3.2.54 \(\int x (a+b \text {ArcCos}(c x))^3 \, dx\) [154]

Optimal. Leaf size=125 \[ \frac {3 b^3 x \sqrt {1-c^2 x^2}}{8 c}-\frac {3}{4} b^2 x^2 (a+b \text {ArcCos}(c x))-\frac {3 b x \sqrt {1-c^2 x^2} (a+b \text {ArcCos}(c x))^2}{4 c}-\frac {(a+b \text {ArcCos}(c x))^3}{4 c^2}+\frac {1}{2} x^2 (a+b \text {ArcCos}(c x))^3-\frac {3 b^3 \text {ArcSin}(c x)}{8 c^2} \]

[Out]

-3/4*b^2*x^2*(a+b*arccos(c*x))-1/4*(a+b*arccos(c*x))^3/c^2+1/2*x^2*(a+b*arccos(c*x))^3-3/8*b^3*arcsin(c*x)/c^2
+3/8*b^3*x*(-c^2*x^2+1)^(1/2)/c-3/4*b*x*(a+b*arccos(c*x))^2*(-c^2*x^2+1)^(1/2)/c

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Rubi [A]
time = 0.15, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {4724, 4796, 4738, 327, 222} \begin {gather*} -\frac {3}{4} b^2 x^2 (a+b \text {ArcCos}(c x))-\frac {3 b x \sqrt {1-c^2 x^2} (a+b \text {ArcCos}(c x))^2}{4 c}-\frac {(a+b \text {ArcCos}(c x))^3}{4 c^2}+\frac {1}{2} x^2 (a+b \text {ArcCos}(c x))^3-\frac {3 b^3 \text {ArcSin}(c x)}{8 c^2}+\frac {3 b^3 x \sqrt {1-c^2 x^2}}{8 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(a + b*ArcCos[c*x])^3,x]

[Out]

(3*b^3*x*Sqrt[1 - c^2*x^2])/(8*c) - (3*b^2*x^2*(a + b*ArcCos[c*x]))/4 - (3*b*x*Sqrt[1 - c^2*x^2]*(a + b*ArcCos
[c*x])^2)/(4*c) - (a + b*ArcCos[c*x])^3/(4*c^2) + (x^2*(a + b*ArcCos[c*x])^3)/2 - (3*b^3*ArcSin[c*x])/(8*c^2)

Rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 4724

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcCo
s[c*x])^n/(d*(m + 1))), x] + Dist[b*c*(n/(d*(m + 1))), Int[(d*x)^(m + 1)*((a + b*ArcCos[c*x])^(n - 1)/Sqrt[1 -
c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 4738

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(-(b*c*(n + 1))^(-1)
)*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcCos[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && E
qQ[c^2*d + e, 0] && NeQ[n, -1]

Rule 4796

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[f
*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcCos[c*x])^n/(e*(m + 2*p + 1))), x] + (Dist[f^2*((m - 1)/(c^2*(m
+ 2*p + 1))), Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcCos[c*x])^n, x], x] - Dist[b*f*(n/(c*(m + 2*p + 1)))*S
imp[(d + e*x^2)^p/(1 - c^2*x^2)^p], Int[(f*x)^(m - 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x],
x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m, 1] && NeQ[m + 2*p + 1, 0]

Rubi steps

\begin {align*} \int x \left (a+b \cos ^{-1}(c x)\right )^3 \, dx &=\frac {1}{2} x^2 \left (a+b \cos ^{-1}(c x)\right )^3+\frac {1}{2} (3 b c) \int \frac {x^2 \left (a+b \cos ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {3 b x \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{4 c}+\frac {1}{2} x^2 \left (a+b \cos ^{-1}(c x)\right )^3-\frac {1}{2} \left (3 b^2\right ) \int x \left (a+b \cos ^{-1}(c x)\right ) \, dx+\frac {(3 b) \int \frac {\left (a+b \cos ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{4 c}\\ &=-\frac {3}{4} b^2 x^2 \left (a+b \cos ^{-1}(c x)\right )-\frac {3 b x \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{4 c}-\frac {\left (a+b \cos ^{-1}(c x)\right )^3}{4 c^2}+\frac {1}{2} x^2 \left (a+b \cos ^{-1}(c x)\right )^3-\frac {1}{4} \left (3 b^3 c\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx\\ &=\frac {3 b^3 x \sqrt {1-c^2 x^2}}{8 c}-\frac {3}{4} b^2 x^2 \left (a+b \cos ^{-1}(c x)\right )-\frac {3 b x \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{4 c}-\frac {\left (a+b \cos ^{-1}(c x)\right )^3}{4 c^2}+\frac {1}{2} x^2 \left (a+b \cos ^{-1}(c x)\right )^3-\frac {\left (3 b^3\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{8 c}\\ &=\frac {3 b^3 x \sqrt {1-c^2 x^2}}{8 c}-\frac {3}{4} b^2 x^2 \left (a+b \cos ^{-1}(c x)\right )-\frac {3 b x \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{4 c}-\frac {\left (a+b \cos ^{-1}(c x)\right )^3}{4 c^2}+\frac {1}{2} x^2 \left (a+b \cos ^{-1}(c x)\right )^3-\frac {3 b^3 \sin ^{-1}(c x)}{8 c^2}\\ \end {align*}

