3.656 \(\int x^3 \left (1-x^2\right )^{3/2} \cos ^{-1}(x) \, dx\)

Optimal. Leaf size=61 \[ -\frac{x^7}{49}+\frac{8 x^5}{175}-\frac{x^3}{105}+\frac{1}{7} \left (1-x^2\right )^{7/2} \cos ^{-1}(x)-\frac{1}{5} \left (1-x^2\right )^{5/2} \cos ^{-1}(x)-\frac{2 x}{35} \]

[Out]

(-2*x)/35 - x^3/105 + (8*x^5)/175 - x^7/49 - ((1 - x^2)^(5/2)*ArcCos[x])/5 + ((1
- x^2)^(7/2)*ArcCos[x])/7

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Rubi [A]  time = 0.125726, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ -\frac{x^7}{49}+\frac{8 x^5}{175}-\frac{x^3}{105}+\frac{1}{7} \left (1-x^2\right )^{7/2} \cos ^{-1}(x)-\frac{1}{5} \left (1-x^2\right )^{5/2} \cos ^{-1}(x)-\frac{2 x}{35} \]

Antiderivative was successfully verified.

[In]  Int[x^3*(1 - x^2)^(3/2)*ArcCos[x],x]

[Out]

(-2*x)/35 - x^3/105 + (8*x^5)/175 - x^7/49 - ((1 - x^2)^(5/2)*ArcCos[x])/5 + ((1
- x^2)^(7/2)*ArcCos[x])/7

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Rubi in Sympy [A]  time = 7.70813, size = 48, normalized size = 0.79 \[ - \frac{x^{7}}{49} + \frac{8 x^{5}}{175} - \frac{x^{3}}{105} - \frac{2 x}{35} + \frac{\left (- x^{2} + 1\right )^{\frac{7}{2}} \operatorname{acos}{\left (x \right )}}{7} - \frac{\left (- x^{2} + 1\right )^{\frac{5}{2}} \operatorname{acos}{\left (x \right )}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(-x**2+1)**(3/2)*acos(x),x)

[Out]

-x**7/49 + 8*x**5/175 - x**3/105 - 2*x/35 + (-x**2 + 1)**(7/2)*acos(x)/7 - (-x**
2 + 1)**(5/2)*acos(x)/5

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Mathematica [A]  time = 0.052059, size = 47, normalized size = 0.77 \[ -\frac{1}{35} \left (5 x^2+2\right ) \left (1-x^2\right )^{5/2} \cos ^{-1}(x)-\frac{x \left (75 x^6-168 x^4+35 x^2+210\right )}{3675} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3*(1 - x^2)^(3/2)*ArcCos[x],x]

[Out]

-(x*(210 + 35*x^2 - 168*x^4 + 75*x^6))/3675 - ((1 - x^2)^(5/2)*(2 + 5*x^2)*ArcCo
s[x])/35

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Maple [C]  time = 0.378, size = 430, normalized size = 7.1 \[{\frac{i+7\,\arccos \left ( x \right ) }{6272} \left ( 64\,i{x}^{7}-64\,\sqrt{-{x}^{2}+1}{x}^{6}-112\,i{x}^{5}+80\,\sqrt{-{x}^{2}+1}{x}^{4}+56\,i{x}^{3}-24\,{x}^{2}\sqrt{-{x}^{2}+1}-7\,ix+\sqrt{-{x}^{2}+1} \right ) }-{\frac{i+5\,\arccos \left ( x \right ) }{3200} \left ( 16\,i{x}^{5}-16\,\sqrt{-{x}^{2}+1}{x}^{4}-20\,i{x}^{3}+12\,{x}^{2}\sqrt{-{x}^{2}+1}+5\,ix-\sqrt{-{x}^{2}+1} \right ) }-{\frac{i+3\,\arccos \left ( x \right ) }{384} \left ( 4\,i{x}^{3}-4\,{x}^{2}\sqrt{-{x}^{2}+1}-3\,ix+\sqrt{-{x}^{2}+1} \right ) }+{\frac{3\,\arccos \left ( x \right ) +3\,i}{128} \left ( ix-\sqrt{-{x}^{2}+1} \right ) }-{\frac{3\,\arccos \left ( x \right ) -3\,i}{128} \left ( ix+\sqrt{-{x}^{2}+1} \right ) }+{\frac{-i+3\,\arccos \left ( x \right ) }{384} \left ( 4\,i{x}^{3}+4\,{x}^{2}\sqrt{-{x}^{2}+1}-3\,ix-\sqrt{-{x}^{2}+1} \right ) }+{\frac{-i+5\,\arccos \left ( x \right ) }{3200} \left ( 16\,i{x}^{5}+16\,\sqrt{-{x}^{2}+1}{x}^{4}-20\,i{x}^{3}-12\,{x}^{2}\sqrt{-{x}^{2}+1}+5\,ix+\sqrt{-{x}^{2}+1} \right ) }-{\frac{-i+7\,\arccos \left ( x \right ) }{6272} \left ( 64\,i{x}^{7}+64\,\sqrt{-{x}^{2}+1}{x}^{6}-112\,i{x}^{5}-80\,\sqrt{-{x}^{2}+1}{x}^{4}+56\,i{x}^{3}+24\,{x}^{2}\sqrt{-{x}^{2}+1}-7\,ix-\sqrt{-{x}^{2}+1} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(-x^2+1)^(3/2)*arccos(x),x)

