Integrand size = 15, antiderivative size = 70 \[ \int \frac {\sqrt {a-a \cos (x)}}{x^3} \, dx=-\frac {\sqrt {a-a \cos (x)}}{2 x^2}-\frac {\sqrt {a-a \cos (x)} \cot \left (\frac {x}{2}\right )}{4 x}-\frac {1}{8} \sqrt {a-a \cos (x)} \csc \left (\frac {x}{2}\right ) \text {Si}\left (\frac {x}{2}\right ) \] Output:
-1/2*(a-a*cos(x))^(1/2)/x^2-1/4*(a-a*cos(x))^(1/2)*cot(1/2*x)/x-1/8*(a-a*c os(x))^(1/2)*csc(1/2*x)*Si(1/2*x)
Time = 0.16 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.64 \[ \int \frac {\sqrt {a-a \cos (x)}}{x^3} \, dx=-\frac {\sqrt {a-a \cos (x)} \left (4+2 x \cot \left (\frac {x}{2}\right )+x^2 \csc \left (\frac {x}{2}\right ) \text {Si}\left (\frac {x}{2}\right )\right )}{8 x^2} \] Input:
Integrate[Sqrt[a - a*Cos[x]]/x^3,x]
Output:
-1/8*(Sqrt[a - a*Cos[x]]*(4 + 2*x*Cot[x/2] + x^2*Csc[x/2]*SinIntegral[x/2] ))/x^2
Time = 0.44 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.83, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3042, 3800, 3042, 3778, 3042, 3778, 25, 3042, 3780}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\sqrt {a-a \cos (x)}}{x^3} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sqrt {a-a \sin \left (x+\frac {\pi }{2}\right )}}{x^3}dx\) |
\(\Big \downarrow \) 3800 |
\(\displaystyle \csc \left (\frac {x}{2}\right ) \sqrt {a-a \cos (x)} \int \frac {\sin \left (\frac {x}{2}\right )}{x^3}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \csc \left (\frac {x}{2}\right ) \sqrt {a-a \cos (x)} \int \frac {\sin \left (\frac {x}{2}\right )}{x^3}dx\) |
\(\Big \downarrow \) 3778 |
\(\displaystyle \csc \left (\frac {x}{2}\right ) \sqrt {a-a \cos (x)} \left (\frac {1}{4} \int \frac {\cos \left (\frac {x}{2}\right )}{x^2}dx-\frac {\sin \left (\frac {x}{2}\right )}{2 x^2}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \csc \left (\frac {x}{2}\right ) \sqrt {a-a \cos (x)} \left (\frac {1}{4} \int \frac {\sin \left (\frac {x}{2}+\frac {\pi }{2}\right )}{x^2}dx-\frac {\sin \left (\frac {x}{2}\right )}{2 x^2}\right )\) |
\(\Big \downarrow \) 3778 |
\(\displaystyle \csc \left (\frac {x}{2}\right ) \sqrt {a-a \cos (x)} \left (\frac {1}{4} \left (\frac {1}{2} \int -\frac {\sin \left (\frac {x}{2}\right )}{x}dx-\frac {\cos \left (\frac {x}{2}\right )}{x}\right )-\frac {\sin \left (\frac {x}{2}\right )}{2 x^2}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \csc \left (\frac {x}{2}\right ) \sqrt {a-a \cos (x)} \left (\frac {1}{4} \left (-\frac {1}{2} \int \frac {\sin \left (\frac {x}{2}\right )}{x}dx-\frac {\cos \left (\frac {x}{2}\right )}{x}\right )-\frac {\sin \left (\frac {x}{2}\right )}{2 x^2}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \csc \left (\frac {x}{2}\right ) \sqrt {a-a \cos (x)} \left (\frac {1}{4} \left (-\frac {1}{2} \int \frac {\sin \left (\frac {x}{2}\right )}{x}dx-\frac {\cos \left (\frac {x}{2}\right )}{x}\right )-\frac {\sin \left (\frac {x}{2}\right )}{2 x^2}\right )\) |
