Integrand size = 8, antiderivative size = 82 \[ \int \arccos (a+b x)^3 \, dx=\frac {6 \sqrt {1-(a+b x)^2}}{b}-\frac {6 (a+b x) \arccos (a+b x)}{b}-\frac {3 \sqrt {1-(a+b x)^2} \arccos (a+b x)^2}{b}+\frac {(a+b x) \arccos (a+b x)^3}{b} \] Output:
6*(1-(b*x+a)^2)^(1/2)/b-6*(b*x+a)*arccos(b*x+a)/b-3*(1-(b*x+a)^2)^(1/2)*ar ccos(b*x+a)^2/b+(b*x+a)*arccos(b*x+a)^3/b
Time = 0.03 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.90 \[ \int \arccos (a+b x)^3 \, dx=\frac {6 \sqrt {1-(a+b x)^2}-6 (a+b x) \arccos (a+b x)-3 \sqrt {1-(a+b x)^2} \arccos (a+b x)^2+(a+b x) \arccos (a+b x)^3}{b} \] Input:
Integrate[ArcCos[a + b*x]^3,x]
Output:
(6*Sqrt[1 - (a + b*x)^2] - 6*(a + b*x)*ArcCos[a + b*x] - 3*Sqrt[1 - (a + b *x)^2]*ArcCos[a + b*x]^2 + (a + b*x)*ArcCos[a + b*x]^3)/b
Time = 0.34 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.96, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5303, 5131, 5183, 5131, 241}
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.
\(\displaystyle \int \arccos (a+b x)^3 \, dx\) |
\(\Big \downarrow \) 5303 |
\(\displaystyle \frac {\int \arccos (a+b x)^3d(a+b x)}{b}\) |
\(\Big \downarrow \) 5131 |
\(\displaystyle \frac {3 \int \frac {(a+b x) \arccos (a+b x)^2}{\sqrt {1-(a+b x)^2}}d(a+b x)+(a+b x) \arccos (a+b x)^3}{b}\) |
\(\Big \downarrow \) 5183 |
\(\displaystyle \frac {3 \left (-2 \int \arccos (a+b x)d(a+b x)-\sqrt {1-(a+b x)^2} \arccos (a+b x)^2\right )+(a+b x) \arccos (a+b x)^3}{b}\) |
\(\Big \downarrow \) 5131 |
\(\displaystyle \frac {3 \left (-2 \left (\int \frac {a+b x}{\sqrt {1-(a+b x)^2}}d(a+b x)+(a+b x) \arccos (a+b x)\right )-\sqrt {1-(a+b x)^2} \arccos (a+b x)^2\right )+(a+b x) \arccos (a+b x)^3}{b}\) |
\(\Big \downarrow \) 241 |
\(\displaystyle \frac {(a+b x) \arccos (a+b x)^3+3 \left (-\sqrt {1-(a+b x)^2} \arccos (a+b x)^2-2 \left ((a+b x) \arccos (a+b x)-\sqrt {1-(a+b x)^2}\right )\right )}{b}\) |
Input:
Int[ArcCos[a + b*x]^3,x]
Output:
((a + b*x)*ArcCos[a + b*x]^3 + 3*(-(Sqrt[1 - (a + b*x)^2]*ArcCos[a + b*x]^ 2) - 2*(-Sqrt[1 - (a + b*x)^2] + (a + b*x)*ArcCos[a + b*x])))/b
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ (2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
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]
Int[((a_.) + ArcCos[(c_) + (d_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[1/d Subst[Int[(a + b*ArcCos[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, n}, x]
Time = 0.15 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.87
method | result | size |
derivativedivides | \(\frac {\arccos \left (b x +a \right )^{3} \left (b x +a \right )-3 \arccos \left (b x +a \right )^{2} \sqrt {1-\left (b x +a \right )^{2}}+6 \sqrt {1-\left (b x +a \right )^{2}}-6 \left (b x +a \right ) \arccos \left (b x +a \right )}{b}\) | \(71\) |
default | \(\frac {\arccos \left (b x +a \right )^{3} \left (b x +a \right )-3 \arccos \left (b x +a \right )^{2} \sqrt {1-\left (b x +a \right )^{2}}+6 \sqrt {1-\left (b x +a \right )^{2}}-6 \left (b x +a \right ) \arccos \left (b x +a \right )}{b}\) | \(71\) |
orering | \(\frac {\left (b x +a \right ) \arccos \left (b x +a \right )^{3}}{b}+\frac {3 \left (b^{2} x^{2}+2 a b x +a^{2}-2\right ) \arccos \left (b x +a \right )^{2}}{b \sqrt {1-\left (b x +a \right )^{2}}}-\frac {2 \left (b x +a \right ) \left (b x +a +1\right ) \left (b x +a -1\right ) \left (\frac {6 \arccos \left (b x +a \right ) b^{2}}{1-\left (b x +a \right )^{2}}-\frac {3 \arccos \left (b x +a \right )^{2} b^{2} \left (b x +a \right )}{\left (1-\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}\right )}{b^{3}}-\frac {\left (b x +a +1\right )^{2} \left (b x +a -1\right )^{2} \left (-\frac {6 b^{3}}{\left (1-\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}+\frac {18 \arccos \left (b x +a \right ) b^{3} \left (b x +a \right )}{\left (1-\left (b x +a \right )^{2}\right )^{2}}-\frac {9 \arccos \left (b x +a \right )^{2} b^{3} \left (b x +a \right )^{2}}{\left (1-\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}-\frac {3 \arccos \left (b x +a \right )^{2} b^{3}}{\left (1-\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}\right )}{b^{4}}\) | \(268\) |
