3.149 \(\int \frac{\sqrt{a+a \cos (c+d x)}}{x^3} \, dx\)

Optimal. Leaf size=151 \[ -\frac{1}{8} d^2 \cos \left (\frac{c}{2}\right ) \text{CosIntegral}\left (\frac{d x}{2}\right ) \sec \left (\frac{c}{2}+\frac{d x}{2}\right ) \sqrt{a \cos (c+d x)+a}+\frac{1}{8} d^2 \sin \left (\frac{c}{2}\right ) \text{Si}\left (\frac{d x}{2}\right ) \sec \left (\frac{c}{2}+\frac{d x}{2}\right ) \sqrt{a \cos (c+d x)+a}-\frac{\sqrt{a \cos (c+d x)+a}}{2 x^2}+\frac{d \tan \left (\frac{c}{2}+\frac{d x}{2}\right ) \sqrt{a \cos (c+d x)+a}}{4 x} \]

[Out]

-Sqrt[a + a*Cos[c + d*x]]/(2*x^2) - (d^2*Cos[c/2]*Sqrt[a + a*Cos[c + d*x]]*CosIntegral[(d*x)/2]*Sec[c/2 + (d*x
)/2])/8 + (d^2*Sqrt[a + a*Cos[c + d*x]]*Sec[c/2 + (d*x)/2]*Sin[c/2]*SinIntegral[(d*x)/2])/8 + (d*Sqrt[a + a*Co
s[c + d*x]]*Tan[c/2 + (d*x)/2])/(4*x)

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Rubi [A]  time = 0.161713, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {3319, 3297, 3303, 3299, 3302} \[ -\frac{1}{8} d^2 \cos \left (\frac{c}{2}\right ) \text{CosIntegral}\left (\frac{d x}{2}\right ) \sec \left (\frac{c}{2}+\frac{d x}{2}\right ) \sqrt{a \cos (c+d x)+a}+\frac{1}{8} d^2 \sin \left (\frac{c}{2}\right ) \text{Si}\left (\frac{d x}{2}\right ) \sec \left (\frac{c}{2}+\frac{d x}{2}\right ) \sqrt{a \cos (c+d x)+a}-\frac{\sqrt{a \cos (c+d x)+a}}{2 x^2}+\frac{d \tan \left (\frac{c}{2}+\frac{d x}{2}\right ) \sqrt{a \cos (c+d x)+a}}{4 x} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + a*Cos[c + d*x]]/x^3,x]

[Out]

-Sqrt[a + a*Cos[c + d*x]]/(2*x^2) - (d^2*Cos[c/2]*Sqrt[a + a*Cos[c + d*x]]*CosIntegral[(d*x)/2]*Sec[c/2 + (d*x
)/2])/8 + (d^2*Sqrt[a + a*Cos[c + d*x]]*Sec[c/2 + (d*x)/2]*Sin[c/2]*SinIntegral[(d*x)/2])/8 + (d*Sqrt[a + a*Co
s[c + d*x]]*Tan[c/2 + (d*x)/2])/(4*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 3297

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[((c + d*x)^(m + 1)*Sin[e + f*x])/(d*(
m + 1)), x] - Dist[f/(d*(m + 1)), Int[(c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[
m, -1]

