\(\int \frac {\sin (a+b \sqrt {c+d x})}{e+f x} \, dx\) [190]

Optimal result

Integrand size = 22, antiderivative size = 238 \[ \int \frac {\sin \left (a+b \sqrt {c+d x}\right )}{e+f x} \, dx=\frac {\operatorname {CosIntegral}\left (\frac {b \sqrt {-d e+c f}}{\sqrt {f}}+b \sqrt {c+d x}\right ) \sin \left (a-\frac {b \sqrt {-d e+c f}}{\sqrt {f}}\right )}{f}+\frac {\operatorname {CosIntegral}\left (\frac {b \sqrt {-d e+c f}}{\sqrt {f}}-b \sqrt {c+d x}\right ) \sin \left (a+\frac {b \sqrt {-d e+c f}}{\sqrt {f}}\right )}{f}-\frac {\cos \left (a+\frac {b \sqrt {-d e+c f}}{\sqrt {f}}\right ) \text {Si}\left (\frac {b \sqrt {-d e+c f}}{\sqrt {f}}-b \sqrt {c+d x}\right )}{f}+\frac {\cos \left (a-\frac {b \sqrt {-d e+c f}}{\sqrt {f}}\right ) \text {Si}\left (\frac {b \sqrt {-d e+c f}}{\sqrt {f}}+b \sqrt {c+d x}\right )}{f} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.37 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.00 \[ \int \frac {\sin \left (a+b \sqrt {c+d x}\right )}{e+f x} \, dx=\frac {i e^{-i \left (a+\frac {b \sqrt {-d e+c f}}{\sqrt {f}}\right )} \left (\operatorname {ExpIntegralEi}\left (-i b \left (-\frac {\sqrt {-d e+c f}}{\sqrt {f}}+\sqrt {c+d x}\right )\right )-e^{2 i \left (a+\frac {b \sqrt {-d e+c f}}{\sqrt {f}}\right )} \operatorname {ExpIntegralEi}\left (i b \left (-\frac {\sqrt {-d e+c f}}{\sqrt {f}}+\sqrt {c+d x}\right )\right )+e^{\frac {2 i b \sqrt {-d e+c f}}{\sqrt {f}}} \operatorname {ExpIntegralEi}\left (-i b \left (\frac {\sqrt {-d e+c f}}{\sqrt {f}}+\sqrt {c+d x}\right )\right )-e^{2 i a} \operatorname {ExpIntegralEi}\left (i b \left (\frac {\sqrt {-d e+c f}}{\sqrt {f}}+\sqrt {c+d x}\right )\right )\right )}{2 f} \] Input:

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

Output:

((I/2)*(ExpIntegralEi[(-I)*b*(-(Sqrt[-(d*e) + c*f]/Sqrt[f]) + Sqrt[c + d*x 
])] - E^((2*I)*(a + (b*Sqrt[-(d*e) + c*f])/Sqrt[f]))*ExpIntegralEi[I*b*(-( 
Sqrt[-(d*e) + c*f]/Sqrt[f]) + Sqrt[c + d*x])] + E^(((2*I)*b*Sqrt[-(d*e) + 
c*f])/Sqrt[f])*ExpIntegralEi[(-I)*b*(Sqrt[-(d*e) + c*f]/Sqrt[f] + Sqrt[c + 
 d*x])] - E^((2*I)*a)*ExpIntegralEi[I*b*(Sqrt[-(d*e) + c*f]/Sqrt[f] + Sqrt 
[c + d*x])]))/(E^(I*(a + (b*Sqrt[-(d*e) + c*f])/Sqrt[f]))*f)
 

Rubi [A] (verified)

Time = 0.89 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.08, 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.

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

\(\Big \downarrow \) 3912

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(2*((d*CosIntegral[(b*Sqrt[-(d*e) + c*f])/Sqrt[f] + b*Sqrt[c + d*x]]*Sin[a 
 - (b*Sqrt[-(d*e) + c*f])/Sqrt[f]])/(2*f) + (d*CosIntegral[(b*Sqrt[-(d*e) 
+ c*f])/Sqrt[f] - b*Sqrt[c + d*x]]*Sin[a + (b*Sqrt[-(d*e) + c*f])/Sqrt[f]] 
)/(2*f) - (d*Cos[a + (b*Sqrt[-(d*e) + c*f])/Sqrt[f]]*SinIntegral[(b*Sqrt[- 
(d*e) + c*f])/Sqrt[f] - b*Sqrt[c + d*x]])/(2*f) + (d*Cos[a - (b*Sqrt[-(d*e 
) + c*f])/Sqrt[f]]*SinIntegral[(b*Sqrt[-(d*e) + c*f])/Sqrt[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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(792\) vs. \(2(197)=394\).

