3.36 \(\int x \cos ^{-1}(a x)^4 \, dx\)

Optimal. Leaf size=112 \[ -\frac {x \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^3}{a}+\frac {3 x \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{2 a}-\frac {\cos ^{-1}(a x)^4}{4 a^2}+\frac {3 \cos ^{-1}(a x)^2}{4 a^2}+\frac {1}{2} x^2 \cos ^{-1}(a x)^4-\frac {3}{2} x^2 \cos ^{-1}(a x)^2+\frac {3 x^2}{4} \]

[Out]

3/4*x^2+3/4*arccos(a*x)^2/a^2-3/2*x^2*arccos(a*x)^2-1/4*arccos(a*x)^4/a^2+1/2*x^2*arccos(a*x)^4+3/2*x*arccos(a
*x)*(-a^2*x^2+1)^(1/2)/a-x*arccos(a*x)^3*(-a^2*x^2+1)^(1/2)/a

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Rubi [A]  time = 0.24, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4628, 4708, 4642, 30} \[ -\frac {x \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^3}{a}+\frac {3 x \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{2 a}-\frac {\cos ^{-1}(a x)^4}{4 a^2}+\frac {3 \cos ^{-1}(a x)^2}{4 a^2}+\frac {1}{2} x^2 \cos ^{-1}(a x)^4-\frac {3}{2} x^2 \cos ^{-1}(a x)^2+\frac {3 x^2}{4} \]

Antiderivative was successfully verified.

[In]

Int[x*ArcCos[a*x]^4,x]

[Out]

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

Rule 30

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

Rule 4628

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 4642

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

Rule 4708

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

Rubi steps

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

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Mathematica [A]  time = 0.05, size = 96, normalized size = 0.86 \[ \frac {3 a^2 x^2+\left (2 a^2 x^2-1\right ) \cos ^{-1}(a x)^4-4 a x \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^3+\left (3-6 a^2 x^2\right ) \cos ^{-1}(a x)^2+6 a x \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{4 a^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x*ArcCos[a*x]^4,x]

[Out]

(3*a^2*x^2 + 6*a*x*Sqrt[1 - a^2*x^2]*ArcCos[a*x] + (3 - 6*a^2*x^2)*ArcCos[a*x]^2 - 4*a*x*Sqrt[1 - a^2*x^2]*Arc
Cos[a*x]^3 + (-1 + 2*a^2*x^2)*ArcCos[a*x]^4)/(4*a^2)

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fricas [A]  time = 0.44, size = 82, normalized size = 0.73 \[ \frac {{\left (2 \, a^{2} x^{2} - 1\right )} \arccos \left (a x\right )^{4} + 3 \, a^{2} x^{2} - 3 \, {\left (2 \, a^{2} x^{2} - 1\right )} \arccos \left (a x\right )^{2} - 2 \, {\left (2 \, a x \arccos \left (a x\right )^{3} - 3 \, a x \arccos \left (a x\right )\right )} \sqrt {-a^{2} x^{2} + 1}}{4 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arccos(a*x)^4,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.19, size = 101, normalized size = 0.90 \[ \frac {1}{2} \, x^{2} \arccos \left (a x\right )^{4} - \frac {3}{2} \, x^{2} \arccos \left (a x\right )^{2} - \frac {\sqrt {-a^{2} x^{2} + 1} x \arccos \left (a x\right )^{3}}{a} + \frac {3}{4} \, x^{2} - \frac {\arccos \left (a x\right )^{4}}{4 \, a^{2}} + \frac {3 \, \sqrt {-a^{2} x^{2} + 1} x \arccos \left (a x\right )}{2 \, a} + \frac {3 \, \arccos \left (a x\right )^{2}}{4 \, a^{2}} - \frac {3}{8 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arccos(a*x)^4,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.06, size = 113, normalized size = 1.01 \[ \frac {\frac {a^{2} x^{2} \arccos \left (a x \right )^{4}}{2}-\arccos \left (a x \right )^{3} \left (a x \sqrt {-a^{2} x^{2}+1}+\arccos \left (a x \right )\right )-\frac {3 a^{2} x^{2} \arccos \left (a x \right )^{2}}{2}+\frac {3 \arccos \left (a x \right ) \left (a x \sqrt {-a^{2} x^{2}+1}+\arccos \left (a x \right )\right )}{2}-\frac {3 \arccos \left (a x \right )^{2}}{4}+\frac {3 a^{2} x^{2}}{4}-\frac {3}{4}+\frac {3 \arccos \left (a x \right )^{4}}{4}}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*arccos(a*x)^4,x)

[Out]

1/a^2*(1/2*a^2*x^2*arccos(a*x)^4-arccos(a*x)^3*(a*x*(-a^2*x^2+1)^(1/2)+arccos(a*x))-3/2*a^2*x^2*arccos(a*x)^2+
3/2*arccos(a*x)*(a*x*(-a^2*x^2+1)^(1/2)+arccos(a*x))-3/4*arccos(a*x)^2+3/4*a^2*x^2-3/4+3/4*arccos(a*x)^4)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, x^{2} \arctan \left (\sqrt {a x + 1} \sqrt {-a x + 1}, a x\right )^{4} - 2 \, a \int \frac {\sqrt {a x + 1} \sqrt {-a x + 1} x^{2} \arctan \left (\sqrt {a x + 1} \sqrt {-a x + 1}, a x\right )^{3}}{a^{2} x^{2} - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arccos(a*x)^4,x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,{\mathrm {acos}\left (a\,x\right )}^4 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*acos(a*x)^4,x)

[Out]

int(x*acos(a*x)^4, x)

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sympy [A]  time = 1.74, size = 110, normalized size = 0.98 \[ \begin {cases} \frac {x^{2} \operatorname {acos}^{4}{\left (a x \right )}}{2} - \frac {3 x^{2} \operatorname {acos}^{2}{\left (a x \right )}}{2} + \frac {3 x^{2}}{4} - \frac {x \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}^{3}{\left (a x \right )}}{a} + \frac {3 x \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}{\left (a x \right )}}{2 a} - \frac {\operatorname {acos}^{4}{\left (a x \right )}}{4 a^{2}} + \frac {3 \operatorname {acos}^{2}{\left (a x \right )}}{4 a^{2}} & \text {for}\: a \neq 0 \\\frac {\pi ^{4} x^{2}}{32} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*acos(a*x)**4,x)

[Out]

Piecewise((x**2*acos(a*x)**4/2 - 3*x**2*acos(a*x)**2/2 + 3*x**2/4 - x*sqrt(-a**2*x**2 + 1)*acos(a*x)**3/a + 3*
x*sqrt(-a**2*x**2 + 1)*acos(a*x)/(2*a) - acos(a*x)**4/(4*a**2) + 3*acos(a*x)**2/(4*a**2), Ne(a, 0)), (pi**4*x*
*2/32, True))

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