3.34 \(\int x^3 \cos ^{-1}(a x)^4 \, dx\)

Optimal. Leaf size=198 \[ \frac{45 x^2}{128 a^2}-\frac{x^3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{4 a}+\frac{3 x^3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{32 a}-\frac{9 x^2 \cos ^{-1}(a x)^2}{16 a^2}-\frac{3 x \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{8 a^3}+\frac{45 x \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{64 a^3}-\frac{3 \cos ^{-1}(a x)^4}{32 a^4}+\frac{45 \cos ^{-1}(a x)^2}{128 a^4}+\frac{1}{4} x^4 \cos ^{-1}(a x)^4-\frac{3}{16} x^4 \cos ^{-1}(a x)^2+\frac{3 x^4}{128} \]

[Out]

(45*x^2)/(128*a^2) + (3*x^4)/128 + (45*x*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(64*a^3) + (3*x^3*Sqrt[1 - a^2*x^2]*Ar
cCos[a*x])/(32*a) + (45*ArcCos[a*x]^2)/(128*a^4) - (9*x^2*ArcCos[a*x]^2)/(16*a^2) - (3*x^4*ArcCos[a*x]^2)/16 -
(3*x*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^3)/(8*a^3) - (x^3*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^3)/(4*a) - (3*ArcCos[a*x]^
4)/(32*a^4) + (x^4*ArcCos[a*x]^4)/4

________________________________________________________________________________________

Rubi [A]  time = 0.519584, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4628, 4708, 4642, 30} \[ \frac{45 x^2}{128 a^2}-\frac{x^3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{4 a}+\frac{3 x^3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{32 a}-\frac{9 x^2 \cos ^{-1}(a x)^2}{16 a^2}-\frac{3 x \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{8 a^3}+\frac{45 x \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{64 a^3}-\frac{3 \cos ^{-1}(a x)^4}{32 a^4}+\frac{45 \cos ^{-1}(a x)^2}{128 a^4}+\frac{1}{4} x^4 \cos ^{-1}(a x)^4-\frac{3}{16} x^4 \cos ^{-1}(a x)^2+\frac{3 x^4}{128} \]

Antiderivative was successfully verified.

[In]

Int[x^3*ArcCos[a*x]^4,x]

[Out]

(45*x^2)/(128*a^2) + (3*x^4)/128 + (45*x*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(64*a^3) + (3*x^3*Sqrt[1 - a^2*x^2]*Ar
cCos[a*x])/(32*a) + (45*ArcCos[a*x]^2)/(128*a^4) - (9*x^2*ArcCos[a*x]^2)/(16*a^2) - (3*x^4*ArcCos[a*x]^2)/16 -
(3*x*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^3)/(8*a^3) - (x^3*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^3)/(4*a) - (3*ArcCos[a*x]^
4)/(32*a^4) + (x^4*ArcCos[a*x]^4)/4

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 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]

