Integrand size = 6, antiderivative size = 69 \[ \int \arccos (a x)^4 \, dx=24 x+\frac {24 \sqrt {1-a^2 x^2} \arccos (a x)}{a}-12 x \arccos (a x)^2-\frac {4 \sqrt {1-a^2 x^2} \arccos (a x)^3}{a}+x \arccos (a x)^4 \] Output:
24*x+24*(-a^2*x^2+1)^(1/2)*arccos(a*x)/a-12*x*arccos(a*x)^2-4*(-a^2*x^2+1) ^(1/2)*arccos(a*x)^3/a+x*arccos(a*x)^4
Time = 0.02 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00 \[ \int \arccos (a x)^4 \, dx=24 x+\frac {24 \sqrt {1-a^2 x^2} \arccos (a x)}{a}-12 x \arccos (a x)^2-\frac {4 \sqrt {1-a^2 x^2} \arccos (a x)^3}{a}+x \arccos (a x)^4 \] Input:
Integrate[ArcCos[a*x]^4,x]
Output:
24*x + (24*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/a - 12*x*ArcCos[a*x]^2 - (4*Sqrt [1 - a^2*x^2]*ArcCos[a*x]^3)/a + x*ArcCos[a*x]^4
Time = 0.46 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.23, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {5131, 5183, 5131, 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.
\(\displaystyle \int \arccos (a x)^4 \, dx\) |
\(\Big \downarrow \) 5131 |
\(\displaystyle 4 a \int \frac {x \arccos (a x)^3}{\sqrt {1-a^2 x^2}}dx+x \arccos (a x)^4\) |
\(\Big \downarrow \) 5183 |
\(\displaystyle 4 a \left (-\frac {3 \int \arccos (a x)^2dx}{a}-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^3}{a^2}\right )+x \arccos (a x)^4\) |
\(\Big \downarrow \) 5131 |
\(\displaystyle 4 a \left (-\frac {3 \left (2 a \int \frac {x \arccos (a x)}{\sqrt {1-a^2 x^2}}dx+x \arccos (a x)^2\right )}{a}-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^3}{a^2}\right )+x \arccos (a x)^4\) |
\(\Big \downarrow \) 5183 |
\(\displaystyle 4 a \left (-\frac {3 \left (2 a \left (-\frac {\int 1dx}{a}-\frac {\sqrt {1-a^2 x^2} \arccos (a x)}{a^2}\right )+x \arccos (a x)^2\right )}{a}-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^3}{a^2}\right )+x \arccos (a x)^4\) |
\(\Big \downarrow \) 24 |
\(\displaystyle 4 a \left (-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^3}{a^2}-\frac {3 \left (2 a \left (-\frac {\sqrt {1-a^2 x^2} \arccos (a x)}{a^2}-\frac {x}{a}\right )+x \arccos (a x)^2\right )}{a}\right )+x \arccos (a x)^4\) |
Input:
Int[ArcCos[a*x]^4,x]
Output:
x*ArcCos[a*x]^4 + 4*a*(-((Sqrt[1 - a^2*x^2]*ArcCos[a*x]^3)/a^2) - (3*(x*Ar cCos[a*x]^2 + 2*a*(-(x/a) - (Sqrt[1 - a^2*x^2]*ArcCos[a*x])/a^2)))/a)
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*Ar cCos[c*x])^n, x] + Simp[b*c*n Int[x*((a + b*ArcCos[c*x])^(n - 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
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]
Time = 0.34 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.97
method | result | size |
derivativedivides | \(\frac {a x \arccos \left (a x \right )^{4}-4 \arccos \left (a x \right )^{3} \sqrt {-a^{2} x^{2}+1}-12 a x \arccos \left (a x \right )^{2}+24 a x +24 \arccos \left (a x \right ) \sqrt {-a^{2} x^{2}+1}}{a}\) | \(67\) |
default | \(\frac {a x \arccos \left (a x \right )^{4}-4 \arccos \left (a x \right )^{3} \sqrt {-a^{2} x^{2}+1}-12 a x \arccos \left (a x \right )^{2}+24 a x +24 \arccos \left (a x \right ) \sqrt {-a^{2} x^{2}+1}}{a}\) | \(67\) |
orering | \(x \arccos \left (a x \right )^{4}-\frac {8 \arccos \left (a x \right )^{3}}{a \sqrt {-a^{2} x^{2}+1}}+\frac {\left (5 a^{2} x^{2}-2\right ) x \left (\frac {12 \arccos \left (a x \right )^{2} a^{2}}{-a^{2} x^{2}+1}-\frac {4 \arccos \left (a x \right )^{3} a^{3} x}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{a^{2}}+\frac {\left (a x -1\right ) \left (a x +1\right ) \left (5 a^{2} x^{2}+1\right ) \left (-\frac {24 \arccos \left (a x \right ) a^{3}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {36 \arccos \left (a x \right )^{2} a^{4} x}{\left (-a^{2} x^{2}+1\right )^{2}}-\frac {12 \arccos \left (a x \right )^{3} a^{5} x^{2}}{\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}-\frac {4 \arccos \left (a x \right )^{3} a^{3}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{a^{4}}+\frac {x \left (a x -1\right )^{2} \left (a x +1\right )^{2} \left (\frac {24 a^{4}}{\left (-a^{2} x^{2}+1\right )^{2}}-\frac {144 \arccos \left (a x \right ) a^{5} x}{\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}+\frac {180 \arccos \left (a x \right )^{2} a^{6} x^{2}}{\left (-a^{2} x^{2}+1\right )^{3}}+\frac {48 \arccos \left (a x \right )^{2} a^{4}}{\left (-a^{2} x^{2}+1\right )^{2}}-\frac {60 \arccos \left (a x \right )^{3} a^{7} x^{3}}{\left (-a^{2} x^{2}+1\right )^{\frac {7}{2}}}-\frac {36 \arccos \left (a x \right )^{3} a^{5} x}{\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}\right )}{a^{4}}\) | \(373\) |
