3.155 \(\int (a+b \cos ^{-1}(c x))^3 \, dx\)

Optimal. Leaf size=82 \[ -6 a b^2 x-\frac{3 b \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{c}+x \left (a+b \cos ^{-1}(c x)\right )^3+\frac{6 b^3 \sqrt{1-c^2 x^2}}{c}-6 b^3 x \cos ^{-1}(c x) \]

[Out]

-6*a*b^2*x + (6*b^3*Sqrt[1 - c^2*x^2])/c - 6*b^3*x*ArcCos[c*x] - (3*b*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x])^2)
/c + x*(a + b*ArcCos[c*x])^3

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Rubi [A]  time = 0.107572, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {4620, 4678, 261} \[ -6 a b^2 x-\frac{3 b \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{c}+x \left (a+b \cos ^{-1}(c x)\right )^3+\frac{6 b^3 \sqrt{1-c^2 x^2}}{c}-6 b^3 x \cos ^{-1}(c x) \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCos[c*x])^3,x]

[Out]

-6*a*b^2*x + (6*b^3*Sqrt[1 - c^2*x^2])/c - 6*b^3*x*ArcCos[c*x] - (3*b*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x])^2)
/c + x*(a + b*ArcCos[c*x])^3

Rule 4620

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcCos[c*x])^n, x] + Dist[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]

Rule 4678

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] - Dist[(b*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p + 1
)*(1 - c^2*x^2)^FracPart[p]), Int[(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]

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin{align*} \int \left (a+b \cos ^{-1}(c x)\right )^3 \, dx &=x \left (a+b \cos ^{-1}(c x)\right )^3+(3 b c) \int \frac{x \left (a+b \cos ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}} \, dx\\ &=-\frac{3 b \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{c}+x \left (a+b \cos ^{-1}(c x)\right )^3-\left (6 b^2\right ) \int \left (a+b \cos ^{-1}(c x)\right ) \, dx\\ &=-6 a b^2 x-\frac{3 b \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{c}+x \left (a+b \cos ^{-1}(c x)\right )^3-\left (6 b^3\right ) \int \cos ^{-1}(c x) \, dx\\ &=-6 a b^2 x-6 b^3 x \cos ^{-1}(c x)-\frac{3 b \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{c}+x \left (a+b \cos ^{-1}(c x)\right )^3-\left (6 b^3 c\right ) \int \frac{x}{\sqrt{1-c^2 x^2}} \, dx\\ &=-6 a b^2 x+\frac{6 b^3 \sqrt{1-c^2 x^2}}{c}-6 b^3 x \cos ^{-1}(c x)-\frac{3 b \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{c}+x \left (a+b \cos ^{-1}(c x)\right )^3\\ \end{align*}

Mathematica [A]  time = 0.113723, size = 128, normalized size = 1.56 \[ \frac{-3 b \left (a^2-2 b^2\right ) \sqrt{1-c^2 x^2}+3 b \cos ^{-1}(c x) \left (a^2 c x-2 a b \sqrt{1-c^2 x^2}-2 b^2 c x\right )+a c x \left (a^2-6 b^2\right )+3 b^2 \cos ^{-1}(c x)^2 \left (a c x-b \sqrt{1-c^2 x^2}\right )+b^3 c x \cos ^{-1}(c x)^3}{c} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcCos[c*x])^3,x]

[Out]

(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

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Maple [A]  time = 0.047, size = 134, normalized size = 1.6 \begin{align*}{\frac{1}{c} \left ( cx{a}^{3}+{b}^{3} \left ( cx \left ( \arccos \left ( cx \right ) \right ) ^{3}-3\, \left ( \arccos \left ( cx \right ) \right ) ^{2}\sqrt{-{c}^{2}{x}^{2}+1}+6\,\sqrt{-{c}^{2}{x}^{2}+1}-6\,cx\arccos \left ( cx \right ) \right ) +3\,a{b}^{2} \left ( cx \left ( \arccos \left ( cx \right ) \right ) ^{2}-2\,cx-2\,\arccos \left ( cx \right ) \sqrt{-{c}^{2}{x}^{2}+1} \right ) +3\,{a}^{2}b \left ( cx\arccos \left ( cx \right ) -\sqrt{-{c}^{2}{x}^{2}+1} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccos(c*x))^3,x)

[Out]

1/c*(c*x*a^3+b^3*(c*x*arccos(c*x)^3-3*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*(c*x*arccos(c*x)^2-2*c*x-2*arccos(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
)))

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Maxima [A]  time = 1.44053, size = 194, normalized size = 2.37 \begin{align*} 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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccos(c*x))^3,x, algorithm="maxima")

[Out]

b^3*x*arccos(c*x)^3 + 3*a*b^2*x*arccos(c*x)^2 - 3*(sqrt(-c^2*x^2 + 1)*arccos(c*x)^2/c + 2*(c*x*arccos(c*x) - s
qrt(-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

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Fricas [A]  time = 2.29421, size = 262, normalized size = 3.2 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccos(c*x))^3,x, algorithm="fricas")

[Out]

(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

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Sympy [A]  time = 0.770118, size = 165, normalized size = 2.01 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acos(c*x))**3,x)

[Out]

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)*acos(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))

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Giac [A]  time = 1.14788, size = 203, normalized size = 2.48 \begin{align*} 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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccos(c*x))^3,x, algorithm="giac")

[Out]

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