Integrand size = 12, antiderivative size = 89 \[ \int x (a+a \cos (x))^{3/2} \, dx=\frac {16}{3} a \sqrt {a+a \cos (x)}+\frac {8}{9} a \cos ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cos (x)}+\frac {4}{3} a x \cos \left (\frac {x}{2}\right ) \sqrt {a+a \cos (x)} \sin \left (\frac {x}{2}\right )+\frac {8}{3} a x \sqrt {a+a \cos (x)} \tan \left (\frac {x}{2}\right ) \] Output:
16/3*a*(a+a*cos(x))^(1/2)+8/9*a*cos(1/2*x)^2*(a+a*cos(x))^(1/2)+4/3*a*x*co s(1/2*x)*(a+a*cos(x))^(1/2)*sin(1/2*x)+8/3*a*x*(a+a*cos(x))^(1/2)*tan(1/2* x)
Time = 0.12 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.51 \[ \int x (a+a \cos (x))^{3/2} \, dx=\frac {1}{9} a \sqrt {a (1+\cos (x))} \left (52+4 \cos (x)+3 x \sec \left (\frac {x}{2}\right ) \sin \left (\frac {3 x}{2}\right )+27 x \tan \left (\frac {x}{2}\right )\right ) \] Input:
Integrate[x*(a + a*Cos[x])^(3/2),x]
Output:
(a*Sqrt[a*(1 + Cos[x])]*(52 + 4*Cos[x] + 3*x*Sec[x/2]*Sin[(3*x)/2] + 27*x* Tan[x/2]))/9
Time = 0.41 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.82, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {3042, 3800, 3042, 3791, 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.
\(\displaystyle \int x (a \cos (x)+a)^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int x \left (a \sin \left (x+\frac {\pi }{2}\right )+a\right )^{3/2}dx\) |
\(\Big \downarrow \) 3800 |
\(\displaystyle 2 a \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \int x \cos ^3\left (\frac {x}{2}\right )dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 2 a \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \int x \sin \left (\frac {x}{2}+\frac {\pi }{2}\right )^3dx\) |
\(\Big \downarrow \) 3791 |
\(\displaystyle 2 a \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \left (\frac {2}{3} \int x \cos \left (\frac {x}{2}\right )dx+\frac {4}{9} \cos ^3\left (\frac {x}{2}\right )+\frac {2}{3} x \sin \left (\frac {x}{2}\right ) \cos ^2\left (\frac {x}{2}\right )\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 2 a \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \left (\frac {2}{3} \int x \sin \left (\frac {x}{2}+\frac {\pi }{2}\right )dx+\frac {4}{9} \cos ^3\left (\frac {x}{2}\right )+\frac {2}{3} x \sin \left (\frac {x}{2}\right ) \cos ^2\left (\frac {x}{2}\right )\right )\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle 2 a \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \left (\frac {2}{3} \left (2 \int -\sin \left (\frac {x}{2}\right )dx+2 x \sin \left (\frac {x}{2}\right )\right )+\frac {4}{9} \cos ^3\left (\frac {x}{2}\right )+\frac {2}{3} x \sin \left (\frac {x}{2}\right ) \cos ^2\left (\frac {x}{2}\right )\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle 2 a \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \left (\frac {2}{3} \left (2 x \sin \left (\frac {x}{2}\right )-2 \int \sin \left (\frac {x}{2}\right )dx\right )+\frac {4}{9} \cos ^3\left (\frac {x}{2}\right )+\frac {2}{3} x \sin \left (\frac {x}{2}\right ) \cos ^2\left (\frac {x}{2}\right )\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 2 a \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \left (\frac {2}{3} \left (2 x \sin \left (\frac {x}{2}\right )-2 \int \sin \left (\frac {x}{2}\right )dx\right )+\frac {4}{9} \cos ^3\left (\frac {x}{2}\right )+\frac {2}{3} x \sin \left (\frac {x}{2}\right ) \cos ^2\left (\frac {x}{2}\right )\right )\) |