Rule 3303

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

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 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{\sqrt{a+a \cos (c+d x)}}{x^3} \, dx &=\left (\sqrt{a+a \cos (c+d x)} \csc \left (\frac{1}{2} \left (c+\frac{\pi }{2}\right )+\frac{\pi }{4}+\frac{d x}{2}\right )\right ) \int \frac{\sin \left (\frac{1}{2} \left (c+\frac{\pi }{2}\right )+\frac{\pi }{4}+\frac{d x}{2}\right )}{x^3} \, dx\\ &=-\frac{\sqrt{a+a \cos (c+d x)}}{2 x^2}-\frac{1}{4} \left (d \sqrt{a+a \cos (c+d x)} \csc \left (\frac{1}{2} \left (c+\frac{\pi }{2}\right )+\frac{\pi }{4}+\frac{d x}{2}\right )\right ) \int \frac{\sin \left (\frac{c}{2}+\frac{d x}{2}\right )}{x^2} \, dx\\ &=-\frac{\sqrt{a+a \cos (c+d x)}}{2 x^2}+\frac{d \sqrt{a+a \cos (c+d x)} \tan \left (\frac{c}{2}+\frac{d x}{2}\right )}{4 x}-\frac{1}{8} \left (d^2 \sqrt{a+a \cos (c+d x)} \csc \left (\frac{1}{2} \left (c+\frac{\pi }{2}\right )+\frac{\pi }{4}+\frac{d x}{2}\right )\right ) \int \frac{\cos \left (\frac{c}{2}+\frac{d x}{2}\right )}{x} \, dx\\ &=-\frac{\sqrt{a+a \cos (c+d x)}}{2 x^2}+\frac{d \sqrt{a+a \cos (c+d x)} \tan \left (\frac{c}{2}+\frac{d x}{2}\right )}{4 x}-\frac{1}{8} \left (d^2 \cos \left (\frac{c}{2}\right ) \sqrt{a+a \cos (c+d x)} \csc \left (\frac{1}{2} \left (c+\frac{\pi }{2}\right )+\frac{\pi }{4}+\frac{d x}{2}\right )\right ) \int \frac{\cos \left (\frac{d x}{2}\right )}{x} \, dx+\frac{1}{8} \left (d^2 \sqrt{a+a \cos (c+d x)} \csc \left (\frac{1}{2} \left (c+\frac{\pi }{2}\right )+\frac{\pi }{4}+\frac{d x}{2}\right ) \sin \left (\frac{c}{2}\right )\right ) \int \frac{\sin \left (\frac{d x}{2}\right )}{x} \, dx\\ &=-\frac{\sqrt{a+a \cos (c+d x)}}{2 x^2}-\frac{1}{8} d^2 \cos \left (\frac{c}{2}\right ) \sqrt{a+a \cos (c+d x)} \text{Ci}\left (\frac{d x}{2}\right ) \sec \left (\frac{c}{2}+\frac{d x}{2}\right )+\frac{1}{8} d^2 \sqrt{a+a \cos (c+d x)} \sec \left (\frac{c}{2}+\frac{d x}{2}\right ) \sin \left (\frac{c}{2}\right ) \text{Si}\left (\frac{d x}{2}\right )+\frac{d \sqrt{a+a \cos (c+d x)} \tan \left (\frac{c}{2}+\frac{d x}{2}\right )}{4 x}\\ \end{align*}

Mathematica [A]  time = 0.252596, size = 98, normalized size = 0.65 \[ \frac{\sqrt{a (\cos (c+d x)+1)} \left (-d^2 x^2 \cos \left (\frac{c}{2}\right ) \text{CosIntegral}\left (\frac{d x}{2}\right ) \sec \left (\frac{1}{2} (c+d x)\right )+d^2 x^2 \sin \left (\frac{c}{2}\right ) \text{Si}\left (\frac{d x}{2}\right ) \sec \left (\frac{1}{2} (c+d x)\right )+2 d x \tan \left (\frac{1}{2} (c+d x)\right )-4\right )}{8 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + a*Cos[c + d*x]]/x^3,x]

[Out]

(Sqrt[a*(1 + Cos[c + d*x])]*(-4 - d^2*x^2*Cos[c/2]*CosIntegral[(d*x)/2]*Sec[(c + d*x)/2] + d^2*x^2*Sec[(c + d*
x)/2]*Sin[c/2]*SinIntegral[(d*x)/2] + 2*d*x*Tan[(c + d*x)/2]))/(8*x^2)

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Maple [F]  time = 0.223, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}}\sqrt{a+\cos \left ( dx+c \right ) a}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+cos(d*x+c)*a)^(1/2)/x^3,x)

[Out]

int((a+cos(d*x+c)*a)^(1/2)/x^3,x)

