Integrand size = 10, antiderivative size = 82 \[ \int \frac {x^2}{\arccos (a x)^3} \, dx=\frac {x^2 \sqrt {1-a^2 x^2}}{2 a \arccos (a x)^2}-\frac {x}{a^2 \arccos (a x)}+\frac {3 x^3}{2 \arccos (a x)}+\frac {\text {Si}(\arccos (a x))}{8 a^3}+\frac {9 \text {Si}(3 \arccos (a x))}{8 a^3} \]
-x/a^2/arccos(a*x)+3/2*x^3/arccos(a*x)+1/8*Si(arccos(a*x))/a^3+9/8*Si(3*ar ccos(a*x))/a^3+1/2*x^2*(-a^2*x^2+1)^(1/2)/a/arccos(a*x)^2
Time = 0.10 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.79 \[ \int \frac {x^2}{\arccos (a x)^3} \, dx=\frac {\frac {4 a x \left (a x \sqrt {1-a^2 x^2}+\left (-2+3 a^2 x^2\right ) \arccos (a x)\right )}{\arccos (a x)^2}+\text {Si}(\arccos (a x))+9 \text {Si}(3 \arccos (a x))}{8 a^3} \]
((4*a*x*(a*x*Sqrt[1 - a^2*x^2] + (-2 + 3*a^2*x^2)*ArcCos[a*x]))/ArcCos[a*x ]^2 + SinIntegral[ArcCos[a*x]] + 9*SinIntegral[3*ArcCos[a*x]])/(8*a^3)
Time = 0.75 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.24, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {5145, 5223, 5135, 3042, 3780, 5147, 4906, 2009}
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.
\(\displaystyle \int \frac {x^2}{\arccos (a x)^3} \, dx\) |
\(\Big \downarrow \) 5145 |
\(\displaystyle -\frac {\int \frac {x}{\sqrt {1-a^2 x^2} \arccos (a x)^2}dx}{a}+\frac {3}{2} a \int \frac {x^3}{\sqrt {1-a^2 x^2} \arccos (a x)^2}dx+\frac {x^2 \sqrt {1-a^2 x^2}}{2 a \arccos (a x)^2}\) |
\(\Big \downarrow \) 5223 |
\(\displaystyle \frac {3}{2} a \left (\frac {x^3}{a \arccos (a x)}-\frac {3 \int \frac {x^2}{\arccos (a x)}dx}{a}\right )-\frac {\frac {x}{a \arccos (a x)}-\frac {\int \frac {1}{\arccos (a x)}dx}{a}}{a}+\frac {x^2 \sqrt {1-a^2 x^2}}{2 a \arccos (a x)^2}\) |
\(\Big \downarrow \) 5135 |
\(\displaystyle -\frac {\frac {\int \frac {\sqrt {1-a^2 x^2}}{\arccos (a x)}d\arccos (a x)}{a^2}+\frac {x}{a \arccos (a x)}}{a}+\frac {3}{2} a \left (\frac {x^3}{a \arccos (a x)}-\frac {3 \int \frac {x^2}{\arccos (a x)}dx}{a}\right )+\frac {x^2 \sqrt {1-a^2 x^2}}{2 a \arccos (a x)^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\frac {\int \frac {\sin (\arccos (a x))}{\arccos (a x)}d\arccos (a x)}{a^2}+\frac {x}{a \arccos (a x)}}{a}+\frac {3}{2} a \left (\frac {x^3}{a \arccos (a x)}-\frac {3 \int \frac {x^2}{\arccos (a x)}dx}{a}\right )+\frac {x^2 \sqrt {1-a^2 x^2}}{2 a \arccos (a x)^2}\) |
\(\Big \downarrow \) 3780 |
\(\displaystyle \frac {3}{2} a \left (\frac {x^3}{a \arccos (a x)}-\frac {3 \int \frac {x^2}{\arccos (a x)}dx}{a}\right )-\frac {\frac {\text {Si}(\arccos (a x))}{a^2}+\frac {x}{a \arccos (a x)}}{a}+\frac {x^2 \sqrt {1-a^2 x^2}}{2 a \arccos (a x)^2}\) |
\(\Big \downarrow \) 5147 |
\(\displaystyle \frac {3}{2} a \left (\frac {3 \int \frac {a^2 x^2 \sqrt {1-a^2 x^2}}{\arccos (a x)}d\arccos (a x)}{a^4}+\frac {x^3}{a \arccos (a x)}\right )-\frac {\frac {\text {Si}(\arccos (a x))}{a^2}+\frac {x}{a \arccos (a x)}}{a}+\frac {x^2 \sqrt {1-a^2 x^2}}{2 a \arccos (a x)^2}\) |
\(\Big \downarrow \) 4906 |
\(\displaystyle \frac {3}{2} a \left (\frac {3 \int \left (\frac {\sin (3 \arccos (a x))}{4 \arccos (a x)}+\frac {\sqrt {1-a^2 x^2}}{4 \arccos (a x)}\right )d\arccos (a x)}{a^4}+\frac {x^3}{a \arccos (a x)}\right )-\frac {\frac {\text {Si}(\arccos (a x))}{a^2}+\frac {x}{a \arccos (a x)}}{a}+\frac {x^2 \sqrt {1-a^2 x^2}}{2 a \arccos (a x)^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {3}{2} a \left (\frac {3 \left (\frac {1}{4} \text {Si}(\arccos (a x))+\frac {1}{4} \text {Si}(3 \arccos (a x))\right )}{a^4}+\frac {x^3}{a \arccos (a x)}\right )-\frac {\frac {\text {Si}(\arccos (a x))}{a^2}+\frac {x}{a \arccos (a x)}}{a}+\frac {x^2 \sqrt {1-a^2 x^2}}{2 a \arccos (a x)^2}\) |
(x^2*Sqrt[1 - a^2*x^2])/(2*a*ArcCos[a*x]^2) - (x/(a*ArcCos[a*x]) + SinInte gral[ArcCos[a*x]]/a^2)/a + (3*a*(x^3/(a*ArcCos[a*x]) + (3*(SinIntegral[Arc Cos[a*x]]/4 + SinIntegral[3*ArcCos[a*x]]/4))/a^4))/2