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 30

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

Rubi steps

\begin{align*} \int x^3 \cos ^{-1}(a x)^4 \, dx &=\frac{1}{4} x^4 \cos ^{-1}(a x)^4+a \int \frac{x^4 \cos ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{x^3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{4 a}+\frac{1}{4} x^4 \cos ^{-1}(a x)^4-\frac{3}{4} \int x^3 \cos ^{-1}(a x)^2 \, dx+\frac{3 \int \frac{x^2 \cos ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}} \, dx}{4 a}\\ &=-\frac{3}{16} x^4 \cos ^{-1}(a x)^2-\frac{3 x \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{8 a^3}-\frac{x^3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{4 a}+\frac{1}{4} x^4 \cos ^{-1}(a x)^4+\frac{3 \int \frac{\cos ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}} \, dx}{8 a^3}-\frac{9 \int x \cos ^{-1}(a x)^2 \, dx}{8 a^2}-\frac{1}{8} (3 a) \int \frac{x^4 \cos ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{3 x^3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{32 a}-\frac{9 x^2 \cos ^{-1}(a x)^2}{16 a^2}-\frac{3}{16} x^4 \cos ^{-1}(a x)^2-\frac{3 x \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{8 a^3}-\frac{x^3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{4 a}-\frac{3 \cos ^{-1}(a x)^4}{32 a^4}+\frac{1}{4} x^4 \cos ^{-1}(a x)^4+\frac{3 \int x^3 \, dx}{32}-\frac{9 \int \frac{x^2 \cos ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{32 a}-\frac{9 \int \frac{x^2 \cos ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{8 a}\\ &=\frac{3 x^4}{128}+\frac{45 x \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{64 a^3}+\frac{3 x^3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{32 a}-\frac{9 x^2 \cos ^{-1}(a x)^2}{16 a^2}-\frac{3}{16} x^4 \cos ^{-1}(a x)^2-\frac{3 x \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{8 a^3}-\frac{x^3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{4 a}-\frac{3 \cos ^{-1}(a x)^4}{32 a^4}+\frac{1}{4} x^4 \cos ^{-1}(a x)^4-\frac{9 \int \frac{\cos ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{64 a^3}-\frac{9 \int \frac{\cos ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{16 a^3}+\frac{9 \int x \, dx}{64 a^2}+\frac{9 \int x \, dx}{16 a^2}\\ &=\frac{45 x^2}{128 a^2}+\frac{3 x^4}{128}+\frac{45 x \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{64 a^3}+\frac{3 x^3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{32 a}+\frac{45 \cos ^{-1}(a x)^2}{128 a^4}-\frac{9 x^2 \cos ^{-1}(a x)^2}{16 a^2}-\frac{3}{16} x^4 \cos ^{-1}(a x)^2-\frac{3 x \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{8 a^3}-\frac{x^3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{4 a}-\frac{3 \cos ^{-1}(a x)^4}{32 a^4}+\frac{1}{4} x^4 \cos ^{-1}(a x)^4\\ \end{align*}

Mathematica [A]  time = 0.066978, size = 135, normalized size = 0.68 \[ \frac{3 a^2 x^2 \left (a^2 x^2+15\right )+4 \left (8 a^4 x^4-3\right ) \cos ^{-1}(a x)^4-16 a x \sqrt{1-a^2 x^2} \left (2 a^2 x^2+3\right ) \cos ^{-1}(a x)^3-3 \left (8 a^4 x^4+24 a^2 x^2-15\right ) \cos ^{-1}(a x)^2+6 a x \sqrt{1-a^2 x^2} \left (2 a^2 x^2+15\right ) \cos ^{-1}(a x)}{128 a^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*ArcCos[a*x]^4,x]

[Out]

(3*a^2*x^2*(15 + a^2*x^2) + 6*a*x*Sqrt[1 - a^2*x^2]*(15 + 2*a^2*x^2)*ArcCos[a*x] - 3*(-15 + 24*a^2*x^2 + 8*a^4
*x^4)*ArcCos[a*x]^2 - 16*a*x*Sqrt[1 - a^2*x^2]*(3 + 2*a^2*x^2)*ArcCos[a*x]^3 + 4*(-3 + 8*a^4*x^4)*ArcCos[a*x]^
4)/(128*a^4)

