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

Optimal. Leaf size=174 \[ -\frac{1}{8} d^2 \sin \left (\frac{1}{4} (2 c+\pi )\right ) \text{CosIntegral}\left (\frac{d x}{2}\right ) \csc \left (\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (c+d x)+a}-\frac{1}{8} d^2 \cos \left (\frac{1}{4} (2 c+\pi )\right ) \text{Si}\left (\frac{d x}{2}\right ) \csc \left (\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (c+d x)+a}-\frac{\sqrt{a \sin (c+d x)+a}}{2 x^2}-\frac{d \cot \left (\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (c+d x)+a}}{4 x} \]

[Out]

-Sqrt[a + a*Sin[c + d*x]]/(2*x^2) - (d*Cot[c/2 + Pi/4 + (d*x)/2]*Sqrt[a + a*Sin[c + d*x]])/(4*x) - (d^2*CosInt
egral[(d*x)/2]*Csc[c/2 + Pi/4 + (d*x)/2]*Sin[(2*c + Pi)/4]*Sqrt[a + a*Sin[c + d*x]])/8 - (d^2*Cos[(2*c + Pi)/4
]*Csc[c/2 + Pi/4 + (d*x)/2]*Sqrt[a + a*Sin[c + d*x]]*SinIntegral[(d*x)/2])/8

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Rubi [A]  time = 0.193083, antiderivative size = 174, 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} \[ -\frac{1}{8} d^2 \sin \left (\frac{1}{4} (2 c+\pi )\right ) \text{CosIntegral}\left (\frac{d x}{2}\right ) \csc \left (\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (c+d x)+a}-\frac{1}{8} d^2 \cos \left (\frac{1}{4} (2 c+\pi )\right ) \text{Si}\left (\frac{d x}{2}\right ) \csc \left (\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (c+d x)+a}-\frac{\sqrt{a \sin (c+d x)+a}}{2 x^2}-\frac{d \cot \left (\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (c+d x)+a}}{4 x} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + a*Sin[c + d*x]]/x^3,x]

[Out]

-Sqrt[a + a*Sin[c + d*x]]/(2*x^2) - (d*Cot[c/2 + Pi/4 + (d*x)/2]*Sqrt[a + a*Sin[c + d*x]])/(4*x) - (d^2*CosInt
egral[(d*x)/2]*Csc[c/2 + Pi/4 + (d*x)/2]*Sin[(2*c + Pi)/4]*Sqrt[a + a*Sin[c + d*x]])/8 - (d^2*Cos[(2*c + Pi)/4
]*Csc[c/2 + Pi/4 + (d*x)/2]*Sqrt[a + a*Sin[c + d*x]]*SinIntegral[(d*x)/2])/8

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

Mathematica [A]  time = 0.327245, size = 153, normalized size = 0.88 \[ -\frac{\sqrt{a (\sin (c+d x)+1)} \left (d^2 x^2 \left (\sin \left (\frac{c}{2}\right )+\cos \left (\frac{c}{2}\right )\right ) \text{CosIntegral}\left (\frac{d x}{2}\right )+d^2 x^2 \left (\cos \left (\frac{c}{2}\right )-\sin \left (\frac{c}{2}\right )\right ) \text{Si}\left (\frac{d x}{2}\right )-2 d x \sin \left (\frac{1}{2} (c+d x)\right )+4 \sin \left (\frac{1}{2} (c+d x)\right )+2 d x \cos \left (\frac{1}{2} (c+d x)\right )+4 \cos \left (\frac{1}{2} (c+d x)\right )\right )}{8 x^2 \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.054, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}}\sqrt{a+a\sin \left ( dx+c \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a \sin \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*sin(d*x+c))^(1/2)/x^3,x, algorithm="maxima")

[Out]

integrate(sqrt(a*sin(d*x + c) + a)/x^3, 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*sin(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 (\sin{\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*sin(d*x+c))**(1/2)/x**3,x)

[Out]

Integral(sqrt(a*(sin(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 \sin \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*sin(d*x+c))^(1/2)/x^3,x, algorithm="giac")

[Out]

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