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

Optimal. Leaf size=109 \[ -\frac{3}{16} a \text{CosIntegral}\left (\frac{x}{2}\right ) \sec \left (\frac{x}{2}\right ) \sqrt{a \cos (x)+a}-\frac{9}{16} a \text{CosIntegral}\left (\frac{3 x}{2}\right ) \sec \left (\frac{x}{2}\right ) \sqrt{a \cos (x)+a}-\frac{a \cos ^2\left (\frac{x}{2}\right ) \sqrt{a \cos (x)+a}}{x^2}+\frac{3 a \sin \left (\frac{x}{2}\right ) \cos \left (\frac{x}{2}\right ) \sqrt{a \cos (x)+a}}{2 x} \]

[Out]

-((a*Cos[x/2]^2*Sqrt[a + a*Cos[x]])/x^2) - (3*a*Sqrt[a + a*Cos[x]]*CosIntegral[x/2]*Sec[x/2])/16 - (9*a*Sqrt[a
+ a*Cos[x]]*CosIntegral[(3*x)/2]*Sec[x/2])/16 + (3*a*Cos[x/2]*Sqrt[a + a*Cos[x]]*Sin[x/2])/(2*x)

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Rubi [A]  time = 0.165469, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3319, 3314, 3302, 3312} \[ -\frac{3}{16} a \text{CosIntegral}\left (\frac{x}{2}\right ) \sec \left (\frac{x}{2}\right ) \sqrt{a \cos (x)+a}-\frac{9}{16} a \text{CosIntegral}\left (\frac{3 x}{2}\right ) \sec \left (\frac{x}{2}\right ) \sqrt{a \cos (x)+a}-\frac{a \cos ^2\left (\frac{x}{2}\right ) \sqrt{a \cos (x)+a}}{x^2}+\frac{3 a \sin \left (\frac{x}{2}\right ) \cos \left (\frac{x}{2}\right ) \sqrt{a \cos (x)+a}}{2 x} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Cos[x])^(3/2)/x^3,x]

[Out]

-((a*Cos[x/2]^2*Sqrt[a + a*Cos[x]])/x^2) - (3*a*Sqrt[a + a*Cos[x]]*CosIntegral[x/2]*Sec[x/2])/16 - (9*a*Sqrt[a
+ a*Cos[x]]*CosIntegral[(3*x)/2]*Sec[x/2])/16 + (3*a*Cos[x/2]*Sqrt[a + a*Cos[x]]*Sin[x/2])/(2*x)

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])

Rule 3314

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[((c + d*x)^(m + 1)*(b*Si
n[e + f*x])^n)/(d*(m + 1)), x] + (Dist[(b^2*f^2*n*(n - 1))/(d^2*(m + 1)*(m + 2)), Int[(c + d*x)^(m + 2)*(b*Sin
[e + f*x])^(n - 2), x], x] - Dist[(f^2*n^2)/(d^2*(m + 1)*(m + 2)), Int[(c + d*x)^(m + 2)*(b*Sin[e + f*x])^n, x
], x] - Simp[(b*f*n*(c + d*x)^(m + 2)*Cos[e + f*x]*(b*Sin[e + f*x])^(n - 1))/(d^2*(m + 1)*(m + 2)), x]) /; Fre
eQ[{b, c, d, e, f}, x] && GtQ[n, 1] && LtQ[m, -2]

