Integrand size = 8, antiderivative size = 97 \[ \int \frac {x}{\arccos (a x)^4} \, dx=\frac {x \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^3}-\frac {1}{6 a^2 \arccos (a x)^2}+\frac {x^2}{3 \arccos (a x)^2}-\frac {2 x \sqrt {1-a^2 x^2}}{3 a \arccos (a x)}+\frac {2 \operatorname {CosIntegral}(2 \arccos (a x))}{3 a^2} \]
-1/6/a^2/arccos(a*x)^2+1/3*x^2/arccos(a*x)^2+2/3*Ci(2*arccos(a*x))/a^2+1/3 *x*(-a^2*x^2+1)^(1/2)/a/arccos(a*x)^3-2/3*x*(-a^2*x^2+1)^(1/2)/a/arccos(a* x)
Time = 0.09 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.89 \[ \int \frac {x}{\arccos (a x)^4} \, dx=\frac {2 a x \sqrt {1-a^2 x^2}+\left (-1+2 a^2 x^2\right ) \arccos (a x)-4 a x \sqrt {1-a^2 x^2} \arccos (a x)^2+4 \arccos (a x)^3 \operatorname {CosIntegral}(2 \arccos (a x))}{6 a^2 \arccos (a x)^3} \]
(2*a*x*Sqrt[1 - a^2*x^2] + (-1 + 2*a^2*x^2)*ArcCos[a*x] - 4*a*x*Sqrt[1 - a ^2*x^2]*ArcCos[a*x]^2 + 4*ArcCos[a*x]^3*CosIntegral[2*ArcCos[a*x]])/(6*a^2 *ArcCos[a*x]^3)
Time = 0.62 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.10, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {5145, 5153, 5223, 5143, 25, 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.
\(\displaystyle \int \frac {x}{\arccos (a x)^4} \, dx\) |
\(\Big \downarrow \) 5145 |
\(\displaystyle -\frac {\int \frac {1}{\sqrt {1-a^2 x^2} \arccos (a x)^3}dx}{3 a}+\frac {2}{3} a \int \frac {x^2}{\sqrt {1-a^2 x^2} \arccos (a x)^3}dx+\frac {x \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^3}\) |
\(\Big \downarrow \) 5153 |
\(\displaystyle \frac {2}{3} a \int \frac {x^2}{\sqrt {1-a^2 x^2} \arccos (a x)^3}dx+\frac {x \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^3}-\frac {1}{6 a^2 \arccos (a x)^2}\) |
\(\Big \downarrow \) 5223 |
\(\displaystyle \frac {2}{3} a \left (\frac {x^2}{2 a \arccos (a x)^2}-\frac {\int \frac {x}{\arccos (a x)^2}dx}{a}\right )+\frac {x \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^3}-\frac {1}{6 a^2 \arccos (a x)^2}\) |
\(\Big \downarrow \) 5143 |
\(\displaystyle \frac {2}{3} a \left (\frac {x^2}{2 a \arccos (a x)^2}-\frac {\frac {\int -\frac {\cos (2 \arccos (a x))}{\arccos (a x)}d\arccos (a x)}{a^2}+\frac {x \sqrt {1-a^2 x^2}}{a \arccos (a x)}}{a}\right )+\frac {x \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^3}-\frac {1}{6 a^2 \arccos (a x)^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {2}{3} a \left (\frac {x^2}{2 a \arccos (a x)^2}-\frac {\frac {x \sqrt {1-a^2 x^2}}{a \arccos (a x)}-\frac {\int \frac {\cos (2 \arccos (a x))}{\arccos (a x)}d\arccos (a x)}{a^2}}{a}\right )+\frac {x \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^3}-\frac {1}{6 a^2 \arccos (a x)^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2}{3} a \left (\frac {x^2}{2 a \arccos (a x)^2}-\frac {\frac {x \sqrt {1-a^2 x^2}}{a \arccos (a x)}-\frac {\int \frac {\sin \left (2 \arccos (a x)+\frac {\pi }{2}\right )}{\arccos (a x)}d\arccos (a x)}{a^2}}{a}\right )+\frac {x \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^3}-\frac {1}{6 a^2 \arccos (a x)^2}\) |
\(\Big \downarrow \) 3783 |
\(\displaystyle \frac {2}{3} a \left (\frac {x^2}{2 a \arccos (a x)^2}-\frac {\frac {x \sqrt {1-a^2 x^2}}{a \arccos (a x)}-\frac {\operatorname {CosIntegral}(2 \arccos (a x))}{a^2}}{a}\right )+\frac {x \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^3}-\frac {1}{6 a^2 \arccos (a x)^2}\) |