3.1.62.3.1 Defintions of rubi rules used
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinInte gral[e + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*e - c*f, 0]
Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b _.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin[a + b*x ]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IG tQ[p, 0]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[-(b*c)^(-1) Subst[Int[x^n*Sin[-a/b + x/b], x], x, a + b*ArcCos[c*x]], x] /; FreeQ[{a, b, c, n}, x]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[( -x^m)*Sqrt[1 - c^2*x^2]*((a + b*ArcCos[c*x])^(n + 1)/(b*c*(n + 1))), x] + ( -Simp[c*((m + 1)/(b*(n + 1))) Int[x^(m + 1)*((a + b*ArcCos[c*x])^(n + 1)/ Sqrt[1 - c^2*x^2]), x], x] + Simp[m/(b*c*(n + 1)) Int[x^(m - 1)*((a + b*A rcCos[c*x])^(n + 1)/Sqrt[1 - c^2*x^2]), x], x]) /; FreeQ[{a, b, c}, x] && I GtQ[m, 0] && LtQ[n, -2]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[- (b*c^(m + 1))^(-1) Subst[Int[x^n*Cos[-a/b + x/b]^m*Sin[-a/b + x/b], x], x , a + b*ArcCos[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]
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]
Time = 0.70 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00
method | result | size |
derivativedivides | \(\frac {\frac {\sin \left (3 \arccos \left (a x \right )\right )}{8 \arccos \left (a x \right )^{2}}+\frac {3 \cos \left (3 \arccos \left (a x \right )\right )}{8 \arccos \left (a x \right )}+\frac {9 \,\operatorname {Si}\left (3 \arccos \left (a x \right )\right )}{8}+\frac {\sqrt {-a^{2} x^{2}+1}}{8 \arccos \left (a x \right )^{2}}+\frac {a x}{8 \arccos \left (a x \right )}+\frac {\operatorname {Si}\left (\arccos \left (a x \right )\right )}{8}}{a^{3}}\) | \(82\) |
default | \(\frac {\frac {\sin \left (3 \arccos \left (a x \right )\right )}{8 \arccos \left (a x \right )^{2}}+\frac {3 \cos \left (3 \arccos \left (a x \right )\right )}{8 \arccos \left (a x \right )}+\frac {9 \,\operatorname {Si}\left (3 \arccos \left (a x \right )\right )}{8}+\frac {\sqrt {-a^{2} x^{2}+1}}{8 \arccos \left (a x \right )^{2}}+\frac {a x}{8 \arccos \left (a x \right )}+\frac {\operatorname {Si}\left (\arccos \left (a x \right )\right )}{8}}{a^{3}}\) | \(82\) |
1/a^3*(1/8/arccos(a*x)^2*sin(3*arccos(a*x))+3/8/arccos(a*x)*cos(3*arccos(a *x))+9/8*Si(3*arccos(a*x))+1/8*(-a^2*x^2+1)^(1/2)/arccos(a*x)^2+1/8/arccos (a*x)*a*x+1/8*Si(arccos(a*x)))
\[ \int \frac {x^2}{\arccos (a x)^3} \, dx=\int { \frac {x^{2}}{\arccos \left (a x\right )^{3}} \,d x } \]
\[ \int \frac {x^2}{\arccos (a x)^3} \, dx=\int \frac {x^{2}}{\operatorname {acos}^{3}{\left (a x \right )}}\, dx \]
\[ \int \frac {x^2}{\arccos (a x)^3} \, dx=\int { \frac {x^{2}}{\arccos \left (a x\right )^{3}} \,d x } \]
1/2*(sqrt(a*x + 1)*sqrt(-a*x + 1)*a*x^2 - arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)^2*integrate((9*a^2*x^2 - 2)/arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1 ), a*x), x) + (3*a^2*x^3 - 2*x)*arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x) )/(a^2*arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)^2)
Time = 0.30 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.88 \[ \int \frac {x^2}{\arccos (a x)^3} \, dx=\frac {3 \, x^{3}}{2 \, \arccos \left (a x\right )} + \frac {\sqrt {-a^{2} x^{2} + 1} x^{2}}{2 \, a \arccos \left (a x\right )^{2}} - \frac {x}{a^{2} \arccos \left (a x\right )} + \frac {9 \, \operatorname {Si}\left (3 \, \arccos \left (a x\right )\right )}{8 \, a^{3}} + \frac {\operatorname {Si}\left (\arccos \left (a x\right )\right )}{8 \, a^{3}} \]
3/2*x^3/arccos(a*x) + 1/2*sqrt(-a^2*x^2 + 1)*x^2/(a*arccos(a*x)^2) - x/(a^ 2*arccos(a*x)) + 9/8*sin_integral(3*arccos(a*x))/a^3 + 1/8*sin_integral(ar ccos(a*x))/a^3
Timed out. \[ \int \frac {x^2}{\arccos (a x)^3} \, dx=\int \frac {x^2}{{\mathrm {acos}\left (a\,x\right )}^3} \,d x \]