Integrand size = 18, antiderivative size = 84 \[ \int \frac {\sqrt {a+a \cos (c+d x)}}{x} \, dx=\cos \left (\frac {c}{2}\right ) \sqrt {a+a \cos (c+d x)} \operatorname {CosIntegral}\left (\frac {d x}{2}\right ) \sec \left (\frac {c}{2}+\frac {d x}{2}\right )-\sqrt {a+a \cos (c+d x)} \sec \left (\frac {c}{2}+\frac {d x}{2}\right ) \sin \left (\frac {c}{2}\right ) \text {Si}\left (\frac {d x}{2}\right ) \] Output:
cos(1/2*c)*(a+a*cos(d*x+c))^(1/2)*Ci(1/2*d*x)*sec(1/2*d*x+1/2*c)-(a+a*cos( d*x+c))^(1/2)*sec(1/2*d*x+1/2*c)*sin(1/2*c)*Si(1/2*d*x)
Time = 0.35 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.65 \[ \int \frac {\sqrt {a+a \cos (c+d x)}}{x} \, dx=\sqrt {a (1+\cos (c+d x))} \sec \left (\frac {1}{2} (c+d x)\right ) \left (\cos \left (\frac {c}{2}\right ) \operatorname {CosIntegral}\left (\frac {d x}{2}\right )-\sin \left (\frac {c}{2}\right ) \text {Si}\left (\frac {d x}{2}\right )\right ) \] Input:
Integrate[Sqrt[a + a*Cos[c + d*x]]/x,x]
Output:
Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(Cos[c/2]*CosIntegral[(d*x)/2] - Sin[c/2]*SinIntegral[(d*x)/2])
Time = 0.46 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.69, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {3042, 3800, 3042, 3784, 3042, 3780, 3783}
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.
\(\displaystyle \int \frac {\sqrt {a \cos (c+d x)+a}}{x} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sqrt {a \sin \left (c+d x+\frac {\pi }{2}\right )+a}}{x}dx\) |
\(\Big \downarrow \) 3800 |
\(\displaystyle \sec \left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cos (c+d x)+a} \int \frac {\cos \left (\frac {c}{2}+\frac {d x}{2}\right )}{x}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sec \left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cos (c+d x)+a} \int \frac {\sin \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{2}\right )}{x}dx\) |
\(\Big \downarrow \) 3784 |
\(\displaystyle \sec \left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cos (c+d x)+a} \left (\cos \left (\frac {c}{2}\right ) \int \frac {\cos \left (\frac {d x}{2}\right )}{x}dx-\sin \left (\frac {c}{2}\right ) \int \frac {\sin \left (\frac {d x}{2}\right )}{x}dx\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sec \left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cos (c+d x)+a} \left (\cos \left (\frac {c}{2}\right ) \int \frac {\sin \left (\frac {d x}{2}+\frac {\pi }{2}\right )}{x}dx-\sin \left (\frac {c}{2}\right ) \int \frac {\sin \left (\frac {d x}{2}\right )}{x}dx\right )\) |
\(\Big \downarrow \) 3780 |
\(\displaystyle \sec \left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cos (c+d x)+a} \left (\cos \left (\frac {c}{2}\right ) \int \frac {\sin \left (\frac {d x}{2}+\frac {\pi }{2}\right )}{x}dx-\sin \left (\frac {c}{2}\right ) \text {Si}\left (\frac {d x}{2}\right )\right )\) |
\(\Big \downarrow \) 3783 |
\(\displaystyle \sec \left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cos (c+d x)+a} \left (\cos \left (\frac {c}{2}\right ) \operatorname {CosIntegral}\left (\frac {d x}{2}\right )-\sin \left (\frac {c}{2}\right ) \text {Si}\left (\frac {d x}{2}\right )\right )\) |
Input:
Int[Sqrt[a + a*Cos[c + d*x]]/x,x]
Output:
Sqrt[a + a*Cos[c + d*x]]*Sec[c/2 + (d*x)/2]*(Cos[c/2]*CosIntegral[(d*x)/2] - Sin[c/2]*SinIntegral[(d*x)/2])
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[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosInte gral[e - Pi/2 + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* e - c*f)/d] Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[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]
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])
\[\int \frac {\sqrt {a +a \cos \left (d x +c \right )}}{x}d x\]
Input:
int((a+a*cos(d*x+c))^(1/2)/x,x)
Output:
int((a+a*cos(d*x+c))^(1/2)/x,x)
Exception generated. \[ \int \frac {\sqrt {a+a \cos (c+d x)}}{x} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a+a*cos(d*x+c))^(1/2)/x,x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (has polynomial part)
\[ \int \frac {\sqrt {a+a \cos (c+d x)}}{x} \, dx=\int \frac {\sqrt {a \left (\cos {\left (c + d x \right )} + 1\right )}}{x}\, dx \] Input:
integrate((a+a*cos(d*x+c))**(1/2)/x,x)
Output:
Integral(sqrt(a*(cos(c + d*x) + 1))/x, x)
Result contains complex when optimal does not.
