\(\int x^2 (a+a \cos (x))^{3/2} \, dx\) [165]

Optimal result

Integrand size = 14, antiderivative size = 145 \[ \int x^2 (a+a \cos (x))^{3/2} \, dx=\frac {32}{3} a x \sqrt {a+a \cos (x)}+\frac {16}{9} a x \cos ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cos (x)}+\frac {4}{3} a x^2 \cos \left (\frac {x}{2}\right ) \sqrt {a+a \cos (x)} \sin \left (\frac {x}{2}\right )-\frac {224}{9} a \sqrt {a+a \cos (x)} \tan \left (\frac {x}{2}\right )+\frac {8}{3} a x^2 \sqrt {a+a \cos (x)} \tan \left (\frac {x}{2}\right )+\frac {32}{27} a \sqrt {a+a \cos (x)} \sin ^2\left (\frac {x}{2}\right ) \tan \left (\frac {x}{2}\right ) \] Output:

32/3*a*x*(a+a*cos(x))^(1/2)+16/9*a*x*cos(1/2*x)^2*(a+a*cos(x))^(1/2)+4/3*a 
*x^2*cos(1/2*x)*(a+a*cos(x))^(1/2)*sin(1/2*x)-224/9*a*(a+a*cos(x))^(1/2)*t 
an(1/2*x)+8/3*a*x^2*(a+a*cos(x))^(1/2)*tan(1/2*x)+32/27*a*(a+a*cos(x))^(1/ 
2)*sin(1/2*x)^2*tan(1/2*x)
 

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.37 \[ \int x^2 (a+a \cos (x))^{3/2} \, dx=\frac {2}{27} a \sqrt {a (1+\cos (x))} \left (156 x+\left (-328+45 x^2\right ) \tan \left (\frac {x}{2}\right )+\cos (x) \left (12 x+\left (-8+9 x^2\right ) \tan \left (\frac {x}{2}\right )\right )\right ) \] Input:

Integrate[x^2*(a + a*Cos[x])^(3/2),x]
 

Output:

(2*a*Sqrt[a*(1 + Cos[x])]*(156*x + (-328 + 45*x^2)*Tan[x/2] + Cos[x]*(12*x 
 + (-8 + 9*x^2)*Tan[x/2])))/27
 

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.79, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.929, Rules used = {3042, 3800, 3042, 3792, 3042, 3113, 2009, 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.

\(\displaystyle \int x^2 (a \cos (x)+a)^{3/2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int x^2 \left (a \sin \left (x+\frac {\pi }{2}\right )+a\right )^{3/2}dx\)

\(\Big \downarrow \) 3800

\(\displaystyle 2 a \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \int x^2 \cos ^3\left (\frac {x}{2}\right )dx\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 a \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \int x^2 \sin \left (\frac {x}{2}+\frac {\pi }{2}\right )^3dx\)

\(\Big \downarrow \) 3792

\(\displaystyle 2 a \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \left (\frac {2}{3} \int x^2 \cos \left (\frac {x}{2}\right )dx-\frac {8}{9} \int \cos ^3\left (\frac {x}{2}\right )dx+\frac {2}{3} x^2 \sin \left (\frac {x}{2}\right ) \cos ^2\left (\frac {x}{2}\right )+\frac {8}{9} x \cos ^3\left (\frac {x}{2}\right )\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 a \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \left (\frac {2}{3} \int x^2 \sin \left (\frac {x}{2}+\frac {\pi }{2}\right )dx-\frac {8}{9} \int \sin \left (\frac {x}{2}+\frac {\pi }{2}\right )^3dx+\frac {2}{3} x^2 \sin \left (\frac {x}{2}\right ) \cos ^2\left (\frac {x}{2}\right )+\frac {8}{9} x \cos ^3\left (\frac {x}{2}\right )\right )\)

