Integrand size = 15, antiderivative size = 72 \[ \int x^3 \sqrt {a-a \cos (x)} \, dx=-96 \sqrt {a-a \cos (x)}+12 x^2 \sqrt {a-a \cos (x)}+48 x \sqrt {a-a \cos (x)} \cot \left (\frac {x}{2}\right )-2 x^3 \sqrt {a-a \cos (x)} \cot \left (\frac {x}{2}\right ) \] Output:
-96*(a-a*cos(x))^(1/2)+12*x^2*(a-a*cos(x))^(1/2)+48*x*(a-a*cos(x))^(1/2)*c ot(1/2*x)-2*x^3*(a-a*cos(x))^(1/2)*cot(1/2*x)
Time = 0.17 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.47 \[ \int x^3 \sqrt {a-a \cos (x)} \, dx=-2 \sqrt {a-a \cos (x)} \left (-6 \left (-8+x^2\right )+x \left (-24+x^2\right ) \cot \left (\frac {x}{2}\right )\right ) \] Input:
Integrate[x^3*Sqrt[a - a*Cos[x]],x]
Output:
-2*Sqrt[a - a*Cos[x]]*(-6*(-8 + x^2) + x*(-24 + x^2)*Cot[x/2])
Time = 0.51 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.89, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.733, Rules used = {3042, 3800, 3042, 3777, 3042, 3777, 25, 3042, 3777, 3042, 3117}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x^3 \sqrt {a-a \cos (x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int x^3 \sqrt {a-a \sin \left (x+\frac {\pi }{2}\right )}dx\) |
\(\Big \downarrow \) 3800 |
\(\displaystyle \csc \left (\frac {x}{2}\right ) \sqrt {a-a \cos (x)} \int x^3 \sin \left (\frac {x}{2}\right )dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \csc \left (\frac {x}{2}\right ) \sqrt {a-a \cos (x)} \int x^3 \sin \left (\frac {x}{2}\right )dx\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \csc \left (\frac {x}{2}\right ) \sqrt {a-a \cos (x)} \left (6 \int x^2 \cos \left (\frac {x}{2}\right )dx-2 x^3 \cos \left (\frac {x}{2}\right )\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \csc \left (\frac {x}{2}\right ) \sqrt {a-a \cos (x)} \left (6 \int x^2 \sin \left (\frac {x}{2}+\frac {\pi }{2}\right )dx-2 x^3 \cos \left (\frac {x}{2}\right )\right )\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \csc \left (\frac {x}{2}\right ) \sqrt {a-a \cos (x)} \left (6 \left (4 \int -x \sin \left (\frac {x}{2}\right )dx+2 x^2 \sin \left (\frac {x}{2}\right )\right )-2 x^3 \cos \left (\frac {x}{2}\right )\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \csc \left (\frac {x}{2}\right ) \sqrt {a-a \cos (x)} \left (6 \left (2 x^2 \sin \left (\frac {x}{2}\right )-4 \int x \sin \left (\frac {x}{2}\right )dx\right )-2 x^3 \cos \left (\frac {x}{2}\right )\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \csc \left (\frac {x}{2}\right ) \sqrt {a-a \cos (x)} \left (6 \left (2 x^2 \sin \left (\frac {x}{2}\right )-4 \int x \sin \left (\frac {x}{2}\right )dx\right )-2 x^3 \cos \left (\frac {x}{2}\right )\right )\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \csc \left (\frac {x}{2}\right ) \sqrt {a-a \cos (x)} \left (6 \left (2 x^2 \sin \left (\frac {x}{2}\right )-4 \left (2 \int \cos \left (\frac {x}{2}\right )dx-2 x \cos \left (\frac {x}{2}\right )\right )\right )-2 x^3 \cos \left (\frac {x}{2}\right )\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \csc \left (\frac {x}{2}\right ) \sqrt {a-a \cos (x)} \left (6 \left (2 x^2 \sin \left (\frac {x}{2}\right )-4 \left (2 \int \sin \left (\frac {x}{2}+\frac {\pi }{2}\right )dx-2 x \cos \left (\frac {x}{2}\right )\right )\right )-2 x^3 \cos \left (\frac {x}{2}\right )\right )\) |
\(\Big \downarrow \) 3117 |
\(\displaystyle \csc \left (\frac {x}{2}\right ) \sqrt {a-a \cos (x)} \left (6 \left (2 x^2 \sin \left (\frac {x}{2}\right )-4 \left (4 \sin \left (\frac {x}{2}\right )-2 x \cos \left (\frac {x}{2}\right )\right )\right )-2 x^3 \cos \left (\frac {x}{2}\right )\right )\) |
Input:
Int[x^3*Sqrt[a - a*Cos[x]],x]
Output:
Sqrt[a - a*Cos[x]]*Csc[x/2]*(-2*x^3*Cos[x/2] + 6*(2*x^2*Sin[x/2] - 4*(-2*x *Cos[x/2] + 4*Sin[x/2])))
Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( -(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*C os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(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])
Result contains complex when optimal does not.
