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

Optimal. Leaf size=79 \[ -\frac {3}{4} a \text {Si}\left (\frac {x}{2}\right ) \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a}-\frac {3}{4} a \text {Si}\left (\frac {3 x}{2}\right ) \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a}-\frac {2 a \cos ^2\left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a}}{x} \]

[Out]

-2*a*cos(1/2*x)^2*(a+a*cos(x))^(1/2)/x-3/4*a*sec(1/2*x)*Si(1/2*x)*(a+a*cos(x))^(1/2)-3/4*a*sec(1/2*x)*Si(3/2*x
)*(a+a*cos(x))^(1/2)

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Rubi [A]  time = 0.13, antiderivative size = 79, 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, 3313, 3299} \[ -\frac {3}{4} a \text {Si}\left (\frac {x}{2}\right ) \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a}-\frac {3}{4} a \text {Si}\left (\frac {3 x}{2}\right ) \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a}-\frac {2 a \cos ^2\left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a}}{x} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Cos[x])^(3/2)/x^2,x]

[Out]

(-2*a*Cos[x/2]^2*Sqrt[a + a*Cos[x]])/x - (3*a*Sqrt[a + a*Cos[x]]*Sec[x/2]*SinIntegral[x/2])/4 - (3*a*Sqrt[a +
a*Cos[x]]*Sec[x/2]*SinIntegral[(3*x)/2])/4

Rule 3299

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

Rule 3313

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Simp[((c + d*x)^(m + 1)*Sin[e + f*x]^
n)/(d*(m + 1)), x] - Dist[(f*n)/(d*(m + 1)), Int[ExpandTrigReduce[(c + d*x)^(m + 1), Cos[e + f*x]*Sin[e + f*x]
^(n - 1), x], x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && GeQ[m, -2] && 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^2} \, 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^2} \, dx\\ &=-\frac {2 a \cos ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cos (x)}}{x}+\left (3 a \sqrt {a+a \cos (x)} \sec \left (\frac {x}{2}\right )\right ) \int \left (-\frac {\sin \left (\frac {x}{2}\right )}{4 x}-\frac {\sin \left (\frac {3 x}{2}\right )}{4 x}\right ) \, dx\\ &=-\frac {2 a \cos ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cos (x)}}{x}-\frac {1}{4} \left (3 a \sqrt {a+a \cos (x)} \sec \left (\frac {x}{2}\right )\right ) \int \frac {\sin \left (\frac {x}{2}\right )}{x} \, dx-\frac {1}{4} \left (3 a \sqrt {a+a \cos (x)} \sec \left (\frac {x}{2}\right )\right ) \int \frac {\sin \left (\frac {3 x}{2}\right )}{x} \, dx\\ &=-\frac {2 a \cos ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cos (x)}}{x}-\frac {3}{4} a \sqrt {a+a \cos (x)} \sec \left (\frac {x}{2}\right ) \text {Si}\left (\frac {x}{2}\right )-\frac {3}{4} a \sqrt {a+a \cos (x)} \sec \left (\frac {x}{2}\right ) \text {Si}\left (\frac {3 x}{2}\right )\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 53, normalized size = 0.67 \[ -\frac {a \sec \left (\frac {x}{2}\right ) \sqrt {a (\cos (x)+1)} \left (3 x \text {Si}\left (\frac {x}{2}\right )+3 x \text {Si}\left (\frac {3 x}{2}\right )+8 \cos ^3\left (\frac {x}{2}\right )\right )}{4 x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Cos[x])^(3/2)/x^2,x]

[Out]

-1/4*(a*Sqrt[a*(1 + Cos[x])]*Sec[x/2]*(8*Cos[x/2]^3 + 3*x*SinIntegral[x/2] + 3*x*SinIntegral[(3*x)/2]))/x

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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^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 = 0.42, size = 62, normalized size = 0.78 \[ -\frac {\sqrt {2} {\left (3 \, a x \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right ) \operatorname {Si}\left (\frac {3}{2} \, x\right ) + 3 \, a x \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right ) \operatorname {Si}\left (\frac {1}{2} \, x\right ) + 2 \, a \cos \left (\frac {3}{2} \, x\right ) \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right ) + 6 \, a \cos \left (\frac {1}{2} \, x\right ) \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right )\right )} \sqrt {a}}{4 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*sqrt(2)*(3*a*x*sgn(cos(1/2*x))*sin_integral(3/2*x) + 3*a*x*sgn(cos(1/2*x))*sin_integral(1/2*x) + 2*a*cos(
3/2*x)*sgn(cos(1/2*x)) + 6*a*cos(1/2*x)*sgn(cos(1/2*x)))*sqrt(a)/x

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [C]  time = 1.33, size = 37, normalized size = 0.47 \[ -\frac {1}{8} \, \sqrt {2} a^{\frac {3}{2}} {\left (3 i \, \Gamma \left (-1, \frac {3}{2} i \, x\right ) + 3 i \, \Gamma \left (-1, \frac {1}{2} i \, x\right ) - 3 i \, \Gamma \left (-1, -\frac {1}{2} i \, x\right ) - 3 i \, \Gamma \left (-1, -\frac {3}{2} i \, x\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*sqrt(2)*a^(3/2)*(3*I*gamma(-1, 3/2*I*x) + 3*I*gamma(-1, 1/2*I*x) - 3*I*gamma(-1, -1/2*I*x) - 3*I*gamma(-1
, -3/2*I*x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a*cos(x))^(3/2)/x^2,x)

[Out]

int((a + a*cos(x))^(3/2)/x^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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