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Mathematica [A]
time = 0.10, size = 185, normalized size = 1.48 \begin {gather*} \frac {c x \left (4 a^3 c x-6 a b^2 c x-6 a^2 b \sqrt {1-c^2 x^2}+3 b^3 \sqrt {1-c^2 x^2}\right )-6 b c x \left (-2 a^2 c x+b^2 c x+2 a b \sqrt {1-c^2 x^2}\right ) \text {ArcCos}(c x)-6 b^2 \left (a-2 a c^2 x^2+b c x \sqrt {1-c^2 x^2}\right ) \text {ArcCos}(c x)^2+2 b^3 \left (-1+2 c^2 x^2\right ) \text {ArcCos}(c x)^3+\left (6 a^2 b-3 b^3\right ) \text {ArcSin}(c x)}{8 c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*ArcCos[c*x])^3,x]

[Out]

(c*x*(4*a^3*c*x - 6*a*b^2*c*x - 6*a^2*b*Sqrt[1 - c^2*x^2] + 3*b^3*Sqrt[1 - c^2*x^2]) - 6*b*c*x*(-2*a^2*c*x + b
^2*c*x + 2*a*b*Sqrt[1 - c^2*x^2])*ArcCos[c*x] - 6*b^2*(a - 2*a*c^2*x^2 + b*c*x*Sqrt[1 - c^2*x^2])*ArcCos[c*x]^
2 + 2*b^3*(-1 + 2*c^2*x^2)*ArcCos[c*x]^3 + (6*a^2*b - 3*b^3)*ArcSin[c*x])/(8*c^2)

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Maple [A]
time = 0.06, size = 211, normalized size = 1.69

