Integrand size = 8, antiderivative size = 63 \[ \int \frac {x}{\arccos (a x)^3} \, dx=\frac {x \sqrt {1-a^2 x^2}}{2 a \arccos (a x)^2}-\frac {1}{2 a^2 \arccos (a x)}+\frac {x^2}{\arccos (a x)}+\frac {\text {Si}(2 \arccos (a x))}{a^2} \]
-1/2/a^2/arccos(a*x)+x^2/arccos(a*x)+Si(2*arccos(a*x))/a^2+1/2*x*(-a^2*x^2 +1)^(1/2)/a/arccos(a*x)^2
Time = 0.04 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00 \[ \int \frac {x}{\arccos (a x)^3} \, dx=\frac {x \sqrt {1-a^2 x^2}}{2 a \arccos (a x)^2}+\frac {-1+2 a^2 x^2}{2 a^2 \arccos (a x)}+\frac {\text {Si}(2 \arccos (a x))}{a^2} \]
(x*Sqrt[1 - a^2*x^2])/(2*a*ArcCos[a*x]^2) + (-1 + 2*a^2*x^2)/(2*a^2*ArcCos [a*x]) + SinIntegral[2*ArcCos[a*x]]/a^2
Time = 0.65 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.10, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5145, 5153, 5223, 5147, 4906, 27, 3042, 3780}
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.
\(\displaystyle \int \frac {x}{\arccos (a x)^3} \, dx\) |
\(\Big \downarrow \) 5145 |
\(\displaystyle -\frac {\int \frac {1}{\sqrt {1-a^2 x^2} \arccos (a x)^2}dx}{2 a}+a \int \frac {x^2}{\sqrt {1-a^2 x^2} \arccos (a x)^2}dx+\frac {x \sqrt {1-a^2 x^2}}{2 a \arccos (a x)^2}\) |
\(\Big \downarrow \) 5153 |
\(\displaystyle a \int \frac {x^2}{\sqrt {1-a^2 x^2} \arccos (a x)^2}dx+\frac {x \sqrt {1-a^2 x^2}}{2 a \arccos (a x)^2}-\frac {1}{2 a^2 \arccos (a x)}\) |
\(\Big \downarrow \) 5223 |
\(\displaystyle a \left (\frac {x^2}{a \arccos (a x)}-\frac {2 \int \frac {x}{\arccos (a x)}dx}{a}\right )+\frac {x \sqrt {1-a^2 x^2}}{2 a \arccos (a x)^2}-\frac {1}{2 a^2 \arccos (a x)}\) |
\(\Big \downarrow \) 5147 |
\(\displaystyle a \left (\frac {2 \int \frac {a x \sqrt {1-a^2 x^2}}{\arccos (a x)}d\arccos (a x)}{a^3}+\frac {x^2}{a \arccos (a x)}\right )+\frac {x \sqrt {1-a^2 x^2}}{2 a \arccos (a x)^2}-\frac {1}{2 a^2 \arccos (a x)}\) |
\(\Big \downarrow \) 4906 |
\(\displaystyle a \left (\frac {2 \int \frac {\sin (2 \arccos (a x))}{2 \arccos (a x)}d\arccos (a x)}{a^3}+\frac {x^2}{a \arccos (a x)}\right )+\frac {x \sqrt {1-a^2 x^2}}{2 a \arccos (a x)^2}-\frac {1}{2 a^2 \arccos (a x)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle a \left (\frac {\int \frac {\sin (2 \arccos (a x))}{\arccos (a x)}d\arccos (a x)}{a^3}+\frac {x^2}{a \arccos (a x)}\right )+\frac {x \sqrt {1-a^2 x^2}}{2 a \arccos (a x)^2}-\frac {1}{2 a^2 \arccos (a x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a \left (\frac {\int \frac {\sin (2 \arccos (a x))}{\arccos (a x)}d\arccos (a x)}{a^3}+\frac {x^2}{a \arccos (a x)}\right )+\frac {x \sqrt {1-a^2 x^2}}{2 a \arccos (a x)^2}-\frac {1}{2 a^2 \arccos (a x)}\) |
\(\Big \downarrow \) 3780 |
\(\displaystyle a \left (\frac {\text {Si}(2 \arccos (a x))}{a^3}+\frac {x^2}{a \arccos (a x)}\right )+\frac {x \sqrt {1-a^2 x^2}}{2 a \arccos (a x)^2}-\frac {1}{2 a^2 \arccos (a x)}\) |
(x*Sqrt[1 - a^2*x^2])/(2*a*ArcCos[a*x]^2) - 1/(2*a^2*ArcCos[a*x]) + a*(x^2 /(a*ArcCos[a*x]) + SinIntegral[2*ArcCos[a*x]]/a^3)
3.1.63.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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_)^(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_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S ymbol] :> Simp[(-(b*c*(n + 1))^(-1))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2] ]*(a + b*ArcCos[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^ 2*d + e, 0] && NeQ[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]
Time = 0.61 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.68
method | result | size |
derivativedivides | \(\frac {\frac {\sin \left (2 \arccos \left (a x \right )\right )}{4 \arccos \left (a x \right )^{2}}+\frac {\cos \left (2 \arccos \left (a x \right )\right )}{2 \arccos \left (a x \right )}+\operatorname {Si}\left (2 \arccos \left (a x \right )\right )}{a^{2}}\) | \(43\) |
default | \(\frac {\frac {\sin \left (2 \arccos \left (a x \right )\right )}{4 \arccos \left (a x \right )^{2}}+\frac {\cos \left (2 \arccos \left (a x \right )\right )}{2 \arccos \left (a x \right )}+\operatorname {Si}\left (2 \arccos \left (a x \right )\right )}{a^{2}}\) | \(43\) |
1/a^2*(1/4/arccos(a*x)^2*sin(2*arccos(a*x))+1/2/arccos(a*x)*cos(2*arccos(a *x))+Si(2*arccos(a*x)))
\[ \int \frac {x}{\arccos (a x)^3} \, dx=\int { \frac {x}{\arccos \left (a x\right )^{3}} \,d x } \]
\[ \int \frac {x}{\arccos (a x)^3} \, dx=\int \frac {x}{\operatorname {acos}^{3}{\left (a x \right )}}\, dx \]
\[ \int \frac {x}{\arccos (a x)^3} \, dx=\int { \frac {x}{\arccos \left (a x\right )^{3}} \,d x } \]
-1/2*(4*a^2*arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)^2*integrate(x/arcta n2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x), x) - sqrt(a*x + 1)*sqrt(-a*x + 1)*a *x - (2*a^2*x^2 - 1)*arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x))/(a^2*arct an2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)^2)
Time = 0.30 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.90 \[ \int \frac {x}{\arccos (a x)^3} \, dx=\frac {x^{2}}{\arccos \left (a x\right )} + \frac {\operatorname {Si}\left (2 \, \arccos \left (a x\right )\right )}{a^{2}} + \frac {\sqrt {-a^{2} x^{2} + 1} x}{2 \, a \arccos \left (a x\right )^{2}} - \frac {1}{2 \, a^{2} \arccos \left (a x\right )} \]
x^2/arccos(a*x) + sin_integral(2*arccos(a*x))/a^2 + 1/2*sqrt(-a^2*x^2 + 1) *x/(a*arccos(a*x)^2) - 1/2/(a^2*arccos(a*x))
Timed out. \[ \int \frac {x}{\arccos (a x)^3} \, dx=\int \frac {x}{{\mathrm {acos}\left (a\,x\right )}^3} \,d x \]