3.2.55 \(\int (a+b \text {ArcCos}(c x))^3 \, dx\) [155]

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

[Out]

-6*a*b^2*x-6*b^3*x*arccos(c*x)+x*(a+b*arccos(c*x))^3+6*b^3*(-c^2*x^2+1)^(1/2)/c-3*b*(a+b*arccos(c*x))^2*(-c^2*
x^2+1)^(1/2)/c

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Rubi [A]
time = 0.08, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4716, 4768, 267} \begin {gather*} -\frac {3 b \sqrt {1-c^2 x^2} (a+b \text {ArcCos}(c x))^2}{c}+x (a+b \text {ArcCos}(c x))^3-6 a b^2 x-6 b^3 x \text {ArcCos}(c x)+\frac {6 b^3 \sqrt {1-c^2 x^2}}{c} \end {gather*}

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 267

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]

Rule 4716

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 4768

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/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^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]

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*}

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Mathematica [A]
time = 0.07, size = 128, normalized size = 1.56 \begin {gather*} \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 ) \text {ArcCos}(c x)+3 b^2 \left (a c x-b \sqrt {1-c^2 x^2}\right ) \text {ArcCos}(c x)^2+b^3 c x \text {ArcCos}(c x)^3}{c} \end {gather*}

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 = 0.08, size = 134, normalized size = 1.63

method	result	size
derivativedivides	\(\frac {c x \,a^{3}+b^{3} \left (c x \arccos \left (c x \right )^{3}-3 \arccos \left (c x \right )^{2} \sqrt {-c^{2} x^{2}+1}+6 \sqrt {-c^{2} x^{2}+1}-6 c x \arccos \left (c x \right )\right )+3 a \,b^{2} \left (c x \arccos \left (c x \right )^{2}-2 c x -2 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )+3 a^{2} b \left (c x \arccos \left (c x \right )-\sqrt {-c^{2} x^{2}+1}\right )}{c}\)	\(134\)
default	\(\frac {c x \,a^{3}+b^{3} \left (c x \arccos \left (c x \right )^{3}-3 \arccos \left (c x \right )^{2} \sqrt {-c^{2} x^{2}+1}+6 \sqrt {-c^{2} x^{2}+1}-6 c x \arccos \left (c x \right )\right )+3 a \,b^{2} \left (c x \arccos \left (c x \right )^{2}-2 c x -2 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )+3 a^{2} b \left (c x \arccos \left (c x \right )-\sqrt {-c^{2} x^{2}+1}\right )}{c}\)	\(134\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.47, size = 144, normalized size = 1.76 \begin {gather*} 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 {gather*}

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 = 1.62, size = 108, normalized size = 1.32 \begin {gather*} \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 {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 165 vs. \(2 (76) = 152\).
time = 0.17, size = 165, normalized size = 2.01 \begin {gather*} \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 {gather*}

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 = 0.43, size = 150, normalized size = 1.83 \begin {gather*} 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 {gather*}

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

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Mupad [B]
time = 0.47, size = 164, normalized size = 2.00 \begin {gather*} \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 . \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*acos(c*x))^3,x)

[Out]

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)

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