Time = 0.85 (sec) , antiderivative size = 793, normalized size of antiderivative = 3.33

method	result	size
derivativedivides	\(\frac {-\frac {b^{2} \left (a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}\right ) \left (-\operatorname {Si}\left (-\sqrt {d x +c}\, b -a +\frac {a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right ) \cos \left (\frac {a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right )+\operatorname {Ci}\left (\sqrt {d x +c}\, b +a -\frac {a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right ) \sin \left (\frac {a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right )\right )}{f^{2} \left (-\frac {a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}+a \right )}+\frac {b^{2} \left (-a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}\right ) \left (-\operatorname {Si}\left (-\sqrt {d x +c}\, b -a -\frac {-a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right ) \cos \left (\frac {-a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right )-\operatorname {Ci}\left (\sqrt {d x +c}\, b +a +\frac {-a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right ) \sin \left (\frac {-a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right )\right )}{f^{2} \left (\frac {-a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}+a \right )}-2 b^{2} a \left (-\frac {-\operatorname {Si}\left (-\sqrt {d x +c}\, b -a +\frac {a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right ) \cos \left (\frac {a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right )+\operatorname {Ci}\left (\sqrt {d x +c}\, b +a -\frac {a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right ) \sin \left (\frac {a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right )}{2 f \left (-\frac {a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}+a \right )}-\frac {-\operatorname {Si}\left (-\sqrt {d x +c}\, b -a -\frac {-a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right ) \cos \left (\frac {-a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right )-\operatorname {Ci}\left (\sqrt {d x +c}\, b +a +\frac {-a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right ) \sin \left (\frac {-a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right )}{2 f \left (\frac {-a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}+a \right )}\right )}{b^{2}}\)	\(793\)
default	\(\frac {-\frac {b^{2} \left (a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}\right ) \left (-\operatorname {Si}\left (-\sqrt {d x +c}\, b -a +\frac {a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right ) \cos \left (\frac {a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right )+\operatorname {Ci}\left (\sqrt {d x +c}\, b +a -\frac {a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right ) \sin \left (\frac {a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right )\right )}{f^{2} \left (-\frac {a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}+a \right )}+\frac {b^{2} \left (-a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}\right ) \left (-\operatorname {Si}\left (-\sqrt {d x +c}\, b -a -\frac {-a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right ) \cos \left (\frac {-a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right )-\operatorname {Ci}\left (\sqrt {d x +c}\, b +a +\frac {-a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right ) \sin \left (\frac {-a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right )\right )}{f^{2} \left (\frac {-a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}+a \right )}-2 b^{2} a \left (-\frac {-\operatorname {Si}\left (-\sqrt {d x +c}\, b -a +\frac {a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right ) \cos \left (\frac {a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right )+\operatorname {Ci}\left (\sqrt {d x +c}\, b +a -\frac {a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right ) \sin \left (\frac {a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right )}{2 f \left (-\frac {a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}+a \right )}-\frac {-\operatorname {Si}\left (-\sqrt {d x +c}\, b -a -\frac {-a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right ) \cos \left (\frac {-a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right )-\operatorname {Ci}\left (\sqrt {d x +c}\, b +a +\frac {-a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right ) \sin \left (\frac {-a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right )}{2 f \left (\frac {-a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}+a \right )}\right )}{b^{2}}\)	\(793\)

Input:

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

Output:

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.09 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.05 \[ \int \frac {\sin \left (a+b \sqrt {c+d x}\right )}{e+f x} \, dx=\frac {-i \, {\rm Ei}\left (i \, \sqrt {d x + c} b - \sqrt {\frac {b^{2} d e - b^{2} c f}{f}}\right ) e^{\left (i \, a + \sqrt {\frac {b^{2} d e - b^{2} c f}{f}}\right )} - i \, {\rm Ei}\left (i \, \sqrt {d x + c} b + \sqrt {\frac {b^{2} d e - b^{2} c f}{f}}\right ) e^{\left (i \, a - \sqrt {\frac {b^{2} d e - b^{2} c f}{f}}\right )} + i \, {\rm Ei}\left (-i \, \sqrt {d x + c} b - \sqrt {\frac {b^{2} d e - b^{2} c f}{f}}\right ) e^{\left (-i \, a + \sqrt {\frac {b^{2} d e - b^{2} c f}{f}}\right )} + i \, {\rm Ei}\left (-i \, \sqrt {d x + c} b + \sqrt {\frac {b^{2} d e - b^{2} c f}{f}}\right ) e^{\left (-i \, a - \sqrt {\frac {b^{2} d e - b^{2} c f}{f}}\right )}}{2 \, f} \] Input:

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

Output:

1/2*(-I*Ei(I*sqrt(d*x + c)*b - sqrt((b^2*d*e - b^2*c*f)/f))*e^(I*a + sqrt( 
(b^2*d*e - b^2*c*f)/f)) - I*Ei(I*sqrt(d*x + c)*b + sqrt((b^2*d*e - b^2*c*f 
)/f))*e^(I*a - sqrt((b^2*d*e - b^2*c*f)/f)) + I*Ei(-I*sqrt(d*x + c)*b - sq 
rt((b^2*d*e - b^2*c*f)/f))*e^(-I*a + sqrt((b^2*d*e - b^2*c*f)/f)) + I*Ei(- 
I*sqrt(d*x + c)*b + sqrt((b^2*d*e - b^2*c*f)/f))*e^(-I*a - sqrt((b^2*d*e - 
 b^2*c*f)/f)))/f
 

Sympy [F]

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

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

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

Output:

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