Integrand size = 18, antiderivative size = 101 \[ \int \frac {\sqrt {a+a \sin (c+d x)}}{x} \, dx=\operatorname {CosIntegral}\left (\frac {d x}{2}\right ) \csc \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) \sin \left (\frac {1}{4} (2 c+\pi )\right ) \sqrt {a+a \sin (c+d x)}+\cos \left (\frac {1}{4} (2 c+\pi )\right ) \csc \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) \sqrt {a+a \sin (c+d x)} \text {Si}\left (\frac {d x}{2}\right ) \] Output:
Ci(1/2*d*x)*csc(1/2*c+1/4*Pi+1/2*d*x)*sin(1/2*c+1/4*Pi)*(a+a*sin(d*x+c))^( 1/2)+cos(1/2*c+1/4*Pi)*csc(1/2*c+1/4*Pi+1/2*d*x)*(a+a*sin(d*x+c))^(1/2)*Si (1/2*d*x)
Time = 0.44 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.82 \[ \int \frac {\sqrt {a+a \sin (c+d x)}}{x} \, dx=\frac {\sqrt {a (1+\sin (c+d x))} \left (\operatorname {CosIntegral}\left (\frac {d x}{2}\right ) \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right )+\left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \text {Si}\left (\frac {d x}{2}\right )\right )}{\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )} \] Input:
Integrate[Sqrt[a + a*Sin[c + d*x]]/x,x]
Output:
(Sqrt[a*(1 + Sin[c + d*x])]*(CosIntegral[(d*x)/2]*(Cos[c/2] + Sin[c/2]) + (Cos[c/2] - Sin[c/2])*SinIntegral[(d*x)/2]))/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])
Time = 0.50 (sec) , antiderivative size = 70, 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.
\(\displaystyle \int \frac {\sqrt {a \sin (c+d x)+a}}{x} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sqrt {a \sin (c+d x)+a}}{x}dx\) |
\(\Big \downarrow \) 3800 |
\(\displaystyle \csc \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (c+d x)+a} \int \frac {\sin \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{x}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \csc \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (c+d x)+a} \int \frac {\sin \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{x}dx\) |
\(\Big \downarrow \) 3784 |
\(\displaystyle \csc \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (c+d x)+a} \left (\sin \left (\frac {1}{4} (2 c+\pi )\right ) \int \frac {\cos \left (\frac {d x}{2}\right )}{x}dx+\cos \left (\frac {1}{4} (2 c+\pi )\right ) \int \frac {\sin \left (\frac {d x}{2}\right )}{x}dx\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \csc \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (c+d x)+a} \left (\sin \left (\frac {1}{4} (2 c+\pi )\right ) \int \frac {\sin \left (\frac {d x}{2}+\frac {\pi }{2}\right )}{x}dx+\cos \left (\frac {1}{4} (2 c+\pi )\right ) \int \frac {\sin \left (\frac {d x}{2}\right )}{x}dx\right )\) |
\(\Big \downarrow \) 3780 |
\(\displaystyle \csc \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (c+d x)+a} \left (\sin \left (\frac {1}{4} (2 c+\pi )\right ) \int \frac {\sin \left (\frac {d x}{2}+\frac {\pi }{2}\right )}{x}dx+\cos \left (\frac {1}{4} (2 c+\pi )\right ) \text {Si}\left (\frac {d x}{2}\right )\right )\) |
\(\Big \downarrow \) 3783 |
\(\displaystyle \csc \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (c+d x)+a} \left (\sin \left (\frac {1}{4} (2 c+\pi )\right ) \operatorname {CosIntegral}\left (\frac {d x}{2}\right )+\cos \left (\frac {1}{4} (2 c+\pi )\right ) \text {Si}\left (\frac {d x}{2}\right )\right )\) |
