Integrand size = 10, antiderivative size = 79 \[ \int (a+b \arccos (c x))^3 \, dx=\frac {6 b^3 \sqrt {1-c^2 x^2}}{c}-6 b^2 x (a+b \arccos (c x))-\frac {3 b \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{c}+x (a+b \arccos (c x))^3 \] Output:
6*b^3*(-c^2*x^2+1)^(1/2)/c-6*b^2*x*(a+b*arccos(c*x))-3*b*(-c^2*x^2+1)^(1/2 )*(a+b*arccos(c*x))^2/c+x*(a+b*arccos(c*x))^3
Time = 0.04 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.62 \[ \int (a+b \arccos (c x))^3 \, dx=\frac {a \left (a^2-6 b^2\right ) c x-3 b \left (a^2-2 b^2\right ) \sqrt {1-c^2 x^2}+3 b \left (a^2 c x-2 b^2 c x-2 a b \sqrt {1-c^2 x^2}\right ) \arccos (c x)+3 b^2 \left (a c x-b \sqrt {1-c^2 x^2}\right ) \arccos (c x)^2+b^3 c x \arccos (c x)^3}{c} \] Input:
Integrate[(a + b*ArcCos[c*x])^3,x]
Output:
(a*(a^2 - 6*b^2)*c*x - 3*b*(a^2 - 2*b^2)*Sqrt[1 - c^2*x^2] + 3*b*(a^2*c*x - 2*b^2*c*x - 2*a*b*Sqrt[1 - c^2*x^2])*ArcCos[c*x] + 3*b^2*(a*c*x - b*Sqrt [1 - c^2*x^2])*ArcCos[c*x]^2 + b^3*c*x*ArcCos[c*x]^3)/c
Time = 0.35 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.06, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5131, 5183, 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.
\(\displaystyle \int (a+b \arccos (c x))^3 \, dx\) |
\(\Big \downarrow \) 5131 |
\(\displaystyle 3 b c \int \frac {x (a+b \arccos (c x))^2}{\sqrt {1-c^2 x^2}}dx+x (a+b \arccos (c x))^3\) |
\(\Big \downarrow \) 5183 |
\(\displaystyle 3 b c \left (-\frac {2 b \int (a+b \arccos (c x))dx}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{c^2}\right )+x (a+b \arccos (c x))^3\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 3 b c \left (-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{c^2}-\frac {2 b \left (a x+b x \arccos (c x)-\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c}\right )+x (a+b \arccos (c x))^3\) |
Input:
Int[(a + b*ArcCos[c*x])^3,x]
Output:
x*(a + b*ArcCos[c*x])^3 + 3*b*c*(-((Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x])^ 2)/c^2) - (2*b*(a*x - (b*Sqrt[1 - c^2*x^2])/c + b*x*ArcCos[c*x]))/c)
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 = 0.00 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.70
method | result | size |
derivativedivides | \(\frac {c x \,a^{3}+b^{3} \left (\arccos \left (c x \right )^{3} c x -3 \arccos \left (c x \right )^{2} \sqrt {-c^{2} x^{2}+1}+6 \sqrt {-c^{2} x^{2}+1}-6 c x \arccos \left (c x \right )\right )+3 a \,b^{2} \left (\arccos \left (c x \right )^{2} c x -2 c x -2 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )+3 a^{2} b \left (c x \arccos \left (c x \right )-\sqrt {-c^{2} x^{2}+1}\right )}{c}\) | \(134\) |