\(\Big \downarrow \) 3118 |
\(\displaystyle 2 a \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \left (\frac {4}{9} \cos ^3\left (\frac {x}{2}\right )+\frac {2}{3} x \sin \left (\frac {x}{2}\right ) \cos ^2\left (\frac {x}{2}\right )+\frac {2}{3} \left (2 x \sin \left (\frac {x}{2}\right )+4 \cos \left (\frac {x}{2}\right )\right )\right )\) |
Input:
Int[x*(a + a*Cos[x])^(3/2),x]
Output:
2*a*Sqrt[a + a*Cos[x]]*Sec[x/2]*((4*Cos[x/2]^3)/9 + (2*x*Cos[x/2]^2*Sin[x/ 2])/3 + (2*(4*Cos[x/2] + 2*x*Sin[x/2]))/3)
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_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[d*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Simp[b*(c + d*x)*Cos[e + f*x ]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^2*((n - 1)/n) Int[(c + d* x)*(b*Sin[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1]
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])
\[\int x \left (a +a \cos \left (x \right )\right )^{\frac {3}{2}}d x\]
Input:
int(x*(a+a*cos(x))^(3/2),x)
Output:
int(x*(a+a*cos(x))^(3/2),x)
Exception generated. \[ \int x (a+a \cos (x))^{3/2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x*(a+a*cos(x))^(3/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (has polynomial part)
\[ \int x (a+a \cos (x))^{3/2} \, dx=\int x \left (a \left (\cos {\left (x \right )} + 1\right )\right )^{\frac {3}{2}}\, dx \] Input:
integrate(x*(a+a*cos(x))**(3/2),x)
Output:
Integral(x*(a*(cos(x) + 1))**(3/2), x)
Time = 0.12 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.54 \[ \int x (a+a \cos (x))^{3/2} \, dx=\frac {1}{9} \, {\left (3 \, \sqrt {2} a x \sin \left (\frac {3}{2} \, x\right ) + 27 \, \sqrt {2} a x \sin \left (\frac {1}{2} \, x\right ) + 2 \, \sqrt {2} a \cos \left (\frac {3}{2} \, x\right ) + 54 \, \sqrt {2} a \cos \left (\frac {1}{2} \, x\right )\right )} \sqrt {a} \] Input:
integrate(x*(a+a*cos(x))^(3/2),x, algorithm="maxima")
Output:
1/9*(3*sqrt(2)*a*x*sin(3/2*x) + 27*sqrt(2)*a*x*sin(1/2*x) + 2*sqrt(2)*a*co s(3/2*x) + 54*sqrt(2)*a*cos(1/2*x))*sqrt(a)
Time = 0.39 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.66 \[ \int x (a+a \cos (x))^{3/2} \, dx=\frac {1}{9} \, \sqrt {2} {\left (3 \, a x \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right ) \sin \left (\frac {3}{2} \, x\right ) + 27 \, a x \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right ) \sin \left (\frac {1}{2} \, x\right ) + 2 \, a \cos \left (\frac {3}{2} \, x\right ) \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right ) + 54 \, a \cos \left (\frac {1}{2} \, x\right ) \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right )\right )} \sqrt {a} \] Input:
integrate(x*(a+a*cos(x))^(3/2),x, algorithm="giac")
Output:
1/9*sqrt(2)*(3*a*x*sgn(cos(1/2*x))*sin(3/2*x) + 27*a*x*sgn(cos(1/2*x))*sin (1/2*x) + 2*a*cos(3/2*x)*sgn(cos(1/2*x)) + 54*a*cos(1/2*x)*sgn(cos(1/2*x)) )*sqrt(a)
Timed out. \[ \int x (a+a \cos (x))^{3/2} \, dx=\int x\,{\left (a+a\,\cos \left (x\right )\right )}^{3/2} \,d x \] Input:
int(x*(a + a*cos(x))^(3/2),x)
Output:
int(x*(a + a*cos(x))^(3/2), x)
\[ \int x (a+a \cos (x))^{3/2} \, dx=\sqrt {a}\, a \left (\int \sqrt {\cos \left (x \right )+1}\, \cos \left (x \right ) x d x +\int \sqrt {\cos \left (x \right )+1}\, x d x \right ) \] Input:
int(x*(a+a*cos(x))^(3/2),x)
Output:
sqrt(a)*a*(int(sqrt(cos(x) + 1)*cos(x)*x,x) + int(sqrt(cos(x) + 1)*x,x))