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

Optimal. Leaf size=55 \[ \frac{3}{2} a \text{CosIntegral}\left (\frac{x}{2}\right ) \sec \left (\frac{x}{2}\right ) \sqrt{a \cos (x)+a}+\frac{1}{2} a \text{CosIntegral}\left (\frac{3 x}{2}\right ) \sec \left (\frac{x}{2}\right ) \sqrt{a \cos (x)+a} \]

[Out]

(3*a*Sqrt[a + a*Cos[x]]*CosIntegral[x/2]*Sec[x/2])/2 + (a*Sqrt[a + a*Cos[x]]*CosIntegral[(3*x)/2]*Sec[x/2])/2

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Rubi [A]  time = 0.127667, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {3319, 3312, 3302} \[ \frac{3}{2} a \text{CosIntegral}\left (\frac{x}{2}\right ) \sec \left (\frac{x}{2}\right ) \sqrt{a \cos (x)+a}+\frac{1}{2} a \text{CosIntegral}\left (\frac{3 x}{2}\right ) \sec \left (\frac{x}{2}\right ) \sqrt{a \cos (x)+a} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Cos[x])^(3/2)/x,x]

[Out]

(3*a*Sqrt[a + a*Cos[x]]*CosIntegral[x/2]*Sec[x/2])/2 + (a*Sqrt[a + a*Cos[x]]*CosIntegral[(3*x)/2]*Sec[x/2])/2

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 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])
)

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]

Rubi steps

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

Mathematica [A]  time = 0.0199382, size = 36, normalized size = 0.65 \[ \frac{1}{2} a \left (3 \text{CosIntegral}\left (\frac{x}{2}\right )+\text{CosIntegral}\left (\frac{3 x}{2}\right )\right ) \sec \left (\frac{x}{2}\right ) \sqrt{a (\cos (x)+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Cos[x])^(3/2)/x,x]

[Out]

(a*Sqrt[a*(1 + Cos[x])]*(3*CosIntegral[x/2] + CosIntegral[(3*x)/2])*Sec[x/2])/2

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Maple [F]  time = 0.089, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x} \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,x)

[Out]

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

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Maxima [C]  time = 2.42917, size = 39, normalized size = 0.71 \begin{align*} \frac{1}{4} \, \sqrt{2} a^{\frac{3}{2}}{\left ({\rm Ei}\left (\frac{3}{2} i \, x\right ) + 3 \,{\rm Ei}\left (\frac{1}{2} i \, x\right ) + 3 \,{\rm Ei}\left (-\frac{1}{2} i \, x\right ) +{\rm Ei}\left (-\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,x, algorithm="maxima")

[Out]

1/4*sqrt(2)*a^(3/2)*(Ei(3/2*I*x) + 3*Ei(1/2*I*x) + 3*Ei(-1/2*I*x) + Ei(-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,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,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}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a*cos(x) + a)^(3/2)/x, x)