Integrand size = 16, antiderivative size = 213 \[ \int \frac {\sin (c+d x)}{a+b x^2} \, dx=-\frac {\operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 \sqrt {-a} \sqrt {b}}+\frac {\operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 \sqrt {-a} \sqrt {b}}-\frac {\cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 \sqrt {-a} \sqrt {b}}-\frac {\cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 \sqrt {-a} \sqrt {b}} \]
1/2*cos(c+d*(-a)^(1/2)/b^(1/2))*Si(d*x-d*(-a)^(1/2)/b^(1/2))/(-a)^(1/2)/b^ (1/2)-1/2*cos(c-d*(-a)^(1/2)/b^(1/2))*Si(d*x+d*(-a)^(1/2)/b^(1/2))/(-a)^(1 /2)/b^(1/2)-1/2*Ci(d*x+d*(-a)^(1/2)/b^(1/2))*sin(c-d*(-a)^(1/2)/b^(1/2))/( -a)^(1/2)/b^(1/2)+1/2*Ci(-d*x+d*(-a)^(1/2)/b^(1/2))*sin(c+d*(-a)^(1/2)/b^( 1/2))/(-a)^(1/2)/b^(1/2)
Result contains complex when optimal does not.
Time = 0.40 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.77 \[ \int \frac {\sin (c+d x)}{a+b x^2} \, dx=\frac {e^{-i c-\frac {\sqrt {a} d}{\sqrt {b}}} \left (e^{\frac {2 \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )-\operatorname {ExpIntegralEi}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )+e^{2 i c} \left (e^{\frac {2 \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )-\operatorname {ExpIntegralEi}\left (\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )\right )\right )}{4 \sqrt {a} \sqrt {b}} \]
(E^((-I)*c - (Sqrt[a]*d)/Sqrt[b])*(E^((2*Sqrt[a]*d)/Sqrt[b])*ExpIntegralEi [-((Sqrt[a]*d)/Sqrt[b]) - I*d*x] - ExpIntegralEi[(Sqrt[a]*d)/Sqrt[b] - I*d *x] + E^((2*I)*c)*(E^((2*Sqrt[a]*d)/Sqrt[b])*ExpIntegralEi[-((Sqrt[a]*d)/S qrt[b]) + I*d*x] - ExpIntegralEi[(Sqrt[a]*d)/Sqrt[b] + I*d*x])))/(4*Sqrt[a ]*Sqrt[b])
Time = 0.44 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3814, 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.
\(\displaystyle \int \frac {\sin (c+d x)}{a+b x^2} \, dx\) |
\(\Big \downarrow \) 3814 |
\(\displaystyle \int \left (\frac {\sqrt {-a} \sin (c+d x)}{2 a \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sqrt {-a} \sin (c+d x)}{2 a \left (\sqrt {-a}+\sqrt {b} x\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 \sqrt {-a} \sqrt {b}}+\frac {\sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 \sqrt {-a} \sqrt {b}}-\frac {\cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 \sqrt {-a} \sqrt {b}}-\frac {\cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 \sqrt {-a} \sqrt {b}}\) |
-1/2*(CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b] ])/(Sqrt[-a]*Sqrt[b]) + (CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sin[c + ( Sqrt[-a]*d)/Sqrt[b]])/(2*Sqrt[-a]*Sqrt[b]) - (Cos[c + (Sqrt[-a]*d)/Sqrt[b] ]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*Sqrt[-a]*Sqrt[b]) - (Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*Sqrt[-a ]*Sqrt[b])
3.1.61.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*Sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Int [ExpandIntegrand[Sin[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])
Time = 0.26 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.06
method | result | size |
