3.1.15 \(\int x \text {ArcCos}(a x)^2 \, dx\) [15]

Optimal. Leaf size=60 \[ -\frac {x^2}{4}-\frac {x \sqrt {1-a^2 x^2} \text {ArcCos}(a x)}{2 a}-\frac {\text {ArcCos}(a x)^2}{4 a^2}+\frac {1}{2} x^2 \text {ArcCos}(a x)^2 \]

[Out]

-1/4*x^2-1/4*arccos(a*x)^2/a^2+1/2*x^2*arccos(a*x)^2-1/2*x*arccos(a*x)*(-a^2*x^2+1)^(1/2)/a

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Rubi [A]
time = 0.06, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4724, 4796, 4738, 30} \begin {gather*} -\frac {x \sqrt {1-a^2 x^2} \text {ArcCos}(a x)}{2 a}-\frac {\text {ArcCos}(a x)^2}{4 a^2}+\frac {1}{2} x^2 \text {ArcCos}(a x)^2-\frac {x^2}{4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*ArcCos[a*x]^2,x]

[Out]

-1/4*x^2 - (x*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(2*a) - ArcCos[a*x]^2/(4*a^2) + (x^2*ArcCos[a*x]^2)/2

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 \cos ^{-1}(a x)^2 \, dx &=\frac {1}{2} x^2 \cos ^{-1}(a x)^2+a \int \frac {x^2 \cos ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {x \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{2 a}+\frac {1}{2} x^2 \cos ^{-1}(a x)^2-\frac {\int x \, dx}{2}+\frac {\int \frac {\cos ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{2 a}\\ &=-\frac {x^2}{4}-\frac {x \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{2 a}-\frac {\cos ^{-1}(a x)^2}{4 a^2}+\frac {1}{2} x^2 \cos ^{-1}(a x)^2\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 57, normalized size = 0.95 \begin {gather*} -\frac {x^2}{4}-\frac {x \sqrt {1-a^2 x^2} \text {ArcCos}(a x)}{2 a}+\frac {\left (-1+2 a^2 x^2\right ) \text {ArcCos}(a x)^2}{4 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*ArcCos[a*x]^2,x]

[Out]

-1/4*x^2 - (x*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(2*a) + ((-1 + 2*a^2*x^2)*ArcCos[a*x]^2)/(4*a^2)

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Maple [A]
time = 0.06, size = 63, normalized size = 1.05

method	result	size
derivativedivides	\(\frac {\frac {a^{2} x^{2} \arccos \left (a x \right )^{2}}{2}-\frac {\arccos \left (a x \right ) \left (a x \sqrt {-a^{2} x^{2}+1}+\arccos \left (a x \right )\right )}{2}+\frac {\arccos \left (a x \right )^{2}}{4}-\frac {a^{2} x^{2}}{4}+\frac {1}{4}}{a^{2}}\)	\(63\)
default	\(\frac {\frac {a^{2} x^{2} \arccos \left (a x \right )^{2}}{2}-\frac {\arccos \left (a x \right ) \left (a x \sqrt {-a^{2} x^{2}+1}+\arccos \left (a x \right )\right )}{2}+\frac {\arccos \left (a x \right )^{2}}{4}-\frac {a^{2} x^{2}}{4}+\frac {1}{4}}{a^{2}}\)	\(63\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*arccos(a*x)^2,x,method=_RETURNVERBOSE)

[Out]

1/a^2*(1/2*a^2*x^2*arccos(a*x)^2-1/2*arccos(a*x)*(a*x*(-a^2*x^2+1)^(1/2)+arccos(a*x))+1/4*arccos(a*x)^2-1/4*a^
2*x^2+1/4)

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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*arccos(a*x)^2,x, algorithm="maxima")

[Out]

1/2*x^2*arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)^2 - a*integrate(sqrt(a*x + 1)*sqrt(-a*x + 1)*x^2*arctan2(sq
rt(a*x + 1)*sqrt(-a*x + 1), a*x)/(a^2*x^2 - 1), x)

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Fricas [A]
time = 3.43, size = 51, normalized size = 0.85 \begin {gather*} -\frac {a^{2} x^{2} + 2 \, \sqrt {-a^{2} x^{2} + 1} a x \arccos \left (a x\right ) - {\left (2 \, a^{2} x^{2} - 1\right )} \arccos \left (a x\right )^{2}}{4 \, a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arccos(a*x)^2,x, algorithm="fricas")

[Out]

-1/4*(a^2*x^2 + 2*sqrt(-a^2*x^2 + 1)*a*x*arccos(a*x) - (2*a^2*x^2 - 1)*arccos(a*x)^2)/a^2

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Sympy [A]
time = 0.13, size = 58, normalized size = 0.97 \begin {gather*} \begin {cases} \frac {x^{2} \operatorname {acos}^{2}{\left (a x \right )}}{2} - \frac {x^{2}}{4} - \frac {x \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}{\left (a x \right )}}{2 a} - \frac {\operatorname {acos}^{2}{\left (a x \right )}}{4 a^{2}} & \text {for}\: a \neq 0 \\\frac {\pi ^{2} x^{2}}{8} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*acos(a*x)**2,x)

[Out]

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

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Giac [A]
time = 0.44, size = 55, normalized size = 0.92 \begin {gather*} \frac {1}{2} \, x^{2} \arccos \left (a x\right )^{2} - \frac {1}{4} \, x^{2} - \frac {\sqrt {-a^{2} x^{2} + 1} x \arccos \left (a x\right )}{2 \, a} - \frac {\arccos \left (a x\right )^{2}}{4 \, a^{2}} + \frac {1}{8 \, a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arccos(a*x)^2,x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int x\,{\mathrm {acos}\left (a\,x\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*acos(a*x)^2,x)

[Out]

int(x*acos(a*x)^2, x)

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