Time = 0.16 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.73 \[ \int \frac {\sqrt {a+a \cos (c+d x)}}{x} \, dx=-\frac {1}{2} \, {\left ({\left (\sqrt {2} E_{1}\left (\frac {1}{2} i \, d x\right ) + \sqrt {2} E_{1}\left (-\frac {1}{2} i \, d x\right )\right )} \cos \left (\frac {1}{2} \, c\right ) - {\left (i \, \sqrt {2} E_{1}\left (\frac {1}{2} i \, d x\right ) - i \, \sqrt {2} E_{1}\left (-\frac {1}{2} i \, d x\right )\right )} \sin \left (\frac {1}{2} \, c\right )\right )} \sqrt {a} \] Input:
integrate((a+a*cos(d*x+c))^(1/2)/x,x, algorithm="maxima")
Output:
-1/2*((sqrt(2)*exp_integral_e(1, 1/2*I*d*x) + sqrt(2)*exp_integral_e(1, -1 /2*I*d*x))*cos(1/2*c) - (I*sqrt(2)*exp_integral_e(1, 1/2*I*d*x) - I*sqrt(2 )*exp_integral_e(1, -1/2*I*d*x))*sin(1/2*c))*sqrt(a)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.39 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.98 \[ \int \frac {\sqrt {a+a \cos (c+d x)}}{x} \, dx=-\frac {\sqrt {2} {\left (\Re \left ( \operatorname {Ci}\left (\frac {1}{2} \, d x\right ) \right ) \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \tan \left (\frac {1}{4} \, c\right )^{2} + \Re \left ( \operatorname {Ci}\left (-\frac {1}{2} \, d x\right ) \right ) \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \tan \left (\frac {1}{4} \, c\right )^{2} + 2 \, \Im \left ( \operatorname {Ci}\left (\frac {1}{2} \, d x\right ) \right ) \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \tan \left (\frac {1}{4} \, c\right ) - 2 \, \Im \left ( \operatorname {Ci}\left (-\frac {1}{2} \, d x\right ) \right ) \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \tan \left (\frac {1}{4} \, c\right ) + 4 \, \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \operatorname {Si}\left (\frac {1}{2} \, d x\right ) \tan \left (\frac {1}{4} \, c\right ) - \Re \left ( \operatorname {Ci}\left (\frac {1}{2} \, d x\right ) \right ) \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) - \Re \left ( \operatorname {Ci}\left (-\frac {1}{2} \, d x\right ) \right ) \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sqrt {a}}{2 \, {\left (\tan \left (\frac {1}{4} \, c\right )^{2} + 1\right )}} \] Input:
integrate((a+a*cos(d*x+c))^(1/2)/x,x, algorithm="giac")
Output:
-1/2*sqrt(2)*(real_part(cos_integral(1/2*d*x))*sgn(cos(1/2*d*x + 1/2*c))*t an(1/4*c)^2 + real_part(cos_integral(-1/2*d*x))*sgn(cos(1/2*d*x + 1/2*c))* tan(1/4*c)^2 + 2*imag_part(cos_integral(1/2*d*x))*sgn(cos(1/2*d*x + 1/2*c) )*tan(1/4*c) - 2*imag_part(cos_integral(-1/2*d*x))*sgn(cos(1/2*d*x + 1/2*c ))*tan(1/4*c) + 4*sgn(cos(1/2*d*x + 1/2*c))*sin_integral(1/2*d*x)*tan(1/4* c) - real_part(cos_integral(1/2*d*x))*sgn(cos(1/2*d*x + 1/2*c)) - real_par t(cos_integral(-1/2*d*x))*sgn(cos(1/2*d*x + 1/2*c)))*sqrt(a)/(tan(1/4*c)^2 + 1)
Timed out. \[ \int \frac {\sqrt {a+a \cos (c+d x)}}{x} \, dx=\int \frac {\sqrt {a+a\,\cos \left (c+d\,x\right )}}{x} \,d x \] Input:
int((a + a*cos(c + d*x))^(1/2)/x,x)
Output:
int((a + a*cos(c + d*x))^(1/2)/x, x)
\[ \int \frac {\sqrt {a+a \cos (c+d x)}}{x} \, dx=\sqrt {a}\, \left (\int \frac {\sqrt {\cos \left (d x +c \right )+1}}{x}d x \right ) \] Input:
int((a+a*cos(d*x+c))^(1/2)/x,x)
Output:
sqrt(a)*int(sqrt(cos(c + d*x) + 1)/x,x)