(n + 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ [{a, b, c}, x] && LtQ[n, -1]
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] + Simp[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]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^ 2)^(p_.), x_Symbol] :> Simp[(-(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] && IGtQ[m, 0]
Time = 0.05 (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 | \(\frac {\frac {\sqrt {-a^{2} x^{2}+1}}{3 \arccos \left (a x \right )^{3}}+\frac {a x}{6 \arccos \left (a x \right )^{2}}-\frac {\sqrt {-a^{2} x^{2}+1}}{6 \arccos \left (a x \right )}+\frac {\operatorname {Ci}\left (\arccos \left (a x \right )\right )}{6}}{a}\) | \(63\) |
Input:
int(1/arccos(a*x)^4,x,method=_RETURNVERBOSE)
Output:
1/a*(1/3/arccos(a*x)^3*(-a^2*x^2+1)^(1/2)+1/6*a*x/arccos(a*x)^2-1/6/arccos (a*x)*(-a^2*x^2+1)^(1/2)+1/6*Ci(arccos(a*x)))
\[ \int \frac {1}{\arccos (a x)^4} \, dx=\int { \frac {1}{\arccos \left (a x\right )^{4}} \,d x } \] Input:
integrate(1/arccos(a*x)^4,x, algorithm="fricas")
Output:
integral(arccos(a*x)^(-4), x)
\[ \int \frac {1}{\arccos (a x)^4} \, dx=\int \frac {1}{\operatorname {acos}^{4}{\left (a x \right )}}\, dx \] Input:
integrate(1/acos(a*x)**4,x)
Output:
Integral(acos(a*x)**(-4), x)
\[ \int \frac {1}{\arccos (a x)^4} \, dx=\int { \frac {1}{\arccos \left (a x\right )^{4}} \,d x } \] Input:
integrate(1/arccos(a*x)^4,x, algorithm="maxima")
Output:
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)
Time = 0.12 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\arccos (a x)^4} \, dx=\frac {\operatorname {Ci}\left (\arccos \left (a x\right )\right )}{6 \, a} + \frac {x}{6 \, \arccos \left (a x\right )^{2}} - \frac {\sqrt {-a^{2} x^{2} + 1}}{6 \, a \arccos \left (a x\right )} + \frac {\sqrt {-a^{2} x^{2} + 1}}{3 \, a \arccos \left (a x\right )^{3}} \] Input:
integrate(1/arccos(a*x)^4,x, algorithm="giac")
Output:
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)
Timed out. \[ \int \frac {1}{\arccos (a x)^4} \, dx=\int \frac {1}{{\mathrm {acos}\left (a\,x\right )}^4} \,d x \] Input:
int(1/acos(a*x)^4,x)
Output:
int(1/acos(a*x)^4, x)
\[ \int \frac {1}{\arccos (a x)^4} \, dx=\int \frac {1}{\mathit {acos} \left (a x \right )^{4}}d x \] Input:
int(1/acos(a*x)^4,x)
Output:
int(1/acos(a*x)**4,x)