________________________________________________________________________________________

Maple [A]  time = 0.06, size = 207, normalized size = 1.1 \begin{align*}{\frac{1}{{a}^{4}} \left ({\frac{{a}^{4}{x}^{4} \left ( \arccos \left ( ax \right ) \right ) ^{4}}{4}}-{\frac{ \left ( \arccos \left ( ax \right ) \right ) ^{3}}{8} \left ( 2\,{a}^{3}{x}^{3}\sqrt{-{a}^{2}{x}^{2}+1}+3\,ax\sqrt{-{a}^{2}{x}^{2}+1}+3\,\arccos \left ( ax \right ) \right ) }-{\frac{3\,{a}^{4}{x}^{4} \left ( \arccos \left ( ax \right ) \right ) ^{2}}{16}}+{\frac{3\,\arccos \left ( ax \right ) }{64} \left ( 2\,{a}^{3}{x}^{3}\sqrt{-{a}^{2}{x}^{2}+1}+3\,ax\sqrt{-{a}^{2}{x}^{2}+1}+3\,\arccos \left ( ax \right ) \right ) }-{\frac{45\, \left ( \arccos \left ( ax \right ) \right ) ^{2}}{128}}+{\frac{3\,{a}^{4}{x}^{4}}{128}}+{\frac{45\,{a}^{2}{x}^{2}}{128}}-{\frac{9\,{a}^{2}{x}^{2} \left ( \arccos \left ( ax \right ) \right ) ^{2}}{16}}+{\frac{9\,\arccos \left ( ax \right ) }{16} \left ( ax\sqrt{-{a}^{2}{x}^{2}+1}+\arccos \left ( ax \right ) \right ) }-{\frac{9}{32}}+{\frac{9\, \left ( \arccos \left ( ax \right ) \right ) ^{4}}{32}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*arccos(a*x)^4,x)

[Out]

1/a^4*(1/4*a^4*x^4*arccos(a*x)^4-1/8*arccos(a*x)^3*(2*a^3*x^3*(-a^2*x^2+1)^(1/2)+3*a*x*(-a^2*x^2+1)^(1/2)+3*ar
ccos(a*x))-3/16*a^4*x^4*arccos(a*x)^2+3/64*arccos(a*x)*(2*a^3*x^3*(-a^2*x^2+1)^(1/2)+3*a*x*(-a^2*x^2+1)^(1/2)+
3*arccos(a*x))-45/128*arccos(a*x)^2+3/128*a^4*x^4+45/128*a^2*x^2-9/16*a^2*x^2*arccos(a*x)^2+9/16*arccos(a*x)*(
a*x*(-a^2*x^2+1)^(1/2)+arccos(a*x))-9/32+9/32*arccos(a*x)^4)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{4} \, x^{4} \arctan \left (\sqrt{a x + 1} \sqrt{-a x + 1}, a x\right )^{4} - a \int \frac{\sqrt{a x + 1} \sqrt{-a x + 1} x^{4} \arctan \left (\sqrt{a x + 1} \sqrt{-a x + 1}, a x\right )^{3}}{a^{2} x^{2} - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arccos(a*x)^4,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 2.32852, size = 292, normalized size = 1.47 \begin{align*} \frac{3 \, a^{4} x^{4} + 4 \,{\left (8 \, a^{4} x^{4} - 3\right )} \arccos \left (a x\right )^{4} + 45 \, a^{2} x^{2} - 3 \,{\left (8 \, a^{4} x^{4} + 24 \, a^{2} x^{2} - 15\right )} \arccos \left (a x\right )^{2} - 2 \, \sqrt{-a^{2} x^{2} + 1}{\left (8 \,{\left (2 \, a^{3} x^{3} + 3 \, a x\right )} \arccos \left (a x\right )^{3} - 3 \,{\left (2 \, a^{3} x^{3} + 15 \, a x\right )} \arccos \left (a x\right )\right )}}{128 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arccos(a*x)^4,x, algorithm="fricas")

[Out]

1/128*(3*a^4*x^4 + 4*(8*a^4*x^4 - 3)*arccos(a*x)^4 + 45*a^2*x^2 - 3*(8*a^4*x^4 + 24*a^2*x^2 - 15)*arccos(a*x)^
2 - 2*sqrt(-a^2*x^2 + 1)*(8*(2*a^3*x^3 + 3*a*x)*arccos(a*x)^3 - 3*(2*a^3*x^3 + 15*a*x)*arccos(a*x)))/a^4