Time = 0.70 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.19
method | result | size |
risch | \(-\frac {i \sqrt {2}\, \sqrt {-a \left ({\mathrm e}^{i x}-1\right )^{2} {\mathrm e}^{-i x}}\, \left (6 i x^{2} {\mathrm e}^{i x}+x^{3} {\mathrm e}^{i x}-6 i x^{2}+x^{3}-48 i {\mathrm e}^{i x}-24 x \,{\mathrm e}^{i x}+48 i-24 x \right )}{{\mathrm e}^{i x}-1}\) | \(86\) |
Input:
int(x^3*(a-a*cos(x))^(1/2),x,method=_RETURNVERBOSE)
Output:
-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*exp(I*x)-24*x*exp(I*x)+48*I-24*x)
Exception generated. \[ \int x^3 \sqrt {a-a \cos (x)} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^3*(a-a*cos(x))^(1/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (has polynomial part)
\[ \int x^3 \sqrt {a-a \cos (x)} \, dx=\int x^{3} \sqrt {- a \left (\cos {\left (x \right )} - 1\right )}\, dx \] Input:
integrate(x**3*(a-a*cos(x))**(1/2),x)
Output:
Integral(x**3*sqrt(-a*(cos(x) - 1)), x)
Leaf count of result is larger than twice the leaf count of optimal. 129 vs. \(2 (60) = 120\).
Time = 0.12 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.79 \[ \int x^3 \sqrt {a-a \cos (x)} \, dx=-{\left ({\left (6 \, \sqrt {2} x^{2} - 6 \, {\left (\sqrt {2} x^{2} - 8 \, \sqrt {2}\right )} \cos \left (x\right ) - {\left (\sqrt {2} x^{3} - 24 \, \sqrt {2} x\right )} \sin \left (x\right ) - 48 \, \sqrt {2}\right )} \cos \left (\frac {1}{2} \, \pi + \frac {1}{2} \, \arctan \left (\sin \left (x\right ), \cos \left (x\right )\right )\right ) + {\left (\sqrt {2} x^{3} + {\left (\sqrt {2} x^{3} - 24 \, \sqrt {2} x\right )} \cos \left (x\right ) - 6 \, {\left (\sqrt {2} x^{2} - 8 \, \sqrt {2}\right )} \sin \left (x\right ) - 24 \, \sqrt {2} x\right )} \sin \left (\frac {1}{2} \, \pi + \frac {1}{2} \, \arctan \left (\sin \left (x\right ), \cos \left (x\right )\right )\right )\right )} \sqrt {a} \] Input:
integrate(x^3*(a-a*cos(x))^(1/2),x, algorithm="maxima")
Output:
-((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)
Time = 0.34 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.76 \[ \int x^3 \sqrt {a-a \cos (x)} \, dx=-2 \, \sqrt {2} {\left ({\left (x^{3} \mathrm {sgn}\left (\sin \left (\frac {1}{2} \, x\right )\right ) - 24 \, x \mathrm {sgn}\left (\sin \left (\frac {1}{2} \, x\right )\right )\right )} \cos \left (\frac {1}{2} \, x\right ) - 6 \, {\left (x^{2} \mathrm {sgn}\left (\sin \left (\frac {1}{2} \, x\right )\right ) - 8 \, \mathrm {sgn}\left (\sin \left (\frac {1}{2} \, x\right )\right )\right )} \sin \left (\frac {1}{2} \, x\right )\right )} \sqrt {a} \] Input:
integrate(x^3*(a-a*cos(x))^(1/2),x, algorithm="giac")
Output:
-2*sqrt(2)*((x^3*sgn(sin(1/2*x)) - 24*x*sgn(sin(1/2*x)))*cos(1/2*x) - 6*(x ^2*sgn(sin(1/2*x)) - 8*sgn(sin(1/2*x)))*sin(1/2*x))*sqrt(a)
Time = 40.64 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.28 \[ \int x^3 \sqrt {a-a \cos (x)} \, dx=\frac {2\,\sqrt {a}\,\sqrt {1-\cos \left (x\right )}\,\left (24\,x+\cos \left (x\right )\,48{}\mathrm {i}-48\,\sin \left (x\right )-x^2\,\cos \left (x\right )\,6{}\mathrm {i}-x^3\,\cos \left (x\right )+6\,x^2\,\sin \left (x\right )-x^3\,\sin \left (x\right )\,1{}\mathrm {i}+24\,x\,\cos \left (x\right )+x\,\sin \left (x\right )\,24{}\mathrm {i}+x^2\,6{}\mathrm {i}-x^3-48{}\mathrm {i}\right )}{\sin \left (x\right )-\cos \left (x\right )\,1{}\mathrm {i}+1{}\mathrm {i}} \] Input:
int(x^3*(a - a*cos(x))^(1/2),x)
Output:
(2*a^(1/2)*(1 - cos(x))^(1/2)*(24*x + cos(x)*48i - 48*sin(x) - x^2*cos(x)* 6i - x^3*cos(x) + 6*x^2*sin(x) - x^3*sin(x)*1i + 24*x*cos(x) + x*sin(x)*24 i + x^2*6i - x^3 - 48i))/(sin(x) - cos(x)*1i + 1i)
\[ \int x^3 \sqrt {a-a \cos (x)} \, dx=\sqrt {a}\, \left (\int \sqrt {-\cos \left (x \right )+1}\, x^{3}d x \right ) \] Input:
int(x^3*(a-a*cos(x))^(1/2),x)
Output:
sqrt(a)*int(sqrt( - cos(x) + 1)*x**3,x)