Integrand size = 10, antiderivative size = 120 \[ \int x^4 \arccos (a x)^2 \, dx=-\frac {16 x}{75 a^4}-\frac {8 x^3}{225 a^2}-\frac {2 x^5}{125}-\frac {16 \sqrt {1-a^2 x^2} \arccos (a x)}{75 a^5}-\frac {8 x^2 \sqrt {1-a^2 x^2} \arccos (a x)}{75 a^3}-\frac {2 x^4 \sqrt {1-a^2 x^2} \arccos (a x)}{25 a}+\frac {1}{5} x^5 \arccos (a x)^2 \]
-16/75*x/a^4-8/225*x^3/a^2-2/125*x^5+1/5*x^5*arccos(a*x)^2-16/75*arccos(a* x)*(-a^2*x^2+1)^(1/2)/a^5-8/75*x^2*arccos(a*x)*(-a^2*x^2+1)^(1/2)/a^3-2/25 *x^4*arccos(a*x)*(-a^2*x^2+1)^(1/2)/a
Time = 0.04 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.68 \[ \int x^4 \arccos (a x)^2 \, dx=-\frac {16 x}{75 a^4}-\frac {8 x^3}{225 a^2}-\frac {2 x^5}{125}-\frac {2 \sqrt {1-a^2 x^2} \left (8+4 a^2 x^2+3 a^4 x^4\right ) \arccos (a x)}{75 a^5}+\frac {1}{5} x^5 \arccos (a x)^2 \]
(-16*x)/(75*a^4) - (8*x^3)/(225*a^2) - (2*x^5)/125 - (2*Sqrt[1 - a^2*x^2]* (8 + 4*a^2*x^2 + 3*a^4*x^4)*ArcCos[a*x])/(75*a^5) + (x^5*ArcCos[a*x]^2)/5
Time = 0.49 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.18, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5139, 5211, 15, 5211, 15, 5183, 24}
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.
\(\displaystyle \int x^4 \arccos (a x)^2 \, dx\) |
\(\Big \downarrow \) 5139 |
\(\displaystyle \frac {2}{5} a \int \frac {x^5 \arccos (a x)}{\sqrt {1-a^2 x^2}}dx+\frac {1}{5} x^5 \arccos (a x)^2\) |
\(\Big \downarrow \) 5211 |
\(\displaystyle \frac {2}{5} a \left (\frac {4 \int \frac {x^3 \arccos (a x)}{\sqrt {1-a^2 x^2}}dx}{5 a^2}-\frac {\int x^4dx}{5 a}-\frac {x^4 \sqrt {1-a^2 x^2} \arccos (a x)}{5 a^2}\right )+\frac {1}{5} x^5 \arccos (a x)^2\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \frac {2}{5} a \left (\frac {4 \int \frac {x^3 \arccos (a x)}{\sqrt {1-a^2 x^2}}dx}{5 a^2}-\frac {x^4 \sqrt {1-a^2 x^2} \arccos (a x)}{5 a^2}-\frac {x^5}{25 a}\right )+\frac {1}{5} x^5 \arccos (a x)^2\) |
\(\Big \downarrow \) 5211 |
\(\displaystyle \frac {2}{5} a \left (\frac {4 \left (\frac {2 \int \frac {x \arccos (a x)}{\sqrt {1-a^2 x^2}}dx}{3 a^2}-\frac {\int x^2dx}{3 a}-\frac {x^2 \sqrt {1-a^2 x^2} \arccos (a x)}{3 a^2}\right )}{5 a^2}-\frac {x^4 \sqrt {1-a^2 x^2} \arccos (a x)}{5 a^2}-\frac {x^5}{25 a}\right )+\frac {1}{5} x^5 \arccos (a x)^2\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \frac {2}{5} a \left (\frac {4 \left (\frac {2 \int \frac {x \arccos (a x)}{\sqrt {1-a^2 x^2}}dx}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \arccos (a x)}{3 a^2}-\frac {x^3}{9 a}\right )}{5 a^2}-\frac {x^4 \sqrt {1-a^2 x^2} \arccos (a x)}{5 a^2}-\frac {x^5}{25 a}\right )+\frac {1}{5} x^5 \arccos (a x)^2\) |
\(\Big \downarrow \) 5183 |