[Out]

1/6272*(I+7*arccos(x))*(64*I*x^7-64*(-x^2+1)^(1/2)*x^6-112*I*x^5+80*(-x^2+1)^(1/
2)*x^4+56*I*x^3-24*x^2*(-x^2+1)^(1/2)-7*I*x+(-x^2+1)^(1/2))-1/3200*(I+5*arccos(x
))*(16*I*x^5-16*(-x^2+1)^(1/2)*x^4-20*I*x^3+12*x^2*(-x^2+1)^(1/2)+5*I*x-(-x^2+1)
^(1/2))-1/384*(I+3*arccos(x))*(4*I*x^3-4*x^2*(-x^2+1)^(1/2)-3*I*x+(-x^2+1)^(1/2)
)+3/128*(arccos(x)+I)*(I*x-(-x^2+1)^(1/2))-3/128*(arccos(x)-I)*(I*x+(-x^2+1)^(1/
2))+1/384*(-I+3*arccos(x))*(4*I*x^3+4*x^2*(-x^2+1)^(1/2)-3*I*x-(-x^2+1)^(1/2))+1
/3200*(-I+5*arccos(x))*(16*I*x^5+16*(-x^2+1)^(1/2)*x^4-20*I*x^3-12*x^2*(-x^2+1)^
(1/2)+5*I*x+(-x^2+1)^(1/2))-1/6272*(-I+7*arccos(x))*(64*I*x^7+64*(-x^2+1)^(1/2)*
x^6-112*I*x^5-80*(-x^2+1)^(1/2)*x^4+56*I*x^3+24*x^2*(-x^2+1)^(1/2)-7*I*x-(-x^2+1
)^(1/2))

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Maxima [A]  time = 1.54128, size = 66, normalized size = 1.08 \[ -\frac{1}{49} \, x^{7} + \frac{8}{175} \, x^{5} - \frac{1}{105} \, x^{3} - \frac{1}{35} \,{\left (5 \,{\left (-x^{2} + 1\right )}^{\frac{5}{2}} x^{2} + 2 \,{\left (-x^{2} + 1\right )}^{\frac{5}{2}}\right )} \arccos \left (x\right ) - \frac{2}{35} \, x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x^2 + 1)^(3/2)*x^3*arccos(x),x, algorithm="maxima")

[Out]

-1/49*x^7 + 8/175*x^5 - 1/105*x^3 - 1/35*(5*(-x^2 + 1)^(5/2)*x^2 + 2*(-x^2 + 1)^
(5/2))*arccos(x) - 2/35*x

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Fricas [A]  time = 0.229817, size = 63, normalized size = 1.03 \[ -\frac{1}{49} \, x^{7} + \frac{8}{175} \, x^{5} - \frac{1}{105} \, x^{3} - \frac{1}{35} \,{\left (5 \, x^{6} - 8 \, x^{4} + x^{2} + 2\right )} \sqrt{-x^{2} + 1} \arccos \left (x\right ) - \frac{2}{35} \, x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x^2 + 1)^(3/2)*x^3*arccos(x),x, algorithm="fricas")

[Out]

-1/49*x^7 + 8/175*x^5 - 1/105*x^3 - 1/35*(5*x^6 - 8*x^4 + x^2 + 2)*sqrt(-x^2 + 1
)*arccos(x) - 2/35*x

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(-x**2+1)**(3/2)*acos(x),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.230798, size = 81, normalized size = 1.33 \[ -\frac{1}{49} \, x^{7} + \frac{8}{175} \, x^{5} - \frac{1}{105} \, x^{3} - \frac{1}{35} \,{\left (5 \,{\left (x^{2} - 1\right )}^{3} \sqrt{-x^{2} + 1} + 7 \,{\left (x^{2} - 1\right )}^{2} \sqrt{-x^{2} + 1}\right )} \arccos \left (x\right ) - \frac{2}{35} \, x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x^2 + 1)^(3/2)*x^3*arccos(x),x, algorithm="giac")

[Out]

-1/49*x^7 + 8/175*x^5 - 1/105*x^3 - 1/35*(5*(x^2 - 1)^3*sqrt(-x^2 + 1) + 7*(x^2
- 1)^2*sqrt(-x^2 + 1))*arccos(x) - 2/35*x