method	result	size
derivativedivides	\(\frac {\frac {c^{2} x^{2} a^{3}}{2}+b^{3} \left (\frac {c^{2} x^{2} \arccos \left (c x \right )^{3}}{2}-\frac {3 \arccos \left (c x \right )^{2} \left (c x \sqrt {-c^{2} x^{2}+1}+\arccos \left (c x \right )\right )}{4}-\frac {3 c^{2} x^{2} \arccos \left (c x \right )}{4}+\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arccos \left (c x \right )}{8}+\frac {\arccos \left (c x \right )^{3}}{2}\right )+3 a \,b^{2} \left (\frac {c^{2} x^{2} \arccos \left (c x \right )^{2}}{2}-\frac {\arccos \left (c x \right ) \left (c x \sqrt {-c^{2} x^{2}+1}+\arccos \left (c x \right )\right )}{2}+\frac {\arccos \left (c x \right )^{2}}{4}-\frac {c^{2} x^{2}}{4}+\frac {1}{4}\right )+3 a^{2} b \left (\frac {c^{2} x^{2} \arccos \left (c x \right )}{2}-\frac {c x \sqrt {-c^{2} x^{2}+1}}{4}+\frac {\arcsin \left (c x \right )}{4}\right )}{c^{2}}\)	\(211\)
default	\(\frac {\frac {c^{2} x^{2} a^{3}}{2}+b^{3} \left (\frac {c^{2} x^{2} \arccos \left (c x \right )^{3}}{2}-\frac {3 \arccos \left (c x \right )^{2} \left (c x \sqrt {-c^{2} x^{2}+1}+\arccos \left (c x \right )\right )}{4}-\frac {3 c^{2} x^{2} \arccos \left (c x \right )}{4}+\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arccos \left (c x \right )}{8}+\frac {\arccos \left (c x \right )^{3}}{2}\right )+3 a \,b^{2} \left (\frac {c^{2} x^{2} \arccos \left (c x \right )^{2}}{2}-\frac {\arccos \left (c x \right ) \left (c x \sqrt {-c^{2} x^{2}+1}+\arccos \left (c x \right )\right )}{2}+\frac {\arccos \left (c x \right )^{2}}{4}-\frac {c^{2} x^{2}}{4}+\frac {1}{4}\right )+3 a^{2} b \left (\frac {c^{2} x^{2} \arccos \left (c x \right )}{2}-\frac {c x \sqrt {-c^{2} x^{2}+1}}{4}+\frac {\arcsin \left (c x \right )}{4}\right )}{c^{2}}\)	\(211\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^2*(1/2*c^2*x^2*a^3+b^3*(1/2*c^2*x^2*arccos(c*x)^3-3/4*arccos(c*x)^2*(c*x*(-c^2*x^2+1)^(1/2)+arccos(c*x))-3
/4*c^2*x^2*arccos(c*x)+3/8*c*x*(-c^2*x^2+1)^(1/2)+3/8*arccos(c*x)+1/2*arccos(c*x)^3)+3*a*b^2*(1/2*c^2*x^2*arcc
os(c*x)^2-1/2*arccos(c*x)*(c*x*(-c^2*x^2+1)^(1/2)+arccos(c*x))+1/4*arccos(c*x)^2-1/4*c^2*x^2+1/4)+3*a^2*b*(1/2
*c^2*x^2*arccos(c*x)-1/4*c*x*(-c^2*x^2+1)^(1/2)+1/4*arcsin(c*x)))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b^3*x^2*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)^3 + 1/2*a^3*x^2 + 3/4*(2*x^2*arccos(c*x) - c*(sqrt(-c^2
*x^2 + 1)*x/c^2 - arcsin(c*x)/c^3))*a^2*b - integrate(3/2*(sqrt(c*x + 1)*sqrt(-c*x + 1)*b^3*c*x^2*arctan2(sqrt
(c*x + 1)*sqrt(-c*x + 1), c*x)^2 - 2*(a*b^2*c^2*x^3 - a*b^2*x)*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)^2)/(
c^2*x^2 - 1), x)

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Fricas [A]
time = 1.83, size = 169, normalized size = 1.35 \begin {gather*} \frac {2 \, {\left (2 \, a^{3} - 3 \, a b^{2}\right )} c^{2} x^{2} + 2 \, {\left (2 \, b^{3} c^{2} x^{2} - b^{3}\right )} \arccos \left (c x\right )^{3} + 6 \, {\left (2 \, a b^{2} c^{2} x^{2} - a b^{2}\right )} \arccos \left (c x\right )^{2} + 3 \, {\left (2 \, {\left (2 \, a^{2} b - b^{3}\right )} c^{2} x^{2} - 2 \, a^{2} b + b^{3}\right )} \arccos \left (c x\right ) - 3 \, {\left (2 \, b^{3} c x \arccos \left (c x\right )^{2} + 4 \, a b^{2} c x \arccos \left (c x\right ) + {\left (2 \, a^{2} b - b^{3}\right )} c x\right )} \sqrt {-c^{2} x^{2} + 1}}{8 \, c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(2*(2*a^3 - 3*a*b^2)*c^2*x^2 + 2*(2*b^3*c^2*x^2 - b^3)*arccos(c*x)^3 + 6*(2*a*b^2*c^2*x^2 - a*b^2)*arccos(
c*x)^2 + 3*(2*(2*a^2*b - b^3)*c^2*x^2 - 2*a^2*b + b^3)*arccos(c*x) - 3*(2*b^3*c*x*arccos(c*x)^2 + 4*a*b^2*c*x*
arccos(c*x) + (2*a^2*b - b^3)*c*x)*sqrt(-c^2*x^2 + 1))/c^2