________________________________________________________________________________________

Sympy [A]  time = 8.28441, size = 197, normalized size = 0.99 \begin{align*} \begin{cases} \frac{x^{4} \operatorname{acos}^{4}{\left (a x \right )}}{4} - \frac{3 x^{4} \operatorname{acos}^{2}{\left (a x \right )}}{16} + \frac{3 x^{4}}{128} - \frac{x^{3} \sqrt{- a^{2} x^{2} + 1} \operatorname{acos}^{3}{\left (a x \right )}}{4 a} + \frac{3 x^{3} \sqrt{- a^{2} x^{2} + 1} \operatorname{acos}{\left (a x \right )}}{32 a} - \frac{9 x^{2} \operatorname{acos}^{2}{\left (a x \right )}}{16 a^{2}} + \frac{45 x^{2}}{128 a^{2}} - \frac{3 x \sqrt{- a^{2} x^{2} + 1} \operatorname{acos}^{3}{\left (a x \right )}}{8 a^{3}} + \frac{45 x \sqrt{- a^{2} x^{2} + 1} \operatorname{acos}{\left (a x \right )}}{64 a^{3}} - \frac{3 \operatorname{acos}^{4}{\left (a x \right )}}{32 a^{4}} + \frac{45 \operatorname{acos}^{2}{\left (a x \right )}}{128 a^{4}} & \text{for}\: a \neq 0 \\\frac{\pi ^{4} x^{4}}{64} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*acos(a*x)**4,x)

[Out]

Piecewise((x**4*acos(a*x)**4/4 - 3*x**4*acos(a*x)**2/16 + 3*x**4/128 - x**3*sqrt(-a**2*x**2 + 1)*acos(a*x)**3/
(4*a) + 3*x**3*sqrt(-a**2*x**2 + 1)*acos(a*x)/(32*a) - 9*x**2*acos(a*x)**2/(16*a**2) + 45*x**2/(128*a**2) - 3*
x*sqrt(-a**2*x**2 + 1)*acos(a*x)**3/(8*a**3) + 45*x*sqrt(-a**2*x**2 + 1)*acos(a*x)/(64*a**3) - 3*acos(a*x)**4/
(32*a**4) + 45*acos(a*x)**2/(128*a**4), Ne(a, 0)), (pi**4*x**4/64, True))

________________________________________________________________________________________

Giac [A]  time = 1.20515, size = 234, normalized size = 1.18 \begin{align*} \frac{1}{4} \, x^{4} \arccos \left (a x\right )^{4} - \frac{3}{16} \, x^{4} \arccos \left (a x\right )^{2} - \frac{\sqrt{-a^{2} x^{2} + 1} x^{3} \arccos \left (a x\right )^{3}}{4 \, a} + \frac{3}{128} \, x^{4} + \frac{3 \, \sqrt{-a^{2} x^{2} + 1} x^{3} \arccos \left (a x\right )}{32 \, a} - \frac{9 \, x^{2} \arccos \left (a x\right )^{2}}{16 \, a^{2}} - \frac{3 \, \sqrt{-a^{2} x^{2} + 1} x \arccos \left (a x\right )^{3}}{8 \, a^{3}} + \frac{45 \, x^{2}}{128 \, a^{2}} - \frac{3 \, \arccos \left (a x\right )^{4}}{32 \, a^{4}} + \frac{45 \, \sqrt{-a^{2} x^{2} + 1} x \arccos \left (a x\right )}{64 \, a^{3}} + \frac{45 \, \arccos \left (a x\right )^{2}}{128 \, a^{4}} - \frac{189}{1024 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arccos(a*x)^4,x, algorithm="giac")

[Out]

1/4*x^4*arccos(a*x)^4 - 3/16*x^4*arccos(a*x)^2 - 1/4*sqrt(-a^2*x^2 + 1)*x^3*arccos(a*x)^3/a + 3/128*x^4 + 3/32
*sqrt(-a^2*x^2 + 1)*x^3*arccos(a*x)/a - 9/16*x^2*arccos(a*x)^2/a^2 - 3/8*sqrt(-a^2*x^2 + 1)*x*arccos(a*x)^3/a^
3 + 45/128*x^2/a^2 - 3/32*arccos(a*x)^4/a^4 + 45/64*sqrt(-a^2*x^2 + 1)*x*arccos(a*x)/a^3 + 45/128*arccos(a*x)^
2/a^4 - 189/1024/a^4