Integrand size = 14, antiderivative size = 68 \[ \int x^3 \sqrt {a+a \cos (x)} \, dx=-96 \sqrt {a+a \cos (x)}+12 x^2 \sqrt {a+a \cos (x)}-48 x \sqrt {a+a \cos (x)} \tan \left (\frac {x}{2}\right )+2 x^3 \sqrt {a+a \cos (x)} \tan \left (\frac {x}{2}\right ) \] 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)*t an(1/2*x)+2*x^3*(a+a*cos(x))^(1/2)*tan(1/2*x)
Time = 0.16 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.49 \[ \int x^3 \sqrt {a+a \cos (x)} \, dx=2 \sqrt {a (1+\cos (x))} \left (6 \left (-8+x^2\right )+x \left (-24+x^2\right ) \tan \left (\frac {x}{2}\right )\right ) \] Input:
Integrate[x^3*Sqrt[a + a*Cos[x]],x]
Output:
2*Sqrt[a*(1 + Cos[x])]*(6*(-8 + x^2) + x*(-24 + x^2)*Tan[x/2])
Time = 0.51 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.93, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {3042, 3800, 3042, 3777, 25, 3042, 3777, 3042, 3777, 25, 3042, 3118}
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.
\(\displaystyle \int x^3 \sqrt {a \cos (x)+a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int x^3 \sqrt {a \sin \left (x+\frac {\pi }{2}\right )+a}dx\) |
\(\Big \downarrow \) 3800 |
\(\displaystyle \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \int x^3 \cos \left (\frac {x}{2}\right )dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \int x^3 \sin \left (\frac {x}{2}+\frac {\pi }{2}\right )dx\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \left (6 \int -x^2 \sin \left (\frac {x}{2}\right )dx+2 x^3 \sin \left (\frac {x}{2}\right )\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \left (2 x^3 \sin \left (\frac {x}{2}\right )-6 \int x^2 \sin \left (\frac {x}{2}\right )dx\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \left (2 x^3 \sin \left (\frac {x}{2}\right )-6 \int x^2 \sin \left (\frac {x}{2}\right )dx\right )\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \left (2 x^3 \sin \left (\frac {x}{2}\right )-6 \left (4 \int x \cos \left (\frac {x}{2}\right )dx-2 x^2 \cos \left (\frac {x}{2}\right )\right )\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \left (2 x^3 \sin \left (\frac {x}{2}\right )-6 \left (4 \int x \sin \left (\frac {x}{2}+\frac {\pi }{2}\right )dx-2 x^2 \cos \left (\frac {x}{2}\right )\right )\right )\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \left (2 x^3 \sin \left (\frac {x}{2}\right )-6 \left (4 \left (2 \int -\sin \left (\frac {x}{2}\right )dx+2 x \sin \left (\frac {x}{2}\right )\right )-2 x^2 \cos \left (\frac {x}{2}\right )\right )\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \left (2 x^3 \sin \left (\frac {x}{2}\right )-6 \left (4 \left (2 x \sin \left (\frac {x}{2}\right )-2 \int \sin \left (\frac {x}{2}\right )dx\right )-2 x^2 \cos \left (\frac {x}{2}\right )\right )\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \left (2 x^3 \sin \left (\frac {x}{2}\right )-6 \left (4 \left (2 x \sin \left (\frac {x}{2}\right )-2 \int \sin \left (\frac {x}{2}\right )dx\right )-2 x^2 \cos \left (\frac {x}{2}\right )\right )\right )\) |
\(\Big \downarrow \) 3118 |
\(\displaystyle \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \left (2 x^3 \sin \left (\frac {x}{2}\right )-6 \left (4 \left (2 x \sin \left (\frac {x}{2}\right )+4 \cos \left (\frac {x}{2}\right )\right )-2 x^2 \cos \left (\frac {x}{2}\right )\right )\right )\) |
Input:
Int[x^3*Sqrt[a + a*Cos[x]],x]
Output:
Sqrt[a + a*Cos[x]]*Sec[x/2]*(2*x^3*Sin[x/2] - 6*(-2*x^2*Cos[x/2] + 4*(4*Co s[x/2] + 2*x*Sin[x/2])))
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 = 0.45 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.28
method | result | size |
risch | \(-\frac {i \sqrt {2}\, \sqrt {a \left ({\mathrm e}^{i x}+1\right )^{2} {\mathrm e}^{-i x}}\, \left (6 i x^{2} {\mathrm e}^{i x}+x^{3} {\mathrm e}^{i x}+6 i x^{2}-x^{3}-48 i {\mathrm e}^{i x}-24 x \,{\mathrm e}^{i x}-48 i+24 x \right )}{{\mathrm e}^{i x}+1}\) | \(87\) |
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)
Time = 0.11 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.71 \[ \int x^3 \sqrt {a+a \cos (x)} \, dx=2 \, {\left (\sqrt {2} x^{3} \sin \left (\frac {1}{2} \, x\right ) + 6 \, \sqrt {2} x^{2} \cos \left (\frac {1}{2} \, x\right ) - 24 \, \sqrt {2} x \sin \left (\frac {1}{2} \, x\right ) - 48 \, \sqrt {2} \cos \left (\frac {1}{2} \, x\right )\right )} \sqrt {a} \] Input:
integrate(x^3*(a+a*cos(x))^(1/2),x, algorithm="maxima")
Output:
2*(sqrt(2)*x^3*sin(1/2*x) + 6*sqrt(2)*x^2*cos(1/2*x) - 24*sqrt(2)*x*sin(1/ 2*x) - 48*sqrt(2)*cos(1/2*x))*sqrt(a)
Time = 0.37 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.81 \[ \int x^3 \sqrt {a+a \cos (x)} \, dx=2 \, \sqrt {2} {\left (6 \, {\left (x^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right ) - 8 \, \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right )\right )} \cos \left (\frac {1}{2} \, x\right ) + {\left (x^{3} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right ) - 24 \, x \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right )\right )} \sin \left (\frac {1}{2} \, x\right )\right )} \sqrt {a} \] Input:
integrate(x^3*(a+a*cos(x))^(1/2),x, algorithm="giac")
Output:
2*sqrt(2)*(6*(x^2*sgn(cos(1/2*x)) - 8*sgn(cos(1/2*x)))*cos(1/2*x) + (x^3*s gn(cos(1/2*x)) - 24*x*sgn(cos(1/2*x)))*sin(1/2*x))*sqrt(a)
Time = 40.67 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.34 \[ \int x^3 \sqrt {a+a \cos (x)} \, dx=\frac {2\,\sqrt {a}\,\sqrt {\cos \left (x\right )+1}\,\left (24\,x-\cos \left (x\right )\,48{}\mathrm {i}+48\,\sin \left (x\right )+x^2\,\cos \left (x\right )\,6{}\mathrm {i}+x^3\,\cos \left (x\right )-6\,x^2\,\sin \left (x\right )+x^3\,\sin \left (x\right )\,1{}\mathrm {i}-24\,x\,\cos \left (x\right )-x\,\sin \left (x\right )\,24{}\mathrm {i}+x^2\,6{}\mathrm {i}-x^3-48{}\mathrm {i}\right )}{\cos \left (x\right )\,1{}\mathrm {i}-\sin \left (x\right )+1{}\mathrm {i}} \] Input:
int(x^3*(a + a*cos(x))^(1/2),x)
Output:
(2*a^(1/2)*(cos(x) + 1)^(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))/(cos(x)*1i - sin(x) + 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)