Integrand size = 22, antiderivative size = 350 \[ \int \frac {\sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{(e+f x)^2} \, dx=-\frac {b d \cos \left (a+\frac {b \sqrt {f}}{\sqrt {-d e+c f}}\right ) \operatorname {CosIntegral}\left (\frac {b \sqrt {f}}{\sqrt {-d e+c f}}-\frac {b}{\sqrt {c+d x}}\right )}{2 \sqrt {f} (-d e+c f)^{3/2}}+\frac {b d \cos \left (a-\frac {b \sqrt {f}}{\sqrt {-d e+c f}}\right ) \operatorname {CosIntegral}\left (\frac {b \sqrt {f}}{\sqrt {-d e+c f}}+\frac {b}{\sqrt {c+d x}}\right )}{2 \sqrt {f} (-d e+c f)^{3/2}}+\frac {(c+d x) \sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{(d e-c f) (e+f x)}-\frac {b d \sin \left (a+\frac {b \sqrt {f}}{\sqrt {-d e+c f}}\right ) \text {Si}\left (\frac {b \sqrt {f}}{\sqrt {-d e+c f}}-\frac {b}{\sqrt {c+d x}}\right )}{2 \sqrt {f} (-d e+c f)^{3/2}}-\frac {b d \sin \left (a-\frac {b \sqrt {f}}{\sqrt {-d e+c f}}\right ) \text {Si}\left (\frac {b \sqrt {f}}{\sqrt {-d e+c f}}+\frac {b}{\sqrt {c+d x}}\right )}{2 \sqrt {f} (-d e+c f)^{3/2}} \]
(d*x+c)*sin(a+b/(d*x+c)^(1/2))/(-c*f+d*e)/(f*x+e)+1/2*b*d*Ci(b*f^(1/2)/(c* f-d*e)^(1/2)+b/(d*x+c)^(1/2))*cos(a-b*f^(1/2)/(c*f-d*e)^(1/2))/(c*f-d*e)^( 3/2)/f^(1/2)-1/2*b*d*Ci(b*f^(1/2)/(c*f-d*e)^(1/2)-b/(d*x+c)^(1/2))*cos(a+b *f^(1/2)/(c*f-d*e)^(1/2))/(c*f-d*e)^(3/2)/f^(1/2)-1/2*b*d*Si(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))/(c*f-d*e)^( 3/2)/f^(1/2)-1/2*b*d*Si(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))/(c*f-d*e)^(3/2)/f^(1/2)
\[ \int \frac {\sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{(e+f x)^2} \, dx=\int \frac {\sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{(e+f x)^2} \, dx \]
Time = 0.98 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.06, 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.
| \(\displaystyle \int \frac {\sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{(e+f x)^2} \, dx\) |
| \(\Big \downarrow \) 3912 |
| \(\displaystyle -\frac {2 \int \frac {d^2 \sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{\sqrt {c+d x} \left (f+\frac {d \left (e-\frac {c f}{d}\right )}{c+d x}\right )^2}d\frac {1}{\sqrt {c+d x}}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -2 d \int \frac {\sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{\sqrt {c+d x} \left (f+\frac {d e-c f}{c+d x}\right )^2}d\frac {1}{\sqrt {c+d x}}\) |
| \(\Big \downarrow \) 3822 |
| \(\displaystyle -2 d \left (\frac {b \int \frac {\cos \left (a+\frac {b}{\sqrt {c+d x}}\right )}{f+\frac {d e-c f}{c+d x}}d\frac {1}{\sqrt {c+d x}}}{2 (d e-c f)}-\frac {\sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{2 (d e-c f) \left (\frac {d e-c f}{c+d x}+f\right )}\right )\) |
| \(\Big \downarrow \) 3815 |
| \(\displaystyle -2 d \left (\frac {b \int \left (\frac {\cos \left (a+\frac {b}{\sqrt {c+d x}}\right )}{2 \sqrt {f} \left (\sqrt {f}-\frac {\sqrt {c f-d e}}{\sqrt {c+d x}}\right )}+\frac {\cos \left (a+\frac {b}{\sqrt {c+d x}}\right )}{2 \sqrt {f} \left (\sqrt {f}+\frac {\sqrt {c f-d e}}{\sqrt {c+d x}}\right )}\right )d\frac {1}{\sqrt {c+d x}}}{2 (d e-c f)}-\frac {\sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{2 (d e-c f) \left (\frac {d e-c f}{c+d x}+f\right )}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 d \left (\frac {b \left (-\frac {\cos \left (a+\frac {b \sqrt {f}}{\sqrt {c f-d e}}\right ) \operatorname {CosIntegral}\left (\frac {b \sqrt {f}}{\sqrt {c f-d e}}-\frac {b}{\sqrt {c+d