default | \(\frac {c x \,a^{3}+b^{3} \left (\arccos \left (c x \right )^{3} c x -3 \arccos \left (c x \right )^{2} \sqrt {-c^{2} x^{2}+1}+6 \sqrt {-c^{2} x^{2}+1}-6 c x \arccos \left (c x \right )\right )+3 a \,b^{2} \left (\arccos \left (c x \right )^{2} c x -2 c x -2 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )+3 a^{2} b \left (c x \arccos \left (c x \right )-\sqrt {-c^{2} x^{2}+1}\right )}{c}\) | \(134\) |
parts | \(x \,a^{3}+\frac {b^{3} \left (\arccos \left (c x \right )^{3} c x -3 \arccos \left (c x \right )^{2} \sqrt {-c^{2} x^{2}+1}+6 \sqrt {-c^{2} x^{2}+1}-6 c x \arccos \left (c x \right )\right )}{c}+\frac {3 a^{2} b \left (c x \arccos \left (c x \right )-\sqrt {-c^{2} x^{2}+1}\right )}{c}+\frac {3 a \,b^{2} \left (\arccos \left (c x \right )^{2} c x -2 c x -2 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )}{c}\) | \(138\) |
orering | \(x \left (a +b \arccos \left (c x \right )\right )^{3}+\frac {3 \left (c^{2} x^{2}-2\right ) \left (a +b \arccos \left (c x \right )\right )^{2} b}{c \sqrt {-c^{2} x^{2}+1}}-\frac {2 x \left (c x -1\right ) \left (c x +1\right ) \left (\frac {6 \left (a +b \arccos \left (c x \right )\right ) b^{2} c^{2}}{-c^{2} x^{2}+1}-\frac {3 \left (a +b \arccos \left (c x \right )\right )^{2} b \,c^{3} x}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{c^{2}}-\frac {\left (c x -1\right )^{2} \left (c x +1\right )^{2} \left (-\frac {6 b^{3} c^{3}}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {18 \left (a +b \arccos \left (c x \right )\right ) b^{2} c^{4} x}{\left (-c^{2} x^{2}+1\right )^{2}}-\frac {9 \left (a +b \arccos \left (c x \right )\right )^{2} b \,c^{5} x^{2}}{\left (-c^{2} x^{2}+1\right )^{\frac {5}{2}}}-\frac {3 \left (a +b \arccos \left (c x \right )\right )^{2} b \,c^{3}}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{c^{4}}\) | \(253\) |
Input:
int((a+b*arccos(c*x))^3,x,method=_RETURNVERBOSE)
Output:
1/c*(c*x*a^3+b^3*(arccos(c*x)^3*c*x-3*arccos(c*x)^2*(-c^2*x^2+1)^(1/2)+6*( -c^2*x^2+1)^(1/2)-6*c*x*arccos(c*x))+3*a*b^2*(arccos(c*x)^2*c*x-2*c*x-2*ar ccos(c*x)*(-c^2*x^2+1)^(1/2))+3*a^2*b*(c*x*arccos(c*x)-(-c^2*x^2+1)^(1/2)) )
Time = 0.12 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.37 \[ \int (a+b \arccos (c x))^3 \, dx=\frac {b^{3} c x \arccos \left (c x\right )^{3} + 3 \, a b^{2} c x \arccos \left (c x\right )^{2} + 3 \, {\left (a^{2} b - 2 \, b^{3}\right )} c x \arccos \left (c x\right ) + {\left (a^{3} - 6 \, a b^{2}\right )} c x - 3 \, {\left (b^{3} \arccos \left (c x\right )^{2} + 2 \, a b^{2} \arccos \left (c x\right ) + a^{2} b - 2 \, b^{3}\right )} \sqrt {-c^{2} x^{2} + 1}}{c} \] Input:
integrate((a+b*arccos(c*x))^3,x, algorithm="fricas")
Output:
(b^3*c*x*arccos(c*x)^3 + 3*a*b^2*c*x*arccos(c*x)^2 + 3*(a^2*b - 2*b^3)*c*x *arccos(c*x) + (a^3 - 6*a*b^2)*c*x - 3*(b^3*arccos(c*x)^2 + 2*a*b^2*arccos (c*x) + a^2*b - 2*b^3)*sqrt(-c^2*x^2 + 1))/c
Leaf count of result is larger than twice the leaf count of optimal. 165 vs. \(2 (71) = 142\).