derivativedivides | \(d \left (-\frac {\operatorname {Si}\left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}+c b}{b}\right )+\operatorname {Ci}\left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}+c b}{b}\right )}{2 b \left (-\frac {d \sqrt {-a b}+c b}{b}+c \right )}-\frac {\operatorname {Si}\left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}-c b}{b}\right )-\operatorname {Ci}\left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}-c b}{b}\right )}{2 b \left (\frac {d \sqrt {-a b}-c b}{b}+c \right )}\right )\) | \(225\) |
default | \(d \left (-\frac {\operatorname {Si}\left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}+c b}{b}\right )+\operatorname {Ci}\left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}+c b}{b}\right )}{2 b \left (-\frac {d \sqrt {-a b}+c b}{b}+c \right )}-\frac {\operatorname {Si}\left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}-c b}{b}\right )-\operatorname {Ci}\left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}-c b}{b}\right )}{2 b \left (\frac {d \sqrt {-a b}-c b}{b}+c \right )}\right )\) | \(225\) |
risch | \(\frac {\sqrt {a b}\, {\mathrm e}^{-\frac {i c b +d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (-\frac {i c b +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{4 a b}-\frac {\sqrt {a b}\, {\mathrm e}^{-\frac {i c b -d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (-\frac {i c b -d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{4 a b}-\frac {\sqrt {a b}\, {\mathrm e}^{\frac {i c b +d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (\frac {i c b +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{4 a b}+\frac {\sqrt {a b}\, {\mathrm e}^{\frac {i c b -d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (\frac {i c b -d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{4 a b}\) | \(262\) |
d*(-1/2/b/(-(d*(-a*b)^(1/2)+c*b)/b+c)*(Si(d*x+c-(d*(-a*b)^(1/2)+c*b)/b)*co s((d*(-a*b)^(1/2)+c*b)/b)+Ci(d*x+c-(d*(-a*b)^(1/2)+c*b)/b)*sin((d*(-a*b)^( 1/2)+c*b)/b))-1/2/b/((d*(-a*b)^(1/2)-c*b)/b+c)*(Si(d*x+c+(d*(-a*b)^(1/2)-c *b)/b)*cos((d*(-a*b)^(1/2)-c*b)/b)-Ci(d*x+c+(d*(-a*b)^(1/2)-c*b)/b)*sin((d *(-a*b)^(1/2)-c*b)/b)))
Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.88 \[ \int \frac {\sin (c+d x)}{a+b x^2} \, dx=\frac {\sqrt {\frac {a d^{2}}{b}} {\rm Ei}\left (i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} - \sqrt {\frac {a d^{2}}{b}} {\rm Ei}\left (i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c - \sqrt {\frac {a d^{2}}{b}}\right )} + \sqrt {\frac {a d^{2}}{b}} {\rm Ei}\left (-i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} - \sqrt {\frac {a d^{2}}{b}} {\rm Ei}\left (-i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c - \sqrt {\frac {a d^{2}}{b}}\right )}}{4 \, a d} \]
1/4*(sqrt(a*d^2/b)*Ei(I*d*x - sqrt(a*d^2/b))*e^(I*c + sqrt(a*d^2/b)) - sqr t(a*d^2/b)*Ei(I*d*x + sqrt(a*d^2/b))*e^(I*c - sqrt(a*d^2/b)) + sqrt(a*d^2/ b)*Ei(-I*d*x - sqrt(a*d^2/b))*e^(-I*c + sqrt(a*d^2/b)) - sqrt(a*d^2/b)*Ei( -I*d*x + sqrt(a*d^2/b))*e^(-I*c - sqrt(a*d^2/b)))/(a*d)
\[ \int \frac {\sin (c+d x)}{a+b x^2} \, dx=\int \frac {\sin {\left (c + d x \right )}}{a + b x^{2}}\, dx \]
\[ \int \frac {\sin (c+d x)}{a+b x^2} \, dx=\int { \frac {\sin \left (d x + c\right )}{b x^{2} + a} \,d x } \]
\[ \int \frac {\sin (c+d x)}{a+b x^2} \, dx=\int { \frac {\sin \left (d x + c\right )}{b x^{2} + a} \,d x } \]
Timed out. \[ \int \frac {\sin (c+d x)}{a+b x^2} \, dx=\int \frac {\sin \left (c+d\,x\right )}{b\,x^2+a} \,d x \]