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Maxima [C]  time = 2.51852, size = 313, normalized size = 2.07 \begin{align*} -\frac{{\left (4 \,{\left (E_{3}\left (\frac{1}{2} i \, d x\right ) + E_{3}\left (-\frac{1}{2} i \, d x\right )\right )} \cos \left (\frac{1}{2} \, c\right )^{3} + 4 \,{\left (E_{3}\left (\frac{1}{2} i \, d x\right ) + E_{3}\left (-\frac{1}{2} i \, d x\right )\right )} \cos \left (\frac{1}{2} \, c\right ) \sin \left (\frac{1}{2} \, c\right )^{2} -{\left (4 i \, E_{3}\left (\frac{1}{2} i \, d x\right ) - 4 i \, E_{3}\left (-\frac{1}{2} i \, d x\right )\right )} \sin \left (\frac{1}{2} \, c\right )^{3} + 4 \,{\left (E_{3}\left (\frac{1}{2} i \, d x\right ) + E_{3}\left (-\frac{1}{2} i \, d x\right )\right )} \cos \left (\frac{1}{2} \, c\right ) -{\left ({\left (4 i \, E_{3}\left (\frac{1}{2} i \, d x\right ) - 4 i \, E_{3}\left (-\frac{1}{2} i \, d x\right )\right )} \cos \left (\frac{1}{2} \, c\right )^{2} + 4 i \, E_{3}\left (\frac{1}{2} i \, d x\right ) - 4 i \, E_{3}\left (-\frac{1}{2} i \, d x\right )\right )} \sin \left (\frac{1}{2} \, c\right )\right )} \sqrt{a} d^{2}}{8 \,{\left ({\left (\sqrt{2} \cos \left (\frac{1}{2} \, c\right )^{2} + \sqrt{2} \sin \left (\frac{1}{2} \, c\right )^{2}\right )}{\left (d x + c\right )}^{2} - 2 \,{\left (\sqrt{2} \cos \left (\frac{1}{2} \, c\right )^{2} + \sqrt{2} \sin \left (\frac{1}{2} \, c\right )^{2}\right )}{\left (d x + c\right )} c +{\left (\sqrt{2} \cos \left (\frac{1}{2} \, c\right )^{2} + \sqrt{2} \sin \left (\frac{1}{2} \, c\right )^{2}\right )} c^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*cos(d*x+c))^(1/2)/x^3,x, algorithm="maxima")

[Out]

-1/8*(4*(exp_integral_e(3, 1/2*I*d*x) + exp_integral_e(3, -1/2*I*d*x))*cos(1/2*c)^3 + 4*(exp_integral_e(3, 1/2
*I*d*x) + exp_integral_e(3, -1/2*I*d*x))*cos(1/2*c)*sin(1/2*c)^2 - (4*I*exp_integral_e(3, 1/2*I*d*x) - 4*I*exp
_integral_e(3, -1/2*I*d*x))*sin(1/2*c)^3 + 4*(exp_integral_e(3, 1/2*I*d*x) + exp_integral_e(3, -1/2*I*d*x))*co
s(1/2*c) - ((4*I*exp_integral_e(3, 1/2*I*d*x) - 4*I*exp_integral_e(3, -1/2*I*d*x))*cos(1/2*c)^2 + 4*I*exp_inte
gral_e(3, 1/2*I*d*x) - 4*I*exp_integral_e(3, -1/2*I*d*x))*sin(1/2*c))*sqrt(a)*d^2/((sqrt(2)*cos(1/2*c)^2 + sqr
t(2)*sin(1/2*c)^2)*(d*x + c)^2 - 2*(sqrt(2)*cos(1/2*c)^2 + sqrt(2)*sin(1/2*c)^2)*(d*x + c)*c + (sqrt(2)*cos(1/
2*c)^2 + sqrt(2)*sin(1/2*c)^2)*c^2)

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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(d*x+c))^(1/2)/x^3,x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*cos(d*x+c))**(1/2)/x**3,x)

[Out]

Integral(sqrt(a*(cos(c + d*x) + 1))/x**3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*cos(d*x+c))^(1/2)/x^3,x, algorithm="giac")

[Out]

integrate(sqrt(a*cos(d*x + c) + a)/x^3, x)