x}}\right )}{2 \sqrt {f} \sqrt {c f-d e}}+\frac {\cos \left (a-\frac {b \sqrt {f}}{\sqrt {c f-d e}}\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {f} b}{\sqrt {c f-d e}}+\frac {b}{\sqrt {c+d x}}\right )}{2 \sqrt {f} \sqrt {c f-d e}}-\frac {\sin \left (a+\frac {b \sqrt {f}}{\sqrt {c f-d e}}\right ) \text {Si}\left (\frac {b \sqrt {f}}{\sqrt {c f-d e}}-\frac {b}{\sqrt {c+d x}}\right )}{2 \sqrt {f} \sqrt {c f-d e}}-\frac {\sin \left (a-\frac {b \sqrt {f}}{\sqrt {c f-d e}}\right ) \text {Si}\left (\frac {\sqrt {f} b}{\sqrt {c f-d e}}+\frac {b}{\sqrt {c+d x}}\right )}{2 \sqrt {f} \sqrt {c f-d e}}\right )}{2 (d e-c f)}-\frac {\sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{2 (d e-c f) \left (\frac {d e-c f}{c+d x}+f\right )}\right )\) |
-2*d*(-1/2*Sin[a + b/Sqrt[c + d*x]]/((d*e - c*f)*(f + (d*e - c*f)/(c + d*x ))) + (b*(-1/2*(Cos[a + (b*Sqrt[f])/Sqrt[-(d*e) + c*f]]*CosIntegral[(b*Sqr t[f])/Sqrt[-(d*e) + c*f] - b/Sqrt[c + d*x]])/(Sqrt[f]*Sqrt[-(d*e) + c*f]) + (Cos[a - (b*Sqrt[f])/Sqrt[-(d*e) + c*f]]*CosIntegral[(b*Sqrt[f])/Sqrt[-( d*e) + c*f] + b/Sqrt[c + d*x]])/(2*Sqrt[f]*Sqrt[-(d*e) + c*f]) - (Sin[a + (b*Sqrt[f])/Sqrt[-(d*e) + c*f]]*SinIntegral[(b*Sqrt[f])/Sqrt[-(d*e) + c*f] - b/Sqrt[c + d*x]])/(2*Sqrt[f]*Sqrt[-(d*e) + c*f]) - (Sin[a - (b*Sqrt[f]) /Sqrt[-(d*e) + c*f]]*SinIntegral[(b*Sqrt[f])/Sqrt[-(d*e) + c*f] + b/Sqrt[c + d*x]])/(2*Sqrt[f]*Sqrt[-(d*e) + c*f])))/(2*(d*e - c*f)))
3.3.1.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
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])
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]
Leaf count of result is larger than twice the leaf count of optimal. \(2733\) vs. \(2(284)=568\).
Time = 0.96 (sec) , antiderivative size = 2734, normalized size of antiderivative = 7.81
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(2734\) |
| default | \(\text {Expression too large to display}\) | \(2734\) |
-2*d*b^2*(sin(a+b/(d*x+c)^(1/2))*(-1/2*a/f/b^2*(a+b/(d*x+c)^(1/2))+1/2*(a^ 2*c*f-a^2*d*e-b^2*f)/f/b^2/(c*f-d*e))/(a^2*c*f-a^2*d*e-2*a*c*f*(a+b/(d*x+c )^(1/2))+2*a*d*e*(a+b/(d*x+c)^(1/2))+c*f*(a+b/(d*x+c)^(1/2))^2-d*e*(a+b/(d *x+c)^(1/2))^2-b^2*f)+1/4*a/f/b^2/(a*c*f-a*d*e-c*f*(a*c*f-a*d*e+(b^2*c*f^2 -b^2*d*e*f)^(1/2))/(c*f-d*e)+d*e*(a*c*f-a*d*e+(b^2*c*f^2-b^2*d*e*f)^(1/2)) /(c*f-d*e))*(-Si(-b/(d*x+c)^(1/2)-a+(a*c*f-a*d*e+(b^2*c*f^2-b^2*d*e*f)^(1/ 2))/(c*f-d*e))*cos((a*c*f-a*d*e+(b^2*c*f^2-b^2*d*e*f)^(1/2))/(c*f-d*e))+Ci (b/(d*x+c)^(1/2)+a-(a*c*f-a*d*e+(b^2*c*f^2-b^2*d*e*f)^(1/2))/(c*f-d*e))*si n((a*c*f-a*d*e+(b^2*c*f^2-b^2*d*e*f)^(1/2))/(c*f-d*e)))+1/4*a/f/b^2/(a*c*f -a*d*e+c*f*(-a*c*f+a*d*e+(b^2*c*f^2-b^2*d*e*f)^(1/2))/(c*f-d*e)-d*e*(-a*c* f+a*d*e+(b^2*c*f^2-b^2*d*e*f)^(1/2))/(c*f-d*e))*(Si(b/(d*x+c)^(1/2)+a+(-a* c*f+a*d*e+(b^2*c*f^2-b^2*d*e*f)^(1/2))/(c*f-d*e))*cos((-a*c*f+a*d*e+(b^2*c *f^2-b^2*d*e*f)^(1/2))/(c*f-d*e))-Ci(b/(d*x+c)^(1/2)+a+(-a*c*f+a*d*e+(b^2* c*f^2-b^2*d*e*f)^(1/2))/(c*f-d*e))*sin((-a*c*f+a*d*e+(b^2*c*f^2-b^2*d*e*f) ^(1/2))/(c*f-d*e)))+1/4*(a^2*c*f-a^2*d*e-a*c*f*(a*c*f-a*d*e+(b^2*c*f^2-b^2 *d*e*f)^(1/2))/(c*f-d*e)+a*d*e*(a*c*f-a*d*e+(b^2*c*f^2-b^2*d*e*f)^(1/2))/( c*f-d*e)-b^2*f)/b^2/(c*f-d*e)/f/(a*c*f-a*d*e-c*f*(a*c*f-a*d*e+(b^2*c*f^2-b ^2*d*e*f)^(1/2))/(c*f-d*e)+d*e*(a*c*f-a*d*e+(b^2*c*f^2-b^2*d*e*f)^(1/2))/( c*f-d*e))*(Si(-b/(d*x+c)^(1/2)-a+(a*c*f-a*d*e+(b^2*c*f^2-b^2*d*e*f)^(1/2)) /(c*f-d*e))*sin((a*c*f-a*d*e+(b^2*c*f^2-b^2*d*e*f)^(1/2))/(c*f-d*e))+Ci...
Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 448, normalized size of antiderivative = 1.28 \[ \int \frac {\sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{(e+f x)^2} \, dx=-\frac {{\left (i \, d f x + i \, d e\right )} \sqrt {\frac {b^{2} f}{d e - c f}} {\rm Ei}\left (-\frac {\sqrt {\frac {b^{2} f}{d e - c f}} {\left (d x + c\right )} - i \, \sqrt {d x + c} b}{d x + c}\right ) e^{\left (i \, a + \sqrt {\frac {b^{2} f}{d e - c f}}\right )} + {\left (-i \, d f x - i \, d e\right )} \sqrt {\frac {b^{2} f}{d e - c f}} {\rm Ei}\left (\frac {\sqrt {\frac {b^{2} f}{d e - c f}} {\left (d x + c\right )} + i \, \sqrt {d x + c} b}{d x + c}\right ) e^{\left (i \, a - \sqrt {\frac {b^{2} f}{d e - c f}}\right )} + {\left (-i \, d f x - i \, d e\right )} \sqrt {\frac {b^{2} f}{d e - c f}} {\rm Ei}\left (-\frac {\sqrt {\frac {b^{2} f}{d e - c f}} {\left (d x + c\right )} + i \, \sqrt {d x + c} b}{d x + c}\right ) e^{\left (-i \, a + \sqrt {\frac {b^{2} f}{d e - c f}}\right )} + {\left (i \, d f x + i \, d e\right )} \sqrt {\frac {b^{2} f}{d e - c f}} {\rm Ei}\left (\frac {\sqrt {\frac {b^{2} f}{d e - c f}} {\left (d x + c\right )} - i \, \sqrt {d x + c} b}{d x + c}\right ) e^{\left (-i \, a - \sqrt {\frac {b^{2} f}{d e - c f}}\right )} - 4 \, {\left (d f x + c f\right )} \sin \left (\frac {a d x + a c + \sqrt {d x + c} b}{d x + c}\right )}{4 \, {\left (d e^{2} f - c e f^{2} + {\left (d e f^{2} - c f^{3}\right )} x\right )}} \]
-1/4*((I*d*f*x + I*d*e)*sqrt(b^2*f/(d*e - c*f))*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*d*f*x - I*d*e)*sqrt(b^2*f/(d*e - c*f))*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*d*f*x - I*d*e)*sqrt(b^2*f/(d*e - c*f))*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*d*f*x + I*d*e)*sqrt(b^2*f/(d*e - c*f))*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*(d*f*x + c*f)*sin((a*d*x + a*c + sqrt(d*x + c)*b)/(d*x + c)) )/(d*e^2*f - c*e*f^2 + (d*e*f^2 - c*f^3)*x)
\[ \int \frac {\sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{(e+f x)^2} \, dx=\int \frac {\sin {\left (a + \frac {b}{\sqrt {c + d x}} \right )}}{\left (e + f x\right )^{2}}\, dx \]
\[ \int \frac {\sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{(e+f x)^2} \, dx=\int { \frac {\sin \left (a + \frac {b}{\sqrt {d x + c}}\right )}{{\left (f x + e\right )}^{2}} \,d x } \]
\[ \int \frac {\sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{(e+f x)^2} \, dx=\int { \frac {\sin \left (a + \frac {b}{\sqrt {d x + c}}\right )}{{\left (f x + e\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{(e+f x)^2} \, dx=\int \frac {\sin \left (a+\frac {b}{\sqrt {c+d\,x}}\right )}{{\left (e+f\,x\right )}^2} \,d x \]