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

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

[Out]

3/2*a*Ci(1/2*x)*sec(1/2*x)*(a+a*cos(x))^(1/2)+1/2*a*Ci(3/2*x)*sec(1/2*x)*(a+a*cos(x))^(1/2)

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Rubi [A]  time = 0.13, antiderivative size = 55, normalized size of antiderivative = 1.00, 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 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])
)

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 \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*}

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Mathematica [A]  time = 0.02, size = 36, normalized size = 0.65 \[ \frac {1}{2} a \left (3 \text {Ci}\left (\frac {x}{2}\right )+\text {Ci}\left (\frac {3 x}{2}\right )\right ) \sec \left (\frac {x}{2}\right ) \sqrt {a (\cos (x)+1)} \]

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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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((a+a*cos(x))^(3/2)/x,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 = 0.42, size = 32, normalized size = 0.58 \[ \frac {1}{2} \, \sqrt {2} {\left (a \operatorname {Ci}\left (\frac {3}{2} \, x\right ) \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right ) + 3 \, a \operatorname {Ci}\left (\frac {1}{2} \, x\right ) \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right )\right )} \sqrt {a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(2)*(a*cos_integral(3/2*x)*sgn(cos(1/2*x)) + 3*a*cos_integral(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 \[ \int \frac {\left (a +a \cos \relax (x )\right )^{\frac {3}{2}}}{x}\, dx \]

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 = 1.18, size = 29, normalized size = 0.53 \[ \frac {1}{4} \, \sqrt {2} a^{\frac {3}{2}} {\left ({\rm Ei}\left (\frac {3}{2} i \, x\right ) + 3 \, {\rm Ei}\left (\frac {1}{2} i \, x\right ) + 3 \, {\rm Ei}\left (-\frac {1}{2} i \, x\right ) + {\rm Ei}\left (-\frac {3}{2} i \, x\right )\right )} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (a+a\,\cos \relax (x)\right )}^{3/2}}{x} \,d x \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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