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

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

[Out]

-a*cos(1/2*x)^2*(a+a*cos(x))^(1/2)/x^2-3/16*a*Ci(1/2*x)*sec(1/2*x)*(a+a*cos(x))^(1/2)-9/16*a*Ci(3/2*x)*sec(1/2
*x)*(a+a*cos(x))^(1/2)+3/2*a*cos(1/2*x)*sin(1/2*x)*(a+a*cos(x))^(1/2)/x

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Rubi [A]  time = 0.17, antiderivative size = 109, normalized size of antiderivative = 1.00, 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 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 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 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^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*}

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Mathematica [A]  time = 0.05, size = 66, normalized size = 0.61 \[ -\frac {(a (\cos (x)+1))^{3/2} \left (3 x^2 \text {Ci}\left (\frac {x}{2}\right ) \sec ^3\left (\frac {x}{2}\right )+9 x^2 \text {Ci}\left (\frac {3 x}{2}\right ) \sec ^3\left (\frac {x}{2}\right )-24 x \tan \left (\frac {x}{2}\right )+16\right )}{32 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Cos[x])^(3/2)/x^3,x]

[Out]

-1/32*((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]))/x^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^3,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 = 0.52, size = 92, normalized size = 0.84 \[ -\frac {\sqrt {2} {\left (9 \, a x^{2} \operatorname {Ci}\left (\frac {3}{2} \, x\right ) \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right ) + 3 \, a x^{2} \operatorname {Ci}\left (\frac {1}{2} \, x\right ) \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right ) - 6 \, a x \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right ) \sin \left (\frac {3}{2} \, x\right ) - 6 \, a x \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right ) \sin \left (\frac {1}{2} \, x\right ) + 4 \, a \cos \left (\frac {3}{2} \, x\right ) \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right ) + 12 \, a \cos \left (\frac {1}{2} \, x\right ) \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right )\right )} \sqrt {a}}{16 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*sqrt(2)*(9*a*x^2*cos_integral(3/2*x)*sgn(cos(1/2*x)) + 3*a*x^2*cos_integral(1/2*x)*sgn(cos(1/2*x)) - 6*a
*x*sgn(cos(1/2*x))*sin(3/2*x) - 6*a*x*sgn(cos(1/2*x))*sin(1/2*x) + 4*a*cos(3/2*x)*sgn(cos(1/2*x)) + 12*a*cos(1
/2*x)*sgn(cos(1/2*x)))*sqrt(a)/x^2

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maple [F]  time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +a \cos \relax (x )\right )^{\frac {3}{2}}}{x^{3}}\, dx \]

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 = 1.09, size = 33, normalized size = 0.30 \[ \frac {3}{16} \, \sqrt {2} a^{\frac {3}{2}} {\left (3 \, \Gamma \left (-2, \frac {3}{2} i \, x\right ) + \Gamma \left (-2, \frac {1}{2} i \, x\right ) + \Gamma \left (-2, -\frac {1}{2} i \, x\right ) + 3 \, \Gamma \left (-2, -\frac {3}{2} i \, x\right )\right )} \]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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