Input:
int(arccos(b*x+a)^3,x,method=_RETURNVERBOSE)
Output:
1/b*(arccos(b*x+a)^3*(b*x+a)-3*arccos(b*x+a)^2*(1-(b*x+a)^2)^(1/2)+6*(1-(b *x+a)^2)^(1/2)-6*(b*x+a)*arccos(b*x+a))
Time = 0.16 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.80 \[ \int \arccos (a+b x)^3 \, dx=\frac {{\left (b x + a\right )} \arccos \left (b x + a\right )^{3} - 6 \, {\left (b x + a\right )} \arccos \left (b x + a\right ) - 3 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (\arccos \left (b x + a\right )^{2} - 2\right )}}{b} \] Input:
integrate(arccos(b*x+a)^3,x, algorithm="fricas")
Output:
((b*x + a)*arccos(b*x + a)^3 - 6*(b*x + a)*arccos(b*x + a) - 3*sqrt(-b^2*x ^2 - 2*a*b*x - a^2 + 1)*(arccos(b*x + a)^2 - 2))/b
Time = 0.15 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.33 \[ \int \arccos (a+b x)^3 \, dx=\begin {cases} \frac {a \operatorname {acos}^{3}{\left (a + b x \right )}}{b} - \frac {6 a \operatorname {acos}{\left (a + b x \right )}}{b} + x \operatorname {acos}^{3}{\left (a + b x \right )} - 6 x \operatorname {acos}{\left (a + b x \right )} - \frac {3 \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {acos}^{2}{\left (a + b x \right )}}{b} + \frac {6 \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{b} & \text {for}\: b \neq 0 \\x \operatorname {acos}^{3}{\left (a \right )} & \text {otherwise} \end {cases} \] Input:
integrate(acos(b*x+a)**3,x)
Output:
Piecewise((a*acos(a + b*x)**3/b - 6*a*acos(a + b*x)/b + x*acos(a + b*x)**3 - 6*x*acos(a + b*x) - 3*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*acos(a + b* x)**2/b + 6*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)/b, Ne(b, 0)), (x*acos(a) **3, True))
\[ \int \arccos (a+b x)^3 \, dx=\int { \arccos \left (b x + a\right )^{3} \,d x } \] Input:
integrate(arccos(b*x+a)^3,x, algorithm="maxima")
Output:
x*arctan2(sqrt(b*x + a + 1)*sqrt(-b*x - a + 1), b*x + a)^3 - 3*b*integrate (sqrt(b*x + a + 1)*sqrt(-b*x - a + 1)*x*arctan2(sqrt(b*x + a + 1)*sqrt(-b* x - a + 1), b*x + a)^2/(b^2*x^2 + 2*a*b*x + a^2 - 1), x)
Time = 0.13 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.95 \[ \int \arccos (a+b x)^3 \, dx=\frac {{\left (b x + a\right )} \arccos \left (b x + a\right )^{3}}{b} - \frac {3 \, \sqrt {-{\left (b x + a\right )}^{2} + 1} \arccos \left (b x + a\right )^{2}}{b} - \frac {6 \, {\left (b x + a\right )} \arccos \left (b x + a\right )}{b} + \frac {6 \, \sqrt {-{\left (b x + a\right )}^{2} + 1}}{b} \] Input:
integrate(arccos(b*x+a)^3,x, algorithm="giac")
Output:
(b*x + a)*arccos(b*x + a)^3/b - 3*sqrt(-(b*x + a)^2 + 1)*arccos(b*x + a)^2 /b - 6*(b*x + a)*arccos(b*x + a)/b + 6*sqrt(-(b*x + a)^2 + 1)/b
Time = 0.33 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.73 \[ \int \arccos (a+b x)^3 \, dx=-\frac {\left (3\,{\mathrm {acos}\left (a+b\,x\right )}^2-6\right )\,\sqrt {1-{\left (a+b\,x\right )}^2}}{b}-\frac {\left (6\,\mathrm {acos}\left (a+b\,x\right )-{\mathrm {acos}\left (a+b\,x\right )}^3\right )\,\left (a+b\,x\right )}{b} \] Input:
int(acos(a + b*x)^3,x)
Output:
- ((3*acos(a + b*x)^2 - 6)*(1 - (a + b*x)^2)^(1/2))/b - ((6*acos(a + b*x) - acos(a + b*x)^3)*(a + b*x))/b
Time = 0.27 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.21 \[ \int \arccos (a+b x)^3 \, dx=\frac {\mathit {acos} \left (b x +a \right )^{3} a +\mathit {acos} \left (b x +a \right )^{3} b x -3 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \mathit {acos} \left (b x +a \right )^{2}-6 \mathit {acos} \left (b x +a \right ) a -6 \mathit {acos} \left (b x +a \right ) b x +6 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{b} \] Input:
int(acos(b*x+a)^3,x)
Output:
(acos(a + b*x)**3*a + acos(a + b*x)**3*b*x - 3*sqrt( - a**2 - 2*a*b*x - b* *2*x**2 + 1)*acos(a + b*x)**2 - 6*acos(a + b*x)*a - 6*acos(a + b*x)*b*x + 6*sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1))/b