\(\int \frac {\sin (a+\frac {b}{\sqrt {c+d x}})}{e+f x} \, dx\) [200]

Optimal result

Integrand size = 22, antiderivative size = 276 \[ \int \frac {\sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{e+f x} \, dx=-\frac {2 \operatorname {CosIntegral}\left (\frac {b}{\sqrt {c+d x}}\right ) \sin (a)}{f}+\frac {\operatorname {CosIntegral}\left (\frac {b \sqrt {f}}{\sqrt {-d e+c f}}+\frac {b}{\sqrt {c+d x}}\right ) \sin \left (a-\frac {b \sqrt {f}}{\sqrt {-d e+c f}}\right )}{f}+\frac {\operatorname {CosIntegral}\left (\frac {b \sqrt {f}}{\sqrt {-d e+c f}}-\frac {b}{\sqrt {c+d x}}\right ) \sin \left (a+\frac {b \sqrt {f}}{\sqrt {-d e+c f}}\right )}{f}-\frac {2 \cos (a) \text {Si}\left (\frac {b}{\sqrt {c+d x}}\right )}{f}-\frac {\cos \left (a+\frac {b \sqrt {f}}{\sqrt {-d e+c f}}\right ) \text {Si}\left (\frac {b \sqrt {f}}{\sqrt {-d e+c f}}-\frac {b}{\sqrt {c+d x}}\right )}{f}+\frac {\cos \left (a-\frac {b \sqrt {f}}{\sqrt {-d e+c f}}\right ) \text {Si}\left (\frac {b \sqrt {f}}{\sqrt {-d e+c f}}+\frac {b}{\sqrt {c+d x}}\right )}{f} \] Output:

-2*Ci(b/(d*x+c)^(1/2))*sin(a)/f+Ci(b*f^(1/2)/(c*f-d*e)^(1/2)+b/(d*x+c)^(1/ 
2))*sin(a-b*f^(1/2)/(c*f-d*e)^(1/2))/f+Ci(b*f^(1/2)/(c*f-d*e)^(1/2)-b/(d*x 
+c)^(1/2))*sin(a+b*f^(1/2)/(c*f-d*e)^(1/2))/f-2*cos(a)*Si(b/(d*x+c)^(1/2)) 
/f-cos(a+b*f^(1/2)/(c*f-d*e)^(1/2))*Si(b*f^(1/2)/(c*f-d*e)^(1/2)-b/(d*x+c) 
^(1/2))/f+cos(a-b*f^(1/2)/(c*f-d*e)^(1/2))*Si(b*f^(1/2)/(c*f-d*e)^(1/2)+b/ 
(d*x+c)^(1/2))/f
 

Mathematica [F]

\[ \int \frac {\sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{e+f x} \, dx=\int \frac {\sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{e+f x} \, dx \] Input:

Integrate[Sin[a + b/Sqrt[c + d*x]]/(e + f*x),x]
 

Output:

Integrate[Sin[a + b/Sqrt[c + d*x]]/(e + f*x), x]
 

Rubi [A] (verified)

Time = 1.12 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.07, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {3912, 2009}

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.

\(\displaystyle \int \frac {\sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{e+f x} \, dx\)

\(\Big \downarrow \) 3912

\(\displaystyle -\frac {2 \int \left (\frac {d \sqrt {c+d x} \sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{f}-\frac {d (d e-c f) \sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{f \sqrt {c+d x} \left (f+\frac {d e-c f}{c+d x}\right )}\right )d\frac {1}{\sqrt {c+d x}}}{d}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 \left (-\frac {d \sin \left (a-\frac {b \sqrt {f}}{\sqrt {c f-d e}}\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {f} b}{\sqrt {c f-d e}}+\frac {b}{\sqrt {c+d x}}\right )}{2 f}-\frac {d \sin \left (a+\frac {b \sqrt {f}}{\sqrt {c f-d e}}\right ) \operatorname {CosIntegral}\left (\frac {b \sqrt {f}}{\sqrt {c f-d e}}-\frac {b}{\sqrt {c+d x}}\right )}{2 f}+\frac {d \sin (a) \operatorname {CosIntegral}\left (\frac {b}{\sqrt {c+d x}}\right )}{f}+\frac {d \cos \left (a+\frac {b \sqrt {f}}{\sqrt {c f-d e}}\right ) \text {Si}\left (\frac {b \sqrt {f}}{\sqrt {c f-d e}}-\frac {b}{\sqrt {c+d x}}\right )}{2 f}-\frac {d \cos \left (a-\frac {b \sqrt {f}}{\sqrt {c f-d e}}\right ) \text {Si}\left (\frac {\sqrt {f} b}{\sqrt {c f-d e}}+\frac {b}{\sqrt {c+d x}}\right )}{2 f}+\frac {d \cos (a) \text {Si}\left (\frac {b}{\sqrt {c+d x}}\right )}{f}\right )}{d}\)