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 269 vs. \(2 (116) = 232\).
time = 0.27, size = 269, normalized size = 2.15 \begin {gather*} \begin {cases} \frac {a^{3} x^{2}}{2} + \frac {3 a^{2} b x^{2} \operatorname {acos}{\left (c x \right )}}{2} - \frac {3 a^{2} b x \sqrt {- c^{2} x^{2} + 1}}{4 c} - \frac {3 a^{2} b \operatorname {acos}{\left (c x \right )}}{4 c^{2}} + \frac {3 a b^{2} x^{2} \operatorname {acos}^{2}{\left (c x \right )}}{2} - \frac {3 a b^{2} x^{2}}{4} - \frac {3 a b^{2} x \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{2 c} - \frac {3 a b^{2} \operatorname {acos}^{2}{\left (c x \right )}}{4 c^{2}} + \frac {b^{3} x^{2} \operatorname {acos}^{3}{\left (c x \right )}}{2} - \frac {3 b^{3} x^{2} \operatorname {acos}{\left (c x \right )}}{4} - \frac {3 b^{3} x \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}^{2}{\left (c x \right )}}{4 c} + \frac {3 b^{3} x \sqrt {- c^{2} x^{2} + 1}}{8 c} - \frac {b^{3} \operatorname {acos}^{3}{\left (c x \right )}}{4 c^{2}} + \frac {3 b^{3} \operatorname {acos}{\left (c x \right )}}{8 c^{2}} & \text {for}\: c \neq 0 \\\frac {x^{2} \left (a + \frac {\pi b}{2}\right )^{3}}{2} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**3*x**2/2 + 3*a**2*b*x**2*acos(c*x)/2 - 3*a**2*b*x*sqrt(-c**2*x**2 + 1)/(4*c) - 3*a**2*b*acos(c*x
)/(4*c**2) + 3*a*b**2*x**2*acos(c*x)**2/2 - 3*a*b**2*x**2/4 - 3*a*b**2*x*sqrt(-c**2*x**2 + 1)*acos(c*x)/(2*c)
- 3*a*b**2*acos(c*x)**2/(4*c**2) + b**3*x**2*acos(c*x)**3/2 - 3*b**3*x**2*acos(c*x)/4 - 3*b**3*x*sqrt(-c**2*x*
*2 + 1)*acos(c*x)**2/(4*c) + 3*b**3*x*sqrt(-c**2*x**2 + 1)/(8*c) - b**3*acos(c*x)**3/(4*c**2) + 3*b**3*acos(c*
x)/(8*c**2), Ne(c, 0)), (x**2*(a + pi*b/2)**3/2, True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 231 vs. \(2 (109) = 218\).
time = 0.45, size = 231, normalized size = 1.85 \begin {gather*} \frac {1}{2} \, b^{3} x^{2} \arccos \left (c x\right )^{3} + \frac {3}{2} \, a b^{2} x^{2} \arccos \left (c x\right )^{2} + \frac {3}{2} \, a^{2} b x^{2} \arccos \left (c x\right ) - \frac {3}{4} \, b^{3} x^{2} \arccos \left (c x\right ) - \frac {3 \, \sqrt {-c^{2} x^{2} + 1} b^{3} x \arccos \left (c x\right )^{2}}{4 \, c} + \frac {1}{2} \, a^{3} x^{2} - \frac {3}{4} \, a b^{2} x^{2} - \frac {3 \, \sqrt {-c^{2} x^{2} + 1} a b^{2} x \arccos \left (c x\right )}{2 \, c} - \frac {b^{3} \arccos \left (c x\right )^{3}}{4 \, c^{2}} - \frac {3 \, \sqrt {-c^{2} x^{2} + 1} a^{2} b x}{4 \, c} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} b^{3} x}{8 \, c} - \frac {3 \, a b^{2} \arccos \left (c x\right )^{2}}{4 \, c^{2}} - \frac {3 \, a^{2} b \arccos \left (c x\right )}{4 \, c^{2}} + \frac {3 \, b^{3} \arccos \left (c x\right )}{8 \, c^{2}} + \frac {3 \, a b^{2}}{8 \, c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b^3*x^2*arccos(c*x)^3 + 3/2*a*b^2*x^2*arccos(c*x)^2 + 3/2*a^2*b*x^2*arccos(c*x) - 3/4*b^3*x^2*arccos(c*x)
- 3/4*sqrt(-c^2*x^2 + 1)*b^3*x*arccos(c*x)^2/c + 1/2*a^3*x^2 - 3/4*a*b^2*x^2 - 3/2*sqrt(-c^2*x^2 + 1)*a*b^2*x*
arccos(c*x)/c - 1/4*b^3*arccos(c*x)^3/c^2 - 3/4*sqrt(-c^2*x^2 + 1)*a^2*b*x/c + 3/8*sqrt(-c^2*x^2 + 1)*b^3*x/c
- 3/4*a*b^2*arccos(c*x)^2/c^2 - 3/4*a^2*b*arccos(c*x)/c^2 + 3/8*b^3*arccos(c*x)/c^2 + 3/8*a*b^2/c^2

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x\,{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^3 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b*acos(c*x))^3,x)

[Out]

int(x*(a + b*acos(c*x))^3, x)

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