3.157 \(\int x^3 \sqrt{a-a \cos (x)} \, dx\)

Optimal. Leaf size=72 \[ 12 x^2 \sqrt{a-a \cos (x)}-2 x^3 \cot \left (\frac{x}{2}\right ) \sqrt{a-a \cos (x)}-96 \sqrt{a-a \cos (x)}+48 x \cot \left (\frac{x}{2}\right ) \sqrt{a-a \cos (x)} \]

[Out]

-96*Sqrt[a - a*Cos[x]] + 12*x^2*Sqrt[a - a*Cos[x]] + 48*x*Sqrt[a - a*Cos[x]]*Cot[x/2] - 2*x^3*Sqrt[a - a*Cos[x
]]*Cot[x/2]

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Rubi [A]  time = 0.114764, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3319, 3296, 2637} \[ 12 x^2 \sqrt{a-a \cos (x)}-2 x^3 \cot \left (\frac{x}{2}\right ) \sqrt{a-a \cos (x)}-96 \sqrt{a-a \cos (x)}+48 x \cot \left (\frac{x}{2}\right ) \sqrt{a-a \cos (x)} \]

Antiderivative was successfully verified.

[In]

Int[x^3*Sqrt[a - a*Cos[x]],x]

[Out]

-96*Sqrt[a - a*Cos[x]] + 12*x^2*Sqrt[a - a*Cos[x]] + 48*x*Sqrt[a - a*Cos[x]]*Cot[x/2] - 2*x^3*Sqrt[a - a*Cos[x
]]*Cot[x/2]

Rule 3319

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[((2*a)^IntPart[n
]*(a + b*Sin[e + f*x])^FracPart[n])/Sin[e/2 + (a*Pi)/(4*b) + (f*x)/2]^(2*FracPart[n]), Int[(c + d*x)^m*Sin[e/2
+ (a*Pi)/(4*b) + (f*x)/2]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[n
+ 1/2] && (GtQ[n, 0] || IGtQ[m, 0])

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int x^3 \sqrt{a-a \cos (x)} \, dx &=\left (\sqrt{a-a \cos (x)} \csc \left (\frac{x}{2}\right )\right ) \int x^3 \sin \left (\frac{x}{2}\right ) \, dx\\ &=-2 x^3 \sqrt{a-a \cos (x)} \cot \left (\frac{x}{2}\right )+\left (6 \sqrt{a-a \cos (x)} \csc \left (\frac{x}{2}\right )\right ) \int x^2 \cos \left (\frac{x}{2}\right ) \, dx\\ &=12 x^2 \sqrt{a-a \cos (x)}-2 x^3 \sqrt{a-a \cos (x)} \cot \left (\frac{x}{2}\right )-\left (24 \sqrt{a-a \cos (x)} \csc \left (\frac{x}{2}\right )\right ) \int x \sin \left (\frac{x}{2}\right ) \, dx\\ &=12 x^2 \sqrt{a-a \cos (x)}+48 x \sqrt{a-a \cos (x)} \cot \left (\frac{x}{2}\right )-2 x^3 \sqrt{a-a \cos (x)} \cot \left (\frac{x}{2}\right )-\left (48 \sqrt{a-a \cos (x)} \csc \left (\frac{x}{2}\right )\right ) \int \cos \left (\frac{x}{2}\right ) \, dx\\ &=-96 \sqrt{a-a \cos (x)}+12 x^2 \sqrt{a-a \cos (x)}+48 x \sqrt{a-a \cos (x)} \cot \left (\frac{x}{2}\right )-2 x^3 \sqrt{a-a \cos (x)} \cot \left (\frac{x}{2}\right )\\ \end{align*}

Mathematica [A]  time = 0.051273, size = 34, normalized size = 0.47 \[ -2 \left (x \left (x^2-24\right ) \cot \left (\frac{x}{2}\right )-6 \left (x^2-8\right )\right ) \sqrt{a-a \cos (x)} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*Sqrt[a - a*Cos[x]],x]

[Out]

-2*Sqrt[a - a*Cos[x]]*(-6*(-8 + x^2) + x*(-24 + x^2)*Cot[x/2])

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Maple [C]  time = 0.163, size = 86, normalized size = 1.2 \begin{align*}{\frac{-i\sqrt{2} \left ( 6\,i{x}^{2}{{\rm e}^{ix}}+{x}^{3}{{\rm e}^{ix}}-6\,i{x}^{2}+{x}^{3}-48\,i{{\rm e}^{ix}}-24\,x{{\rm e}^{ix}}+48\,i-24\,x \right ) }{{{\rm e}^{ix}}-1}\sqrt{-a \left ({{\rm e}^{ix}}-1 \right ) ^{2}{{\rm e}^{-ix}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a-a*cos(x))^(1/2),x)

[Out]

-I*2^(1/2)*(-a*(exp(I*x)-1)^2*exp(-I*x))^(1/2)/(exp(I*x)-1)*(6*I*x^2*exp(I*x)+x^3*exp(I*x)-6*I*x^2+x^3-48*I*ex
p(I*x)-24*x*exp(I*x)+48*I-24*x)

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Maxima [B]  time = 1.97059, size = 174, normalized size = 2.42 \begin{align*} -{\left ({\left (6 \, \sqrt{2} x^{2} - 6 \,{\left (\sqrt{2} x^{2} - 8 \, \sqrt{2}\right )} \cos \left (x\right ) -{\left (\sqrt{2} x^{3} - 24 \, \sqrt{2} x\right )} \sin \left (x\right ) - 48 \, \sqrt{2}\right )} \cos \left (\frac{1}{2} \, \pi + \frac{1}{2} \, \arctan \left (\sin \left (x\right ), \cos \left (x\right )\right )\right ) +{\left (\sqrt{2} x^{3} +{\left (\sqrt{2} x^{3} - 24 \, \sqrt{2} x\right )} \cos \left (x\right ) - 6 \,{\left (\sqrt{2} x^{2} - 8 \, \sqrt{2}\right )} \sin \left (x\right ) - 24 \, \sqrt{2} x\right )} \sin \left (\frac{1}{2} \, \pi + \frac{1}{2} \, \arctan \left (\sin \left (x\right ), \cos \left (x\right )\right )\right )\right )} \sqrt{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a-a*cos(x))^(1/2),x, algorithm="maxima")

[Out]

-((6*sqrt(2)*x^2 - 6*(sqrt(2)*x^2 - 8*sqrt(2))*cos(x) - (sqrt(2)*x^3 - 24*sqrt(2)*x)*sin(x) - 48*sqrt(2))*cos(
1/2*pi + 1/2*arctan2(sin(x), cos(x))) + (sqrt(2)*x^3 + (sqrt(2)*x^3 - 24*sqrt(2)*x)*cos(x) - 6*(sqrt(2)*x^2 -
8*sqrt(2))*sin(x) - 24*sqrt(2)*x)*sin(1/2*pi + 1/2*arctan2(sin(x), cos(x))))*sqrt(a)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a-a*cos(x))^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \sqrt{- a \left (\cos{\left (x \right )} - 1\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(a-a*cos(x))**(1/2),x)

[Out]

Integral(x**3*sqrt(-a*(cos(x) - 1)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{-a \cos \left (x\right ) + a} x^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a-a*cos(x))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(-a*cos(x) + a)*x^3, x)