Integrand size = 22, antiderivative size = 238 \[ \int \frac {\sin \left (a+b \sqrt {c+d x}\right )}{e+f x} \, dx=\frac {\operatorname {CosIntegral}\left (\frac {b \sqrt {-d e+c f}}{\sqrt {f}}+b \sqrt {c+d x}\right ) \sin \left (a-\frac {b \sqrt {-d e+c f}}{\sqrt {f}}\right )}{f}+\frac {\operatorname {CosIntegral}\left (\frac {b \sqrt {-d e+c f}}{\sqrt {f}}-b \sqrt {c+d x}\right ) \sin \left (a+\frac {b \sqrt {-d e+c f}}{\sqrt {f}}\right )}{f}-\frac {\cos \left (a+\frac {b \sqrt {-d e+c f}}{\sqrt {f}}\right ) \text {Si}\left (\frac {b \sqrt {-d e+c f}}{\sqrt {f}}-b \sqrt {c+d x}\right )}{f}+\frac {\cos \left (a-\frac {b \sqrt {-d e+c f}}{\sqrt {f}}\right ) \text {Si}\left (\frac {b \sqrt {-d e+c f}}{\sqrt {f}}+b \sqrt {c+d x}\right )}{f} \]
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+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
Result contains complex when optimal does not.
Time = 1.37 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.00 \[ \int \frac {\sin \left (a+b \sqrt {c+d x}\right )}{e+f x} \, dx=\frac {i e^{-i \left (a+\frac {b \sqrt {-d e+c f}}{\sqrt {f}}\right )} \left (\operatorname {ExpIntegralEi}\left (-i b \left (-\frac {\sqrt {-d e+c f}}{\sqrt {f}}+\sqrt {c+d x}\right )\right )-e^{2 i \left (a+\frac {b \sqrt {-d e+c f}}{\sqrt {f}}\right )} \operatorname {ExpIntegralEi}\left (i b \left (-\frac {\sqrt {-d e+c f}}{\sqrt {f}}+\sqrt {c+d x}\right )\right )+e^{\frac {2 i b \sqrt {-d e+c f}}{\sqrt {f}}} \operatorname {ExpIntegralEi}\left (-i b \left (\frac {\sqrt {-d e+c f}}{\sqrt {f}}+\sqrt {c+d x}\right )\right )-e^{2 i a} \operatorname {ExpIntegralEi}\left (i b \left (\frac {\sqrt {-d e+c f}}{\sqrt {f}}+\sqrt {c+d x}\right )\right )\right )}{2 f} \]
((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)
Time = 0.86 (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 |
\(\displaystyle \frac {2 \left (\frac {d \sin \left (a-\frac {b \sqrt {c f-d e}}{\sqrt {f}}\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {c f-d e} b}{\sqrt {f}}+\sqrt {c+d x} b\right )}{2 f}+\frac {d \sin \left (a+\frac {b \sqrt {c f-d e}}{\sqrt {f}}\right ) \operatorname {CosIntegral}\left (\frac {b \sqrt {c f-d e}}{\sqrt {f}}-b \sqrt {c+d x}\right )}{2 f}-\frac {d \cos \left (a+\frac {b \sqrt {c f-d e}}{\sqrt {f}}\right ) \text {Si}\left (\frac {b \sqrt {c f-d e}}{\sqrt {f}}-b \sqrt {c+d x}\right )}{2 f}+\frac {d \cos \left (a-\frac {b \sqrt {c f-d e}}{\sqrt {f}}\right ) \text {Si}\left (\frac {\sqrt {c f-d e} b}{\sqrt {f}}+\sqrt {c+d x} b\right )}{2 f}\right )}{d}\) |
(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
3.2.90.3.1 Defintions of rubi rules used
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. \(792\) vs. \(2(197)=394\).
Time = 0.25 (sec) , antiderivative size = 793, normalized size of antiderivative = 3.33
method | result | size |
derivativedivides | \(\frac {-\frac {b^{2} \left (f a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}\right ) \left (-\operatorname {Si}\left (-b \sqrt {d x +c}-a +\frac {f a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right ) \cos \left (\frac {f a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right )+\operatorname {Ci}\left (b \sqrt {d x +c}+a -\frac {f a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right ) \sin \left (\frac {f a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right )\right )}{f^{2} \left (-\frac {f a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}+a \right )}+\frac {b^{2} \left (-f a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}\right ) \left (-\operatorname {Si}\left (-b \sqrt {d x +c}-a -\frac {-f a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right ) \cos \left (\frac {-f a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right )-\operatorname {Ci}\left (b \sqrt {d x +c}+a +\frac {-f a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right ) \sin \left (\frac {-f a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right )\right )}{f^{2} \left (\frac {-f a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}+a \right )}-2 b^{2} a \left (-\frac {-\operatorname {Si}\left (-b \sqrt {d x +c}-a +\frac {f a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right ) \cos \left (\frac {f a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right )+\operatorname {Ci}\left (b \sqrt {d x +c}+a -\frac {f a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right ) \sin \left (\frac {f a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right )}{2 f \left (-\frac {f a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}+a \right )}-\frac {-\operatorname {Si}\left (-b \sqrt {d x +c}-a -\frac {-f a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right ) \cos \left (\frac {-f a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right )-\operatorname {Ci}\left (b \sqrt {d x +c}+a +\frac {-f a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right ) \sin \left (\frac {-f a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right )}{2 f \left (\frac {-f a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}+a \right )}\right )}{b^{2}}\) | \(793\) |
