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

Optimal. Leaf size=89 \[ \frac {8}{9} a \cos ^2\left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a}+\frac {16}{3} a \sqrt {a \cos (x)+a}+\frac {4}{3} a x \sin \left (\frac {x}{2}\right ) \cos \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a}+\frac {8}{3} a x \tan \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \]

[Out]

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*cos(1/2*x)*sin(1/2*x)*(a+a*cos(x))^(1/
2)+8/3*a*x*(a+a*cos(x))^(1/2)*tan(1/2*x)

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Rubi [A]  time = 0.07, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3319, 3310, 3296, 2638} \[ \frac {8}{9} a \cos ^2\left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a}+\frac {16}{3} a \sqrt {a \cos (x)+a}+\frac {4}{3} a x \sin \left (\frac {x}{2}\right ) \cos \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a}+\frac {8}{3} a x \tan \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + a*Cos[x])^(3/2),x]

[Out]

(16*a*Sqrt[a + a*Cos[x]])/3 + (8*a*Cos[x/2]^2*Sqrt[a + a*Cos[x]])/9 + (4*a*x*Cos[x/2]*Sqrt[a + a*Cos[x]]*Sin[x
/2])/3 + (8*a*x*Sqrt[a + a*Cos[x]]*Tan[x/2])/3

Rule 2638

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

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 3310

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] + (Dist[(b^2*(n - 1))/n, Int[(c + d*x)*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[(b*(c + d*x)*Cos[e + f*
x]*(b*Sin[e + f*x])^(n - 1))/(f*n), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1]

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])

Rubi steps

\begin {align*} \int x (a+a \cos (x))^{3/2} \, dx &=\left (2 a \sqrt {a+a \cos (x)} \sec \left (\frac {x}{2}\right )\right ) \int x \cos ^3\left (\frac {x}{2}\right ) \, dx\\ &=\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 {1}{3} \left (4 a \sqrt {a+a \cos (x)} \sec \left (\frac {x}{2}\right )\right ) \int x \cos \left (\frac {x}{2}\right ) \, dx\\ &=\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 )-\frac {1}{3} \left (8 a \sqrt {a+a \cos (x)} \sec \left (\frac {x}{2}\right )\right ) \int \sin \left (\frac {x}{2}\right ) \, 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 )\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 45, normalized size = 0.51 \[ \frac {1}{9} a \sqrt {a (\cos (x)+1)} \left (4 \cos (x)+27 x \tan \left (\frac {x}{2}\right )+3 x \sin \left (\frac {3 x}{2}\right ) \sec \left (\frac {x}{2}\right )+52\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + a*Cos[x])^(3/2),x]

[Out]

(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

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

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giac [A]  time = 0.48, size = 59, normalized size = 0.66 \[ \frac {1}{9} \, \sqrt {2} {\left (3 \, a x \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right ) \sin \left (\frac {3}{2} \, x\right ) + 27 \, a x \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right ) \sin \left (\frac {1}{2} \, x\right ) + 2 \, a \cos \left (\frac {3}{2} \, x\right ) \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right ) + 54 \, a \cos \left (\frac {1}{2} \, x\right ) \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right )\right )} \sqrt {a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)

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maple [F]  time = 0.06, size = 0, normalized size = 0.00 \[ \int x \left (a +a \cos \relax (x )\right )^{\frac {3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.07, size = 48, normalized size = 0.54 \[ \frac {1}{9} \, {\left (3 \, \sqrt {2} a x \sin \left (\frac {3}{2} \, x\right ) + 27 \, \sqrt {2} a x \sin \left (\frac {1}{2} \, x\right ) + 2 \, \sqrt {2} a \cos \left (\frac {3}{2} \, x\right ) + 54 \, \sqrt {2} a \cos \left (\frac {1}{2} \, x\right )\right )} \sqrt {a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*(3*sqrt(2)*a*x*sin(3/2*x) + 27*sqrt(2)*a*x*sin(1/2*x) + 2*sqrt(2)*a*cos(3/2*x) + 54*sqrt(2)*a*cos(1/2*x))*
sqrt(a)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,{\left (a+a\,\cos \relax (x)\right )}^{3/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + a*cos(x))^(3/2),x)

[Out]

int(x*(a + a*cos(x))^(3/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x \left (a \left (\cos {\relax (x )} + 1\right )\right )^{\frac {3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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