\(\int \frac {\sin (a+b \sqrt {c+d x})}{(e+f x)^2} \, dx\) [191]

Optimal result

Integrand size = 22, antiderivative size = 339 \[ \int \frac {\sin \left (a+b \sqrt {c+d x}\right )}{(e+f x)^2} \, dx=\frac {b d \cos \left (a+\frac {b \sqrt {-d e+c f}}{\sqrt {f}}\right ) \operatorname {CosIntegral}\left (\frac {b \sqrt {-d e+c f}}{\sqrt {f}}-b \sqrt {c+d x}\right )}{2 f^{3/2} \sqrt {-d e+c f}}-\frac {b d \cos \left (a-\frac {b \sqrt {-d e+c f}}{\sqrt {f}}\right ) \operatorname {CosIntegral}\left (\frac {b \sqrt {-d e+c f}}{\sqrt {f}}+b \sqrt {c+d x}\right )}{2 f^{3/2} \sqrt {-d e+c f}}-\frac {\sin \left (a+b \sqrt {c+d x}\right )}{f (e+f x)}+\frac {b d \sin \left (a+\frac {b \sqrt {-d e+c f}}{\sqrt {f}}\right ) \text {Si}\left (\frac {b \sqrt {-d e+c f}}{\sqrt {f}}-b \sqrt {c+d x}\right )}{2 f^{3/2} \sqrt {-d e+c f}}+\frac {b d \sin \left (a-\frac {b \sqrt {-d e+c f}}{\sqrt {f}}\right ) \text {Si}\left (\frac {b \sqrt {-d e+c f}}{\sqrt {f}}+b \sqrt {c+d x}\right )}{2 f^{3/2} \sqrt {-d e+c f}} \] Output:

1/2*b*d*cos(a+b*(c*f-d*e)^(1/2)/f^(1/2))*Ci(b*(c*f-d*e)^(1/2)/f^(1/2)-b*(d 
*x+c)^(1/2))/f^(3/2)/(c*f-d*e)^(1/2)-1/2*b*d*cos(a-b*(c*f-d*e)^(1/2)/f^(1/ 
2))*Ci(b*(c*f-d*e)^(1/2)/f^(1/2)+b*(d*x+c)^(1/2))/f^(3/2)/(c*f-d*e)^(1/2)- 
sin(a+b*(d*x+c)^(1/2))/f/(f*x+e)-1/2*b*d*sin(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^(3/2)/(c*f-d*e)^(1/2)+1/2 
*b*d*sin(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^(3/2)/(c*f-d*e)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 3.96 (sec) , antiderivative size = 397, normalized size of antiderivative = 1.17 \[ \int \frac {\sin \left (a+b \sqrt {c+d x}\right )}{(e+f x)^2} \, dx=\frac {i d e^{-i a} \left (-\frac {2 e^{-i b \sqrt {c+d x}} \sqrt {f}}{d e+d f x}-\frac {i b e^{-\frac {i b \sqrt {-d e+c f}}{\sqrt {f}}} \operatorname {ExpIntegralEi}\left (-i b \left (-\frac {\sqrt {-d e+c f}}{\sqrt {f}}+\sqrt {c+d x}\right )\right )}{\sqrt {-d e+c f}}+\frac {i b e^{\frac {i b \sqrt {-d e+c f}}{\sqrt {f}}} \operatorname {ExpIntegralEi}\left (-i b \left (\frac {\sqrt {-d e+c f}}{\sqrt {f}}+\sqrt {c+d x}\right )\right )}{\sqrt {-d e+c f}}+e^{2 i a} \left (\frac {2 e^{i b \sqrt {c+d x}} \sqrt {f}}{d e+d f x}-\frac {i b e^{\frac {i b \sqrt {-d e+c f}}{\sqrt {f}}} \operatorname {ExpIntegralEi}\left (i b \left (-\frac {\sqrt {-d e+c f}}{\sqrt {f}}+\sqrt {c+d x}\right )\right )}{\sqrt {-d e+c f}}+\frac {i b e^{-\frac {i b \sqrt {-d e+c f}}{\sqrt {f}}} \operatorname {ExpIntegralEi}\left (i b \left (\frac {\sqrt {-d e+c f}}{\sqrt {f}}+\sqrt {c+d x}\right )\right )}{\sqrt {-d e+c f}}\right )\right )}{4 f^{3/2}} \] Input:

Integrate[Sin[a + b*Sqrt[c + d*x]]/(e + f*x)^2,x]
 

Output:

((I/4)*d*((-2*Sqrt[f])/(E^(I*b*Sqrt[c + d*x])*(d*e + d*f*x)) - (I*b*ExpInt 
egralEi[(-I)*b*(-(Sqrt[-(d*e) + c*f]/Sqrt[f]) + Sqrt[c + d*x])])/(E^((I*b* 
Sqrt[-(d*e) + c*f])/Sqrt[f])*Sqrt[-(d*e) + c*f]) + (I*b*E^((I*b*Sqrt[-(d*e 
) + c*f])/Sqrt[f])*ExpIntegralEi[(-I)*b*(Sqrt[-(d*e) + c*f]/Sqrt[f] + Sqrt 
[c + d*x])])/Sqrt[-(d*e) + c*f] + E^((2*I)*a)*((2*E^(I*b*Sqrt[c + d*x])*Sq 
rt[f])/(d*e + d*f*x) - (I*b*E^((I*b*Sqrt[-(d*e) + c*f])/Sqrt[f])*ExpIntegr 
alEi[I*b*(-(Sqrt[-(d*e) + c*f]/Sqrt[f]) + Sqrt[c + d*x])])/Sqrt[-(d*e) + c 
*f] + (I*b*ExpIntegralEi[I*b*(Sqrt[-(d*e) + c*f]/Sqrt[f] + Sqrt[c + d*x])] 
)/(E^((I*b*Sqrt[-(d*e) + c*f])/Sqrt[f])*Sqrt[-(d*e) + c*f]))))/(E^(I*a)*f^ 
(3/2))
 

Rubi [A] (verified)

Time = 1.03 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.05, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {3912, 27, 3822, 3815, 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.

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

\(\Big \downarrow \) 3912

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3822

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

\(\Big \downarrow \) 3815

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

\(\Big \downarrow \) 2009

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

Input:

Int[Sin[a + b*Sqrt[c + d*x]]/(e + f*x)^2,x]
 

Output:

2*d*(-1/2*Sin[a + b*Sqrt[c + d*x]]/(f*(d*e - c*f + f*(c + d*x))) + (b*((Co 
s[a + (b*Sqrt[-(d*e) + c*f])/Sqrt[f]]*CosIntegral[(b*Sqrt[-(d*e) + c*f])/S 
qrt[f] - b*Sqrt[c + d*x]])/(2*Sqrt[f]*Sqrt[-(d*e) + c*f]) - (Cos[a - (b*Sq 
rt[-(d*e) + c*f])/Sqrt[f]]*CosIntegral[(b*Sqrt[-(d*e) + c*f])/Sqrt[f] + b* 
Sqrt[c + d*x]])/(2*Sqrt[f]*Sqrt[-(d*e) + c*f]) + (Sin[a + (b*Sqrt[-(d*e) + 
 c*f])/Sqrt[f]]*SinIntegral[(b*Sqrt[-(d*e) + c*f])/Sqrt[f] - b*Sqrt[c + d* 
x]])/(2*Sqrt[f]*Sqrt[-(d*e) + c*f]) + (Sin[a - (b*Sqrt[-(d*e) + c*f])/Sqrt 
[f]]*SinIntegral[(b*Sqrt[-(d*e) + c*f])/Sqrt[f] + b*Sqrt[c + d*x]])/(2*Sqr 
t[f]*Sqrt[-(d*e) + c*f])))/(2*f))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

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

rule 3815

Int[Cos[(c_.) + (d_.)*(x_)]*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int 
[ExpandIntegrand[Cos[c + d*x], (a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d}, 
 x] && ILtQ[p, 0] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ[p, -1])
 

rule 3822

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*Sin[(c_.) + (d_.)*(x_) 
], x_Symbol] :> Simp[e^m*(a + b*x^n)^(p + 1)*(Sin[c + d*x]/(b*n*(p + 1))), 
x] - Simp[d*(e^m/(b*n*(p + 1)))   Int[(a + b*x^n)^(p + 1)*Cos[c + d*x], x], 
 x] /; FreeQ[{a, b, c, d, e, m, n}, x] && ILtQ[p, -1] && EqQ[m, n - 1] && ( 
IntegerQ[n] || GtQ[e, 0])
 

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. \(1830\) vs. \(2(273)=546\).

Time = 0.95 (sec) , antiderivative size = 1831, normalized size of antiderivative = 5.40

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1831\)
default	\(\text {Expression too large to display}\)	\(1831\)

Input:

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

Output:

2*d/b^2*(sin(a+(d*x+c)^(1/2)*b)*(-1/2*a*b^2/(c*f-d*e)*(a+(d*x+c)^(1/2)*b)+ 
1/2*b^2*(-b^2*c*f+b^2*d*e+a^2*f)/(c*f-d*e)/f)/(-c*f*b^2+d*e*b^2+a^2*f-2*a* 
f*(a+(d*x+c)^(1/2)*b)+f*(a+(d*x+c)^(1/2)*b)^2)+1/4*a*b^2/(c*f-d*e)/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))+1/4*a*b^2/(c*f-d*e)/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))+1/4*b^2*(-c 
*f*b^2+d*e*b^2+a^2*f-a*(a*f+(b^2*c*f^2-b^2*d*e*f)^(1/2)))/(c*f-d*e)/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)*sin((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)*cos((a*f+(b^2*c*f^2-b^2* 
d*e*f)^(1/2))/f))+1/4*b^2*(-c*f*b^2+d*e*b^2+a^2*f+a*(-a*f+(b^2*c*f^2-b^2*d 
*e*f)^(1/2)))/(c*f-d*e)/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)*sin((-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)*cos((-a*f+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f))-a*b^4*(sin(a+(d*x+c)^ 
(1/2)*b)*(-1/2/b^2/(c*f-d*e)*(a+(d*x+c)^(1/2)*b)+1/2*a/b^2/(c*f-d*e))/(...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.10 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.23 \[ \int \frac {\sin \left (a+b \sqrt {c+d x}\right )}{(e+f x)^2} \, dx=-\frac {{\left (-i \, d f x - i \, d e\right )} \sqrt {\frac {b^{2} d e - b^{2} c f}{f}} {\rm Ei}\left (i \, \sqrt {d x + c} b - \sqrt {\frac {b^{2} d e - b^{2} c f}{f}}\right ) e^{\left (i \, a + \sqrt {\frac {b^{2} d e - b^{2} c f}{f}}\right )} + {\left (i \, d f x + i \, d e\right )} \sqrt {\frac {b^{2} d e - b^{2} c f}{f}} {\rm Ei}\left (i \, \sqrt {d x + c} b + \sqrt {\frac {b^{2} d e - b^{2} c f}{f}}\right ) e^{\left (i \, a - \sqrt {\frac {b^{2} d e - b^{2} c f}{f}}\right )} + {\left (i \, d f x + i \, d e\right )} \sqrt {\frac {b^{2} d e - b^{2} c f}{f}} {\rm Ei}\left (-i \, \sqrt {d x + c} b - \sqrt {\frac {b^{2} d e - b^{2} c f}{f}}\right ) e^{\left (-i \, a + \sqrt {\frac {b^{2} d e - b^{2} c f}{f}}\right )} + {\left (-i \, d f x - i \, d e\right )} \sqrt {\frac {b^{2} d e - b^{2} c f}{f}} {\rm Ei}\left (-i \, \sqrt {d x + c} b + \sqrt {\frac {b^{2} d e - b^{2} c f}{f}}\right ) e^{\left (-i \, a - \sqrt {\frac {b^{2} d e - b^{2} c f}{f}}\right )} + 4 \, {\left (d e - c f\right )} \sin \left (\sqrt {d x + c} b + a\right )}{4 \, {\left (d e^{2} f - c e f^{2} + {\left (d e f^{2} - c f^{3}\right )} x\right )}} \] Input:

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

Output:

-1/4*((-I*d*f*x - I*d*e)*sqrt((b^2*d*e - b^2*c*f)/f)*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* 
d*f*x + I*d*e)*sqrt((b^2*d*e - b^2*c*f)/f)*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*d*f*x + I* 
d*e)*sqrt((b^2*d*e - b^2*c*f)/f)*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*d*f*x - I*d*e)*sq 
rt((b^2*d*e - b^2*c*f)/f)*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)) + 4*(d*e - c*f)*sin(sqrt(d*x + 
 c)*b + a))/(d*e^2*f - c*e*f^2 + (d*e*f^2 - c*f^3)*x)
 

Sympy [F]

\[ \int \frac {\sin \left (a+b \sqrt {c+d x}\right )}{(e+f x)^2} \, dx=\int \frac {\sin {\left (a + b \sqrt {c + d x} \right )}}{\left (e + f x\right )^{2}}\, dx \] Input:

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

Output:

Integral(sin(a + b*sqrt(c + d*x))/(e + f*x)**2, x)
 

Maxima [F]

\[ \int \frac {\sin \left (a+b \sqrt {c+d x}\right )}{(e+f x)^2} \, dx=\int { \frac {\sin \left (\sqrt {d x + c} b + a\right )}{{\left (f x + e\right )}^{2}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {\sin \left (a+b \sqrt {c+d x}\right )}{(e+f x)^2} \, dx=\int { \frac {\sin \left (\sqrt {d x + c} b + a\right )}{{\left (f x + e\right )}^{2}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sin \left (a+b \sqrt {c+d x}\right )}{(e+f x)^2} \, dx=\int \frac {\sin \left (a+b\,\sqrt {c+d\,x}\right )}{{\left (e+f\,x\right )}^2} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\sin \left (a+b \sqrt {c+d x}\right )}{(e+f x)^2} \, dx=\int \frac {\sin \left (\sqrt {d x +c}\, b +a \right )}{f^{2} x^{2}+2 e f x +e^{2}}d x \] Input:

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

Output:

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