(x*Sqrt[1 - a^2*x^2])/(3*a*ArcCos[a*x]^3) - 1/(6*a^2*ArcCos[a*x]^2) + (2*a *(x^2/(2*a*ArcCos[a*x]^2) - ((x*Sqrt[1 - a^2*x^2])/(a*ArcCos[a*x]) - CosIn tegral[2*ArcCos[a*x]]/a^2)/a))/3
3.1.70.3.1 Defintions of rubi rules used
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_)^(m_.), x_Symbol] :> Simp[( -x^m)*Sqrt[1 - c^2*x^2]*((a + b*ArcCos[c*x])^(n + 1)/(b*c*(n + 1))), x] - S imp[1/(b^2*c^(m + 1)*(n + 1)) Subst[Int[ExpandTrigReduce[x^(n + 1), Cos[- a/b + x/b]^(m - 1)*(m - (m + 1)*Cos[-a/b + x/b]^2), x], x], x, a + b*ArcCos [c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]
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_.)/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 = 60, normalized size of antiderivative = 0.62
method | result | size |
derivativedivides | \(\frac {\frac {\sin \left (2 \arccos \left (a x \right )\right )}{6 \arccos \left (a x \right )^{3}}+\frac {\cos \left (2 \arccos \left (a x \right )\right )}{6 \arccos \left (a x \right )^{2}}-\frac {\sin \left (2 \arccos \left (a x \right )\right )}{3 \arccos \left (a x \right )}+\frac {2 \,\operatorname {Ci}\left (2 \arccos \left (a x \right )\right )}{3}}{a^{2}}\) | \(60\) |
default | \(\frac {\frac {\sin \left (2 \arccos \left (a x \right )\right )}{6 \arccos \left (a x \right )^{3}}+\frac {\cos \left (2 \arccos \left (a x \right )\right )}{6 \arccos \left (a x \right )^{2}}-\frac {\sin \left (2 \arccos \left (a x \right )\right )}{3 \arccos \left (a x \right )}+\frac {2 \,\operatorname {Ci}\left (2 \arccos \left (a x \right )\right )}{3}}{a^{2}}\) | \(60\) |
1/a^2*(1/6/arccos(a*x)^3*sin(2*arccos(a*x))+1/6/arccos(a*x)^2*cos(2*arccos (a*x))-1/3/arccos(a*x)*sin(2*arccos(a*x))+2/3*Ci(2*arccos(a*x)))
\[ \int \frac {x}{\arccos (a x)^4} \, dx=\int { \frac {x}{\arccos \left (a x\right )^{4}} \,d x } \]
\[ \int \frac {x}{\arccos (a x)^4} \, dx=\int \frac {x}{\operatorname {acos}^{4}{\left (a x \right )}}\, dx \]
\[ \int \frac {x}{\arccos (a x)^4} \, dx=\int { \frac {x}{\arccos \left (a x\right )^{4}} \,d x } \]
1/6*(6*a^2*arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)^3*integrate(2/3*(2*a ^2*x^2 - 1)*sqrt(a*x + 1)*sqrt(-a*x + 1)/((a^3*x^2 - a)*arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)), x) - 2*(2*a*x*arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)^2 - a*x)*sqrt(a*x + 1)*sqrt(-a*x + 1) + (2*a^2*x^2 - 1)*arctan2( sqrt(a*x + 1)*sqrt(-a*x + 1), a*x))/(a^2*arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)^3)
Time = 0.28 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.86 \[ \int \frac {x}{\arccos (a x)^4} \, dx=\frac {x^{2}}{3 \, \arccos \left (a x\right )^{2}} - \frac {2 \, \sqrt {-a^{2} x^{2} + 1} x}{3 \, a \arccos \left (a x\right )} + \frac {2 \, \operatorname {Ci}\left (2 \, \arccos \left (a x\right )\right )}{3 \, a^{2}} + \frac {\sqrt {-a^{2} x^{2} + 1} x}{3 \, a \arccos \left (a x\right )^{3}} - \frac {1}{6 \, a^{2} \arccos \left (a x\right )^{2}} \]
1/3*x^2/arccos(a*x)^2 - 2/3*sqrt(-a^2*x^2 + 1)*x/(a*arccos(a*x)) + 2/3*cos _integral(2*arccos(a*x))/a^2 + 1/3*sqrt(-a^2*x^2 + 1)*x/(a*arccos(a*x)^3) - 1/6/(a^2*arccos(a*x)^2)
Timed out. \[ \int \frac {x}{\arccos (a x)^4} \, dx=\int \frac {x}{{\mathrm {acos}\left (a\,x\right )}^4} \,d x \]