\(\Big \downarrow \) 3113

\(\displaystyle 2 a \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \left (\frac {2}{3} \int x^2 \sin \left (\frac {x}{2}+\frac {\pi }{2}\right )dx+\frac {16}{9} \int \left (1-\sin ^2\left (\frac {x}{2}\right )\right )d\left (-\sin \left (\frac {x}{2}\right )\right )+\frac {2}{3} x^2 \sin \left (\frac {x}{2}\right ) \cos ^2\left (\frac {x}{2}\right )+\frac {8}{9} x \cos ^3\left (\frac {x}{2}\right )\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 a \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \left (\frac {2}{3} \int x^2 \sin \left (\frac {x}{2}+\frac {\pi }{2}\right )dx+\frac {2}{3} x^2 \sin \left (\frac {x}{2}\right ) \cos ^2\left (\frac {x}{2}\right )+\frac {16}{9} \left (\frac {1}{3} \sin ^3\left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )+\frac {8}{9} x \cos ^3\left (\frac {x}{2}\right )\right )\)

\(\Big \downarrow \) 3777

\(\displaystyle 2 a \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \left (\frac {2}{3} \left (4 \int -x \sin \left (\frac {x}{2}\right )dx+2 x^2 \sin \left (\frac {x}{2}\right )\right )+\frac {2}{3} x^2 \sin \left (\frac {x}{2}\right ) \cos ^2\left (\frac {x}{2}\right )+\frac {16}{9} \left (\frac {1}{3} \sin ^3\left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )+\frac {8}{9} x \cos ^3\left (\frac {x}{2}\right )\right )\)

\(\Big \downarrow \) 25

\(\displaystyle 2 a \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \left (\frac {2}{3} \left (2 x^2 \sin \left (\frac {x}{2}\right )-4 \int x \sin \left (\frac {x}{2}\right )dx\right )+\frac {2}{3} x^2 \sin \left (\frac {x}{2}\right ) \cos ^2\left (\frac {x}{2}\right )+\frac {16}{9} \left (\frac {1}{3} \sin ^3\left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )+\frac {8}{9} x \cos ^3\left (\frac {x}{2}\right )\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 a \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \left (\frac {2}{3} \left (2 x^2 \sin \left (\frac {x}{2}\right )-4 \int x \sin \left (\frac {x}{2}\right )dx\right )+\frac {2}{3} x^2 \sin \left (\frac {x}{2}\right ) \cos ^2\left (\frac {x}{2}\right )+\frac {16}{9} \left (\frac {1}{3} \sin ^3\left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )+\frac {8}{9} x \cos ^3\left (\frac {x}{2}\right )\right )\)

\(\Big \downarrow \) 3777

\(\displaystyle 2 a \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \left (\frac {2}{3} \left (2 x^2 \sin \left (\frac {x}{2}\right )-4 \left (2 \int \cos \left (\frac {x}{2}\right )dx-2 x \cos \left (\frac {x}{2}\right )\right )\right )+\frac {2}{3} x^2 \sin \left (\frac {x}{2}\right ) \cos ^2\left (\frac {x}{2}\right )+\frac {16}{9} \left (\frac {1}{3} \sin ^3\left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )+\frac {8}{9} x \cos ^3\left (\frac {x}{2}\right )\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 a \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \left (\frac {2}{3} \left (2 x^2 \sin \left (\frac {x}{2}\right )-4 \left (2 \int \sin \left (\frac {x}{2}+\frac {\pi }{2}\right )dx-2 x \cos \left (\frac {x}{2}\right )\right )\right )+\frac {2}{3} x^2 \sin \left (\frac {x}{2}\right ) \cos ^2\left (\frac {x}{2}\right )+\frac {16}{9} \left (\frac {1}{3} \sin ^3\left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )+\frac {8}{9} x \cos ^3\left (\frac {x}{2}\right )\right )\)

\(\Big \downarrow \) 3117

\(\displaystyle 2 a \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \left (\frac {2}{3} x^2 \sin \left (\frac {x}{2}\right ) \cos ^2\left (\frac {x}{2}\right )+\frac {2}{3} \left (2 x^2 \sin \left (\frac {x}{2}\right )-4 \left (4 \sin \left (\frac {x}{2}\right )-2 x \cos \left (\frac {x}{2}\right )\right )\right )+\frac {16}{9} \left (\frac {1}{3} \sin ^3\left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )+\frac {8}{9} x \cos ^3\left (\frac {x}{2}\right )\right )\)

Input:

Int[x^2*(a + a*Cos[x])^(3/2),x]
 

Output:

2*a*Sqrt[a + a*Cos[x]]*Sec[x/2]*((8*x*Cos[x/2]^3)/9 + (2*x^2*Cos[x/2]^2*Si 
n[x/2])/3 + (16*(-Sin[x/2] + Sin[x/2]^3/3))/9 + (2*(2*x^2*Sin[x/2] - 4*(-2 
*x*Cos[x/2] + 4*Sin[x/2])))/3)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3113