Input:

Int[Sin[a + b/Sqrt[c + d*x]]/(e + f*x),x]
 

Output:

(-2*((d*CosIntegral[b/Sqrt[c + d*x]]*Sin[a])/f - (d*CosIntegral[(b*Sqrt[f] 
)/Sqrt[-(d*e) + c*f] + b/Sqrt[c + d*x]]*Sin[a - (b*Sqrt[f])/Sqrt[-(d*e) + 
c*f]])/(2*f) - (d*CosIntegral[(b*Sqrt[f])/Sqrt[-(d*e) + c*f] - b/Sqrt[c + 
d*x]]*Sin[a + (b*Sqrt[f])/Sqrt[-(d*e) + c*f]])/(2*f) + (d*Cos[a]*SinIntegr 
al[b/Sqrt[c + d*x]])/f + (d*Cos[a + (b*Sqrt[f])/Sqrt[-(d*e) + c*f]]*SinInt 
egral[(b*Sqrt[f])/Sqrt[-(d*e) + c*f] - b/Sqrt[c + d*x]])/(2*f) - (d*Cos[a 
- (b*Sqrt[f])/Sqrt[-(d*e) + c*f]]*SinIntegral[(b*Sqrt[f])/Sqrt[-(d*e) + c* 
f] + b/Sqrt[c + d*x]])/(2*f)))/d
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3912

Int[((g_.) + (h_.)*(x_))^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*((e_.) + (f 
_.)*(x_))^(n_)])^(p_.), x_Symbol] :> Simp[1/(n*f)   Subst[Int[ExpandIntegra 
nd[(a + b*Sin[c + d*x])^p, x^(1/n - 1)*(g - e*(h/f) + h*(x^(1/n)/f))^m, x], 
 x], x, (e + f*x)^n], x] /; FreeQ[{a, b, c, d, e, f, g, h, m}, x] && IGtQ[p 
, 0] && IntegerQ[1/n]
 

Maple [A] (verified)

Time = 2.25 (sec) , antiderivative size = 441, normalized size of antiderivative = 1.60