\(\displaystyle \frac {2}{5} a \left (\frac {4 \left (\frac {2 \left (-\frac {\int 1dx}{a}-\frac {\sqrt {1-a^2 x^2} \arccos (a x)}{a^2}\right )}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \arccos (a x)}{3 a^2}-\frac {x^3}{9 a}\right )}{5 a^2}-\frac {x^4 \sqrt {1-a^2 x^2} \arccos (a x)}{5 a^2}-\frac {x^5}{25 a}\right )+\frac {1}{5} x^5 \arccos (a x)^2\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {2}{5} a \left (-\frac {x^4 \sqrt {1-a^2 x^2} \arccos (a x)}{5 a^2}+\frac {4 \left (-\frac {x^2 \sqrt {1-a^2 x^2} \arccos (a x)}{3 a^2}+\frac {2 \left (-\frac {\sqrt {1-a^2 x^2} \arccos (a x)}{a^2}-\frac {x}{a}\right )}{3 a^2}-\frac {x^3}{9 a}\right )}{5 a^2}-\frac {x^5}{25 a}\right )+\frac {1}{5} x^5 \arccos (a x)^2\) |
(x^5*ArcCos[a*x]^2)/5 + (2*a*(-1/25*x^5/a - (x^4*Sqrt[1 - a^2*x^2]*ArcCos[ a*x])/(5*a^2) + (4*(-1/9*x^3/a - (x^2*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(3*a^ 2) + (2*(-(x/a) - (Sqrt[1 - a^2*x^2]*ArcCos[a*x])/a^2))/(3*a^2)))/(5*a^2)) )/5
3.1.12.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcCos[c*x])^n/(d*(m + 1))), x] + Simp[b*c*(n /(d*(m + 1))) Int[(d*x)^(m + 1)*((a + b*ArcCos[c*x])^(n - 1)/Sqrt[1 - c^2 *x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_ .), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcCos[c*x])^n/(2*e*(p + 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] I nt[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcCos[c*x])^n/(e*(m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcCos[c*x])^n, x], x] - S imp[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] Int[(f* x)^(m - 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m , 1] && NeQ[m + 2*p + 1, 0]
Time = 1.08 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.63
method | result | size |
derivativedivides | \(\frac {\frac {a^{5} x^{5} \arccos \left (a x \right )^{2}}{5}-\frac {2 \arccos \left (a x \right ) \left (3 a^{4} x^{4}+4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{75}-\frac {2 a^{5} x^{5}}{125}-\frac {8 a^{3} x^{3}}{225}-\frac {16 a x}{75}}{a^{5}}\) | \(76\) |
default | \(\frac {\frac {a^{5} x^{5} \arccos \left (a x \right )^{2}}{5}-\frac {2 \arccos \left (a x \right ) \left (3 a^{4} x^{4}+4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{75}-\frac {2 a^{5} x^{5}}{125}-\frac {8 a^{3} x^{3}}{225}-\frac {16 a x}{75}}{a^{5}}\) | \(76\) |
1/a^5*(1/5*a^5*x^5*arccos(a*x)^2-2/75*arccos(a*x)*(3*a^4*x^4+4*a^2*x^2+8)* (-a^2*x^2+1)^(1/2)-2/125*a^5*x^5-8/225*a^3*x^3-16/75*a*x)
Time = 0.26 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.63 \[ \int x^4 \arccos (a x)^2 \, dx=\frac {225 \, a^{5} x^{5} \arccos \left (a x\right )^{2} - 18 \, a^{5} x^{5} - 40 \, a^{3} x^{3} - 30 \, {\left (3 \, a^{4} x^{4} + 4 \, a^{2} x^{2} + 8\right )} \sqrt {-a^{2} x^{2} + 1} \arccos \left (a x\right ) - 240 \, a x}{1125 \, a^{5}} \]