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1)   Subst[Int[Exp 
and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] 
 && IGtQ[(n - 1)/2, 0]
 

rule 3117

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

rule 3777

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]
 

rule 3792

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo 
l] :> Simp[d*m*(c + d*x)^(m - 1)*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Sim 
p[b*(c + d*x)^m*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^ 
2*((n - 1)/n)   Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[d^2 
*m*((m - 1)/(f^2*n^2))   Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x]) 
/; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]
 

rule 3800

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])
 

Maple [F]

\[\int x^{2} \left (a +a \cos \left (x \right )\right )^{\frac {3}{2}}d x\]

Input:

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

Output:

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

Fricas [F(-2)]

Exception generated. \[ \int x^2 (a+a \cos (x))^{3/2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^2*(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)
 

Sympy [F]

\[ \int x^2 (a+a \cos (x))^{3/2} \, dx=\int x^{2} \left (a \left (\cos {\left (x \right )} + 1\right )\right )^{\frac {3}{2}}\, dx \] Input:

integrate(x**2*(a+a*cos(x))**(3/2),x)
 

Output:

Integral(x**2*(a*(cos(x) + 1))**(3/2), x)
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.50 \[ \int x^2 (a+a \cos (x))^{3/2} \, dx=\frac {1}{27} \, {\left (81 \, \sqrt {2} a x^{2} \sin \left (\frac {1}{2} \, x\right ) + 12 \, \sqrt {2} a x \cos \left (\frac {3}{2} \, x\right ) + 324 \, \sqrt {2} a x \cos \left (\frac {1}{2} \, x\right ) - 648 \, \sqrt {2} a \sin \left (\frac {1}{2} \, x\right ) + {\left (9 \, \sqrt {2} a x^{2} - 8 \, \sqrt {2} a\right )} \sin \left (\frac {3}{2} \, x\right )\right )} \sqrt {a} \] Input:

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

Output:

1/27*(81*sqrt(2)*a*x^2*sin(1/2*x) + 12*sqrt(2)*a*x*cos(3/2*x) + 324*sqrt(2 
)*a*x*cos(1/2*x) - 648*sqrt(2)*a*sin(1/2*x) + (9*sqrt(2)*a*x^2 - 8*sqrt(2) 
*a)*sin(3/2*x))*sqrt(a)
 

Giac [A] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.59 \[ \int x^2 (a+a \cos (x))^{3/2} \, dx=\frac {1}{27} \, \sqrt {2} {\left (12 \, a x \cos \left (\frac {3}{2} \, x\right ) \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right ) + 324 \, a x \cos \left (\frac {1}{2} \, x\right ) \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right ) + {\left (9 \, a x^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right ) - 8 \, a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right )\right )} \sin \left (\frac {3}{2} \, x\right ) + 81 \, {\left (a x^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right ) - 8 \, a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right )\right )} \sin \left (\frac {1}{2} \, x\right )\right )} \sqrt {a} \] Input:

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

Output:

1/27*sqrt(2)*(12*a*x*cos(3/2*x)*sgn(cos(1/2*x)) + 324*a*x*cos(1/2*x)*sgn(c 
os(1/2*x)) + (9*a*x^2*sgn(cos(1/2*x)) - 8*a*sgn(cos(1/2*x)))*sin(3/2*x) + 
81*(a*x^2*sgn(cos(1/2*x)) - 8*a*sgn(cos(1/2*x)))*sin(1/2*x))*sqrt(a)
 

Mupad [F(-1)]

Timed out. \[ \int x^2 (a+a \cos (x))^{3/2} \, dx=\int x^2\,{\left (a+a\,\cos \left (x\right )\right )}^{3/2} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int x^2 (a+a \cos (x))^{3/2} \, dx=\sqrt {a}\, a \left (\int \sqrt {\cos \left (x \right )+1}\, \cos \left (x \right ) x^{2}d x +\int \sqrt {\cos \left (x \right )+1}\, x^{2}d x \right ) \] Input:

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

Output:

sqrt(a)*a*(int(sqrt(cos(x) + 1)*cos(x)*x**2,x) + int(sqrt(cos(x) + 1)*x**2 
,x))