method	result	size
derivativedivides	\(-2 b^{2} \left (-\frac {-\operatorname {Si}\left (-\frac {b}{\sqrt {d x +c}}-a +\frac {c f a -d e a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{c f -d e}\right ) \cos \left (\frac {c f a -d e a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{c f -d e}\right )+\operatorname {Ci}\left (\frac {b}{\sqrt {d x +c}}+a -\frac {c f a -d e a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{c f -d e}\right ) \sin \left (\frac {c f a -d e a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{c f -d e}\right )}{2 f \,b^{2}}-\frac {\operatorname {Si}\left (\frac {b}{\sqrt {d x +c}}+a +\frac {-c f a +d e a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{c f -d e}\right ) \cos \left (\frac {-c f a +d e a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{c f -d e}\right )-\operatorname {Ci}\left (\frac {b}{\sqrt {d x +c}}+a +\frac {-c f a +d e a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{c f -d e}\right ) \sin \left (\frac {-c f a +d e a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{c f -d e}\right )}{2 f \,b^{2}}+\frac {\operatorname {Ci}\left (\frac {b}{\sqrt {d x +c}}\right ) \sin \left (a \right )+\operatorname {Si}\left (\frac {b}{\sqrt {d x +c}}\right ) \cos \left (a \right )}{f \,b^{2}}\right )\)	\(441\)
default	\(-2 b^{2} \left (-\frac {-\operatorname {Si}\left (-\frac {b}{\sqrt {d x +c}}-a +\frac {c f a -d e a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{c f -d e}\right ) \cos \left (\frac {c f a -d e a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{c f -d e}\right )+\operatorname {Ci}\left (\frac {b}{\sqrt {d x +c}}+a -\frac {c f a -d e a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{c f -d e}\right ) \sin \left (\frac {c f a -d e a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{c f -d e}\right )}{2 f \,b^{2}}-\frac {\operatorname {Si}\left (\frac {b}{\sqrt {d x +c}}+a +\frac {-c f a +d e a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{c f -d e}\right ) \cos \left (\frac {-c f a +d e a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{c f -d e}\right )-\operatorname {Ci}\left (\frac {b}{\sqrt {d x +c}}+a +\frac {-c f a +d e a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{c f -d e}\right ) \sin \left (\frac {-c f a +d e a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{c f -d e}\right )}{2 f \,b^{2}}+\frac {\operatorname {Ci}\left (\frac {b}{\sqrt {d x +c}}\right ) \sin \left (a \right )+\operatorname {Si}\left (\frac {b}{\sqrt {d x +c}}\right ) \cos \left (a \right )}{f \,b^{2}}\right )\)	\(441\)

Input:

int(sin(a+b/(d*x+c)^(1/2))/(f*x+e),x,method=_RETURNVERBOSE)
 

Output:

-2*b^2*(-1/2/f/b^2*(-Si(-b/(d*x+c)^(1/2)-a+(c*f*a-d*e*a+(b^2*c*f^2-b^2*d*e 
*f)^(1/2))/(c*f-d*e))*cos((c*f*a-d*e*a+(b^2*c*f^2-b^2*d*e*f)^(1/2))/(c*f-d 
*e))+Ci(b/(d*x+c)^(1/2)+a-(c*f*a-d*e*a+(b^2*c*f^2-b^2*d*e*f)^(1/2))/(c*f-d 
*e))*sin((c*f*a-d*e*a+(b^2*c*f^2-b^2*d*e*f)^(1/2))/(c*f-d*e)))-1/2/f/b^2*( 
Si(b/(d*x+c)^(1/2)+a+(-c*f*a+d*e*a+(b^2*c*f^2-b^2*d*e*f)^(1/2))/(c*f-d*e)) 
*cos((-c*f*a+d*e*a+(b^2*c*f^2-b^2*d*e*f)^(1/2))/(c*f-d*e))-Ci(b/(d*x+c)^(1 
/2)+a+(-c*f*a+d*e*a+(b^2*c*f^2-b^2*d*e*f)^(1/2))/(c*f-d*e))*sin((-c*f*a+d* 
e*a+(b^2*c*f^2-b^2*d*e*f)^(1/2))/(c*f-d*e)))+1/f/b^2*(Ci(b/(d*x+c)^(1/2))* 
sin(a)+Si(b/(d*x+c)^(1/2))*cos(a)))
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.11 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.12 \[ \int \frac {\sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{e+f x} \, dx=\frac {-i \, {\rm Ei}\left (-\frac {\sqrt {\frac {b^{2} f}{d e - c f}} {\left (d x + c\right )} - i \, \sqrt {d x + c} b}{d x + c}\right ) e^{\left (i \, a + \sqrt {\frac {b^{2} f}{d e - c f}}\right )} - i \, {\rm Ei}\left (\frac {\sqrt {\frac {b^{2} f}{d e - c f}} {\left (d x + c\right )} + i \, \sqrt {d x + c} b}{d x + c}\right ) e^{\left (i \, a - \sqrt {\frac {b^{2} f}{d e - c f}}\right )} + i \, {\rm Ei}\left (-\frac {\sqrt {\frac {b^{2} f}{d e - c f}} {\left (d x + c\right )} + i \, \sqrt {d x + c} b}{d x + c}\right ) e^{\left (-i \, a + \sqrt {\frac {b^{2} f}{d e - c f}}\right )} + i \, {\rm Ei}\left (\frac {\sqrt {\frac {b^{2} f}{d e - c f}} {\left (d x + c\right )} - i \, \sqrt {d x + c} b}{d x + c}\right ) e^{\left (-i \, a - \sqrt {\frac {b^{2} f}{d e - c f}}\right )} - 4 \, \operatorname {Ci}\left (\frac {b}{\sqrt {d x + c}}\right ) \sin \left (a\right ) - 4 \, \cos \left (a\right ) \operatorname {Si}\left (\frac {b}{\sqrt {d x + c}}\right )}{2 \, f} \] Input:

integrate(sin(a+b/(d*x+c)^(1/2))/(f*x+e),x, algorithm="fricas")
 

Output:

1/2*(-I*Ei(-(sqrt(b^2*f/(d*e - c*f))*(d*x + c) - I*sqrt(d*x + c)*b)/(d*x + 
 c))*e^(I*a + sqrt(b^2*f/(d*e - c*f))) - I*Ei((sqrt(b^2*f/(d*e - c*f))*(d* 
x + c) + I*sqrt(d*x + c)*b)/(d*x + c))*e^(I*a - sqrt(b^2*f/(d*e - c*f))) + 
 I*Ei(-(sqrt(b^2*f/(d*e - c*f))*(d*x + c) + I*sqrt(d*x + c)*b)/(d*x + c))* 
e^(-I*a + sqrt(b^2*f/(d*e - c*f))) + I*Ei((sqrt(b^2*f/(d*e - c*f))*(d*x + 
c) - I*sqrt(d*x + c)*b)/(d*x + c))*e^(-I*a - sqrt(b^2*f/(d*e - c*f))) - 4* 
cos_integral(b/sqrt(d*x + c))*sin(a) - 4*cos(a)*sin_integral(b/sqrt(d*x + 
c)))/f
 

Sympy [F]

\[ \int \frac {\sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{e+f x} \, dx=\int \frac {\sin {\left (a + \frac {b}{\sqrt {c + d x}} \right )}}{e + f x}\, dx \] Input:

integrate(sin(a+b/(d*x+c)**(1/2))/(f*x+e),x)
 

Output:

Integral(sin(a + b/sqrt(c + d*x))/(e + f*x), x)
 

Maxima [F]

\[ \int \frac {\sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{e+f x} \, dx=\int { \frac {\sin \left (a + \frac {b}{\sqrt {d x + c}}\right )}{f x + e} \,d x } \] Input:

integrate(sin(a+b/(d*x+c)^(1/2))/(f*x+e),x, algorithm="maxima")
 

Output:

integrate(sin(a + b/sqrt(d*x + c))/(f*x + e), x)
 

Giac [F]

\[ \int \frac {\sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{e+f x} \, dx=\int { \frac {\sin \left (a + \frac {b}{\sqrt {d x + c}}\right )}{f x + e} \,d x } \] Input:

integrate(sin(a+b/(d*x+c)^(1/2))/(f*x+e),x, algorithm="giac")
 

Output:

integrate(sin(a + b/sqrt(d*x + c))/(f*x + e), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{e+f x} \, dx=\int \frac {\sin \left (a+\frac {b}{\sqrt {c+d\,x}}\right )}{e+f\,x} \,d x \] Input:

int(sin(a + b/(c + d*x)^(1/2))/(e + f*x),x)
 

Output:

int(sin(a + b/(c + d*x)^(1/2))/(e + f*x), x)
 

Reduce [F]

\[ \int \frac {\sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{e+f x} \, dx=\int \frac {\sin \left (\frac {\sqrt {d x +c}\, a +b}{\sqrt {d x +c}}\right )}{f x +e}d x \] Input:

int(sin(a+b/(d*x+c)^(1/2))/(f*x+e),x)
 

Output:

int(sin((sqrt(c + d*x)*a + b)/sqrt(c + d*x))/(e + f*x),x)