1/1125*(225*a^5*x^5*arccos(a*x)^2 - 18*a^5*x^5 - 40*a^3*x^3 - 30*(3*a^4*x^ 4 + 4*a^2*x^2 + 8)*sqrt(-a^2*x^2 + 1)*arccos(a*x) - 240*a*x)/a^5
Time = 0.52 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.01 \[ \int x^4 \arccos (a x)^2 \, dx=\begin {cases} \frac {x^{5} \operatorname {acos}^{2}{\left (a x \right )}}{5} - \frac {2 x^{5}}{125} - \frac {2 x^{4} \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}{\left (a x \right )}}{25 a} - \frac {8 x^{3}}{225 a^{2}} - \frac {8 x^{2} \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}{\left (a x \right )}}{75 a^{3}} - \frac {16 x}{75 a^{4}} - \frac {16 \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}{\left (a x \right )}}{75 a^{5}} & \text {for}\: a \neq 0 \\\frac {\pi ^{2} x^{5}}{20} & \text {otherwise} \end {cases} \]
Piecewise((x**5*acos(a*x)**2/5 - 2*x**5/125 - 2*x**4*sqrt(-a**2*x**2 + 1)* acos(a*x)/(25*a) - 8*x**3/(225*a**2) - 8*x**2*sqrt(-a**2*x**2 + 1)*acos(a* x)/(75*a**3) - 16*x/(75*a**4) - 16*sqrt(-a**2*x**2 + 1)*acos(a*x)/(75*a**5 ), Ne(a, 0)), (pi**2*x**5/20, True))
Time = 0.30 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.85 \[ \int x^4 \arccos (a x)^2 \, dx=\frac {1}{5} \, x^{5} \arccos \left (a x\right )^{2} - \frac {2}{75} \, {\left (\frac {3 \, \sqrt {-a^{2} x^{2} + 1} x^{4}}{a^{2}} + \frac {4 \, \sqrt {-a^{2} x^{2} + 1} x^{2}}{a^{4}} + \frac {8 \, \sqrt {-a^{2} x^{2} + 1}}{a^{6}}\right )} a \arccos \left (a x\right ) - \frac {2 \, {\left (9 \, a^{4} x^{5} + 20 \, a^{2} x^{3} + 120 \, x\right )}}{1125 \, a^{4}} \]
1/5*x^5*arccos(a*x)^2 - 2/75*(3*sqrt(-a^2*x^2 + 1)*x^4/a^2 + 4*sqrt(-a^2*x ^2 + 1)*x^2/a^4 + 8*sqrt(-a^2*x^2 + 1)/a^6)*a*arccos(a*x) - 2/1125*(9*a^4* x^5 + 20*a^2*x^3 + 120*x)/a^4
Time = 0.29 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.83 \[ \int x^4 \arccos (a x)^2 \, dx=\frac {1}{5} \, x^{5} \arccos \left (a x\right )^{2} - \frac {2}{125} \, x^{5} - \frac {2 \, \sqrt {-a^{2} x^{2} + 1} x^{4} \arccos \left (a x\right )}{25 \, a} - \frac {8 \, x^{3}}{225 \, a^{2}} - \frac {8 \, \sqrt {-a^{2} x^{2} + 1} x^{2} \arccos \left (a x\right )}{75 \, a^{3}} - \frac {16 \, x}{75 \, a^{4}} - \frac {16 \, \sqrt {-a^{2} x^{2} + 1} \arccos \left (a x\right )}{75 \, a^{5}} \]
1/5*x^5*arccos(a*x)^2 - 2/125*x^5 - 2/25*sqrt(-a^2*x^2 + 1)*x^4*arccos(a*x )/a - 8/225*x^3/a^2 - 8/75*sqrt(-a^2*x^2 + 1)*x^2*arccos(a*x)/a^3 - 16/75* x/a^4 - 16/75*sqrt(-a^2*x^2 + 1)*arccos(a*x)/a^5
Timed out. \[ \int x^4 \arccos (a x)^2 \, dx=\int x^4\,{\mathrm {acos}\left (a\,x\right )}^2 \,d x \]