Input:
int(arccos(a*x)^4,x,method=_RETURNVERBOSE)
Output:
1/a*(a*x*arccos(a*x)^4-4*arccos(a*x)^3*(-a^2*x^2+1)^(1/2)-12*a*x*arccos(a* x)^2+24*a*x+24*arccos(a*x)*(-a^2*x^2+1)^(1/2))
Time = 0.13 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.80 \[ \int \arccos (a x)^4 \, dx=\frac {a x \arccos \left (a x\right )^{4} - 12 \, a x \arccos \left (a x\right )^{2} + 24 \, a x - 4 \, \sqrt {-a^{2} x^{2} + 1} {\left (\arccos \left (a x\right )^{3} - 6 \, \arccos \left (a x\right )\right )}}{a} \] Input:
integrate(arccos(a*x)^4,x, algorithm="fricas")
Output:
(a*x*arccos(a*x)^4 - 12*a*x*arccos(a*x)^2 + 24*a*x - 4*sqrt(-a^2*x^2 + 1)* (arccos(a*x)^3 - 6*arccos(a*x)))/a
Time = 0.16 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.01 \[ \int \arccos (a x)^4 \, dx=\begin {cases} x \operatorname {acos}^{4}{\left (a x \right )} - 12 x \operatorname {acos}^{2}{\left (a x \right )} + 24 x - \frac {4 \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}^{3}{\left (a x \right )}}{a} + \frac {24 \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}{\left (a x \right )}}{a} & \text {for}\: a \neq 0 \\\frac {\pi ^{4} x}{16} & \text {otherwise} \end {cases} \] Input:
integrate(acos(a*x)**4,x)
Output:
Piecewise((x*acos(a*x)**4 - 12*x*acos(a*x)**2 + 24*x - 4*sqrt(-a**2*x**2 + 1)*acos(a*x)**3/a + 24*sqrt(-a**2*x**2 + 1)*acos(a*x)/a, Ne(a, 0)), (pi** 4*x/16, True))
Time = 0.12 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.07 \[ \int \arccos (a x)^4 \, dx=x \arccos \left (a x\right )^{4} - \frac {4 \, \sqrt {-a^{2} x^{2} + 1} \arccos \left (a x\right )^{3}}{a} - 12 \, {\left (\frac {x \arccos \left (a x\right )^{2}}{a} - \frac {2 \, {\left (x + \frac {\sqrt {-a^{2} x^{2} + 1} \arccos \left (a x\right )}{a}\right )}}{a}\right )} a \] Input:
integrate(arccos(a*x)^4,x, algorithm="maxima")
Output:
x*arccos(a*x)^4 - 4*sqrt(-a^2*x^2 + 1)*arccos(a*x)^3/a - 12*(x*arccos(a*x) ^2/a - 2*(x + sqrt(-a^2*x^2 + 1)*arccos(a*x)/a)/a)*a
Time = 0.13 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.94 \[ \int \arccos (a x)^4 \, dx=x \arccos \left (a x\right )^{4} - 12 \, x \arccos \left (a x\right )^{2} - \frac {4 \, \sqrt {-a^{2} x^{2} + 1} \arccos \left (a x\right )^{3}}{a} + 24 \, x + \frac {24 \, \sqrt {-a^{2} x^{2} + 1} \arccos \left (a x\right )}{a} \] Input:
integrate(arccos(a*x)^4,x, algorithm="giac")
Output:
x*arccos(a*x)^4 - 12*x*arccos(a*x)^2 - 4*sqrt(-a^2*x^2 + 1)*arccos(a*x)^3/ a + 24*x + 24*sqrt(-a^2*x^2 + 1)*arccos(a*x)/a
Time = 0.35 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.91 \[ \int \arccos (a x)^4 \, dx=\left \{\begin {array}{cl} \frac {x\,\pi ^4}{16} & \text {\ if\ \ }a=0\\ x\,\left ({\mathrm {acos}\left (a\,x\right )}^4-12\,{\mathrm {acos}\left (a\,x\right )}^2+24\right )+\frac {\sqrt {1-a^2\,x^2}\,\left (24\,\mathrm {acos}\left (a\,x\right )-4\,{\mathrm {acos}\left (a\,x\right )}^3\right )}{a} & \text {\ if\ \ }a\neq 0 \end {array}\right . \] Input:
int(acos(a*x)^4,x)
Output:
piecewise(a == 0, (x*pi^4)/16, a ~= 0, x*(- 12*acos(a*x)^2 + acos(a*x)^4 + 24) + ((- a^2*x^2 + 1)^(1/2)*(24*acos(a*x) - 4*acos(a*x)^3))/a)
Time = 0.21 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.93 \[ \int \arccos (a x)^4 \, dx=\frac {\mathit {acos} \left (a x \right )^{4} a x -4 \sqrt {-a^{2} x^{2}+1}\, \mathit {acos} \left (a x \right )^{3}-12 \mathit {acos} \left (a x \right )^{2} a x +24 \sqrt {-a^{2} x^{2}+1}\, \mathit {acos} \left (a x \right )+24 a x}{a} \] Input:
int(acos(a*x)^4,x)
Output:
(acos(a*x)**4*a*x - 4*sqrt( - a**2*x**2 + 1)*acos(a*x)**3 - 12*acos(a*x)** 2*a*x + 24*sqrt( - a**2*x**2 + 1)*acos(a*x) + 24*a*x)/a