default | \(\frac {-\frac {b^{2} \left (f a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}\right ) \left (-\operatorname {Si}\left (-b \sqrt {d x +c}-a +\frac {f a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right ) \cos \left (\frac {f a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right )+\operatorname {Ci}\left (b \sqrt {d x +c}+a -\frac {f a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right ) \sin \left (\frac {f a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right )\right )}{f^{2} \left (-\frac {f a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}+a \right )}+\frac {b^{2} \left (-f a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}\right ) \left (-\operatorname {Si}\left (-b \sqrt {d x +c}-a -\frac {-f a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right ) \cos \left (\frac {-f a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right )-\operatorname {Ci}\left (b \sqrt {d x +c}+a +\frac {-f a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right ) \sin \left (\frac {-f a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right )\right )}{f^{2} \left (\frac {-f a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}+a \right )}-2 b^{2} a \left (-\frac {-\operatorname {Si}\left (-b \sqrt {d x +c}-a +\frac {f a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right ) \cos \left (\frac {f a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right )+\operatorname {Ci}\left (b \sqrt {d x +c}+a -\frac {f a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right ) \sin \left (\frac {f a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right )}{2 f \left (-\frac {f a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}+a \right )}-\frac {-\operatorname {Si}\left (-b \sqrt {d x +c}-a -\frac {-f a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right ) \cos \left (\frac {-f a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right )-\operatorname {Ci}\left (b \sqrt {d x +c}+a +\frac {-f a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right ) \sin \left (\frac {-f a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right )}{2 f \left (\frac {-f a +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}+a \right )}\right )}{b^{2}}\) | \(793\) |
2/b^2*(-1/2*b^2*(f*a+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f^2/(-(f*a+(b^2*c*f^2-b^ 2*d*e*f)^(1/2))/f+a)*(-Si(-b*(d*x+c)^(1/2)-a+(f*a+(b^2*c*f^2-b^2*d*e*f)^(1 /2))/f)*cos((f*a+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f)+Ci(b*(d*x+c)^(1/2)+a-(f*a +(b^2*c*f^2-b^2*d*e*f)^(1/2))/f)*sin((f*a+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f)) +1/2*b^2*(-f*a+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f^2/((-f*a+(b^2*c*f^2-b^2*d*e* f)^(1/2))/f+a)*(-Si(-b*(d*x+c)^(1/2)-a-(-f*a+(b^2*c*f^2-b^2*d*e*f)^(1/2))/ f)*cos((-f*a+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f)-Ci(b*(d*x+c)^(1/2)+a+(-f*a+(b ^2*c*f^2-b^2*d*e*f)^(1/2))/f)*sin((-f*a+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f))-b ^2*a*(-1/2/f/(-(f*a+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f+a)*(-Si(-b*(d*x+c)^(1/2 )-a+(f*a+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f)*cos((f*a+(b^2*c*f^2-b^2*d*e*f)^(1 /2))/f)+Ci(b*(d*x+c)^(1/2)+a-(f*a+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f)*sin((f*a +(b^2*c*f^2-b^2*d*e*f)^(1/2))/f))-1/2/f/((-f*a+(b^2*c*f^2-b^2*d*e*f)^(1/2) )/f+a)*(-Si(-b*(d*x+c)^(1/2)-a-(-f*a+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f)*cos(( -f*a+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f)-Ci(b*(d*x+c)^(1/2)+a+(-f*a+(b^2*c*f^2 -b^2*d*e*f)^(1/2))/f)*sin((-f*a+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f))))
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.05 \[ \int \frac {\sin \left (a+b \sqrt {c+d x}\right )}{e+f x} \, dx=\frac {-i \, {\rm Ei}\left (i \, \sqrt {d x + c} b - \sqrt {\frac {b^{2} d e - b^{2} c f}{f}}\right ) e^{\left (i \, a + \sqrt {\frac {b^{2} d e - b^{2} c f}{f}}\right )} - i \, {\rm Ei}\left (i \, \sqrt {d x + c} b + \sqrt {\frac {b^{2} d e - b^{2} c f}{f}}\right ) e^{\left (i \, a - \sqrt {\frac {b^{2} d e - b^{2} c f}{f}}\right )} + i \, {\rm Ei}\left (-i \, \sqrt {d x + c} b - \sqrt {\frac {b^{2} d e - b^{2} c f}{f}}\right ) e^{\left (-i \, a + \sqrt {\frac {b^{2} d e - b^{2} c f}{f}}\right )} + i \, {\rm Ei}\left (-i \, \sqrt {d x + c} b + \sqrt {\frac {b^{2} d e - b^{2} c f}{f}}\right ) e^{\left (-i \, a - \sqrt {\frac {b^{2} d e - b^{2} c f}{f}}\right )}}{2 \, f} \]
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
\[ \int \frac {\sin \left (a+b \sqrt {c+d x}\right )}{e+f x} \, dx=\int \frac {\sin {\left (a + b \sqrt {c + d x} \right )}}{e + f x}\, dx \]
\[ \int \frac {\sin \left (a+b \sqrt {c+d x}\right )}{e+f x} \, dx=\int { \frac {\sin \left (\sqrt {d x + c} b + a\right )}{f x + e} \,d x } \]
\[ \int \frac {\sin \left (a+b \sqrt {c+d x}\right )}{e+f x} \, dx=\int { \frac {\sin \left (\sqrt {d x + c} b + a\right )}{f x + e} \,d x } \]
Timed out. \[ \int \frac {\sin \left (a+b \sqrt {c+d x}\right )}{e+f x} \, dx=\int \frac {\sin \left (a+b\,\sqrt {c+d\,x}\right )}{e+f\,x} \,d x \]