Rule 3302

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rule 3312

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rubi steps

\begin{align*} \int \frac{(a+a \cos (x))^{3/2}}{x^3} \, dx &=\left (2 a \sqrt{a+a \cos (x)} \sec \left (\frac{x}{2}\right )\right ) \int \frac{\cos ^3\left (\frac{x}{2}\right )}{x^3} \, dx\\ &=-\frac{a \cos ^2\left (\frac{x}{2}\right ) \sqrt{a+a \cos (x)}}{x^2}+\frac{3 a \cos \left (\frac{x}{2}\right ) \sqrt{a+a \cos (x)} \sin \left (\frac{x}{2}\right )}{2 x}+\frac{1}{2} \left (3 a \sqrt{a+a \cos (x)} \sec \left (\frac{x}{2}\right )\right ) \int \frac{\cos \left (\frac{x}{2}\right )}{x} \, dx-\frac{1}{4} \left (9 a \sqrt{a+a \cos (x)} \sec \left (\frac{x}{2}\right )\right ) \int \frac{\cos ^3\left (\frac{x}{2}\right )}{x} \, dx\\ &=-\frac{a \cos ^2\left (\frac{x}{2}\right ) \sqrt{a+a \cos (x)}}{x^2}+\frac{3}{2} a \sqrt{a+a \cos (x)} \text{Ci}\left (\frac{x}{2}\right ) \sec \left (\frac{x}{2}\right )+\frac{3 a \cos \left (\frac{x}{2}\right ) \sqrt{a+a \cos (x)} \sin \left (\frac{x}{2}\right )}{2 x}-\frac{1}{4} \left (9 a \sqrt{a+a \cos (x)} \sec \left (\frac{x}{2}\right )\right ) \int \left (\frac{3 \cos \left (\frac{x}{2}\right )}{4 x}+\frac{\cos \left (\frac{3 x}{2}\right )}{4 x}\right ) \, dx\\ &=-\frac{a \cos ^2\left (\frac{x}{2}\right ) \sqrt{a+a \cos (x)}}{x^2}+\frac{3}{2} a \sqrt{a+a \cos (x)} \text{Ci}\left (\frac{x}{2}\right ) \sec \left (\frac{x}{2}\right )+\frac{3 a \cos \left (\frac{x}{2}\right ) \sqrt{a+a \cos (x)} \sin \left (\frac{x}{2}\right )}{2 x}-\frac{1}{16} \left (9 a \sqrt{a+a \cos (x)} \sec \left (\frac{x}{2}\right )\right ) \int \frac{\cos \left (\frac{3 x}{2}\right )}{x} \, dx-\frac{1}{16} \left (27 a \sqrt{a+a \cos (x)} \sec \left (\frac{x}{2}\right )\right ) \int \frac{\cos \left (\frac{x}{2}\right )}{x} \, dx\\ &=-\frac{a \cos ^2\left (\frac{x}{2}\right ) \sqrt{a+a \cos (x)}}{x^2}-\frac{3}{16} a \sqrt{a+a \cos (x)} \text{Ci}\left (\frac{x}{2}\right ) \sec \left (\frac{x}{2}\right )-\frac{9}{16} a \sqrt{a+a \cos (x)} \text{Ci}\left (\frac{3 x}{2}\right ) \sec \left (\frac{x}{2}\right )+\frac{3 a \cos \left (\frac{x}{2}\right ) \sqrt{a+a \cos (x)} \sin \left (\frac{x}{2}\right )}{2 x}\\ \end{align*}

Mathematica [A]  time = 0.0548408, size = 66, normalized size = 0.61 \[ -\frac{(a (\cos (x)+1))^{3/2} \left (3 x^2 \text{CosIntegral}\left (\frac{x}{2}\right ) \sec ^3\left (\frac{x}{2}\right )+9 x^2 \text{CosIntegral}\left (\frac{3 x}{2}\right ) \sec ^3\left (\frac{x}{2}\right )-24 x \tan \left (\frac{x}{2}\right )+16\right )}{32 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Cos[x])^(3/2)/x^3,x]

[Out]

-((a*(1 + Cos[x]))^(3/2)*(16 + 3*x^2*CosIntegral[x/2]*Sec[x/2]^3 + 9*x^2*CosIntegral[(3*x)/2]*Sec[x/2]^3 - 24*
x*Tan[x/2]))/(32*x^2)

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Maple [F]  time = 0.092, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}} \left ( a+a\cos \left ( x \right ) \right ) ^{{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [C]  time = 2.24817, size = 45, normalized size = 0.41 \begin{align*} \frac{3}{16} \, \sqrt{2} a^{\frac{3}{2}}{\left (3 \, \Gamma \left (-2, \frac{3}{2} i \, x\right ) + \Gamma \left (-2, \frac{1}{2} i \, x\right ) + \Gamma \left (-2, -\frac{1}{2} i \, x\right ) + 3 \, \Gamma \left (-2, -\frac{3}{2} i \, x\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/16*sqrt(2)*a^(3/2)*(3*gamma(-2, 3/2*I*x) + gamma(-2, 1/2*I*x) + gamma(-2, -1/2*I*x) + 3*gamma(-2, -3/2*I*x))

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a \cos \left (x\right ) + a\right )}^{\frac{3}{2}}}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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