Input:
Int[Sqrt[a + a*Sin[c + d*x]]/x,x]
Output:
Csc[c/2 + Pi/4 + (d*x)/2]*Sqrt[a + a*Sin[c + d*x]]*(CosIntegral[(d*x)/2]*S in[(2*c + Pi)/4] + Cos[(2*c + Pi)/4]*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])
\[\int \frac {\sqrt {a +a \sin \left (d x +c \right )}}{x}d x\]
Input:
int((a+a*sin(d*x+c))^(1/2)/x,x)
Output:
int((a+a*sin(d*x+c))^(1/2)/x,x)
Exception generated. \[ \int \frac {\sqrt {a+a \sin (c+d x)}}{x} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a+a*sin(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)
\[ \int \frac {\sqrt {a+a \sin (c+d x)}}{x} \, dx=\int \frac {\sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )}}{x}\, dx \] Input:
integrate((a+a*sin(d*x+c))**(1/2)/x,x)
Output:
Integral(sqrt(a*(sin(c + d*x) + 1))/x, x)
\[ \int \frac {\sqrt {a+a \sin (c+d x)}}{x} \, dx=\int { \frac {\sqrt {a \sin \left (d x + c\right ) + a}}{x} \,d x } \] Input:
integrate((a+a*sin(d*x+c))^(1/2)/x,x, algorithm="maxima")
Output:
integrate(sqrt(a*sin(d*x + c) + a)/x, x)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.39 (sec) , antiderivative size = 383, normalized size of antiderivative = 3.79 \[ \int \frac {\sqrt {a+a \sin (c+d x)}}{x} \, dx=\text {Too large to display} \] Input:
integrate((a+a*sin(d*x+c))^(1/2)/x,x, algorithm="giac")
Output:
-1/2*sqrt(2)*(imag_part(cos_integral(1/2*d*x))*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*c)^2 - imag_part(cos_integral(-1/2*d*x))*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*c)^2 + real_part(cos_integral(1/2*d*x))*sgn(co s(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*c)^2 + real_part(cos_integral(-1/2*d *x))*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*c)^2 + 2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin_integral(1/2*d*x)*tan(1/4*c)^2 + 2*imag_part(cos_i ntegral(1/2*d*x))*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*c) - 2*imag_ part(cos_integral(-1/2*d*x))*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*c ) - 2*real_part(cos_integral(1/2*d*x))*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c)) *tan(1/4*c) - 2*real_part(cos_integral(-1/2*d*x))*sgn(cos(-1/4*pi + 1/2*d* x + 1/2*c))*tan(1/4*c) + 4*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin_integra l(1/2*d*x)*tan(1/4*c) - imag_part(cos_integral(1/2*d*x))*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c)) + imag_part(cos_integral(-1/2*d*x))*sgn(cos(-1/4*pi + 1 /2*d*x + 1/2*c)) - real_part(cos_integral(1/2*d*x))*sgn(cos(-1/4*pi + 1/2* d*x + 1/2*c)) - real_part(cos_integral(-1/2*d*x))*sgn(cos(-1/4*pi + 1/2*d* x + 1/2*c)) - 2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin_integral(1/2*d*x)) *sqrt(a)/(sqrt(2)*tan(1/4*c)^2 + sqrt(2))
Timed out. \[ \int \frac {\sqrt {a+a \sin (c+d x)}}{x} \, dx=\int \frac {\sqrt {a+a\,\sin \left (c+d\,x\right )}}{x} \,d x \] Input:
int((a + a*sin(c + d*x))^(1/2)/x,x)
Output:
int((a + a*sin(c + d*x))^(1/2)/x, x)
\[ \int \frac {\sqrt {a+a \sin (c+d x)}}{x} \, dx=\sqrt {a}\, \left (\int \frac {\sqrt {\sin \left (d x +c \right )+1}}{x}d x \right ) \] Input:
int((a+a*sin(d*x+c))^(1/2)/x,x)
Output:
sqrt(a)*int(sqrt(sin(c + d*x) + 1)/x,x)