Integrand size = 19, antiderivative size = 250 \[ \int \frac {\sin (c+d x)}{x^2 \left (a+b x^2\right )} \, dx=\frac {d \cos (c) \operatorname {CosIntegral}(d x)}{a}-\frac {\sqrt {b} \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 (-a)^{3/2}}+\frac {\sqrt {b} \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 (-a)^{3/2}}-\frac {\sin (c+d x)}{a x}-\frac {d \sin (c) \text {Si}(d x)}{a}-\frac {\sqrt {b} \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 (-a)^{3/2}}-\frac {\sqrt {b} \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 (-a)^{3/2}} \]
d*Ci(d*x)*cos(c)/a-d*Si(d*x)*sin(c)/a-sin(d*x+c)/a/x+1/2*cos(c+d*(-a)^(1/2 )/b^(1/2))*Si(d*x-d*(-a)^(1/2)/b^(1/2))*b^(1/2)/(-a)^(3/2)-1/2*cos(c-d*(-a )^(1/2)/b^(1/2))*Si(d*x+d*(-a)^(1/2)/b^(1/2))*b^(1/2)/(-a)^(3/2)-1/2*Ci(d* x+d*(-a)^(1/2)/b^(1/2))*sin(c-d*(-a)^(1/2)/b^(1/2))*b^(1/2)/(-a)^(3/2)+1/2 *Ci(-d*x+d*(-a)^(1/2)/b^(1/2))*sin(c+d*(-a)^(1/2)/b^(1/2))*b^(1/2)/(-a)^(3 /2)
Result contains complex when optimal does not.
Time = 0.57 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.95 \[ \int \frac {\sin (c+d x)}{x^2 \left (a+b x^2\right )} \, dx=\frac {\sqrt {b} e^{-i c-\frac {\sqrt {a} d}{\sqrt {b}}} \left (-e^{\frac {2 \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )+\operatorname {ExpIntegralEi}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )\right )}{4 a^{3/2}}+\frac {\sqrt {b} e^{i c-\frac {\sqrt {a} d}{\sqrt {b}}} \left (-e^{\frac {2 \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )+\operatorname {ExpIntegralEi}\left (\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )\right )}{4 a^{3/2}}-\frac {\cos (d x) \sin (c)}{a x}-\frac {\cos (c) \sin (d x)}{a x}+\frac {d (\cos (c) \operatorname {CosIntegral}(d x)-\sin (c) \text {Si}(d x))}{a} \]
(Sqrt[b]*E^((-I)*c - (Sqrt[a]*d)/Sqrt[b])*(-(E^((2*Sqrt[a]*d)/Sqrt[b])*Exp IntegralEi[-((Sqrt[a]*d)/Sqrt[b]) - I*d*x]) + ExpIntegralEi[(Sqrt[a]*d)/Sq rt[b] - I*d*x]))/(4*a^(3/2)) + (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]))/(4*a^(3/2)) - (Cos[d*x]*Sin[c ])/(a*x) - (Cos[c]*Sin[d*x])/(a*x) + (d*(Cos[c]*CosIntegral[d*x] - Sin[c]* SinIntegral[d*x]))/a
Time = 0.62 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3826, 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.
| \(\displaystyle \int \frac {\sin (c+d x)}{x^2 \left (a+b x^2\right )} \, dx\) |
| \(\Big \downarrow \) 3826 |
| \(\displaystyle \int \left (\frac {\sin (c+d x)}{a x^2}-\frac {b \sin (c+d x)}{a \left (a+b x^2\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sqrt {b} \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 (-a)^{3/2}}+\frac {\sqrt {b} \sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 (-a)^{3/2}}-\frac {\sqrt {b} \cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 (-a)^{3/2}}-\frac {\sqrt {b} \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 (-a)^{3/2}}+\frac {d \cos (c) \operatorname {CosIntegral}(d x)}{a}-\frac {d \sin (c) \text {Si}(d x)}{a}-\frac {\sin (c+d x)}{a x}\) |
(d*Cos[c]*CosIntegral[d*x])/a - (Sqrt[b]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(2*(-a)^(3/2)) + (Sqrt[b]*CosIntegra l[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sin[c + (Sqrt[-a]*d)/Sqrt[b]])/(2*(-a)^(3/2) ) - Sin[c + d*x]/(a*x) - (d*Sin[c]*SinIntegral[d*x])/a - (Sqrt[b]*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*(-a)^(3/ 2)) - (Sqrt[b]*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt [b] + d*x])/(2*(-a)^(3/2))
3.1.63.3.1 Defintions of rubi rules used
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*Sin[(c_.) + (d_.)*(x_)], x_Sym bol] :> Int[ExpandIntegrand[Sin[c + d*x], x^m*(a + b*x^n)^p, x], x] /; Free Q[{a, b, c, d, m}, x] && ILtQ[p, 0] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ[p, - 1]) && IntegerQ[m]
Time = 0.33 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.06
| method | result | size |