\(\Big \downarrow \) 3780 |
\(\displaystyle \csc \left (\frac {x}{2}\right ) \sqrt {a-a \cos (x)} \left (\frac {1}{4} \left (-\frac {\text {Si}\left (\frac {x}{2}\right )}{2}-\frac {\cos \left (\frac {x}{2}\right )}{x}\right )-\frac {\sin \left (\frac {x}{2}\right )}{2 x^2}\right )\) |
Input:
Int[Sqrt[a - a*Cos[x]]/x^3,x]
Output:
Sqrt[a - a*Cos[x]]*Csc[x/2]*(-1/2*Sin[x/2]/x^2 + (-(Cos[x/2]/x) - SinInteg ral[x/2]/2)/4)
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] - Simp[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]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinInte gral[e + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*e - c*f, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(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])
\[\int \frac {\sqrt {a -a \cos \left (x \right )}}{x^{3}}d x\]
Input:
int((a-a*cos(x))^(1/2)/x^3,x)
Output:
int((a-a*cos(x))^(1/2)/x^3,x)
Exception generated. \[ \int \frac {\sqrt {a-a \cos (x)}}{x^3} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a-a*cos(x))^(1/2)/x^3,x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (has polynomial part)
\[ \int \frac {\sqrt {a-a \cos (x)}}{x^3} \, dx=\int \frac {\sqrt {- a \left (\cos {\left (x \right )} - 1\right )}}{x^{3}}\, dx \] Input:
integrate((a-a*cos(x))**(1/2)/x**3,x)
Output:
Integral(sqrt(-a*(cos(x) - 1))/x**3, x)
\[ \int \frac {\sqrt {a-a \cos (x)}}{x^3} \, dx=\int { \frac {\sqrt {-a \cos \left (x\right ) + a}}{x^{3}} \,d x } \] Input:
integrate((a-a*cos(x))^(1/2)/x^3,x, algorithm="maxima")
Output:
integrate(sqrt(-a*cos(x) + a)/x^3, x)
Time = 0.35 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.69 \[ \int \frac {\sqrt {a-a \cos (x)}}{x^3} \, dx=-\frac {\sqrt {2} {\left (x^{2} \mathrm {sgn}\left (\sin \left (\frac {1}{2} \, x\right )\right ) \operatorname {Si}\left (\frac {1}{2} \, x\right ) + 2 \, x \cos \left (\frac {1}{2} \, x\right ) \mathrm {sgn}\left (\sin \left (\frac {1}{2} \, x\right )\right ) + 4 \, \mathrm {sgn}\left (\sin \left (\frac {1}{2} \, x\right )\right ) \sin \left (\frac {1}{2} \, x\right )\right )} \sqrt {a}}{8 \, x^{2}} \] Input:
integrate((a-a*cos(x))^(1/2)/x^3,x, algorithm="giac")
Output:
-1/8*sqrt(2)*(x^2*sgn(sin(1/2*x))*sin_integral(1/2*x) + 2*x*cos(1/2*x)*sgn (sin(1/2*x)) + 4*sgn(sin(1/2*x))*sin(1/2*x))*sqrt(a)/x^2
Timed out. \[ \int \frac {\sqrt {a-a \cos (x)}}{x^3} \, dx=\int \frac {\sqrt {a-a\,\cos \left (x\right )}}{x^3} \,d x \] Input:
int((a - a*cos(x))^(1/2)/x^3,x)
Output:
int((a - a*cos(x))^(1/2)/x^3, x)
\[ \int \frac {\sqrt {a-a \cos (x)}}{x^3} \, dx=\sqrt {a}\, \left (\int \frac {\sqrt {-\cos \left (x \right )+1}}{x^{3}}d x \right ) \] Input:
int((a-a*cos(x))^(1/2)/x^3,x)
Output:
sqrt(a)*int(sqrt( - cos(x) + 1)/x**3,x)