Time = 0.13 (sec) , antiderivative size = 165, normalized size of antiderivative = 2.09 \[ \int (a+b \arccos (c x))^3 \, dx=\begin {cases} a^{3} x + 3 a^{2} b x \operatorname {acos}{\left (c x \right )} - \frac {3 a^{2} b \sqrt {- c^{2} x^{2} + 1}}{c} + 3 a b^{2} x \operatorname {acos}^{2}{\left (c x \right )} - 6 a b^{2} x - \frac {6 a b^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{c} + b^{3} x \operatorname {acos}^{3}{\left (c x \right )} - 6 b^{3} x \operatorname {acos}{\left (c x \right )} - \frac {3 b^{3} \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}^{2}{\left (c x \right )}}{c} + \frac {6 b^{3} \sqrt {- c^{2} x^{2} + 1}}{c} & \text {for}\: c \neq 0 \\x \left (a + \frac {\pi b}{2}\right )^{3} & \text {otherwise} \end {cases} \] Input:
integrate((a+b*acos(c*x))**3,x)
Output:
Piecewise((a**3*x + 3*a**2*b*x*acos(c*x) - 3*a**2*b*sqrt(-c**2*x**2 + 1)/c + 3*a*b**2*x*acos(c*x)**2 - 6*a*b**2*x - 6*a*b**2*sqrt(-c**2*x**2 + 1)*ac os(c*x)/c + b**3*x*acos(c*x)**3 - 6*b**3*x*acos(c*x) - 3*b**3*sqrt(-c**2*x **2 + 1)*acos(c*x)**2/c + 6*b**3*sqrt(-c**2*x**2 + 1)/c, Ne(c, 0)), (x*(a + pi*b/2)**3, True))
Time = 0.18 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.82 \[ \int (a+b \arccos (c x))^3 \, dx=b^{3} x \arccos \left (c x\right )^{3} + 3 \, a b^{2} x \arccos \left (c x\right )^{2} - 3 \, {\left (\frac {\sqrt {-c^{2} x^{2} + 1} \arccos \left (c x\right )^{2}}{c} + \frac {2 \, {\left (c x \arccos \left (c x\right ) - \sqrt {-c^{2} x^{2} + 1}\right )}}{c}\right )} b^{3} - 6 \, a b^{2} {\left (x + \frac {\sqrt {-c^{2} x^{2} + 1} \arccos \left (c x\right )}{c}\right )} + a^{3} x + \frac {3 \, {\left (c x \arccos \left (c x\right ) - \sqrt {-c^{2} x^{2} + 1}\right )} a^{2} b}{c} \] Input:
integrate((a+b*arccos(c*x))^3,x, algorithm="maxima")
Output:
b^3*x*arccos(c*x)^3 + 3*a*b^2*x*arccos(c*x)^2 - 3*(sqrt(-c^2*x^2 + 1)*arcc os(c*x)^2/c + 2*(c*x*arccos(c*x) - sqrt(-c^2*x^2 + 1))/c)*b^3 - 6*a*b^2*(x + sqrt(-c^2*x^2 + 1)*arccos(c*x)/c) + a^3*x + 3*(c*x*arccos(c*x) - sqrt(- c^2*x^2 + 1))*a^2*b/c
Time = 0.13 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.90 \[ \int (a+b \arccos (c x))^3 \, dx=b^{3} x \arccos \left (c x\right )^{3} + 3 \, a b^{2} x \arccos \left (c x\right )^{2} + 3 \, a^{2} b x \arccos \left (c x\right ) - 6 \, b^{3} x \arccos \left (c x\right ) - \frac {3 \, \sqrt {-c^{2} x^{2} + 1} b^{3} \arccos \left (c x\right )^{2}}{c} + a^{3} x - 6 \, a b^{2} x - \frac {6 \, \sqrt {-c^{2} x^{2} + 1} a b^{2} \arccos \left (c x\right )}{c} - \frac {3 \, \sqrt {-c^{2} x^{2} + 1} a^{2} b}{c} + \frac {6 \, \sqrt {-c^{2} x^{2} + 1} b^{3}}{c} \] Input:
integrate((a+b*arccos(c*x))^3,x, algorithm="giac")
Output:
b^3*x*arccos(c*x)^3 + 3*a*b^2*x*arccos(c*x)^2 + 3*a^2*b*x*arccos(c*x) - 6* b^3*x*arccos(c*x) - 3*sqrt(-c^2*x^2 + 1)*b^3*arccos(c*x)^2/c + a^3*x - 6*a *b^2*x - 6*sqrt(-c^2*x^2 + 1)*a*b^2*arccos(c*x)/c - 3*sqrt(-c^2*x^2 + 1)*a ^2*b/c + 6*sqrt(-c^2*x^2 + 1)*b^3/c
Time = 0.01 (sec) , antiderivative size = 164, normalized size of antiderivative = 2.08 \[ \int (a+b \arccos (c x))^3 \, dx=\left \{\begin {array}{cl} x\,\left (a^3+\frac {3\,\pi \,a^2\,b}{2}+\frac {3\,\pi ^2\,a\,b^2}{4}+\frac {\pi ^3\,b^3}{8}\right ) & \text {\ if\ \ }c=0\\ a^3\,x-b^3\,x\,\left (6\,\mathrm {acos}\left (c\,x\right )-{\mathrm {acos}\left (c\,x\right )}^3\right )-\frac {3\,a^2\,b\,\left (\sqrt {1-c^2\,x^2}-c\,x\,\mathrm {acos}\left (c\,x\right )\right )}{c}+3\,a\,b^2\,x\,\left ({\mathrm {acos}\left (c\,x\right )}^2-2\right )-\frac {b^3\,\sqrt {1-c^2\,x^2}\,\left (3\,{\mathrm {acos}\left (c\,x\right )}^2-6\right )}{c}-\frac {6\,a\,b^2\,\mathrm {acos}\left (c\,x\right )\,\sqrt {1-c^2\,x^2}}{c} & \text {\ if\ \ }c\neq 0 \end {array}\right . \] Input:
int((a + b*acos(c*x))^3,x)
Output:
piecewise(c == 0, x*(a^3 + (b^3*pi^3)/8 + (3*a*b^2*pi^2)/4 + (3*a^2*b*pi)/ 2), c ~= 0, a^3*x - b^3*x*(6*acos(c*x) - acos(c*x)^3) - (3*a^2*b*((- c^2*x ^2 + 1)^(1/2) - c*x*acos(c*x)))/c + 3*a*b^2*x*(acos(c*x)^2 - 2) - (b^3*(- c^2*x^2 + 1)^(1/2)*(3*acos(c*x)^2 - 6))/c - (6*a*b^2*acos(c*x)*(- c^2*x^2 + 1)^(1/2))/c)
Time = 0.22 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.82 \[ \int (a+b \arccos (c x))^3 \, dx=\frac {\mathit {acos} \left (c x \right )^{3} b^{3} c x -3 \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right )^{2} b^{3}+3 \mathit {acos} \left (c x \right )^{2} a \,b^{2} c x -6 \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) a \,b^{2}+3 \mathit {acos} \left (c x \right ) a^{2} b c x -6 \mathit {acos} \left (c x \right ) b^{3} c x -3 \sqrt {-c^{2} x^{2}+1}\, a^{2} b +6 \sqrt {-c^{2} x^{2}+1}\, b^{3}+a^{3} c x -6 a \,b^{2} c x}{c} \] Input:
int((a+b*acos(c*x))^3,x)
Output:
(acos(c*x)**3*b**3*c*x - 3*sqrt( - c**2*x**2 + 1)*acos(c*x)**2*b**3 + 3*ac os(c*x)**2*a*b**2*c*x - 6*sqrt( - c**2*x**2 + 1)*acos(c*x)*a*b**2 + 3*acos (c*x)*a**2*b*c*x - 6*acos(c*x)*b**3*c*x - 3*sqrt( - c**2*x**2 + 1)*a**2*b + 6*sqrt( - c**2*x**2 + 1)*b**3 + a**3*c*x - 6*a*b**2*c*x)/c