| derivativedivides | \(d \left (-\frac {b \left (-\frac {\operatorname {Si}\left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}+c b}{b}\right )+\operatorname {Ci}\left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}+c b}{b}\right )}{2 b \left (-\frac {d \sqrt {-a b}+c b}{b}+c \right )}-\frac {\operatorname {Si}\left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}-c b}{b}\right )-\operatorname {Ci}\left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}-c b}{b}\right )}{2 b \left (\frac {d \sqrt {-a b}-c b}{b}+c \right )}\right )}{a}+\frac {-\frac {\sin \left (d x +c \right )}{d x}-\operatorname {Si}\left (d x \right ) \sin \left (c \right )+\operatorname {Ci}\left (d x \right ) \cos \left (c \right )}{a}\right )\) | \(266\) |
| default | \(d \left (-\frac {b \left (-\frac {\operatorname {Si}\left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}+c b}{b}\right )+\operatorname {Ci}\left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}+c b}{b}\right )}{2 b \left (-\frac {d \sqrt {-a b}+c b}{b}+c \right )}-\frac {\operatorname {Si}\left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}-c b}{b}\right )-\operatorname {Ci}\left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}-c b}{b}\right )}{2 b \left (\frac {d \sqrt {-a b}-c b}{b}+c \right )}\right )}{a}+\frac {-\frac {\sin \left (d x +c \right )}{d x}-\operatorname {Si}\left (d x \right ) \sin \left (c \right )+\operatorname {Ci}\left (d x \right ) \cos \left (c \right )}{a}\right )\) | \(266\) |
| risch | \(-\frac {d \,\operatorname {Ei}_{1}\left (-i d x \right ) {\mathrm e}^{i c}}{2 a}+\frac {\sqrt {a b}\, {\mathrm e}^{\frac {i c b +d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (\frac {i c b +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{4 a^{2}}-\frac {\sqrt {a b}\, {\mathrm e}^{\frac {i c b -d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (-\frac {-i c b +d \sqrt {a b}+b \left (i d x +i c \right )}{b}\right )}{4 a^{2}}-\frac {d \,\operatorname {Ei}_{1}\left (i d x \right ) {\mathrm e}^{-i c}}{2 a}-\frac {{\mathrm e}^{-\frac {i c b +d \sqrt {a b}}{b}} \sqrt {a b}\, \operatorname {Ei}_{1}\left (-\frac {i c b +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{4 a^{2}}+\frac {\sqrt {a b}\, {\mathrm e}^{-\frac {i c b -d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (\frac {-i c b +d \sqrt {a b}+b \left (i d x +i c \right )}{b}\right )}{4 a^{2}}-\frac {\sin \left (d x +c \right )}{a x}\) | \(296\) |
d*(-b/a*(-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))-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)))+1/a*(-sin(d*x+c)/d/x-Si(d*x)*sin(c)+Ci(d*x)* cos(c)))
Result contains complex when optimal does not.
Time = 0.34 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.94 \[ \int \frac {\sin (c+d x)}{x^2 \left (a+b x^2\right )} \, dx=\frac {4 \, a d^{2} x \cos \left (c\right ) \operatorname {Ci}\left (d x\right ) - 4 \, a d^{2} x \sin \left (c\right ) \operatorname {Si}\left (d x\right ) - \sqrt {\frac {a d^{2}}{b}} b x {\rm Ei}\left (i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} + \sqrt {\frac {a d^{2}}{b}} b x {\rm Ei}\left (i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c - \sqrt {\frac {a d^{2}}{b}}\right )} - \sqrt {\frac {a d^{2}}{b}} b x {\rm Ei}\left (-i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} + \sqrt {\frac {a d^{2}}{b}} b x {\rm Ei}\left (-i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c - \sqrt {\frac {a d^{2}}{b}}\right )} - 4 \, a d \sin \left (d x + c\right )}{4 \, a^{2} d x} \]
1/4*(4*a*d^2*x*cos(c)*cos_integral(d*x) - 4*a*d^2*x*sin(c)*sin_integral(d* x) - sqrt(a*d^2/b)*b*x*Ei(I*d*x - sqrt(a*d^2/b))*e^(I*c + sqrt(a*d^2/b)) + sqrt(a*d^2/b)*b*x*Ei(I*d*x + sqrt(a*d^2/b))*e^(I*c - sqrt(a*d^2/b)) - sqr t(a*d^2/b)*b*x*Ei(-I*d*x - sqrt(a*d^2/b))*e^(-I*c + sqrt(a*d^2/b)) + sqrt( a*d^2/b)*b*x*Ei(-I*d*x + sqrt(a*d^2/b))*e^(-I*c - sqrt(a*d^2/b)) - 4*a*d*s in(d*x + c))/(a^2*d*x)
\[ \int \frac {\sin (c+d x)}{x^2 \left (a+b x^2\right )} \, dx=\int \frac {\sin {\left (c + d x \right )}}{x^{2} \left (a + b x^{2}\right )}\, dx \]
\[ \int \frac {\sin (c+d x)}{x^2 \left (a+b x^2\right )} \, dx=\int { \frac {\sin \left (d x + c\right )}{{\left (b x^{2} + a\right )} x^{2}} \,d x } \]
\[ \int \frac {\sin (c+d x)}{x^2 \left (a+b x^2\right )} \, dx=\int { \frac {\sin \left (d x + c\right )}{{\left (b x^{2} + a\right )} x^{2}} \,d x } \]
Timed out. \[ \int \frac {\sin (c+d x)}{x^2 \left (a+b x^2\right )} \, dx=\int \frac {\sin \left (c+d\,x\right )}{x^2\,\left (b\,x^2+a\right )} \,d x \]